Lecture 7 Monotonicity. September 21, 2008

Transcription

1 Lecture 7 Monotonicity September 21, 2008

2 Outline Introduce several monotonicity properties of vector functions Are satisfied immediately by gradient maps of convex functions In a sense, role of monotonicity in VI/CPs is similar that of convexity in optimization Game theory: Models, Algorithms and Applications 1

3 Focus Motivation from convex optimization and game theoretic problems Definition of monotone mappings and their variants Existence of solutions and convexity of solution sets Convex programming - a special case Game theory: Models, Algorithms and Applications 2

4 Monotone functions Definition 1 Given a set K R n, a mapping F : K R n is said to be (a) pseudo-monotone on K if for all vectors x, y K (x y) T F (y) 0 = (x y) T F (x) 0; (b) monotone on K if for all vectors x, y K (x y) T (F (x) F (y)) 0 (c) strictly monotone on K if for all vectors x, y K, x y (x y) T (F (x) F (y)) > 0 Game theory: Models, Algorithms and Applications 3

5 (d) ξ-monotone on K for some ξ > 1 if there exists a c > 0 such that for all x, y K (x y) T (F (x) F (y)) > c x y ξ (e) strongly monotone (or 2-monotone) on K if there exists a c > 0 such that for all x, y K (x y) T (F (x) F (y)) > c x y 2 Game theory: Models, Algorithms and Applications 4

6 Relationships strongly-monotone = ξ monotone = strictly-monotone = monotone monotone = pseudo-monotone. In addition, for an affine map, F (x) Ax + b and K R n, we have strongly-monotone ξ monotone strictly-monotone A 0 monotonicity A 0 Game theory: Models, Algorithms and Applications 5

7 Vector Mappings and their Jacobians More generally, if F C 1 on an open convex set D, then the following proposition holds: Proposition 1 Let D R n be an open convex set, and let F : D R n be a continuously differentiable function on D. Then the following hold: (a) F is monotone on D if and only if F (x) is positive semidefinite for all x D. (b) F is strictly monotone on D if and only if F (x) is positive definite for all x D. (c) F is strongly monotone on D if and only if F (x) is uniformly positive definite over D, i.e., there exists a c > 0 such that for all x D y T F (x)y c y 2, for all y R n. Proof: Homework. Game theory: Models, Algorithms and Applications 6

8 Why are monotone mappings important? Arise from important classes of optimization/game-theoretic problems Can articulate existence/uniqueness statements for such problems Convergence properties of algorithms may sometimes (but not always) be restricted to such monotone problems Game theory: Models, Algorithms and Applications 7

9 Convex programming problems Consider the optimization problem given by min x θ(x) x K, where K R n is a closed convex set, and θ : D R is a twice-continuously differentiable convex function on an open superset set D of the set K. An optimizer of this problem is given by the solution to VI(K, θ), i.e., x K such that (y x) T θ(x), y K. When θ( ) is a convex function, 2 θ(x) is positive semidefinite everywhere, implying that θ is a monotone vector mapping Therefore VI(K, θ) is a monotone VI NOTE: This is true even when θ is convex and differentiable Game theory: Models, Algorithms and Applications 8

10 Game-theoretic Problems Consider the game-theoretic problem given by agent problems: S i (x i ) min x i θ i (x i ; x i ) x i K i. Proposition 2 Let K i be closed and convex sets, and let θ i be convex and C 1. Then x is a Nash equilibrium if and only if x SOL(K, F ), where K = Π n i=1 K i and F(x) := ( θ i (x)) n i=1. K is a Cartesian product of K i and is also closed and convex Game theory: Models, Algorithms and Applications 9

11 Existence of solutions to monotone VIs Theorem 3 Let K R n be closed convex and F : K R n be continuous. (a) If F is strictly monotone on K, the VI(K, F ) has at most one solution. (b) If F is ξ monotone on K, the VI(K, F ) has a unique solution. (c) If F is Lipschitz continuous and ξ monotone on Ω for some ξ > 1, where K Ω, then there exists a c > 0 such that x x c F nat K (x) 1 ξ 1 for every x Ω, where x is the unique solution to the VI(K, F ). Result in (c) provides an upper bound on the distance from the solution Game theory: Models, Algorithms and Applications 10

