Complex Variational Inequalities Generalized Nash...

Complex Variational Inequalities

Generalized Nash Equilibrium

Problems and Applications

Francisco FacchineiDepartment of Computer, Control, and Management Engineering

University of Rome La Sapienza

Game Theory @ the Universities of Milan IIIPolitecnico di Milano

June 23-24, 2013

Co-authorsWirtinger Derivatives Complex VIs Complex (G)NEPs Applications

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Based on work done with

Gesualdo Scutari, State University of New York at Buffalo, USA

Jong-Shi Pang, University of Illinois at Urbana-Champaign, USA

Daniel P. Palomar, Hong-Kong University of Science and Technology,Hong-Kong

Main MotivationWirtinger Derivatives Complex VIs Complex (G)NEPs Applications

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Main aim: To study algorithms for (Generalized) Nash Equilibrium Problems(GNEPs) whose decision variables are complex

Main difficulty: Consider

min f(z), z ∈ K ⊆ Cn

with f real valued. f is not differentiable in the holomorphic sense. Shouldwe give up the whole (algorithmic) methods based on differential properties?

The answer is NO. This is the topic of this talk.

OutlineWirtinger Derivatives Complex VIs Complex (G)NEPs Applications

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Wirtinger Derivatives

Complex VI and related issues

Complex (G)NEPs and algorithms

Applications in telecommunications

Numerical results

C- vs. R-differentiabilityWirtinger Derivatives Complex VIs Complex (G)NEPs Applications

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f : Cn → C

The scalar field is F, with F = R or F = C

f is F-Frechet differentiable at z if an F-linear function DFf(z) : Cn → C

exists such that, for any h ∈ Cn,

f(z + h) = f(z) +DFf(z)(h) + (‖h‖)

• F = C =⇒ f is holomorphic

• F = R =⇒ ?

f(z) = Φ(z) + jΨ(z) = Φ(x, y) + jΨ(x, y), (z = x+ jy)

• F = R =⇒ Φ and Ψ are differentiable

Wirtinger Derivatives Complex VIs Complex (G)NEPs Applications

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If F = R then R-differentiability, i.e. standard real differentiability of Φ andΨ, is enough for approximation, Taylor expansion, optimality conditions etc.

End of the story? NO

Why? In our applications we need to consider very complicated functions(complex valued matrices that depend on matrices of variables, and a lot ofstructure which has a physical meaning....)

Any original structure is lost;

Calculations are very difficult;

Relating the results to the original problem becomes almost impossible

Wirtinger calculus...is quite indispensable in the function theory of severalvariables(from “Theory of Complex Functions” (1991) by R. Remmert)


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MATHEMATICIANS (Some) ENGINEERS

mainly interested in mainly interested inholomorphic functions real differentiable functions

⇓

Wirtinger calculus is little known outside restricted circlesand its developments are sometime a bit sloppy/shaky


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Let f : Cn → C be given

•∂f(z)

∂z,

1

2

(∂f(z)∂x− j ∂f(z)

∂y

)Wirtinger derivative

•∂f(z)

∂z∗,

1

2

(∂f(z)∂x

+ j ∂f(z)∂y

)Wirtinger conjugate derivative

We also set: ∇zf(x) , (∂f(z)/∂z)T ∇z∗f(x) , (∂f(z)/∂z∗)T

Wirtinger derivatives are easy to compute. If we can write f(z) asan expression using z and z∗ and these expression is holomorphic in z(z∗) when z∗ (z) is fixed, then Wirtinger (conjugate) derivative can beobtained by differentiating in the standard way with respect z (z∗)

f(z) = |z|2. This is not holomorphic but clearly R-differentiable.

f(z) = zz∗ =⇒∂f(z)

∂z= z∗,

∂f(z)

∂z∗= z


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We can write Taylor expansions:

f(z + h) = f(z) +∂f(z)

∂zh+

∂f(z)

∂z∗h∗ + (‖h, h∗‖)

and

f(z + h) = f(z) +∂f(z)

∂zh+

∂f(z)

∂z∗h∗

+1

2

(hH , h∗H

)

∂2f(z)∂z∗∂z

∂2f(z)∂2z∗

∂2f(z)∂2z

∂2f(z)∂z∂z∗

(

hh∗

)+ (‖h, h∗‖2),

Note differences and similarities with the real case....Using these Taylor expansions, the definition of Wirtinger derivatives, sometricks and mimicking proofs valid in the real case, we can extend in anappealing way many classical results


