Nash and Walras Equilibrium via Brouwer_John Geanakoplos.pdf
Four Lectures on Generalized Nash Equilibrium Problems in ...€¦ · Nash Equilibrium Problems in...
Transcript of Four Lectures on Generalized Nash Equilibrium Problems in ...€¦ · Nash Equilibrium Problems in...
Four Lectures on GeneralizedNash Equilibrium Problems in Finite Dimensions
Jong-Shi Pang
Daniel J Epstein Department of Industrial and Systems Engineering
University of Southern California
presented at
Autumn School on Quasi-Variational Inequalities
Theory Algorithms and Applications
Universitat Wurzburg Germany
Monday September 23 through Wednesday September 25 2019
Foreword Brief history of generalized Nash games
Lecture I Formulations and a recent model
Lecture II Existence results with and without convexity
Lecture III Exact penalization theory
Lecture IV Best-response algorithms
2
A brief historical account
Gerard Debreu A social equilibrium existence theorem Proceedings of theNational Academy of Sciences 38(10) 886ndash893 1952
Kenneth J Arrow and Gerard Debreu Existence of an equilibrium fora competitive economy Econometrica Journal of the Econometric Society22(3) 265ndash290 1954
Hukukane Nikaido and Kazuo Isoda Note on noncooperative convexgames Pacific Journal of Mathematics 5(Suppl 1) 807-815 1955
J Ben Rosen Existence and uniqueness of equilibrium points for concaven-person games Econometrica Journal of the Econometric Society 33(3)520ndash534 1965
Alain Bensoussan Points de Nash dans le cas de fontionnelles quadratiques
et jeux differentiels lineaires a N personnes SIAM Journal on Control 12(3)
460ndash499 1974
3
A brief historical account (cont)Patrick T Harker Generalized Nash games and quasi-variationalinequalities European Journal of Operational Research 54(1) 81ndash94 1991
Jong-Shi Pang and Masao Fukushima Quasi-variational inequalitiesgeneralized Nash equilibria and multi-leader-follower games ComputationalManagement Science 2 21-56 2005 [Erratum same journal 6 373-3752009]
Francisco Facchinei and Christian Kanzow Generalized Nashequilibrium problems Annals of Operations Research 175(1)177ndash211 2007
Francisco Facchinei and Jong-Shi Pang Nash equilibria the variationalapproach Chapter 12 in Daniel P Palomar and Yonica C Eldar editorsConvex optimization in signal processing and communications CambridgeUniversity Press pages 443ndash493 2010
Francisco Facchinei Computation of generalized Nash equilibria Recent
advances Part II in Roberto Cominetti Francisco Facchinei and Jean-Bernard
Lasserre Editors and authors Modern Optimization Modeling Techniques
Birkhauser Springer Basel pp 133ndash184 2012
4
Some recent references
Extensive papers by our host Christian Kanzowhttpswwwmathematikuni-wuerzburgdeoptimizationteam
kanzow-christian
Q Ba and JS Pang Exact penalization of generalized Nashequilibrium problems Operations Research (accepted August 2019) inprint
DA Schiro JS Pang and UV Shanbhag On the solution ofaffine generalized Nash equilibrium problems with shared constraints byLemkersquos method Mathematical Programming Series A 142(1ndash2) 1ndash46(2013)
5
Modelling
From single decision maker minusrarr multiple agents
From optimization minusrarr equilibrium
From cooperation minusrarr non-cooperation
From centralized decision making minusrarr distributed responses
Mathematics
From WeierstrassCauchy minusrarr BrouwerBanach
From smooth calculus minusrarr variational analysis
From coordinated descent minusrarr asynchronous computation
From potential function minusrarr non-expansion of mappings
6
Lecture I Formulations and a recent model
930 ndash 1030 AM Monday September 23 2019
7
The Abstract Generalized Nash Equilibrium Problem (GNEP)
N decision makers each (labeled ν = 1 middot middot middot N) with
bull a moving strategy set Ξν(xminusν) sube Rnν and
bull a cost function θν(bull xminusν) Rnν rarr R
both dependent on the rivalsrsquo strategy tuple
xminusν ≜983059xν
prime983060
ν prime ∕=νisin Rminusν ≜
983132
ν prime ∕=ν
Rnν prime
Anticipating rivalsrsquo strategy xminusν player ν solves
minimizexνisinΞν(xminusν)
θν(xν xminusν)
A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that
xlowast ν isin argminxνisinΞν(xlowastminusν)
θν(xν xlowastminusν) forall ν = 1 middot middot middot N
In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy
8
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Foreword Brief history of generalized Nash games
Lecture I Formulations and a recent model
Lecture II Existence results with and without convexity
Lecture III Exact penalization theory
Lecture IV Best-response algorithms
2
A brief historical account
Gerard Debreu A social equilibrium existence theorem Proceedings of theNational Academy of Sciences 38(10) 886ndash893 1952
Kenneth J Arrow and Gerard Debreu Existence of an equilibrium fora competitive economy Econometrica Journal of the Econometric Society22(3) 265ndash290 1954
Hukukane Nikaido and Kazuo Isoda Note on noncooperative convexgames Pacific Journal of Mathematics 5(Suppl 1) 807-815 1955
J Ben Rosen Existence and uniqueness of equilibrium points for concaven-person games Econometrica Journal of the Econometric Society 33(3)520ndash534 1965
Alain Bensoussan Points de Nash dans le cas de fontionnelles quadratiques
et jeux differentiels lineaires a N personnes SIAM Journal on Control 12(3)
460ndash499 1974
3
A brief historical account (cont)Patrick T Harker Generalized Nash games and quasi-variationalinequalities European Journal of Operational Research 54(1) 81ndash94 1991
Jong-Shi Pang and Masao Fukushima Quasi-variational inequalitiesgeneralized Nash equilibria and multi-leader-follower games ComputationalManagement Science 2 21-56 2005 [Erratum same journal 6 373-3752009]