12 Proof: (a) Suppose F is strictly monotone on K. VI(K, F ). Then for all y K, we have Let x, x be two solutions to (y x) T F (x) 0 (y x ) T F (x ) 0. But by substituting x and x for y in first and second expressions, we obtain (x x) T F (x) 0 (x x ) T F (x ) 0. But by adding these inequalities, we obtain (x x) T (F (x ) F (x)) 0, contradicting the strict monotonicity of F. Game theory: Models, Algorithms and Applications 11

13 (b) If F is ξ monotone, we merely need to show that VI(K, F ) has a solution. Its uniqueness follows from F being strictly monotone. For any ξ monotone mapping with ξ > 1, by definition there exists a c > 0 such that x, y K (x y) T (F (x) F (y)) c x y ξ For a fixed y K, it follows that F (x) T (x y) F (y)t (x y) x y ξ + c x ξ x ξ x ξ By letting x with x K, since ξ > 1, it follows lim inf x K x F (x) T (x y) x ξ c > 0. Game theory: Models, Algorithms and Applications 12

14 The existence is now immediate from the following result: Theorem 4 (Corollary from FP(I)) Let K R n be a closed convex set and let F : R n R n be continuous. If there exists an x ref K and a scalar ζ 0 such that lim inf x K, x F (x) T (x x ref ) x ζ > 0, (1) then the VI(K,F) has a nonempty compact solution set. A seen, F satisfies (1) with x ref = y for an arbitrary y K and ζ = ξ. Game theory: Models, Algorithms and Applications 13

15 (c) Let x K be the solution and c > 0 be the constant in ξ-monotonicity For a given x Ω, we define r F nat K (x). Then we have x r = Π K (x F (x)) = (y x + r) T (F (x) r) 0, y K with y = x, we have (x x + r) T (F (x) r) 0 Since x SOL(K, F ) and x r K, it follows (x r x ) T F (x ) 0 By adding the last two inequalities (after some algebra), we obtain (x x ) T (F (x) F (x )) r T (F (x) F (x )). Game theory: Models, Algorithms and Applications 14

16 The ξ monotonicity and Lipschitz continuity (with constant L > 0) of F implies that c x x ξ (x x ) T (F (x) F (x )) L r x x x x ξ 1 L r c x x c r ξ 1, 1 where c = ( L c ) 1 ξ 1. Generally, strict monotonicity is not sufficient to have a solution Example : real line. Consider the equation e x = 0 which has no zero on the Game theory: Models, Algorithms and Applications 15

17 Theorem 5 (Theorem of FP I) Let K R n be closed and convex, and F : K R n be continuous. If F is pseudo-monotone on K, then the three statements (a), (b), (c) of the main result of Lecture 6 are equivalent Proof: It suffices to show that (c) = (a), i.e., that the existence of solution to V I(K, F ) implies that, for some x ref K the set L < is bounded. In particular, we will show that the existence of solution to V I(K, F ) implies that the set L < = {x K F (x) T (x x ref ) < 0} is empty for some x ref K. Consider a solution x, and the set L < = {x K F (x) T (x x ) < 0}. Game theory: Models, Algorithms and Applications 16

18 Since x is solution, we have (x x ) T F (x ) 0 for all x K. By pseudo-monotonicity, it follows that (x x ) T F (x) 0 for all x K, implying that the set L < is empty. Game theory: Models, Algorithms and Applications 17

19 Convexity of solution set Next set of results: Solution set of pseudo-monotone VI is always convex Sufficient condition for such a VI to have a nonempty bounded solution set Need to define recession cones: Definition 2 A recession direction of a set X is a direction d such that for some vector x X, the ray {x + τd : τ 0} is contained in X. The set of all recession directions is denoted by X and called the recession cone of X. Game theory: Models, Algorithms and Applications 18

20 Instances of X If there is a nonzero w with w X, then X is unbounded If X is polyhedral, i.e., X = {x : Ax b}, then X = {d R n : Ad 0}. Theorem 6 Let K R n be a closed convex set and F : K R n be a continuous mapping. Also, let F be pseudo-monotone on K. Then the following hold: (a) The solution set SOL(K, F ) is convex. (b) If there exists a vector x ref K such that F (x ref ) belongs to the interior of the dual cone to K, i.e., F (x ref ) int(k ), then SOL(K, F ) is nonempty and compact. Game theory: Models, Algorithms and Applications 19

21 Proof: (a) Let F be pseudo monotone on K. We claim that the solution set SOL(K, F ) has the following structure: SOL(K, F ) = y K {x K : F (y) T (y x) 0}. (2) We prove this statement by showing that a vector in one of the sets lies in the other. Let x SOL(K, F ). Then, we have (y x ) T F (x ) 0 y K. By the pseudo-monotonicity of F on K, we have F (y) T (y x ) 0, y K, implying that x y K {x K : F (y) T (y x) 0.} Game theory: Models, Algorithms and Applications 20

