Generalized Nash Equilibrium Problem: existence, uniqueness and

Transcription

1 Generalized Nash Equilibrium Problem: existence, uniqueness and reformulations Univ. de Perpignan, France CIMPA-UNESCO school, Delhi November 25 - December 6, 2013

2 Outline of the 7 lectures Generalized Nash Equilibrium a- Reformulations b- Existence of equilibrium Variational inequalities: motivations, definitions a- Motivations, definitions b- Existence of solutions c- Stability d- Uniqueness Quasiconvex optimization a- Classical subdifferential approach b- Normal approach Whenever a set-valued map is not really set-valued... Quasivariational inequalities

3 Outline of lecture 2 I- Classical approaches of quasiconvex analysis II- Normal approach a- First definitions b- Adjusted sublevel sets and normal operator III- Quasiconvex optimization a- Optimality conditions b- Convex constraint case c- Nonconvex constraint case

4 I - Introduction to quasiconvex optimization

5 Quasiconvexity A function f : X IR {+ } is said to be quasiconvex on K if, for all x, y K and all t [0, 1], f (tx + (1 t)y) max{f (x), f (y)}. or for all λ IR, the sublevel set S λ = {x X : f (x) λ} is convex.

6 Quasiconvexity A function f : X IR {+ } is said to be quasiconvex on K if, for all x, y K and all t [0, 1], or f (tx + (1 t)y) max{f (x), f (y)}. for all λ IR, the sublevel set S λ = {x X : f (x) λ} is convex. f differentiable f is quasiconvex iff df is quasimonotone iff df (x)(y x) > 0 df (y)(y x) 0

7 Quasiconvexity A function f : X IR {+ } is said to be quasiconvex on K if, for all x, y K and all t [0, 1], or f (tx + (1 t)y) max{f (x), f (y)}. for all λ IR, the sublevel set S λ = {x X : f (x) λ} is convex. f differentiable f is quasiconvex iff df is quasimonotone iff df (x)(y x) > 0 df (y)(y x) 0 f is quasiconvex iff f is quasimonotone iff x f (x) : x, y x > 0 y f (y), y, y x 0

8 Quasiconvexity A function f : X IR {+ } is said to be quasiconvex on K if, for all x, y K and all t [0, 1], f (tx + (1 t)y) max{f (x), f (y)}. or for all λ IR, the sublevel set S λ = {x X : f (x) λ} is convex. A function f : X IR {+ } is said to be semistrictly quasiconvex on K if, f is quasiconvex and for any x, y K, f (x) < f (y) f (z) < f (y), z [x, y[. convex semistrictly quasiconvex quasiconvex

9 Why not a subdifferential for quasiconvex programming?

10 Why not a subdifferential for quasiconvex programming? No (upper) semicontinuity of f if f is not supposed to be Lipschitz

11 Why not a subdifferential for quasiconvex programming? No (upper) semicontinuity of f if f is not supposed to be Lipschitz No sufficient optimality condition x S str ( f, C) = x arg min f C

12 II - Normal approach a- First definitions

13 A first approach Sublevel set: S λ = {x X : f (x) λ} S > λ = {x X : f (x) < λ} Normal operator: Define N f (x) : X 2 X by N f (x) = N(S f (x), x) = {x X : x, y x 0, y S f (x) }. With the corresponding definition for N > f (x)

14 But... N f (x) = N(S f (x), x) has no upper-semicontinuity properties N > f (x) = N(S > f (x), x) has no quasimonotonicity properties

15 But... N f (x) = N(S f (x), x) has no upper-semicontinuity properties N > f (x) = N(S > f (x), x) has no quasimonotonicity properties Example Define f : R 2 R by { a + b, if a + b 1 f (a, b) = 1, if a + b > 1. Then f is quasiconvex.

16 But... N f (x) = N(S f (x), x) has no upper-semicontinuity properties N > f (x) = N(S > f (x), x) has no quasimonotonicity properties Example Define f : R 2 R by { a + b, if a + b 1 f (a, b) = 1, if a + b > 1. Then f is quasiconvex. Consider x = (10, 0), x = (1, 2), y = (0, 10) and y = (2, 1). We see that x N < (x) and y N < (y) (since a + b < 1 implies (1, 2) (a 10, b) 0 and (2, 1) (a, b 10) 0) while x, y x > 0 and y, y x < 0. Hence N < is not quasimonotone.