Francisco Facchinei and Christian Kanzow Generalized Nashequilibrium problems Annals of Operations Research 175(1)177ndash211 2007
Francisco Facchinei and Jong-Shi Pang Nash equilibria the variationalapproach Chapter 12 in Daniel P Palomar and Yonica C Eldar editorsConvex optimization in signal processing and communications CambridgeUniversity Press pages 443ndash493 2010
Francisco Facchinei Computation of generalized Nash equilibria Recent
advances Part II in Roberto Cominetti Francisco Facchinei and Jean-Bernard
Lasserre Editors and authors Modern Optimization Modeling Techniques
Birkhauser Springer Basel pp 133ndash184 2012
4
Some recent references
Extensive papers by our host Christian Kanzowhttpswwwmathematikuni-wuerzburgdeoptimizationteam
kanzow-christian
Q Ba and JS Pang Exact penalization of generalized Nashequilibrium problems Operations Research (accepted August 2019) inprint
DA Schiro JS Pang and UV Shanbhag On the solution ofaffine generalized Nash equilibrium problems with shared constraints byLemkersquos method Mathematical Programming Series A 142(1ndash2) 1ndash46(2013)
5
Modelling
From single decision maker minusrarr multiple agents
From optimization minusrarr equilibrium
From cooperation minusrarr non-cooperation
From centralized decision making minusrarr distributed responses
Mathematics
From WeierstrassCauchy minusrarr BrouwerBanach
From smooth calculus minusrarr variational analysis
From coordinated descent minusrarr asynchronous computation
From potential function minusrarr non-expansion of mappings
6
Lecture I Formulations and a recent model
930 ndash 1030 AM Monday September 23 2019
7
The Abstract Generalized Nash Equilibrium Problem (GNEP)
N decision makers each (labeled ν = 1 middot middot middot N) with
bull a moving strategy set Ξν(xminusν) sube Rnν and
bull a cost function θν(bull xminusν) Rnν rarr R
both dependent on the rivalsrsquo strategy tuple
xminusν ≜983059xν
prime983060
ν prime ∕=νisin Rminusν ≜
983132
ν prime ∕=ν
Rnν prime
Anticipating rivalsrsquo strategy xminusν player ν solves
minimizexνisinΞν(xminusν)
θν(xν xminusν)
A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that
xlowast ν isin argminxνisinΞν(xlowastminusν)
θν(xν xlowastminusν) forall ν = 1 middot middot middot N
In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy
8
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
A brief historical account
Gerard Debreu A social equilibrium existence theorem Proceedings of theNational Academy of Sciences 38(10) 886ndash893 1952
Kenneth J Arrow and Gerard Debreu Existence of an equilibrium fora competitive economy Econometrica Journal of the Econometric Society22(3) 265ndash290 1954
Hukukane Nikaido and Kazuo Isoda Note on noncooperative convexgames Pacific Journal of Mathematics 5(Suppl 1) 807-815 1955
J Ben Rosen Existence and uniqueness of equilibrium points for concaven-person games Econometrica Journal of the Econometric Society 33(3)520ndash534 1965
Alain Bensoussan Points de Nash dans le cas de fontionnelles quadratiques
et jeux differentiels lineaires a N personnes SIAM Journal on Control 12(3)
460ndash499 1974
3
A brief historical account (cont)Patrick T Harker Generalized Nash games and quasi-variationalinequalities European Journal of Operational Research 54(1) 81ndash94 1991
Jong-Shi Pang and Masao Fukushima Quasi-variational inequalitiesgeneralized Nash equilibria and multi-leader-follower games ComputationalManagement Science 2 21-56 2005 [Erratum same journal 6 373-3752009]
Francisco Facchinei and Christian Kanzow Generalized Nashequilibrium problems Annals of Operations Research 175(1)177ndash211 2007
Francisco Facchinei and Jong-Shi Pang Nash equilibria the variationalapproach Chapter 12 in Daniel P Palomar and Yonica C Eldar editorsConvex optimization in signal processing and communications CambridgeUniversity Press pages 443ndash493 2010
Francisco Facchinei Computation of generalized Nash equilibria Recent
advances Part II in Roberto Cominetti Francisco Facchinei and Jean-Bernard
Lasserre Editors and authors Modern Optimization Modeling Techniques
Birkhauser Springer Basel pp 133ndash184 2012
4
Some recent references
Extensive papers by our host Christian Kanzowhttpswwwmathematikuni-wuerzburgdeoptimizationteam
kanzow-christian
Q Ba and JS Pang Exact penalization of generalized Nashequilibrium problems Operations Research (accepted August 2019) inprint
DA Schiro JS Pang and UV Shanbhag On the solution ofaffine generalized Nash equilibrium problems with shared constraints byLemkersquos method Mathematical Programming Series A 142(1ndash2) 1ndash46(2013)
5
Modelling
From single decision maker minusrarr multiple agents
From optimization minusrarr equilibrium
From cooperation minusrarr non-cooperation
From centralized decision making minusrarr distributed responses
Mathematics
From WeierstrassCauchy minusrarr BrouwerBanach
From smooth calculus minusrarr variational analysis
From coordinated descent minusrarr asynchronous computation
From potential function minusrarr non-expansion of mappings
6
Lecture I Formulations and a recent model
930 ndash 1030 AM Monday September 23 2019
7
The Abstract Generalized Nash Equilibrium Problem (GNEP)
N decision makers each (labeled ν = 1 middot middot middot N) with
bull a moving strategy set Ξν(xminusν) sube Rnν and
bull a cost function θν(bull xminusν) Rnν rarr R
both dependent on the rivalsrsquo strategy tuple
xminusν ≜983059xν
prime983060
ν prime ∕=νisin Rminusν ≜
983132
ν prime ∕=ν
Rnν prime
Anticipating rivalsrsquo strategy xminusν player ν solves
