Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems

Transcription

1 Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems Roberto Lucchetti coauthors: A. Daniilidis, M.A. Goberna, M.A. López, Y. Viossat Dipartimento di Matematica, Politecnico di Milano 13-th Workshop on Well-posedness of Optimization Problems and Related Topics Borovets, September 2011 R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems1 / 21

2 Zero Sum Games An n m matrix P = (p ij ), where p ij R for all i, j, represents a two player, finite, zero-sum game: player one chooses a row i, player two a column j, and the entry p ij of the matrix P so determined is the amount the second player pays to the first one. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems2 / 21

3 Zero Sum Games An n m matrix P = (p ij ), where p ij R for all i, j, represents a two player, finite, zero-sum game: player one chooses a row i, player two a column j, and the entry p ij of the matrix P so determined is the amount the second player pays to the first one R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems2 / 21

4 Zero Sum Games An n m matrix P = (p ij ), where p ij R for all i, j, represents a two player, finite, zero-sum game: player one chooses a row i, player two a column j, and the entry p ij of the matrix P so determined is the amount the second player pays to the first one R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems2 / 21

5 Zero Sum Games , R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems3 / 21

6 Zero Sum Games , Suppose there exist a value v, row ī and column j such that pīj v for all j and p i j v for all i: the first player is able to guarantee himself at least v, the second player can guarantee to pay no more than v. In particular (from the first inequality) pī j v and (from the second inequality) pī j v: pī j = v is the rational outcome of the game, with (ī, j) a pair of optimal strategies for the players. From the second example: Need of mixed strategies. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems3 / 21

7 Zero Sum Games Theorem (von Neumann) A two player, finite, zero sum game as described by a payoff matrix P has equilibrium in mixed strategies. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems4 / 21

8 Zero Sum Games Theorem (von Neumann) A two player, finite, zero sum game as described by a payoff matrix P has equilibrium in mixed strategies. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems4 / 21

9 Finding optimal strategies The first player must choose S n α = (α 1,..., α n ): α 1 p 1j + + α n p nj v, 1 j m, v as large as possible. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems5 / 21

10 Finding optimal strategies The first player must choose S n α = (α 1,..., α n ): α 1 p 1j + + α n p nj v, 1 j m, v as large as possible. With the change of variable (p ij > 0, thus v > 0) x i = α i m i=1 α i = 1 becomes m i=1 x i = 1 v : maximizing v is equivalent to minimizing m i=1 x i. The first player problem: { inf 1n, x such that x 0, P T. x 1 m v. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems5 / 21

11 Finding optimal strategies The first player must choose S n α = (α 1,..., α n ): α 1 p 1j + + α n p nj v, 1 j m, v as large as possible. With the change of variable (p ij > 0, thus v > 0) x i = α i m i=1 α i = 1 becomes m i=1 x i = 1 v : maximizing v is equivalent to minimizing m i=1 x i. The first player problem: { inf 1n, x such that x 0, P T. x 1 m Second player v. { sup 1m, y such that y 0, Py 1 n. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems5 / 21

12 Finite General Games A bimatrix (A, B) = (a ij, b ij ), i = 1,..., n, j = 1,..., m is a finite game in strategic form (a ij, b ij utilities of the players). In the zero sum case b ij = a ij ). Let I = {1,..., n}, J = {1,..., m} and X = I J. Let k denote the standard simplex in R k. A Nash equilibrium is a pair ( p, q), ( p n ), ( q m ) such that p i q j a ij i,j i,j p i q j b ij i,j i,j p i q j a ij p i q j b ij for all p n, q m. If both p, q are extreme points of the simplexes: pure Nash equilibria, if inequalities are strict for p p and q q: strict Nash equilibria. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems6 / 21

13 Finding Nash Equilibria Notation: f (p, q) = i,j p iq j a ij. BR 1 : Y X : BR 1 (y) = Max {f (, y)} BR 2 : X Y : BR 2 (x) = Max {g(x, )}, BR : X Y X Y : BR(x, y) = (BR 1 (y), BR 2 (x)). A Nash equilibrium for a game is a fixed point for BR (Best Reaction). R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems7 / 21

14 Correlated Equilibria A correlated equilibrium is a probability distribution p = (p ij ) on X such that, for all ī I, m j=1 pīj aīj m pīj a ij i I, j=1 and such that, for all j J n p i j b i j i=1 n p i j b ij j J. i=1 R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems8 / 21

15 Example 1 Example ( (6, 6) (2, 7) (7, 2) (0, 0) ). Nash equilibria: (2, 7), (7, 2) (pure), also a mixed providing 14 3 to both. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems9 / 21

16 Example 1 Example ( (6, 6) (2, 7) (7, 2) (0, 0) ). Nash equilibria: (2, 7), (7, 2) (pure), also a mixed providing 14 3 ). ( to both. is a nice correlated equilibrium. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems9 / 21

17 Example 2 Example ( (5, 5) (0, 5) (5, 0) (1, 1) ). (5, 5) is fragile R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 10 / 21

18 Aim of the paper(s) 1 to generalize former results of lower stability in linear inequality systems in two ways: by allowing also restricted perturbations by restricting the solution map to its effective domain R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 11 / 21

19 Aim of the paper(s) 1 to generalize former results of lower stability in linear inequality systems in two ways: by allowing also restricted perturbations by restricting the solution map to its effective domain 2 to apply the machinery to equilibria in games R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 11 / 21