17 But...another example N f (x) = N(S f (x), x) has no upper-semicontinuity properties N > f (x) = N(S > f (x), x) has no quasimonotonicity properties Example Then f is quasiconvex.

18 But...another example N f (x) = N(S f (x), x) has no upper-semicontinuity properties N > f (x) = N(S > f (x), x) has no quasimonotonicity properties Example Then f is quasiconvex. We easily see that N(x) is not upper semicontinuous...

19 But...another example N f (x) = N(S f (x), x) has no upper-semicontinuity properties N > f (x) = N(S > f (x), x) has no quasimonotonicity properties Example Then f is quasiconvex. We easily see that N(x) is not upper semicontinuous... These two operators are essentially adapted to the class of semi-strictly quasiconvex functions. Indeed in this case, for each x dom f \ arg min f, the sets S f (x) and S < f (x) have the same closure and N f (x) = N f < (x).

20 II - Normal approach b- Adjusted sublevel sets and normal operator

21 Definition Adjusted sublevel set For any x dom f, we define Sf a (x) = S f (x) B(S < f (x), ρ x) where ρ x = dist(x, S < f (x) ), if S < f (x) and Sf a(x) = S f (x) if S < f (x) =.

22 Definition Adjusted sublevel set For any x dom f, we define Sf a (x) = S f (x) B(S < f (x), ρ x) where ρ x = dist(x, S < f (x) ), if S < f (x) and Sf a(x) = S f (x) if S < f (x) =. S a f (x) coincides with S f (x) if cl(s > f (x) ) = S f (x)

23 Definition Adjusted sublevel set For any x dom f, we define Sf a (x) = S f (x) B(S < f (x), ρ x) where ρ x = dist(x, S < f (x) ), if S < f (x) and Sf a(x) = S f (x) if S < f (x) =. S a f (x) coincides with S f (x) if cl(s > f (x) ) = S f (x) e.g. f is semistrictly quasiconvex

24 Definition Adjusted sublevel set For any x dom f, we define Sf a (x) = S f (x) B(S < f (x), ρ x) where ρ x = dist(x, S < f (x) ), if S < f (x) and Sf a(x) = S f (x) if S < f (x) =. S a f (x) coincides with S f (x) if cl(s > f (x) ) = S f (x) e.g. f is semistrictly quasiconvex Proposition Let f : X IR {+ } be any function, with domain dom f. Then f is quasiconvex S a f (x) is convex, x dom f.

25 Proof Let us suppose that Sf a (u) is convex for every u dom f. We will show that for any x dom f, S f (x) is convex. If x arg min f then S f (x) = Sf a (x) is convex by assumption. Assume now that x / arg min f and take y, z S f (x). ) (S < f (x), ρx If both y and z belong to B If both y and z do not belong to B, then y, z S a f (x) thus [y, z] S a f (x) S f (x). (S < f (x), ρx ), then f (x) = f (y) = f (z), S < f (z) = S< f (y) = S< f (x) ) and ρ y, ρ z are positive. If, say, ρ y ρ z then y, z B (S < f (y), ρy thus y, z S a f (y) and [y, z] S a f (y) S f (y) = S f (x).

26 Proof Finally, suppose that only one of y, z, say z, belongs to B(S < f (x), ρx ) while y / B(S<, ρx ). Then f (x) f (x) = f (y), S < f (y) = S< and ρy > ρx f (x) ) ) so we have z B (S < f (x), ρx B (S < f (y), ρy and we deduce as before that [y, z] Sf a (y) S f (y) = S f (x). The other implication is straightforward.

27 Adjusted normal operator Adjusted sublevel set: For any x dom f, we define Sf a (x) = S f (x) B(S < f (x), ρ x) where ρ x = dist(x, S < f (x) ), if S < f (x). Ajusted normal operator: N a f (x) = {x X : x, y x 0, y S a f (x)}

28 Example x

29 Example x B(S < f (x), ρ x) S a f (x) = S f (x) B(S < f (x), ρ x)

30 Example S a f (x) = S f (x) B(S < f (x), ρ x) N a f (x) = {x X : x, y x 0, y S a f (x)}

31 An exercice... Let us draw the normal operator value Nf a (x, y) at the points (x, y) = (0.5, 0.5), (x, y) = (0, 1), (x, y) = (1, 0), (x, y) = (1, 2), (x, y) = (1.5, 0) and (x, y) = (0.5, 2).

