Nonconvex Quadratic Problems and Games with …...Nonconvex Quadratic Problems and Games with...

Nonconvex Quadratic Problems and Games withSeparable Constraints

Javier Zazo Ruiz, M.Sc.Advisor: Dr. Santiago Zazo Bello

Universidad Politecnica de MadridEscuela Tecnica Superior de Ingenieros de Telecomunicacion

26 of July 2017

ETSITESCUELA TECNICA SUPERIOR DE INGENIEROS DE TELECOMUNICACIÓN

Outline

1 Nonconvex QPs with separable constraints

2 Squared ranged localization problems

3 Robust worst-case analysis of demand-side management

4 Concluding remarks

Outline

1 Nonconvex QPs with separable constraintsIntroduction to QPsRoadmap to establish strong duality in QPsAlgorithmic framework




Quadratically constrained quadratic problem (QP)

I Let’s consider a general QP:

minx∈Rp

xTA0x + 2bT0 x + c0

s.t. xTAix + 2bTi x + ci ≤ 0 ∀i = 1, . . . , N.

where A0, Ai are symmetric matrices and b0, bi, x ∈ Rp, c0, ci ∈ R.

I If A0 0 and every Ai 0 the problem is convex (≈ easy to solve).

I Otherwise, the problem is non-convex (local minima may exist).

I These problems are generally NP-Hard.

Javier Zazo Dissertation 1 / 44

Partinioning problemsAlso called “Boolean Optimization”

minx

xTA0x

s.t. xi ∈ −1, 1

I The problem is NP-hard (even if A0 0).

I Constraints of the formxi ∈ −1, 1 ⇔ x2

i = 1.

I The MAXCUT.


Polynomial minimization

I Minimize a polynomial over a set of of polynomial inequalities:

min p0(x)

s.t. pi(x) ≤ 0, i = 1, . . . ,m.

I Example:

minx,y,z

x3 − 2xyz + y + 2

s.t. x2 + y2 + z2 − 1 = 0.

Introducing change of variables u = x2, v = yz, we get

min ux− 2vx+ y + 2

s.t. x2 + y2 + z2 − 1 = 0

u− x2 = 0

v − yz = 0.


Polynomial minimizationSix-hump-camel problem

−2 −10

12

−1−0.5

00.5

1

0

5

x1x2

f(x 1

,x2)


Transmit beamforming problemI Determine optimal beams for downlink transmissions.

I The beamformers affect the system performance, causing interference.

minwi,∀i

∑i∈N

wiHwi

s.t.wiTRiiwi

σ2i +

∑j 6=i wj

HRijwj≥ Γi

I The above problem can be relaxed:

minWi,∀i

∑i∈N

tr[Wi]

s.t.tr[RiiWi]

σ2i +

∑j 6=i tr[RijWj ]

≥ Γi

Wi = WiH

Wi 0 ∀i ∈ N

Mats Bengtsson and Bjorn Ottersten, “Optimal and suboptimal transmit beamforming,” in Handbook ofAntennas in Wireless Communications, Lal Chand Godara, Ed., pp. 568–600. CRC Press, 2001.


Semidefinite RelaxationI Given a QP:

minx∈Rp

xTA0x + 2bT0 x + c0

s.t. xTAix + 2bTi x + ci ≤ 0 ∀i = 1, . . . ,m,(1)

we can transform to

minX∈Rp×p,x∈Rp

tr(A0X) + 2bT0 x + c0

s.t. tr(AiX) + 2bTi x + ci ≤ 0 ∀i = 1, . . . ,m

X = xxT .

I We relax the rank constraint to X xTx and reformulate to

minX∈Rp×p,x∈Rp

tr(A0X) + 2bT0 x + c0

s.t. tr(AiX) + 2bTi x + ci ≤ 0 ∀i = 1, . . . ,m[X xxT 1

] 0.

(2)

I Strong duality: problems (1) and (2) attain the same solution.


