Introduction to Convex Optimization, Game Theory and ... · Introduction Optimization Problems: I...
Transcript of Introduction to Convex Optimization, Game Theory and ... · Introduction Optimization Problems: I...
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.
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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
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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
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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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