32 An exercice... Let us draw the normal operator value Nf a (x, y) at the points (x, y) = (0.5, 0.5), (x, y) = (0, 1), (x, y) = (1, 0), (x, y) = (1, 2), (x, y) = (1.5, 0) and (x, y) = (0.5, 2). Operator N a f provide information at any point!!!

33 Subdifferential vs normal operator One can have N a f (x) cone( f (x)) or cone( f (x)) N a f (x) Proposition f is quasiconvex and x dom f If there exists δ > 0 such that 0 L f (B(x, δ)) then [ cone( L f (x)) ] f (x) N a f (x). Proposition f is lsc semistrictly quasiconvex and 0 f (X ) then cone( f (x)) N a f (x). If additionally = and f is Lipschitz then N a f (x) = cone( f (x)).

34 Basic properties of N a f Nonemptyness: Proposition Let f : X IR {+ } be lsc. Assume that rad. continuous on dom f or dom f is convex and ints λ, λ > inf X f. Then f is quasiconvex Nf a (x) \ {0}, x dom f \ arg min f. Quasimonotonicity: The normal operator N a f is always quasimonotone

35 Upper sign-continuity T : X 2 X is said to be upper sign-continuous on K iff for any x, y K, one have : t ]0, 1[, where x t = (1 t)x + ty. inf x, y x 0 x T (x t) = sup x, y x 0 x T (x) upper semi-continuous upper hemicontinuous upper sign-continuous

36 locally upper sign continuity Definition Let T : K 2 X be a set-valued map. T is called locally upper sign-continuous on K if, for any x K there exist a neigh. V x of x and a upper sign-continuous set-valued map Φ x ( ) : V x 2 X with nonempty convex w -compact values such that Φ x (y) T (y) \ {0}, y V x

37 locally upper sign continuity Definition Let T : K 2 X be a set-valued map. T is called locally upper sign-continuous on K if, for any x K there exist a neigh. V x of x and a upper sign-continuous set-valued map Φ x ( ) : V x 2 X with nonempty convex w -compact values such that Φ x (y) T (y) \ {0}, y V x Continuity of normal operator Proposition Let f be lsc quasiconvex function such that int(s λ ), λ > inf f. Then N f is locally upper sign-continuous on dom f \ arg min f.

38 Proposition If f is quasiconvex such that int(s λ ), λ > inf f and f is lsc at x dom f \ arg min f, Then N a f is norm-to-w cone-usc at x. A multivalued map with conical valued T : X 2 X is said to be cone-usc at x dom T if there exists a neighbourhood U of x and a base C (u) of T (u), u U, such that u C (u) is usc at x.

39 Integration of N a f Let f : X IR {+ } quasiconvex Question: Is it possible to characterize the functions g : X IR {+ } quasiconvex such that N a f = N a g?

40 Integration of N a f Let f : X IR {+ } quasiconvex Question: Is it possible to characterize the functions g : X IR {+ } quasiconvex such that N a f = N a g? A first answer: Let C = {g : X IR {+ } cont. semistrictly quasiconvex such that argmin f is included in a closed hyperplane} Then, for any f, g C, N a f = N a g g is N a f \ {0}-pseudoconvex x N a f (x) \ {0} : x, y x 0 g(x) g(y).

41 Integration of N a f Let f : X IR {+ } quasiconvex Question: Is it possible to characterize the functions g : X IR {+ } quasiconvex such that N a f = N a g? A first answer: Let C = {g : X IR {+ } cont. semistrictly quasiconvex such that argmin f is included in a closed hyperplane} Then, for any f, g C, N a f = N a g g is N a f \ {0}-pseudoconvex x N a f (x) \ {0} : x, y x 0 g(x) g(y). General case: open question

42 IV Quasiconvex programming a- Optimality conditions

43 Quasiconvex programming Let f : X IR {+ } and K dom f be a convex subset. (P) find x K : f ( x) = inf x K f (x)

44 Quasiconvex programming Let f : X IR {+ } and K dom f be a convex subset. (P) find x K : f ( x) = inf x K f (x) Perfect case: f convex f : X IR {+ } a proper convex function K a nonempty convex subset of X, x K + C.Q. Then f ( x) = inf x K f (x) x S str ( f, K)