QP with a single quadratic constraintStrong duality I

minx

xTA0x + 2bT0 x + c0

s.t. xTA1x + 2bT1 x + c1 ≤ 0

Strong duality holds provided Slater’s condition holds:

∃x | xTA1x + 2bT1 x + c1 < 0

Application:

I Trust region method: minimization ofunconstrained problems

minx

f(x)

mind

dTBkd+ 2∇f(xk)T d

s.t. ‖d‖ ≤ ∆k,

where Bk is the Hessian of f(xk).


QP with a single equality constraintStrong duality II

minx

xTA0x + 2bT0 x + c0

s.t. g1(x) = xTA1x + 2bT1 x + c1 = 0

Strong duality holds provided A1 6= 0 and

∃x1, x2 | g1(x1) < 0 ∧ g1(x2) > 0.

Application:

I Principal component analysis (PCA)

maxx

xTA0x

s.t. ‖x‖2 = 1.


Outline





QP with separable constraintsI QP with separable constraints:

minx

xTA0x + 2bT0 x + c0

s.t. xTi Aixi + 2bTi xi + ci E 0 ∀i = 1, . . . , N, N ≤ p,

where all constraints are separable and x = [x1, . . . , xN ], E ∈ ≤,= .I Roadmap to show strong duality:

S-propertyStrong alternatives

of SDPs

Strong alternativesof diagonalized SDP

QP w/ separableconstraints

TransformationQP ↔ SDP

Existence ofrank 1 solution


of SDPs






S-property ⇐⇒ strong dualityI Lagrangian: L(x,λ) = f(x) +

∑i gi(x)

Definition (S-property.)

A QP satisfies the S-property if and only if the following statements areequivalent:

I ∃x ∈ Rp | f(x) ≥ 0, for x feasible

I ∃λ ∈ Γ |L(x,λ) ≥ 0 for all x ∈ Rp

Theorem (Necessary and sufficient conditions)

Suppose the QP satisfies the S-property. Let x∗ be a feasible point of the QP.Then, x∗ is a global minimizer of the QP if and only if:

∇xf(x∗) +∑i

λi∇xgi(x∗) = 0

∑i

λigi(x∗) = 0, A0 +

∑i

λiAi 0.

V. Jeyakumar, A. M. Rubinov, and Z.Y. Wu, “Non-convex quadratic minimization problems with quadraticconstraints: global optimality conditions,” Mathematical Programming, vol. 110, no. 3, pp. 521–541, Sept. 2007.


Theorem of strong alternatives


of SDPs





Two systems are strong alternatives if the systems cannot be feasible at thesame time, but one of them must be true.

The following systems are strong alternatives:

I λi ∈ R,A0 + λ1A1 + . . .+ λNAN 0

I Z 0,

tr(A0Z) < 0

tr(AiZ) E 0, ∀i = 1, . . . , N


Simultaneous diagonalization via congruence


of SDPs





A set of matrices A1, A2, . . . , AN is said to be simultaneously diagonalizablevia congruence, if there exists a nonsingular matrix P such that PTAiP isdiagonal for every matrix Ai.

I Introduction of variables (from the QP):

Ai =

[Ai bibTi ci

], P =

P1 0n1×n2 · · · p1

0n2×n1 P2 · · · p2

.... . .

. . ....

0nN×n1 · · · PN pm01×n1 01×n2 · · · 1

I Change of variables: Fi = PTAiP for all i ∈ 0, . . . , N .I F0 is not diagonal necessarily, only the constraints.


Existence of rank 1 solutions


of SDPs





1. System of strong alternatives: Y = P−1ZP−T .λi ∈ R, A0 +

∑i λiAi 0

Z 0, tr(A0Z) < 0 ∧ tr(AiZ) = 0, i ≥ 1

2. System of diagonalized strong alternatives: Y = P−1ZP−T .λi ∈ R, F0 +

∑i λiFi 0

Y 0, tr(F0Y ) < 0 ∧ tr(FiY ) = 0, i ≥ 1

3. The next step is to find a vector y such that

yTF0y ≤ tr(F0Y )

yTFiy = tr(FiY ) ∀i ≥ 1.