20 Some notation Linear inequality systems in R n, with arbitrary index set T, i.e. systems of the form σ = {a tx b t, t T }, a : T R n, b : T R. We shall identify σ with the data (a, b), so that the parametric space is Θ = (R n+1 ) T. The solution set mapping (or feasible set mapping) is F : Θ R n such that F (σ) = {x R n : a tx b t, t T }, with domain dom F = {σ Θ : F (σ) }. Solution set mapping relative to its domain the restriction of F to domf, denoted by F R. When one parameter is fixed we write a subscript, so when a is fixed F a, Θ a, when b is fixed F b, Θ b. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 12 / 21

21 Definitions Definition x is a Slater point for σ if a t x < b t, t T ; x is a strong Slater point (SS) for σ if there exists ρ > 0 such that a t x + ρ b t, t T. σ satisfies the (strong) Slater condition if there is a (strong) Slater point for σ R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 13 / 21

22 Definitions Definition x is a Slater point for σ if a t x < b t, t T ; x is a strong Slater point (SS) for σ if there exists ρ > 0 such that a t x + ρ b t, t T. σ satisfies the (strong) Slater condition if there is a (strong) Slater point for σ Definition A consistent system σ is continuous whenever T is a compact Hausdorff topological space and a : T R n and b : T R are continuous. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 13 / 21

23 Main Reference Theorem Theorem The following are equivalent: 1 σ has a strong Slater point 2 F is lower semicontinuous at σ 3 σ intdomf 4 F is dimensionally stable at σ. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 14 / 21

24 First Results: topological structure of the domains of F, F a, F b Proposition dom F is neither open nor closed F a, F b can be open, closed, or both Complete characterizations in the second case R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 15 / 21

25 Perturbing All data Proposition Given σ domf, the following statements are true: (i) If either σ satisfies SSC or F (σ) is a singleton set, then F R is lsc at σ. (ii) If F R is lsc at σ and F (σ) is not a singleton set, then σ satisfies SSC. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 16 / 21

26 Perturbing All data Proposition Given σ domf, the following statements are true: (i) If either σ satisfies SSC or F (σ) is a singleton set, then F R is lsc at σ. (ii) If F R is lsc at σ and F (σ) is not a singleton set, then σ satisfies SSC. Proposition Let σ domf be a continuous system without trivial inequalities. TFAE: F R is dimensionally stable at σ the SSC holds dim F (σ) = n. Then F R is lsc at σ if and only if dim F (σ) {0, n}. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 16 / 21

27 Perturbing the RHS Proposition Let σ = {a tx b t, t T } domf a. If either 0 n+1 / clconv {(a t, b t ), t T \T 0 } or F (σ) is a singleton set, then F R a is lsc at σ. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 17 / 21

28 Perturbing the RHS Proposition Let σ = {a tx b t, t T } domf a. If either 0 n+1 / clconv {(a t, b t ), t T \T 0 } or F (σ) is a singleton set, then F R a is lsc at σ. Proposition Let σ domf a be a continuous system without trivial inequalities. TFAE: F R a is dimensionally stable at σ SSC holds dim F (σ) = n. As a result, if dim F (σ) {0, n}, then F R a is lsc at σ. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 17 / 21

29 Perturbing the RHS Proposition Let σ = {a tx b t, t T } domf a. If either 0 n+1 / clconv {(a t, b t ), t T \T 0 } or F (σ) is a singleton set, then F R a is lsc at σ. Proposition Let σ domf a be a continuous system without trivial inequalities. TFAE: F R a is dimensionally stable at σ SSC holds dim F (σ) = n. As a result, if dim F (σ) {0, n}, then F R a is lsc at σ. Proposition If T is finite, then F R a is lsc at any σ domf a. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 17 / 21

30 Perturbing the LHS Proposition Let σ domf b be a continuous system without trivial inequalities such that 0 n / F(σ). TFAE: F R b is dimensionally stable at σ SSC holds dim F(σ) = n. Moreover, any of these properties implies that F R b is lsc at σ. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 18 / 21

31 Perturbing the LHS Proposition Let σ domf b be a continuous system without trivial inequalities such that 0 n / F(σ). TFAE: F R b is dimensionally stable at σ SSC holds dim F(σ) = n. Moreover, any of these properties implies that F R b Proposition is lsc at σ. Assume that T is finite and that σ contains no trivial inequality. Then F R b is lsc at σ domf if and only if one of the following alternatives holds: dim F(σ) = n dim F(σ) = 0 F(σ) is a non-singleton set contained in some open ray. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 18 / 21

32 Correlated Equilibria:the Two by Two Case Proposition (the two by two case) In a two by two bimatrix game the following happens: If there are no dominated strategies, then all correlated equilibria are lower stable in presence of dominated strategies, only pure strict Nash equilibria are lower stable R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 19 / 21

33 Correlated Equilibria:the Zero Sum Case Proposition A correlated equilibrium of a zero-sum game is stable within the space of zero-sum games if and only if it is the unique correlated equilibrium of the game. R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 20 / 21

34 References A. Daniilidis, M.A. Goberna, M.A. López, R.L. Lower semicontinuity of the solution set mapping of linear systems relative to their domains, 2011 work in progress R.L., Y. Viossat Stable correlated equilibria: the zero-sum case, unpublished note M.A. Goberna, M.A. López Linear Semi-Infinite Optimization.Wiley, Chichester, England (1998) R. Lucchetti (Politecnico di Milano) Lower Semicontinuity of the Solution Set Mapping in Some Optimization Problems 21 / 21