minimizexνisinΞν(xminusν)
θν(xν xminusν)
A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that
xlowast ν isin argminxνisinΞν(xlowastminusν)
θν(xν xlowastminusν) forall ν = 1 middot middot middot N
In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy
8
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
A brief historical account (cont)Patrick T Harker Generalized Nash games and quasi-variationalinequalities European Journal of Operational Research 54(1) 81ndash94 1991
Jong-Shi Pang and Masao Fukushima Quasi-variational inequalitiesgeneralized Nash equilibria and multi-leader-follower games ComputationalManagement Science 2 21-56 2005 [Erratum same journal 6 373-3752009]
Francisco Facchinei and Christian Kanzow Generalized Nashequilibrium problems Annals of Operations Research 175(1)177ndash211 2007
Francisco Facchinei and Jong-Shi Pang Nash equilibria the variationalapproach Chapter 12 in Daniel P Palomar and Yonica C Eldar editorsConvex optimization in signal processing and communications CambridgeUniversity Press pages 443ndash493 2010
Francisco Facchinei Computation of generalized Nash equilibria Recent
advances Part II in Roberto Cominetti Francisco Facchinei and Jean-Bernard
Lasserre Editors and authors Modern Optimization Modeling Techniques
Birkhauser Springer Basel pp 133ndash184 2012
4
Some recent references
Extensive papers by our host Christian Kanzowhttpswwwmathematikuni-wuerzburgdeoptimizationteam
kanzow-christian
Q Ba and JS Pang Exact penalization of generalized Nashequilibrium problems Operations Research (accepted August 2019) inprint
DA Schiro JS Pang and UV Shanbhag On the solution ofaffine generalized Nash equilibrium problems with shared constraints byLemkersquos method Mathematical Programming Series A 142(1ndash2) 1ndash46(2013)
5
Modelling
From single decision maker minusrarr multiple agents
From optimization minusrarr equilibrium
From cooperation minusrarr non-cooperation
From centralized decision making minusrarr distributed responses
Mathematics
From WeierstrassCauchy minusrarr BrouwerBanach
From smooth calculus minusrarr variational analysis
From coordinated descent minusrarr asynchronous computation
From potential function minusrarr non-expansion of mappings
6
Lecture I Formulations and a recent model
930 ndash 1030 AM Monday September 23 2019
7
The Abstract Generalized Nash Equilibrium Problem (GNEP)
N decision makers each (labeled ν = 1 middot middot middot N) with
bull a moving strategy set Ξν(xminusν) sube Rnν and
bull a cost function θν(bull xminusν) Rnν rarr R
both dependent on the rivalsrsquo strategy tuple
xminusν ≜983059xν
prime983060
ν prime ∕=νisin Rminusν ≜
983132
ν prime ∕=ν
Rnν prime
Anticipating rivalsrsquo strategy xminusν player ν solves
minimizexνisinΞν(xminusν)
θν(xν xminusν)
A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that
xlowast ν isin argminxνisinΞν(xlowastminusν)
θν(xν xlowastminusν) forall ν = 1 middot middot middot N
In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy
8
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Some recent references
Extensive papers by our host Christian Kanzowhttpswwwmathematikuni-wuerzburgdeoptimizationteam
kanzow-christian
Q Ba and JS Pang Exact penalization of generalized Nashequilibrium problems Operations Research (accepted August 2019) inprint
DA Schiro JS Pang and UV Shanbhag On the solution ofaffine generalized Nash equilibrium problems with shared constraints byLemkersquos method Mathematical Programming Series A 142(1ndash2) 1ndash46(2013)
5
Modelling
From single decision maker minusrarr multiple agents
From optimization minusrarr equilibrium
From cooperation minusrarr non-cooperation
From centralized decision making minusrarr distributed responses
Mathematics
From WeierstrassCauchy minusrarr BrouwerBanach
From smooth calculus minusrarr variational analysis
From coordinated descent minusrarr asynchronous computation
From potential function minusrarr non-expansion of mappings
6
Lecture I Formulations and a recent model
930 ndash 1030 AM Monday September 23 2019
7
The Abstract Generalized Nash Equilibrium Problem (GNEP)
N decision makers each (labeled ν = 1 middot middot middot N) with
bull a moving strategy set Ξν(xminusν) sube Rnν and
bull a cost function θν(bull xminusν) Rnν rarr R
both dependent on the rivalsrsquo strategy tuple
xminusν ≜983059xν
prime983060
ν prime ∕=νisin Rminusν ≜
983132
ν prime ∕=ν
Rnν prime
Anticipating rivalsrsquo strategy xminusν player ν solves
minimizexνisinΞν(xminusν)
θν(xν xminusν)
A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that
xlowast ν isin argminxνisinΞν(xlowastminusν)
θν(xν xlowastminusν) forall ν = 1 middot middot middot N
In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy
8
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Modelling
From single decision maker minusrarr multiple agents
From optimization minusrarr equilibrium
From cooperation minusrarr non-cooperation
From centralized decision making minusrarr distributed responses
Mathematics
From WeierstrassCauchy minusrarr BrouwerBanach
From smooth calculus minusrarr variational analysis
From coordinated descent minusrarr asynchronous computation
From potential function minusrarr non-expansion of mappings
6
Lecture I Formulations and a recent model
930 ndash 1030 AM Monday September 23 2019
7
The Abstract Generalized Nash Equilibrium Problem (GNEP)
N decision makers each (labeled ν = 1 middot middot middot N) with
bull a moving strategy set Ξν(xminusν) sube Rnν and
bull a cost function θν(bull xminusν) Rnν rarr R
both dependent on the rivalsrsquo strategy tuple
xminusν ≜983059xν
prime983060
ν prime ∕=νisin Rminusν ≜
