Introduction to Convex Optimization, Game Theoryand Variational Inequalities

Javier Zazo

Technical University of Madrid (UPM)

15th January 2015

Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 1 / 30

Table of Contents

1 Introduction: goal of this talk

2 Preliminaries of Convex Theory

1 Examples, de�nitions, solution characterization

3 Variational Inequalities: a general framework

1 De�nitions, problems of interest, properties

4 Noncooperative Game Theory

1 Nash Equilibrium Problems (NEPs)2 Generalized NEPs (GNEPs)

G. Scutari, D. Palomar, F. Facchinei, and J. Pang, �ConvexOptimization, Game Theory, and Variational Inequality Theory,�IEEE Signal Processing Magazine, vol. 27, no. 3, pp. 35�49, May2010.

Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 2 / 30

Introduction

Optimization Problems:I Linear programming:

maxx

cTx

s.t. Ax ≤ bx ≥ 0

I LASSO problem:

minx‖y −Ax‖2 + λ |x|

I Support Vector Machines

minw,b‖w‖2

s.t. yi(wTxi − b

)≥ 1.

I k-means Clustering:

arg minS

k∑i=1

∑x∈Si

‖x− µi‖2

with S = {S1, . . . , Sk}.Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 3 / 30

Introduction

Game Theory:

I Rough de�nition: Coupled optimization problemsI Players interaction: Distributed modellingI Purpose?: solution concept.I Examples: resource sharing of wireless networks, p2p networks,smart grids

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Game Theory examples (I)

Consider a peer-to-peer (ad-hoc) wireless network with Q users:

Ad-hoc Network (=Interference channel)

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Game Theory examples (II)

Consider a Demand-side-management perspective in a smart grid

Distributed generation, consumtion, storage (=big data)

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Convex Optimization Theory

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Convex Problem

min f(x)

s.t. x ∈ K

K is closed and convex.

f(x) is convex.

Convex set: αx+ (1− α)y ∈ K, for all x, y ∈ K and α ∈ [0, 1].

I Unit ball: K = {x ∈ Rn| ‖x‖ ≤ 1}.I Positive quadrant (cone): K = {x ∈ Rn|xi ≥ 0} .

A

B A

B

A

B

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Convex Problem

1 Finding if a problem is convex: inspection, operations thatpreserve convexity, de�nition

2 Properties of the problem:

1 Convexity f(αx+ (1− α)y)≤αf(x) + (1− α)f(x)2 Strict convexity f(αx+ (1− α)y)<αf(x) + (1− α)f(x)

3 Strong convexity f(αx+ (1−α)y)<αf(x) + (1−α)f(x)− c2 ‖x− y‖

2

strongly convex⇒ strictly convex⇒ convex

x

f (x )

x

f (x )

f (x )

f (y )

S

x y x

f (x )

f (y )

S

Convex Strictly

Convex

Strongly

Convex

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Characterization of Solutions

Minimum: A feasible point x∗ ∈ K is said to be optimal if

f(x∗) ≤ f(x) ∀x ∈ K.

Minimum principle:

(y − x∗)∇f(x∗) ≥ 0 ∀y ∈ K

Unconstrained optimization ⇔ ∇f(x∗) = 0.

Existance and uniqueness

Convex ⇒ Multiple Solutions (convex set)Strictly Convex ⇒ 1 solution (at most)Strongly Convex ⇒ Unique solution

Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 10 / 30

Graphical Interpretation

(y − x∗)∇f(x∗) ≥ 0 ∀y ∈ K

yd = y − x*

·

Feasible Set K

Surface of Equal Cost f (x )

∇f (x*)

x*

d = y − x

·yx

Feasible Set K

∇f (x )

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Example I: Projection

Euclidean Projection

minx‖x− u‖22

s.t. x ∈ K

ΠK (u) ≡ x̂ = arg minx∈K‖x− u‖22

Gradient ProjectionAlgorithm

minx

f (x)

s.t.x ∈ K

xk+1 = ΠK

(xk + α∇xf

(xk))

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Example II: Network Flow Control

maxxi

∑b∈B

Ui(xi)

s.t. ATx ≤ cxi ≥ 0

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Karush-Kuhn-Tucker (KKT)

minx

f(x)

s.t. g(x) ≤ 0

Let's de�ne the Lagrangian:

L (x, λ) = f (x) + µT gl (x)

Optimality criteria: KKT conditions

∇xf(x) + µT∇xg (x) = 0

0 ≤ µ ⊥ −g(x) ≥ 0

Dual problem (assumption: strong duality holds)

q(µ) = minx

f (x) + µT gl (x)

maxµ≥0

q(µ)

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Variational Inequalities

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Variational Inequality Problem

Given a closed and convex set K ∈ Rn,a continuous mapping F : K → Rn,then, the V I (K,F) problem is to �nd a vector such

(y − x?)F(x?) ≥ 0, ∀y ∈ K

Feasible

Set K

· ·yx*

F(x*)

y − x*

Feasible

Set K

··y

x

F(x )

y − x

The importance of VI: that they provide a theory in which to testexistance/uniqueness of solutions, and algorithms to �nd those

solutions!!

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Variational Inequality Examples

Optimization problem: minx∈K f(x)

K = {x ∈ K}F = ∇xf(x)

V I(K, F )

System of equations: �nd an x∗ ∈ Rn such that F (x∗) = 0

K = Rn

V I(Rn, F )

Nonlinear complementarity problem: 0 ≤ µ∗ ⊥ F (µ∗) ≥ 0

K = {µ ≥ 0}V I(Rn+, F )

Non-cooperative Games

V I (K,F) represents a wider range of problems than classicaloptimization whenever F 6= ∇f .