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Assuming it is R-differentiable, f : Cn → R is convex on the convex set Ω ifand only if the following conditions hold for any two z1, z2 ∈ Ω:

f(z2) ≥ f(z1) + 2〈z2 − z1,∇z∗f(z1)〉

〈z2 − z1,∇z∗f(z2)−∇z∗f(z1)〉 ≥ 0

where 〈a, b〉 , Re(aHb)This definition of inner product is essential

Assuming Ω is also closed, a point z ∈ Ω is a solution of the optimizationproblem minz∈Ω f(z), if and only if

〈z − z,∇z∗f(z)〉 ≥ 0, ∀z ∈ Ω


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Wirtinger calculus is powerful, “friendly” and permits to extend in asystematic way very many results valid for differentiable real functions of realvaraibles to R−differentiable (complex) functions of complex variables. Wewill see that Wirtinger calculus will permit us to study in a useful waycomplex Variational Ineqaulities (VIs) and Nash equilibrium problems

However caution should be exercised, and one should not be fooled by formalsimilarities

For example rules for the derivation of composite functions are quitedissimilar....


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Given a convex and closed set K ⊆ Cn and a complex-valued function

F (z) : K → Cn, the complex VI problem V I (K,F ), consists in finding a

point z ∈ K such that

〈y − z, F (z)〉 ≥ 0, ∀y ∈ K

This is certainly not new. What is new is the the use of Wirtinger derivativeto analyze this VI and to get tractable conditions amenable to applications


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We can readily extend the standard definitions of monotonicity and P propertyto the complex-value matrix mapping FC; the aforementioned definitions arein fact formally the same, with the only difference that the scalar product andthe Euclidean norm are replaced with the inner product 〈•, •〉

The non-trivial task is instead to provide conditions easy to check thatguarantee these properties. These conditions are indeed instrumental to studyconvergence of algorithms for complex NEPs


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F (z) : K → Cn is monotone on (convex) K ⊆ C

n if

〈y − z, F (y)− F (z)〉 ≥ 0, ∀y, z ∈ K

Suppose F is R-differentiable, we define its Wirtinger Jacobian as

JF (z) ,

[∂F (z)∂z

∂F (z)∂z∗

∂F ∗(z)∂z

∂F ∗(z)∂z∗

]

F is monotone on (the open set) K if and only if

(dH , d∗H)JF (z)

(dd∗

)≥ 0, ∀z ∈ K, ∀d ∈ C

n


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Suppose Cn ⊃ K = ΠN

i=1Ki, with Ki ⊆ Cni and convex

F : K → Cn is uniform P on K if

maxi=1...N

〈yi − xi, Fi(y)− Fi(x)〉 ≥ c‖y − x‖2

for some positive constant c

It is possible to give conditions based on the Wirtinger Jacobian for theuniform P property (see later)

The definitions and results above can be extended to strictlymonotonote, strongly monotone and other kind of functions

All the usual, expected properties obviously hold: strong monotonicity orthe uniform P property give existence and uniqueness of the solution etc.


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Because of the applications to games, we are interested in distributedalgorithms for the solution of a complex VI. We therefore suppose that thefeasible set K is the cartesian product of closed convex sets of lower dimension

K = ΠNi=1Ki, Ki ⊆ C

ni

and that z = (zi)Ni=1, and F = (F i)Ni=1 are partitioned accordingly

We consider two distribute algorithms for the solution of such a partitionedVI. The first one converges (under adequate conditions) to a solution. Thesecond algorithms permits (under adequate conditions) to select, among allsolutions of the VI, the one that minimizes an additional criterion (Nashequilibrium selection)


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Algorithm 1: Jacobi Distributed Algorithm for VIs

S(S.0) : Choose x0 ∈ K and set k = 0.

(S.1) : If xk is a solution of VI (K,F ) stop.

(S.2) : For i = 1, . . . , N set

xik+1 , a solution of V I(Ki, Fi(xi, x−i

k )

(S.3) : Set xk+1 = (xik+1)Ni=1, k ← k + 1 and go to Step 1.

Gauss-Seidel, asynchronous and inexact versions are easily envisaged

This algorithm converges under a condition which implies (and is close to thefact) that F is uniform P. We do not give the details of the condition here,but will see a concrete example later


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Take a problem with two variables x1 ∈ R and x2 ∈ R

K = [1, 3]︸︷︷︸K1

× [−5, 2]︸︷︷︸K2

, F =

(F 1

F 2

)=

(3x1 − x2

0.5x1 + 4x2

)

If xk = (1, 3), then x1k+1 and x2k+1 are the solutions, respectively, of

VI ([1, 3]︸︷︷︸K1

, 3x1−3︸︷︷︸F 1(x1,x2

k)

), VI ([−5, 2]︸︷︷︸K2

, 0.5 + 4x2︸︷︷︸F 2(x1

k,x2)

)


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Suppose that the complex V I(K,F ) is monotone and has more than onesolution. Denote the solution set by SOL(K,F ) and this set is convex.