983132
ν prime ∕=ν
Rnν prime
Anticipating rivalsrsquo strategy xminusν player ν solves
minimizexνisinΞν(xminusν)
θν(xν xminusν)
A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that
xlowast ν isin argminxνisinΞν(xlowastminusν)
θν(xν xlowastminusν) forall ν = 1 middot middot middot N
In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy
8
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Lecture I Formulations and a recent model
930 ndash 1030 AM Monday September 23 2019
7
The Abstract Generalized Nash Equilibrium Problem (GNEP)
N decision makers each (labeled ν = 1 middot middot middot N) with
bull a moving strategy set Ξν(xminusν) sube Rnν and
bull a cost function θν(bull xminusν) Rnν rarr R
both dependent on the rivalsrsquo strategy tuple
xminusν ≜983059xν
prime983060
ν prime ∕=νisin Rminusν ≜
983132
ν prime ∕=ν
Rnν prime
Anticipating rivalsrsquo strategy xminusν player ν solves
minimizexνisinΞν(xminusν)
θν(xν xminusν)
A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that
xlowast ν isin argminxνisinΞν(xlowastminusν)
θν(xν xlowastminusν) forall ν = 1 middot middot middot N
In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy
8
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The Abstract Generalized Nash Equilibrium Problem (GNEP)
N decision makers each (labeled ν = 1 middot middot middot N) with
bull a moving strategy set Ξν(xminusν) sube Rnν and
bull a cost function θν(bull xminusν) Rnν rarr R
both dependent on the rivalsrsquo strategy tuple
xminusν ≜983059xν
prime983060
ν prime ∕=νisin Rminusν ≜
983132
ν prime ∕=ν
Rnν prime
Anticipating rivalsrsquo strategy xminusν player ν solves
minimizexνisinΞν(xminusν)
θν(xν xminusν)
A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that
xlowast ν isin argminxνisinΞν(xlowastminusν)
θν(xν xlowastminusν) forall ν = 1 middot middot middot N
In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy
8
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Notation
bull x ≜ (xν)Nν=1
bull Ξ(x) ≜N983132
ν=1
Ξν(xminusν)
bull FIXΞ ≜ x x isin Ξ(x)
bull the GNEP (Ξ θ)
Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP
9
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications
Ξν(xminusν) ≜
983099983103
983101xν isin Rnν hν(xν) le 0
private constraints
gν(xν xminusν) le 0
coupled constraints
983100983104
983102
where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν
i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν
and every xminusν the component function gνi (bull xminusν) is convex
bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132
ν=1
Xν and C sub Rn
where n ≜N983131
ν=1
nν are closed and convex
bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems
bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1
each component function gi is convex and Lagrange multipliers for the
constraint g(x) le 0 is the same for all players leading to variational equilibria10
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Quasi variational inequality (QVI) formulation
bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1
is a single-valued map
bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)
(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X
11
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Complementarity formulationbull Each θν(bull xminusν) is differentiable and
Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0
Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103
983105983101
0 le xν perp nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi ge 0
0 le λν perp gν(x) le 0
983100983105983104
983105983102ν = 1 middot middot middot N
yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications
0 le983061
xλ
983062
983167 983166983165 983168denoted z
perp
983091
983109983109983107
983075nablaxνθν(x) +
mν983131
i=1
nablaxνgνi (x)λνi
983076N
ν=1
minus ( gν(x) )Nν=1
983092
983110983110983108
983167 983166983165 983168denoted F(z)
ge 0
12
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
One recent model in transportation e-hailing
Contributions
bull realistic modeling of a topical research problem
bull novel approach for proof of solution existence
bull awaiting design of convergent algorithm
bull opportunity in model refinements and abstraction
J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print
13
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices
Formulation as a GNEMOPECMulti-VI
bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures
bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility
bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple
bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served
14
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Network set-up
N set of nodes in the network
A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K
besides being the destinations of the OD pairs where
customers are dropped off these are also the locations
where the e-HSP drivers initiate their next trip to pick up
other customers
Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the
e-HSP labels
15
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Overall model is to determine
bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K
983154for each type
m destination j and OD pair k
bull travel demands Qmk m isin M k isin K for each e-HSP type m and
OD pair k
bull travel times tij (i j) isin N timesN of shortest path from node i to j
bull vehicular flows hp p isin P on paths in network
16
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Interaction of modules in model
17
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The e-HSP module
The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as