I The last step ensures that the QP shows the S-property.


Results on strong duality

Theorem

Given a QP with separable constraints, suppose Slater’s assumption is satisfiedand that bi ∈ range[Ai] for every i ∈ N . Furthermore, assume there exists adiagonal matrix D whose elements are ±1 such that DF0D is a Z–matrix.Then, the S-property holds.

Theorem

Given a QP with separable constraints, suppose Slater’s assumption is satisfiedand that for some i ∈ N , bi /∈ range[Ai]. Assume that A0 0,(A0)mj = (A0)jm = 0 for any m 6= j, m ∈M , and that there exists a diagonalmatrix D whose terms satisfy

(D)iiDjj(F0)ij ≤ 0 ∀i, j /∈M. (3)

Then, the S-property holds.


Outline





SDP formulation

I Based on SDP relaxation: X xxT

minx

tr[A0X] + 2bT0 x + c0

s.t. tr[AiX] + 2bTi x + ci E 0, ∀i,X = xxT .

⇐⇒

minx

tr[A0X] + 2bT0 x + c0

s.t. tr[AiX] + 2bTi x + ci E 0, ∀i,(X xxT 1

) 0,

I SDP methods scale badly with the size of the problem.

I We explore parallel techniques with optimality guarantees.

I Notice that

A0 +∑i

λiAi 0

I We defineW = λ ∈ Γ | A0 +

∑iλiAi 0 .


Review of dual methods

I Consider a general optimization problem:

minx

f(x)

s.t. g(x) ≤ 0.

I Karush-Kuhn-Tucker conditions (necessary):

∇xf(x) +∇xλTg(x) = 0

λTg(x) = 0

g(x) ≤ 0

λ ≥ 0

⇐⇒

∇xf(x) +∇xλ

Tg(x) = 0

0 ≤ λ ⊥ g(x) ≤ 0.

I Dual function:

q(λ) = minxL(x,λ)

I Dual problem:

maxλ≥0

q(λ)

Solve:

Update: λk+1 = [λk + αkg(xk)]+

arg minx L(x,λ)

Ou

ter

loo

p

Inn

erlo

op


Projected dual (sub)gradient method

I Necessary conditions on the constraints: g(x) = (gi(x))i∈N ,

W 3 λ ⊥ g(x) E 0.

I The complementarity problem (CP) is:

maxλ

λTg(x(λk,xk))

s.t. λ ∈W.

I Projected (sub)gradient:

λk+1 = ΠW [λk + αkg(xk+1)]

where

xk+1 = arg minx∈Rp

L(x,λk) (4)

I Distributed version: useFLEXA to solve (4).

I Algorithm:

Solve:

Update: λk+1 = ΠW [λk + αkg(xk)]

arg minx L(x,λ)O

ute

rlo

op

Inn

erlo

op


FLEXA decomposition IParallel updates

I FLEXA is a decomposition framework to solve non-convex problems.

I FLEXA solves problems of the following type:

minx

N∑i=1

fi(x)

s.t. x ∈ KI The framework uses strongly convex surrogate functions:

Iterates with BCDIterates with surrogate function

G. Scutari, F. Facchinei, P. Song, D. P. Palomar, and J.-S. Pang, “Decomposition by partial linearization:Parallel optimization of multi-agent systems,” IEEE TSP, vol. 62, no. 3, pp. 641–656, Feb. 2014


Distributed proposal based on FLEXA decompositionI Necessary conditions:

0 Z ⊥ A0 +∑i∈N

λiAi 0

I Complementarity problem:

minZ

tr[(A0 +

∑i∈N

λiAi)Z]

s.t. Z 0.

I Gradient scheme:

Zk+1 =[Zk−µk(A0+

∑i∈N

λkiAi)]

+,

where λk are calculated solving anonconvex problem.