We may consider the problem

minz φ(z)

z ∈ SOL(K,F )

Where φ is an R−differentiable convex function.

Assuming the complex V I(K,F ) is monotone, this problem can be solved ina distributed way by solving a sequence of uniform P VIs with the distributedalgorithm showed two slides ago.


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Given zk and εk > 0, zk+1 is the solution of the strongly monotone

VI(K,F + εk∇z∗φ+ α ( • − zk )),

where εk and α are positive constants

In other words zk+1 is such that

〈y − zk+1, F (xk+1) + εk ∇z∗φ(zk+1 + α ( zk+1 − zk ) 〉 ≥ 0

∀y ∈ K

Note that VI (X,F + εk∇φ+ α ( • − zk )) is strongly monotone and if α islarge enough can be solved by the distributed algorithm considered previously


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Algorithm 2: Prox-Tikhonov Algorithm

S(S.0) : Let εk be a sequence of positive scalars tending to zerowith

∑εk = ∞. Let α > 0 be arbitrary. Choose z0 ∈ K and set

k = 0.

(S.1) : If zk is a solution, stop.

(S.2) : Find zk+1 ∈ K as the unique solution of

V I(K,F + εk∇z∗φ+ α ( • − zk )),

(S.3) : Set k ← k + 1 and return to Step 1.


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N players

Each player controls zi ∈ Cni

Set n :=∑N

i=1 ni, z :=

z1...zN

∈ C

n, z−i :=

z1...

zi−1

zi+1

...zN

z = (zi, z−i)


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Nash Equilibrium Problem (NEP)

minz1 θ1(z1, z−1)

z1 ∈ K1

. . .minzi θi(z

i, z−i)

zi ∈ Ki

. . .minzN θN (zN , z−N )

zN ∈ KN

Several “optimizers (or players)”

Every optimizer minimizes a different obj. f.

The obj. f. depend on the variables of the other players


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Si(z−i): Optimal solution set of player i for a given z

−i of the other players

z is a Nash equilibrium if

zi ∈ Si(z−i) for all players i

or, equivalently, if for all players

θi(zi, zi) ≤ θi(y, z

−i), ∀y ∈ Ki

No player can improve by unilaterally deviating from the current situation


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θi(zi, z−i) convex in zi for every z

−i

θi R−differentiable

Ki closed and convex

By using the first order optimality conditions for each player, it is easily seenthat

Solve the NEP ⇔ Solve V I(K,F )

K :=∏N

i=1Ki, F (x) :=

∇z1∗θ1(z)

...∇zN∗θN (z)


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Generalized Nash Equilibrium Problem (GNEP)

minz1 θ1(x1, z−1)

z1 ∈ K1(z−1)

. . .minzi θi(z

i, z−i)

xi ∈ Ki(z−i)

. . .minzN θN (zN , z−N )

zN ∈ KN (z−N )

Usually Ki(z−i) = zi : gi(zi, z−i) ≤ 0

Several “optimizers (or players)”

Every optimizer minimizes a different obj. f.

The obj. f. depend on the variables of the other players

Also the feasible sets depend on the other players’ variables


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Si(z−i): Optimal solution set of player i for a given z

−i of the other players

z is a Nash equilibrium if

zi ∈ Si(z−i) for all players i


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A particular case of the Generalized Nash Equilibrium Problem is the

Jointly Convex Nash Equilibrium Problem

which occurs when we have a closed convex set K ⊆ Cn defined by and

Ki(z−i) , zi ∈ Ki : (z

i, z−i) ∈ K

WhenK , z ∈ C

n : h1(z) ≤ 0, . . . , hm(z) ≤ 0

joint convexity means that hi : Cn → R, i = 1, . . . ,m, are jointly convex in all

variables and the constraints h “belong” to all players


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Geometrically a jointly convex GNEP has a simple interpretation:

set X

·

x = (x1, x

2)

X1(x2)

X2(x1)

x1

x2


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In the jointly convex case a VI can still provide a solution of the generalizedgame.

Solve the GNEP. ⇐ Solve V I(K,F )

Jointly Convex

Not all the solutions of the Jointly Convex GNEP are solutions of the VIThe solutions of the VI are called variational (or normalized) equilibria


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Once a game has been reduced to VI, we can apply the distributed algorithmsseen before.