Rmjk ≜ 983141Rm
jk minus βm3 983141wm
k983167983166983165983168e-HSPm waiting timeincurring loss of revenue
where
983141Rmjk ≜ Fm
Okminus βm
1 ( tjOk+ tOkDk
)983167 983166983165 983168travel time based cost
minus βm2 ( djOk
+ dOkDk)
983167 983166983165 983168travel distance based cost
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168time based revenue
+ αm2 dOkDk983167 983166983165 983168
distance based revenue
Anticipating Rmjk as a parameter e-HSPm company decides on the
allocation zmjk of vehicles from location j to serve OD pair k by solving
18
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
e-HSPrsquos choice module of profit maximization for m isin M
maximizezmjkge0
983131
kisinK
983131
jisinD
983141Rmjk z
mjk
983167 983166983165 983168average trip profit
minusβm3
983093
983095Nm minus983131
kisinK
983131
jisinDzmjktjOk
minus983131
kisinKQm
k tOkDk
983094
983096
983167 983166983165 983168cost due to waiting
subject to983131
kisinKzmjk =
983131
k prime j=Dk prime
Qmk prime
983167 983166983165 983168available e-HSP vehicles for service
for all j isin D
983131
jisinDzmjk ge Qm
k
983167 983166983165 983168OD demands served by e-HSPm
for all k isin K
and983131
kisinK
983131
jisinDzmjk tjOk
983167 983166983165 983168trip hours of vehiclesen route to service calls
+983131
kisinKQm
k tOkDk
983167 983166983165 983168trip hours of vehiclesserving travel demands
le Nm983167983166983165983168
e-HSPm vehicleavailability
19
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The passenger module
An e-HSPm customerrsquos disutility V mk for m isin M
FmOk
+ αm1 ( tOkDk
minus f 0OkDk
)983167 983166983165 983168
time based fare
+ αm2 dOkDk983167 983166983165 983168
distance based fare
+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility
+wmk (disutility due to waiting for e-HSPsrsquo pick up)
For a solo driver the disutility can be expressed as
V 0k = γ01 tOkDk983167 983166983165 983168
travel timebased disutility
+ β02 dOkDk983167 983166983165 983168
distancebased disutility
Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on
travel modes (solo or e-hailing) by solving
20
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Passenger mode choice of disutility minimization
minimizeQm
k ge0
983131
misinM+
983131
kisinKV mk Qm
k
subject to
983131
jisinDzmjk ge Qm
k
shared constraints
for all ( km ) isin K timesM
983131
misinM+
Qmk = Qk983167983166983165983168
given
for all k isin K
where M+ = M cup solo mode
21
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Network congestion module
Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes
0 le tij perp983131
pisinPij
hp minus
983093
983095983131
kisinKδODijk Qk +
983131
(kℓ)isinKtimesKδeminusHSPijkℓ
983131
misinMzmiℓ
983094
983096 ge 0
for all (i j) isin N timesN
0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where
δODijk ≜
9830691 if i = Ok and j = Dk
0 otherwise
983070
δeminusHSPi primej primekℓ ≜
9830691 if i prime = Dk and j prime = Oℓ
0 otherwise
983070
22
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Customer waiting disutility
wmk = γm2
983131
jisinDzmjk tjOk
983131
jisinDzmjk
983167 983166983165 983168average travel time to origin ofOD-pair k from all locations
for all m isin M
Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero
End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable
A penalty approach is employed for proving solution existence23
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Lecture II Existence results with and without convexity
1430 ndash 1530 AM Monday September 23 2019
24
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Recall QVI formulation convex player problems
If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x
lowast) such that
(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)
or equivalently existalowast isin Θ(xlowast) ≜N983132
ν=1
partxνθν(xlowast) such that
(xminus xlowast )⊤alowast =
N983131
ν=1
(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)
where Ξ(x) ≜N983132
ν=1
Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)
25
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K
Proof Apply Kakutani fixed-point theorem to the self-map
y isin 983141K 983041rarrN983132
ν=1
argminxνisinΞ ν(yminusν)
θν(xν yminusν) sube 983141K
Assumptions ensure applicability of this theorem
Applicable to the jointly convex compact case with
graph Ξ ν = C983167983166983165983168shared constraints
times 983141X983167983166983165983168private constraints
for all ν
26
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector
xref ≜983059xrefν
983060N
ν=1isin Ω and a continuous function s Ω rarr Ω such that
ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω
ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where
Llt ≜983083
y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)
( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)
983084
bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded
27
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and
bull Ξν(xminusν) ≜
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν
G(x) le 0983167 983166983165 983168nonconvex
983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168
shared
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1
Given prices (π 983141π) and anticipating xminusν player νrsquos problem
minimizexν isinX ν
983093
983097983097983097983097983097983097983095θν(x
ν xminusν) +
p983131
j=1
πj Gj(xν xminusν) +
983141p983131
j=1
983141πj 983141Gj(xν xminusν)
983167 983166983165 983168denoted Lν(xπ 983141π)
983094
983098983098983098983098983098983098983096
28
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The game G(Θ HG 983141G)