I Algorithm:

Solve:

Update: Zk+1 = [Zk + αk(A0 +∑iAi)]+

arg minx L(x,λ, Z)O

ute

rlo

op

Inn

erlo

op


Distributed proposal based on FLEXA decomposition

I We form the Lagrangian:

L(x,λ, Z) = xTA0x + 2bT0 x + c0 +∑i∈N

λi(xTAix + 2bTi x + ci

)+ tr

[Z(A0 +

∑i∈N

λiAi)],

and derive a modified problem from its Lagrangian:

minxi∈Rni

xTA0x + 2bT0 x + c0

s.t. gi(xi) + tr[ZkAi] E 0.(5)

I Problem (5) is a nonconvex QP with separable constraints.

I We can use FLEXA to solve it in a distributed manner.


Robust least squares IApplication example

I Least squares problem:

minx

‖Ax− b‖2

I Robust least squares:

minx∈Rp

max(∆A,∆b)

‖(A+ ∆A)x− (b+ ∆b)‖2

s.t. ‖(∆A,∆b)‖2F ≤ ρ,

I Our proposal of RLS:

minx∈Rp

max(∆A,∆b)

‖(A+ ∆A)x− (b+ ∆b)‖2

s.t. ‖(∆A):i‖2 ≤ ρi ∀i ∈ 1, . . . , p ,‖∆b‖2 ≤ ρp+1


Robust least squares IISolution proposal

I Use gradient descent on x to solve the minimization problem.

xk+1 = xk − αkx∇f(x)

I Renaming variables: ∆ = (∆A,∆b), H = (A, b) and x = (xT ,−1)T :

max∆

(‖H + ∆)x‖2

s.t. ‖(∆):i‖2 = ρi ∀i ∈ 1, . . . , p+ 1 .

I SDP relaxation:

maxU,∆

tr[UxxT ] + tr[(HT∆ + ∆TH)xxT ] + tr[HTHxxT ]

s.t. Uii = ρi ∀i ∈ 1, . . . , p+ 1 (U ∆T

∆ IN

) 0.

I FLEXA decomposition is also posible.


Robust least squares IIISimulations

0 10 20 30 40

101.4

101.45

Iterations number

Co

st

0 100 200 300 400

101.4

Packet exchanges

Co

st


Outline


2 Squared ranged localization problemsTarget localization problemSensor network localization problem



Target localization problemCentralized solution

I Problem formulation:

minx

∑i∈N

(‖x− si‖2 − d2

i

)2I Centralized approach: rewrite the polynomial as a QP.

I Find optimal solution with the dual problem.

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−5

0

5

10Network nodes

Target position

Stationary solution


Summary of proposed algorithmsTarget localization problem

Convergence Description

Centralized optimal [Beck2008]

Diffusion-ATC suboptimal [Chen2012]

NEXT-Linearized suboptimal [DiLorenzo2016]

ADMM-Exact suboptimal

NEXT-Exact suboptimal

Dual ascent optimal if the S-propertyis satisfied

Distributed dual ascent optimal if the S-propertyis satisfied

Table: Summary of TL algorithms.


SimulationsTL problem: 3 nodes, noiseless case and σ = 1.

0 50 100 150 20010−8

10−2

104

Iteration number

1 N

∑ i(‖xk−si‖2−d2 i)2

Diffusion ATC

NEXT-Linearized

NEXT-Exact

ADMM-Exact

Dual Ascent

0 50 100 150 200101

102

103

Iteration number

1 N

∑ i(‖xk−si‖2−d2 i)2

Diffusion ATC

NEXT-Linearized

NEXT-Exact

ADMM-Exact

Dual Ascent

Optimal


SimulationsTL problem: 15 nodes, noiseless and σ = 1.