45 Quasiconvex programming Let f : X IR {+ } and K dom f be a convex subset. (P) find x K : f ( x) = inf x K f (x) Perfect case: f convex f : X IR {+ } a proper convex function K a nonempty convex subset of X, x K + C.Q. Then f ( x) = inf x K x S str ( f, K) What about f quasiconvex case? x S str ( f ( x), K) = x arg min f K

46 Sufficient optimality condition Theorem f : X IR {+ } quasiconvex, radially cont. on dom f C X such that conv(c) dom f. Suppose that C int(dom f ) or AffC = X. Then x S(N a f \ {0}, C) = x C, f ( x) f (x). where x S(N a f \ {0}, K) means that there exists x N a f ( x) \ {0} such that x, c x 0, c C.

47 Lemma Let f : X IR {+ } be a quasiconvex function, radially continuous on dom f. Then f is N a f \ {0}-pseudoconvex on int(dom f ), that is, x N a f (x) \ {0} : x, y x 0 f (y) f (x). Proof. Let x, y int(dom f ). According to the quasiconvexity of f, Nf a (x) \ {0} is nonempty. Let us suppose that x, y x 0 with x Nf a (x) \ {0}. Let d X be such that x, y n x > 0 for any n, where y n = y + n 1 d ( dom f for n large enough). This implies that y n S < (x) since x N a f f (x) N< (x). f It follows by the radial continuity of f that f (y) f (x).

48 Necessary and Sufficient conditions Proposition Let C be a closed convex subset of X, x C and f : X IR be continuous semistrictly quasiconvex such that int(sf a ( x)) and f ( x) > inf X f. Then the following assertions are equivalent: i) f ( x) = min C f ii) x S str (N a f \ {0}, C) iii) 0 Nf a ( x) \ {0} + NK(C, x).

49 Proposition Let C be a closed convex subset of an Asplund space X and f : X IR be a continuous quasiconvex function. Suppose that either f is sequentially normally sub-compact or C is sequentially normally compact. If x C is such that 0 L f ( x) and 0 f ( x) \ {0} + NK(C, x) then the following assertions are equivalent: i) f ( x) = min C f ii) x L f ( x) \ {0} + NK(C, x) iii) 0 Nf a ( x) \ {0} + NK(C, x)

50 References D. A. & N. Hadjisavvas, Adjusted sublevel sets, normal operator and quasiconvex programming, SIAM J. Optim., 16 (2005), D. Aussel & J. Ye, Quasiconvex minimization on locally finite union of convex sets, J. Optim. Th. Appl., 139 (2008), no. 1, 116. D. Aussel & J. Ye, Quasiconvex programming with starshaped constraint region and application to quasiconvex MPEC, Optimization 55 (2006),

51 Appendix 5 - Normally compactness A subset C is said to be sequentially normally compact at x C if for any sequence (x k ) k C converging to x and any sequence (x k ) k, x k NF (C, x k ) weakly converging to 0, one has x k 0. Examples: X finite dimensional space C epi-lipschitz at x C convex with nonempty interior A function f is said to be sequentially normally subcompact at x C if the sublevel set S f (x) is sequentially normally compact at x. Examples: X finite dimensional space f is locally Lipschitz around x and 0 L f (x) f is quasiconvex with int(s f (x) )

52 Appendix 2 - Limiting Normal Cone Let C be any nonempty subset of X and x C. The limiting normal cone to C at x, denoted by N L (C, x) is defined by N L (C, x) = lim sup N F (C, x ) x x where the Frèchet normal cone N F (C, x ) is defined by { N F (C, x ) = x X x, u x : lim sup u x, u C u x 0 }. The limiting normal cone is closed (but in general non convex)

53 Appendix - Limiting subdifferential One can define the Limiting subdifferential (in the sense of Mordukhovich), and its asymptotic associated form, of a function f : X IR {+ } by L f (x) f (x) = Limsup y f x F f (y) = { x X : (x, 1) N L (epi f, (x, f (x))) } = Limsup f λ F f (y) y x λ 0 = { x X : (x, 0) N L (epi f, (x, f (x))) }.