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Existence of the solution

Given the VI(K, F ), suppose that

1 The set K ⊆ Rn is compact and convex, and

2 The function F : K → Rn be continuous.

Then, the V I (K,F) has a nonempty and compact solution set.

The requirement on K being compact can be very restrictive.Other results exists.

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Monotonicity properties of functions:

1 Monotone:

F (x)− F (y)T (x− y) ≥ 0 ∀x, y ∈ K

2 Strictly monotone:

F (x)− F (y)T (x− y) > 0 ∀x, y ∈ K and x 6= y

3 Strongly monotone:

F (x)− F (y)T (x− y) > c ‖x− y‖2 ∀x, y ∈ K

f ′ (x )

x

f ′ (x )

x

f ′ (x )

x

Monotone Strictly Monotone Strongly Monotone

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Existence and uniqueness of the solutions

1 If F is monotone =⇒ the solution set of the VI(K, F ) is closed andconvex

2 If F is strictly monotone =⇒ the VI admits at most one solution

3 If F is strongly monotone =⇒ the VI admits a unique solution

If the V I(K, F ) corresponds to a optimization problemminx∈K f(x), then

i)f convex ⇐⇒ ∇f monotone

ii)f strictly convex ⇐⇒ ∇f strictly monotone

iii)f strongly convex ⇐⇒ ∇f strongly monotone

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Characterization of the solution

x∗ is a solution of the V I(K, F ) ⇐⇒ x∗ = ΠK (x∗ − F (x∗))

Feasible

Set K

·x* – F(x*)

x* = ΠK (x* – F(x*))

F(x*)

Feasible

Set K

··

x

F(x )

ΠK (x – F(x ))x – F(x )

The �xed-point equation invites for an iterative algorithm

xk+1 = ΠK

(xk − αF (xk)

)Convergence is globally guaranteed under monotonityrequirements.There are also necessary KKT conditions for solutions (as in theconvex problem)

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Game Theory

Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 22 / 30

Non-cooperative Game Theory

Resolution of problems with interacting decision-makers (calledplayers).

G =<∏i

Ki, f >

Noncooperative: sel�sh players try to optimize their own objectivefunction.t

minfi (xi,x−i)

s.t.xi ∈ Kii = 1, ..., Q

where x−i = [x1, . . . , xi−1, xi+1, . . . , xQ]T .

Nash Equilibrium (NE): a point x∗ ∈ K is NE, i�

fi(x?i ,x

?−i)≤ fi

(yi,x

?−i), ∀yi ∈ Ki, ∀i

where K =∏iKi.

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Types of Nash Equilibrium Problems

NE Problems (NEP)

minxifi (xi,x−i)

s.t.xi ∈ Kii = 1, ..., Q

Generalized NEP (GNEP)

minxifi (xi,x−i)

s.t.xi ∈ Ki (x−i)i = 1, ..., Q

GNEP with shared constratins: Ki (x−i) = {xi : g (xi,x−i) ≤ 0}Set K

x2

K2(x1)

K1(x2)

x = (x1, x2)

x1Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 24 / 30

VI Reformulation of the NEP

K =∏iKi and f = (fi(x))Qi=1

Equivalence with VI

Given the game G =< K, f >,1 the strategy ste Ki are closed and convex;

2 the payo� functions fi(xi,x−i) are continuously di�erentiable in xand convex in xi for every �xed x−i.

Then the game G is equivalent to the V I(K,F), whereF(x) = (∇xifi(x))Qi=1.

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Characterization of NE

The minimum principle (NE): For every i ∈ {1, . . . , Q},

(yi − x∗i )T ∇xifi(xi, x∗−i) ≥ 0, ∀yi ∈ Ki

The NE necessary condition can be equivalently expressed with thesolution of VI.

If we can express a game with VI, we can use existence anduniqueness results of VI to infer NE solutions.

Moreover, we have a choice of algorithms to �nd the solution.

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Best Response Algorithm

Let Bi(x−i) be the set of optimal solutions of the ith optimizationproblem

minxifi (xi,x−i)

s.t.xi ∈ Ki

and set B(x) = B1(x−1)× B2(x−2)× · · · × BQ(x−Q)

A point is a NE i�x∗ ∈ B(x∗)

which is another �xed-point equation.

An iterative algorithm of the form

xk+1i = B(xk−i)

with xk−i =(xk+11 , xk+1

2 , . . . , xk+1i−1 , x

ki+1, . . . , x

kQ

)(Gauss-Seidel),

converges if the VI associated to the NEP is strongly monotone.

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Best Response Algorithm

x1

x2

B1(x2)

B2(x1)

•

x1

x2

B1(x2)

B2(x1)

•

•

•

x1

x2

B1(x2)

B2(x1)

x1

x2

B1(x2)

B2(x1)

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Example: Network Flow Problem

Q users, shared constraints GNEP

maxxi

Ui(xi)

s.t. ATx ≤ c

Lagrangian and KKT:

Li (xi,x−i, λi) = Ui (xi) + λT(ATx− c

)∇xiLi(xi, x−i, λ) = 0 ∀i0 ≤ λ? ⊥ −

(ATx? − c

)≥ 0

Variational Inequality V I(K,F):

F(x) = (∇xiUi(xi))Qi=1

Ki(x−i) :{xi ≥ 0|g(xi,x−i) = ATx− c ≤ 0

}Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 29 / 30

Thank you!!

Any questions?

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