0 10 20 30 40 5010−10

10−3

104

Iteration number

1 N

∑ i(‖xk−si‖2−d2 i)2

Diffusion ATC

NEXT-Linearized

NEXT-Exact

ADMM-Exact

Dual Ascent

0 10 20 30 40 50101

101.5

102

Iteration number

1 N

∑ i(‖xk−si‖2−d2 i)2

Diffusion ATC

NEXT-Linearized

NEXT-Exact

ADMM-Exact

Dual Ascent

Optimal


Outline


2 Squared ranged localization problemsTarget localization problemSensor network localization problem



Sensor network localization problem

I Problem formulation:

minx∈RNp

∑i∈Nu

∑j∈Na

i

(‖xi − sj‖2 − d2ij)

2 +∑j∈Nu

i

(‖xi − xj‖2 − d2ij)

2.

I Network example:

−50−40−30−20−10 0 10 20 30 40 50

−20

−10

0

10

20 Anchor nodes

Unknown nodes

Estimates


Summary of proposed algorithmsSNL problem

Convergence Description

FLEXA poly2 suboptimal polynomialapprox. order 2.

FLEXA poly4 suboptimal polynomialapprox. order 4.

Centralized optimal based on SDP(if the S-property is satisfied)

Dual ascent optimal(if the S-property is satisfied)

Distributed dual ascent optimal(if the S-property is satisfied)

Costa et al. suboptimal [Costa2006].


SimulationsN = 17 nodes with u. position. Noiseless and σ = 1. LOW CONNECTIVITY

0 50 100 150 20010−4

10−1

102

Iterations number

RM

SE

FLEXA poly 2

FLEXA poly 4

Costa et.al

Dual ascent

0 50 100 150 200100

101

102

Iterations number

RM

SE

FLEXA poly 2

FLEXA poly 4

Costa et.al

Dual ascent

Optimal solution


SimulationsN = 11 nodes with u. position. Noiseless and σ = 1. HIGH CONNECTIVITY

0 50 100 150 20010−8

10−3

102

Iterations number

RM

SE

FLEXA poly 2

FLEXA poly 4

Costa et.al

Dual ascent

0 50 100 150 200

100

101

Iterations number

RM

SE

FLEXA poly 2

FLEXA poly 4

Costa et.al

Dual ascent

Optimal solution


Outline





Introduction to algorithmic game theory I

I Nash equilibrium problem (NEP)

GNEP :

minxi

fi(xi,x−i)

s.t. xi ∈ Ki,∀i ∈ N ,

I Generalized Nash equilibrium problem (GNEP)

GGNEP :

minxi

fi(xi,x−i)

s.t. xi ∈ Ki(x−i),∀i ∈ N ,

I Equivalence with variational inequalities (VI).

Definition

A pure strategy NE, is a feasible point x∗ that satisfies

fi(x∗i ,x∗−i) ≤ fi(xi,x∗−i), ∀xi ∈ Ki(x−i)

for every player i ∈ N .


Smart grid modelDemand side management

I Techniques for better energy efficiency programs, distributed energygeneration, storage and load management.

I Achieved through economic incentives and behavioral change.

I Allows to lower the energy peak consumption, reduce the investment inpower plants and tuning of energy supply according to demand.

I Smart Grid elements:

1. Supply side: energy producersand the distribution network.

2. Central unit: regulation authority.3. Demand-side: energy consumers.

DistributionNetwork

Central Unit

Demand-Side


Energy Cost function

I Quadratic cost function, typical of thermal energy generating plants.

Ch(L(h), δ(h)) , Kh

(L(h) +

∑n∈D

δn(h))2

Ch(·) Total energyproduction cost.

L(h) Total energy loaddemand.

ln(h) User’s energy loaddemand.

δn(h) Deviation errors.

A.-H. Mohsenian-Rad, V.W.S. Wong, J. Jatskevich, R. Schober, and A. Leon-Garcia, “Autonomousdemand-side management based on game-theoretic energy consumption scheduling for the future smart grid,” IEEETransactions on Smart Grid, vol. 1, no. 3, pp. 320–331, Dec. 2010


User’s energy cost models

I Day-ahead: Price of energy is determined by expected demand anddeviation errors.