We need to verify on concrete examples that the theoretical conditons weobtained using Wirtinger calculus give rise to veirifiable and meaningfulconditions. This is what we do next.


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A B D E

C

Messages are sent from A to B, D to C and D to E

For each communication, three (noisy) channels are available (red, blueand green)

The users allocate power to each channel

Deterioration of the signals is caused by interference on the same color(= frequency) channels and by noise

Aim: allocate power to channels for ”best” transmission of information


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N users transmitting using

Q channels each and allocating

pi(q) power on each channel q

Transmission quality of user q is measured by user’s transmission rate

Ri(pi, p−i) :=

Q∑

q=1

log(1 + sinri(q)),

sinri(q) :=|Hii(q)|

2pi(q)

σ2i (q) +

∑j 6=i |Hij(q)|2pj(q)

Hij(q) :=Hij(q)√

dγij


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Game approachGiven the power allocation p−i of the other users, each user i may want tochoose his power so as to solve the problem

maxpi Ri(pi, p−i)

∑Qq=1 p

i(q) ≤ Pi

pi ≥ 0

Thus defining a real NEP. This model describes well situations in which eachtransmitter and each receiver has one single antenna and there is nopossibility to decide when to transmit, in which direction etc.. If this is not so,i.e. we are in the so called (and more interesting) MIMO case, the modelremains formally similar, but each player has n2

i complex variables, where n1

is the number of transmit antennas of player i. In this setting a realisticproblem for the i-th player is:


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maximizeQi0

Ri(Qi,Q−i) , log det(I+HH

iiR−i(Q−i)−1HiiQi

)

subject to

(a) :(b) :(c) :(d) :

tr (Qi) ≤ Pi,UH

i Qi = 0,

tr(GH

piQiGpi

)≤ Iavepi , λmax

(FHpiQiFpi

)≤ Ipeakpi , p = 1, 2, . . .

Qi ∈ Qi,

•Qi matrix of complex variables•R−i(Q−i) , Rni

+∑j 6=i

HijQjHHij covariance matrix of the noise plus MUI

•Rni∈ C

nRi×nRi 0 covariance matrix of thermal Gaussian zero mean noise

• (b) represents directions along which the user should not transmit• (c) represent a relaxed version of (b)


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By using Wirtinger calculus the above game is readily seen to be equivalent tothe V I(K,F ) where

K is the cartesian product of the players’ feasible sets

F(Q) , (Fi(Q))Ni=1, with

Fi(Q) , −∇Q∗

iRi(Q) = −HH

ii

Rni

+N∑

j=1

HijQjHHij

−1

Hii

I “cheated” a bit: Fi(Q) is a matrix!


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Consider the Υ ∈ RN×N defined by:

Υ ,

1 if i = j

−ρ(H

†Hii HH

ijHijH†ii

)· INNRij if i 6= j,

where A† denotes the Moore–Penrose pseudoinverse of A, ρ(A) is thespectral radius of A, and INNRij is defined as

INNRij ,

ρ

(Rni

+I∑

j=1PjHijH

Hij

)

λleast (Rni)

whereλleast (Rni) is the minimum eigenvalue of the (positive definite) matrix

Rni


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Using Wirtinger calculus, the optimality conditions and ... some effort, we get

(a) If Υ is positive semidefinite then F is monotone

(b) If Υ is a P-matrix then F is a uniformly P-function and the NEP has aunique Nash equilibrium.

Under these conditions, the distributed algorithms we considered previouslyconverge

The interesting point is that the conditions on Υ have a simple physicalinterpretationIt can be checked that the P-matrix condition is satified if one (or both) ofthe following conditions are satified:

the amount of interference each receiver gets is “small enough”

the amount of interference each transmitters generates is “small enough”


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10 20 30 40 50 60 70 80 903.5

4

4.5

5

5.5

6

6.5

7

7.5

8

iteration index

Sum

−ra

te o

f SU

s (b

it/cu

)

Algorithm in [60] NE selection (min MUI), Alg. 7 No NE selection, Alg. 2

Figure 1: Comparison of distributed algorithms solving the game Gmimo: Sum-rate of

the SUs versus iterations.

ConclusionsWirtinger Derivatives Complex VIs Complex (G)NEPs Applications

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We discussed the so called Wirtinger calculus for dealing with complexfunctions of complex variables which are not holomorphic, and also introduceda few new results

Wirtinger calculus allowed us to deal in a succesful way with problems notpreviously solvable, or solved only in heuristic ways with poor performancestandards

Numerical results are very promising

Furhter problems can be analyzed in similar way, for example Femtocellnetworks

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