Upload
others
View
18
Download
0
Embed Size (px)
Citation preview
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
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 4 / 30
Game Theory examples (I)
Consider a peer-to-peer (ad-hoc) wireless network with Q users:
Ad-hoc Network (=Interference channel)
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 5 / 30
Game Theory examples (II)
Consider a Demand-side-management perspective in a smart grid
Distributed generation, consumtion, storage (=big data)
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 6 / 30
Convex Optimization Theory
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 7 / 30
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
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 8 / 30
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
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 9 / 30
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 )
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 11 / 30
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))
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 12 / 30
Example II: Network Flow Control
maxxi
∑b∈B
Ui(xi)
s.t. ATx ≤ cxi ≥ 0
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 13 / 30
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)
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 14 / 30
Variational Inequalities
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 15 / 30
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!!
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 16 / 30
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 .
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 17 / 30
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.
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 18 / 30
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
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 19 / 30
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
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 20 / 30
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)
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 21 / 30
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.
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 23 / 30
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.
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 25 / 30
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.
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 26 / 30
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.
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 27 / 30
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)
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 28 / 30
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?
Javier Zazo (UPM) Convexity, Game Theory, VI 15th January 2015 30 / 30