I Establish a robust price of energy at different times of the day.

fn(l, δ) ,∑h∈H

C(L(h), δ(h)) · ln(h) + δn(h)

L(h) +∑δn(h)

I Real-time: Users pay for the energy they use at the robust energy pricewithin certain usage limits (day ahead cost model).

I Outside of these limits, they pay a surpluss (not necessarily simmetric).

f rtn

(l, δ∗, lrt

)=∑h∈H

C(L(h), δ∗(h))

L(h) +∑δn(h)︸︷︷︸

Robust priceh

·(

lrtn(h)︸︷︷︸User’s demand

+ Ψnh(l, δ∗, lrt)︸︷︷︸Penalization

)


Day-ahead Optimization Process

I Game that aims to minimize the user’s cost:

Gδ :

minln

fn(ln, l−n, δ)

s.t. ln ∈ Ωln ,∀n ∈ D

I Best response:

Bτ (l−n) = arg minln

fn(ln, l−n, δ) +τ

2‖ln − ln‖2

s.t. ln ∈ Ωln ,

I Algorithm:

Solve Bn(l−n) for all n.

Update centroids: ln ← ln

Solve worst-case deviation errors.

Ou

ter

loo

p

Inn

erlo

op


Worst-case analysis of the energy load profiles

I We consider a bounded error:

∆h , δ(h) ∈ R|D| | ‖δ(h)‖22 ≤ α(h) ∀h ∈ H

I We propose the following GNEP with min-max objective functions:

Gm :

minln

maxδn

fn(ln, l−n, δn, , δ−n)

s.t. δ(h) ∈ ∆h, ∀h ∈ Hln ∈ Ωln

∀n ∈ D,

I The maximization problem is a QP: sufficient and necessary conditions.

I The solution is achieved after finding a contraction mapping

Th(δ(h),ah) = ΠXh

(√α(h)

ah +Aδ(h)

‖ah +Aδ(h)‖

),

where ah = L(h) + l(h), A = 11T − I.


Simulation results I

I Real time averaged cost functions over 100 simulations.

I Left axis shows monetary cost over all users.

I The number of users vary from 100 to 2000 users.

I Real time deviations are assumed to be Gaussian with zero mean.

500 1,000 1,500 2,0000

5001,0001,5002,0002,5003,0003,5004,000

Number of users

Ave

rag

eC

ost

Non-Robust

Robust


Simulation results IIConvergence speeds

I Outer loop (minimization game)

0 2 4 6 8 1010−3

10−2

10−1

100

Iteration number

Co

nv.

crit

eria

I Inner loop (contraction mapping)

0 2 4 6 8 1010−15

10−1010−810−5

100

105

Iteration number

Co

nv.

crit

eria


Outline





Conclusions

I Quadratic programs with separable constraints.

1. We study the conditions for QPs with separable constraints toexhibit the S-property.

2. Novel algorithmic framework.

I Squared ranged localization problems: TL and SNL problems

1. Primal methods (ADMM, NEXT, Diffusion)2. Dual methods (centralized and distributed)

I Robust worst-case demand side management

1. Assume deviations in the day-ahead energy cost calculations.2. Propose a min-max game with coupled constraints.3. Propose algorithm with convergence guarantees.


Contributions IQuadratic problems, localization and smart grid

I Javier Zazo, Santiago Zazo, and Sergio Valcarcel Macua, “Non-monotonequadratic potential games with single quadratic constraints,” in IEEEInternational Conference on Acoustics, Speech and Signal Processing(ICASSP), March 2016, pp. 4373–4377.

I Javier Zazo, Sergio Valcarcel Macua, Santiago Zazo, Marina Perez, IvanPerez-Alvarez, Eugenio Jimenez, Laura Cardona, Joaquın HernandezBrito, and Eduardo Quevedo, “Underwater electromagnetic sensornetworks, part II: Localization and network simulations,” Sensors, vol. 16,no. 12, pp. 2176, 2016.