A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that
xlowastν isin argminxν isinX ν
Lν(xν xlowastminusν πlowast 983141π lowast) (1)
for every ν = 1 middot middot middot N and
0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0
Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves
(πlowast 983141π lowast) isin minimize(π983141π)ge 0
π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)
A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by
xlowastν isin argminxν isinX ν capN ν
Lν(xν xlowastminusν πlowast 983141π lowast)
29
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)
where K ≜ Ξ times Rp+ times
N983132
ν=1
Rℓν+ with
Ξ ≜
983099983105983105983103
983105983105983101x = (xν )
Nν=1 isin
N983132
ν=1
Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints
983100983105983105983104
983105983105983102 polyhedral
Φ(xπλ) ≜983091
983109983109983109983109983109983109983107
983091
983107nablaxνθν(x) +
p983131
j=1
πj nablaxνGj(x) +
ℓν983131
i=1
λνi nablahν
i (xν)
983092
983108N
ν=1
VI Lagrangian
Ψ(x) ≜983075
minusG(x)
minusH(x)
983076nonconvexconstraints
983092
983110983110983110983110983110983110983108
30
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Roadmap of the Analysis
For a QNE
bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE
mdash need a Slater condition plus a copositivity assumption
bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE
For a NE (model remains nonconvex)
bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE
mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian
bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument
31
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is
C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast
where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is
F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn
where ΠK is the Euclidean projector onto Kbull F nor
K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor
K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0
32
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set
Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded
(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set
Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded
(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and
(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)
and
C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0
ie if (C(KF xlowast) JF (xlowast)) is a R0 pair
Proof by applying degree theory to the normal map F norK (z)
33
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Existence of QNE
The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν
983060N
ν=1isin Ξ
such that
(Slater condition of nonconvex constraints) Ψ(xref) ≜983061
H(xref)G(xref)
983062lt 0
(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is
copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N
(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p
(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0
983154is bounded (possibly empty)
34
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Uniqueness of QNE
For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First
JΦ(xχ) =
983077A(xχ) JΨ(x)T
minusJΨ(x) 0
983078 where χ ≜ (πλ ) and
A(xχ) ≜ JΘ(x) +
p983131
j=1
πj nabla2Gj(x) + blkdiag
983077ℓν983131
i=1
λνi nabla2hνi (x
ν)
983078N
ν=1983167 983166983165 983168Jacobian of VI Lagrangian
The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)
C(KΦ y) =
983141C(xχ) times983045T (Rp
+π) cap G(x)perp983046times
N983132
ν=1
983147T (Rℓν
+ λν) cap hν(x)perp983148
where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)
The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence
bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0
983154is bounded
bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)
bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)
36
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE
37
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
When is a QNE a LNE
Answer Under a second-order sufficiency condition as in NLP
Let L(2)ν (xπλν) ≜ nabla2
xνθν(x) +
p983131
j=1
πj nabla2xνGj(x) +
ℓν983131
i=1
λνi nabla2hνi (x
ν)
note that
A(xχ) = Diag983059L(2)ν (xπλν)
983060N
ν=1983167 983166983165 983168diagonal blocks
+ Off-diagonal blocks of JΘ(x)
The critical cone of player qrsquos optimization problem
C ν(xlowastπlowast 983141π lowast) ≜
T (X ν xlowastν) cap
983093
983095nablaxνθν(xlowast) +
p983131
j=1
πlowastj nablaxνGj(x
lowast) +
983141p983131
j=1
983141π lowastj nablaxν 983141Gj(x
lowast)
983094
983096perp
bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each
ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)
then (xlowastπlowast 983141π lowast) is a LNE37
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems
minimizexν isinXν and h ν(xν)le 0
Lν(xπ 983141π) (2)
bull no explicit bounds on prices π and 983141πminimize
983141πge 0983141π T 983141G(x) and minimize
πge 0πTG(x)
Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with
minimize983141π isin 983141St
983141π T 983141G(x) and minimizeπ isinSt
πTG(x)
where St ≜
983099983103
983101π ge 0 |p983131
j=1
πj le t
983100983104
983102 and similarly for 983141St
38
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0
Write 983141X ≜N983132
ν=1
Xν and let Λνt ≜
983083λν ge 0 |
ℓν983131
i=1
λνi le ξt
983084 where
ξt ≜
max(micro983141micro)isinSttimes983141St
983099983103
983101maxxisin 983141X
983055983055983055983055983055983055
Q983131
q=1
(xrefν minus xν )TnablaxνLν(x micro 983141micro)
983055983055983055983055983055983055
983100983104
983102
min1leνleN
983061min
1leileℓν(minushνi (x
