nonconvex quadratic problems and games with …...nonconvex quadratic problems and games with...
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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
2 Squared ranged localization problems
3 Robust worst-case analysis of demand-side management
4 Concluding remarks
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.
Javier Zazo Dissertation 2 / 44
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.
Javier Zazo Dissertation 3 / 44
Polynomial minimizationSix-hump-camel problem
−2 −10
12
−1−0.5
00.5
1
0
5
x1x2
f(x 1
,x2)
Javier Zazo Dissertation 4 / 44
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.
Javier Zazo Dissertation 5 / 44
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.
Javier Zazo Dissertation 6 / 44
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).
Javier Zazo Dissertation 7 / 44
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.
Javier Zazo Dissertation 8 / 44
Outline
1 Nonconvex QPs with separable constraintsIntroduction to QPsRoadmap to establish strong duality in QPsAlgorithmic framework
2 Squared ranged localization problems
3 Robust worst-case analysis of demand-side management
4 Concluding remarks
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
S-propertyStrong alternatives
of SDPs
Strong alternativesof diagonalized SDP
QP w/ separableconstraints
TransformationQP ↔ SDP
Existence ofrank 1 solution
Javier Zazo Dissertation 9 / 44
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.
Javier Zazo Dissertation 10 / 44
Theorem of strong alternatives
S-propertyStrong alternatives
of SDPs
Strong alternativesof diagonalized SDP
QP w/ separableconstraints
TransformationQP ↔ SDP
Existence ofrank 1 solution
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
Javier Zazo Dissertation 11 / 44
Simultaneous diagonalization via congruence
S-propertyStrong alternatives
of SDPs
Strong alternativesof diagonalized SDP
QP w/ separableconstraints
TransformationQP ↔ SDP
Existence ofrank 1 solution
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.
Javier Zazo Dissertation 12 / 44
Existence of rank 1 solutions
S-propertyStrong alternatives
of SDPs
Strong alternativesof diagonalized SDP
QP w/ separableconstraints
TransformationQP ↔ SDP
Existence ofrank 1 solution
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.
Javier Zazo Dissertation 13 / 44
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.
Javier Zazo Dissertation 14 / 44
Outline
1 Nonconvex QPs with separable constraintsIntroduction to QPsRoadmap to establish strong duality in QPsAlgorithmic framework
2 Squared ranged localization problems
3 Robust worst-case analysis of demand-side management
4 Concluding remarks
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 .
Javier Zazo Dissertation 15 / 44
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)
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Javier Zazo Dissertation 16 / 44
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
Javier Zazo Dissertation 17 / 44
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
Javier Zazo Dissertation 18 / 44
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
Javier Zazo Dissertation 19 / 44
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.
Javier Zazo Dissertation 20 / 44
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
Javier Zazo Dissertation 21 / 44
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.
Javier Zazo Dissertation 22 / 44
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
Javier Zazo Dissertation 23 / 44
Outline
1 Nonconvex QPs with separable constraints
2 Squared ranged localization problemsTarget localization problemSensor network localization problem
3 Robust worst-case analysis of demand-side management
4 Concluding remarks
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
Javier Zazo Dissertation 24 / 44
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.
Javier Zazo Dissertation 25 / 44
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
Javier Zazo Dissertation 26 / 44
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
Javier Zazo Dissertation 27 / 44
Outline
1 Nonconvex QPs with separable constraints
2 Squared ranged localization problemsTarget localization problemSensor network localization problem
3 Robust worst-case analysis of demand-side management
4 Concluding remarks
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
Javier Zazo Dissertation 28 / 44
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].
Javier Zazo Dissertation 29 / 44
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
Javier Zazo Dissertation 30 / 44
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
Javier Zazo Dissertation 31 / 44
Outline
1 Nonconvex QPs with separable constraints
2 Squared ranged localization problems
3 Robust worst-case analysis of demand-side management
4 Concluding remarks
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 .
Javier Zazo Dissertation 32 / 44
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
Javier Zazo Dissertation 33 / 44
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
Javier Zazo Dissertation 34 / 44
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
)
Javier Zazo Dissertation 35 / 44
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.
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Javier Zazo Dissertation 36 / 44
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.
Javier Zazo Dissertation 37 / 44
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
Javier Zazo Dissertation 38 / 44
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
Javier Zazo Dissertation 39 / 44
Outline
1 Nonconvex QPs with separable constraints
2 Squared ranged localization problems
3 Robust worst-case analysis of demand-side management
4 Concluding remarks
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.
Javier Zazo Dissertation 40 / 44
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.
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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.
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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.
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