I Javier Zazo, Santiago Zazo, and Sergio Valcarcel Macua, “Robustworst-case analysis of demand-side management in smart grids,” IEEETransactions on Smart Grid, vol. 8, no. 2, pp. 662–673, March 2017.


Contributions IIDynamic game theory

I Santiago Zazo, Sergio Valcarcel Macua, Matilde Sanchez-Fernandez, and JavierZazo, “Dynamic potential games with constraints: Fundamentals andapplications in communications,” IEEE Transactions on Signal Processing, vol.64, no. 14, pp. 3806–3821, July 2016.

I Santiago Zazo, Javier Zazo, and Matilde Sanchez-Fernandez, “A controltheoretic approach to solve a constrained uplink power dynamic game,” in 22ndEuropean Signal Processing Conference (EUSIPCO), Sept 2014, pp. 401–405.

I Santiago Zazo, Sergio Valcarcel Macua, Matilde Sanchez-Fernandez, and JavierZazo, “A new framework for solving dynamic scheduling games,” in IEEEInternational Conference on Acoustics, Speech and Signal Processing (ICASSP),April 2015, pp. 2071–2075.

I Sergio Valcarcel Macua, Santiago Zazo, and Javier Zazo, “Learning inconstrained stochastic dynamic potential games,” in IEEE InternationalConference on Acoustics, Speech and Signal Processing (ICASSP), March 2016,pp. 4568–4572.

I Sergio Valcarcel Macua, Javier Zazo, and Santiago Zazo, “Closed-loopstochastic dynamic potential games with parametric policies and constraints,” inLearning, Inference and Control of Multi-Agent Systems workshop in NIPS,2016.


Contributions IIIAlgorithmic game theory, distributed optimization, differential and mean field games

I J. Zazo, S. Zazo, and S. Valcarcel Macua, “Distributed cognitive radio systemswith temperature-interference constraints and overlay scheme,” in 22ndEuropean Signal Processing Conference (EUSIPCO), Sept 2014, pp. 855–859.

I S. Valcarcel Macua, S. Zazo, and J. Zazo, “Distributed black-box optimizationof nonconvex functions,” in IEEE International Conference on Acoustics, Speechand Signal Processing (ICASSP), April 2015, pp. 3591–3595.

I S. Valcarcel Macua, C. Moreno Leon, J. S. Romero, S. S. Pereira, J. Zazo, A.Pages-Zamora, R. Lopez-Valcarce, and S. Zazo, “How to implementdoubly-stochastic matrices for consensus-based distributed algorithms,” in IEEE8th Sensor Array and Multichannel Signal Processing Workshop (SAM), June2014, pp. 333–336.

I J. Parras, S. Zazo, J. del Val, J. Zazo, and S. Valcarcel Macua, “Pursuit-evasiongames: a tractable framework for antijamming games in aerial attacks,”EURASIP Journal on Wireless Communications and Networking, vol. 2017, no.1, pp. 69, 2017 doi:10.1186/s13638-017-0857-8.

I J. Parras, J. del Val, S. Zazo, J. Zazo, and S. Valcarcel Macua, “A newapproach for solving anti-jamming games in stochastic scenarios aspursuit-evasion games,” in IEEE Statistical Signal Processing Workshop (SSP),June 2016, pp. 1–5.

I J. del Val, S. Zazo, S. Valcarcel Macua, J. Zazo, and J. Parras, “Optimal attackand defence of large scale networks using mean field theory,” in 24th EuropeanSignal Processing Conference (EUSIPCO), Aug 2016, pp. 973–977.


http://dx.doi.org/10.1186/s13638-017-0857-8

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Nonconvex Quadratic Problems and Games with …...Nonconvex Quadratic Problems and Games with...

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Transcript of Nonconvex Quadratic Problems and Games with …...Nonconvex Quadratic Problems and Games with...