refν) )
983062
Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)
(b) the matrix L(2)ν (x microλν) is positive definite for all
(x microλν) isin 983141X times St times Λνt
Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization
39
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Bounding the prices and removing truncation
There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies
p983131
j=1
πtj +
983141p983131
j=1
983141π tj +
N983131
ν=1
ℓν983131
i=1
λtνi le ξlowast
If t gt ξlowast then the price-truncation constraints983141p983131
j=1
983141πj le t and
p983131
j=1
πj le t are not binding thus recovering the
price complementarity condition
40
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Existence and Uniqueness of NE
(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N
bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St
bull the matrix L(2)ν (xπλq) is positive definite for all
(xπλν) isin 983141X times S t times Λνt
Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St
(B) If the matrix A(xπλ) is positive definite for all
(xπλ) isin 983141X times St timesN983132
ν=1
Λνt then the x-component of a NE of the game
G(Θ HG 983141G ) is unique
41
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq
0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0
for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq
Dominant player qrsquos optimization problem is
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq perp rq +Nqzq +Mqyq ge 0
The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints
Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Existence of ε-QNEFor every ε gt 0 consider the relaxed problem
minimizezq yq wq
θq(zq yq zminusq)
subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq
0 le yq wq ≜ rq +Nqzq +Mqyq ge 0
and yqi wqi le ε i = 1 middot middot middot ℓq
Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103
983105983105983105983105983105983101
( zq yq )Qq=1 isin
Q983132
q=1
Xq |
Q983131
q=1
983045( zq minus zrefq )Tnablazqθq(z
q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)
983046lt 0
983100983105983105983105983105983105983104
983105983105983105983105983105983102
is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the
multi-leader-follower game exists43
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Lecture III Exact penalization via error boundsand constraint qualifications
1100 ndash noon Tuesday September 24 2019
44
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Recall the GNEP which we denote G(CX θ) where player ν solves
minimizexν isinC ν capXν(xminusν)
θν(xν xminusν)
where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and
983045xν isin C ν and rν(x
ν xminusν) = 0983046
hArr xν isin C ν cap Xν(xminusν)
Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x
ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves
minimizexν isinC ν
θνρX(xν xminusν)
(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ
G(CX θ) hArr NEP(C θρX)
45
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Review (partial) exact penalization of optimization problems
ConsiderminimizexisinW capS
f(x)
where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty
The penalized optimization problem for a ρ gt 0
minimizexisinW
f(x) + ρ rS(x)
where rS(x) is a W -residual function of the set S
Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ
argminxisinS capW
f(x) = argminxisinW
f(x) + ρ rS(x)
46
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Exact penalization of games Preliminary result
bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ
bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν
ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness
ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and
ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)
Then for all ρ gt maxνisin[N ]
Lipνγν every equilibrium solution of the
penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa
47
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Example Consider the shared-constrained GNEP with 2 players
Player 1 minimizex1isinR
x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+
(x2 minus 12 )2
subject to x1 + x2 le 1 | subject to x1 + x2 le 1
The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are
0 = 2x2 (x1 minus 1 ) + λ1
0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0
0 le λ1 perp 1minus x1 minus x2 ge 0
0 le λ2 perp 1minus x1 minus x2 ge 0
This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312
12
983044is not a normalized equilibrium in the sense of Rosen
Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach
48
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Example Consider the shared-constrained GNEP with 2 players
C1 == C2 = [ 0 4 ] private constraints
commonconstraints
983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2
(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2
(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1
4x1 x2
x2
0 x1
g1
g2
g32
249
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
bull ℓ1 penalization is not exact
r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+
bull Squared ℓ2 penalization is not exact
r2(x1 x2)2 =
[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]
2 + [(minusx1 + 2x2 minus 2 )+]2
bull ℓ2 penalization is exact
bull ℓ1 penalization is exact for variational equilibria (VE)
bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with
983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)
and similarly for 983144C2 Further research is needed
bull ℓ2-penalized NE is not VE50
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The jointly convex GNEP (CD θ)