22 Suppose x y K {x K : F (y) T (y x) 0}. Let z K be arbitrary. Define y = τx + (1 τ)z. By convexity of K, y belongs to K for all τ [0, 1]. Then we have F (y) T (y x ) 0 F (τx + (1 τ)z) T (z x ) 0. Letting τ 1, we have F (x ) T (z x ) 0, z K. Hence x SOL(K, F ). For a fixed, but arbitrary y K, the set {x K : F (y) T (y x) 0} Game theory: Models, Algorithms and Applications 21

23 is closed and convex. The intersection of any number of convex sets is convex, giving us convexity of SOL(K, F ). (b) We will show that the given property implies that the set L = {x K F (x) T (x x ref ) 0} is bounded. Then, the nonemptyness and compactness of SOL(K, F ) follows from Prop from FP(I) [main result of Lecture 6]. By the pseudo-monotonicity of F on K, we have that F (x ref ) T (x x ref ) 0 x L Hence, L {x K F (x ref ) T (x x ref ) 0} L, where the set L is closed and convex. Suppose that L is unbounded, then since L K, K is also unbounded. Thus, there exists a nonzero recession direction d K. For such a direction, we have F (x ref ) T d 0. Game theory: Models, Algorithms and Applications 22

24 By the given property, we have F (x ref ) int(k ). Therefore, there exists a small enough δ > 0 such that F (x ref ) δd (K ). Hence 0 d T (F (x ref ) δd) δd T d < 0. The contradiction implies that K is bounded and since L K, we see that L is bounded. Game theory: Models, Algorithms and Applications 23

25 Additional Properties of Solutions for Pseudo-Monotone VI s Proposition 7 [FP-I, Prop 2.3.6] Let F : K R n be pseudo-monotone on a convex set K R n. For any two solutions x 1 and x 2 in SOL(K,F), we have (x 1 x 2 ) T F (x 1 ) = (x 1 x 2 ) T F (x 2 ) = 0. Consequently (x 1 x 2 ) T (F (x 1 ) F (x 2 )) = 0. In addition, F (SOL(K, F )) (SOL(K, F ) ). Proposition 8 [FP-I, Cor 2.3.7] Let K R n be closed convex and let F : D R n be continuously differentiable on the open set D, where K D. If F is monotone on K and F (x) is symmetric for all x K, then F(SOL(K,F)) is a singleton. If x SOL(K, F ) and d (SOL(K, F )) ), then x + d SOL(K, F ). Game theory: Models, Algorithms and Applications 24

26 Application to Convex Optimization Consider the convex problem given by min θ(x) subject to x K, where θ : D R is a twice-continuously differentiable convex function on a closed convex set K R n. Proposition 9 Let K R n be a closed convex set and θ C 2 defined on an open convex set D containing K. If S opt, then for any x S opt, S opt = {x K : θ(x) = θ( x), θ( x) T (x x) = 0}. Game theory: Models, Algorithms and Applications 25

27 Proof: Let S denote {x K : θ(x) = θ( x), θ( x) T (x x) = 0}. We show that S opt S by using Propositions 7 and 8. Since θ( ) is convex, 2 θ is positive-semidefinite over K. Therefore θ is monotone over K and the problem may be cast as VI(K,F), with F monotone, and K closed and convex. By Proposition 7, we have for any two solutions of the problem ˆx and x, (ˆx x) T θ( x) = (ˆx x) T θ(x) = 0 (ˆx x) T ( θ(ˆx) θ( x)) = 0. To claim θ(ˆx) = θ( x), we invoke Proposition 8 with F = θ, which claims that F(SOL(K,F)) is a singleton. Hence θ(ˆx) = θ( x), implying ˆx S. We now show that S S opt. Let x S be arbitrary. Game theory: Models, Algorithms and Applications 26

28 By the convexity of θ at some x S opt, we have by the gradient inequality θ( x) θ(x) ( x x) T θ(x). Since x S, we have θ(x) = θ( x) and θ( x) T (x x) = 0, implying that θ( x) θ(x) ( x x) T ( x) = 0. Therefore, θ(x) θ( x) but x is optimal solution, i.e., θ( x) θ(y) for all y K. Hence, θ(x) = θ( x), showing that x S opt. Game theory: Models, Algorithms and Applications 27