bull C ≜N983132
ν=1
C ν is the product of the private constraint sets
bull for some closed convex subset D of Rn
graph X ν = D for all ν
bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )
Notation
Let T (xS) denote the tangent cone of the set S at x isin S
Let ϕ prime(x dx) ≜ limτdarr0
ϕ(x+ τ dx)minus ϕ(x)
τbe the directional derivative of
ϕ at x along the direction dx
51
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Theorem Suppose
bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν
bull for every x = (xν)Nν=1 isin C D either one of the following holds
ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν
rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422
ndash for some nonzero tangent vector d isin T (xC)
983131
νisin[N ]
rD(bull xminusν) prime(xν d ν) le r primeD(x d)
983167 983166983165 983168sum property of directional derivative
le minusα 983042 d 9830422
then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)
52
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Linear metric regularity
Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that
dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn
This holds if either
bull C capD is bounded and rint(C) cap rint(D) ∕= empty or
bull C and D are polyhedra
Corollary Exact penalization holds for shared constrained GNEP if
bull C and D are linear metrically regular
bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D
53
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Shared finitely representable setsLet C be convex and compact and
D = x isin Rn | g(x) le 0 and h(x) = 0
where h is affine and each gj is convex and differentiable Let
rq(x) ≜983056983056983056983056983056
983075max( g(x) 0 )
h(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) which is differentiable at x isin D
Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then
bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution
Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ
NEP (C θ + ρ rq) rArr GNEP (CD θ)
54
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Non-shared finitely representable sets
Similar setting but with
X ν(xminusν) ≜
983099983105983105983103
983105983105983101
xν isin Rnν | gν(xν xminusν) le 0 and
| hν(x) = 0983167 983166983165 983168linear equalities allowed
983100983105983105983104
983105983105983102
where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let
rνq(x) ≜983056983056983056983056983056
983075max( gν(x) 0 )
hν(x)
983076983056983056983056983056983056q
x isin Rn
for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)
Let rXq(x) ≜ ( rνq(x) )Nν=1
55
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Theorem Suppose there exists α gt 0 such that for every
x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with
x983141ν ∕isin X983141ν(xminus983141ν)
bull for all i isin 1 middot middot middot mν
g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168
uniform descent condition at infeasible x
bull for all j isin 1 middot middot middot pν
h983141νj (x) ∕= 0 rArr
983147sgn h983141ν
j (x)983148nablax983141νh983141ν
j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα
983167 983166983165 983168uniform descent condition at infeasible x
Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)
56
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only
X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0
983052no equalities
For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat
gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0
Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution
EMFCQ is more restrictive than required for the validity of exactpenalization
57
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Lecture IV Best-response algorithms
930 ndash 1030 AM Wednesday September 25 2019
58
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function
y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)
N983131
ν=1
983045θν(x
ν yminusν) + 12 983042x
ν minus yν 9830422983046
Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x
A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot
Theoretically when is this a contraction a non-expansion
Computationally allow distributed player optimization
minimizexνisinΞν(ykminusν)
θν(xν ykminusν) + 1
2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N
each of which is a strongly convex optimization problem
59
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Convergence theoryBasic version requires
bull Ξν(xminusν) = Xν for all ν and
bull the second derivatives nabla2xν primexνθν(x) ≜
983077part2θν(x)
partxνprime
j xνi
983078(nν nν prime )
(ij)=1
exist and are
bounded on 983141X ≜N983132
ν=1
Xν Weakening to a 4-point condition of first
derivatives is possible (Hao 2018 PhD thesis)
A matrix-theoretic criterion Let
ζ νmin ≜ inf
xisin983141Xsmallest eigenvalue of nabla2
xνθν(x)
ξ νν primemax ≜ sup
xisin983141X
983056983056983056nabla2xνxν primeθν(x)
983056983056983056
60
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61
Define the N timesN Z-matrix (all off-diagonal entries are non-positive)
Υ ≜
983093
983097983097983097983097983097983097983097983097983097983097983097983097983095
ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax
minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax
minusξ(Nminus1) 1max minusξ
(Nminus1) 2max middot middot middot ζNminus1
min minusξ(Nminus1)Nmax
minusξN 1max minusξN 2
max middot middot middot minusξN Nminus1max ζNmin
983094
983098983098983098983098983098983098983098983098983098983098983098983098983096
bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration
bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists
61