Calmness in stochastic programming exact penalization and sample - PowerPoint PPT Presentation

Calmness in stochastic programming – exact penalization and sample approximation techniques Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Stochastic Programming and Approximation 21 February 2013, Prague M.Branda (Charles University) SPaA 2013 1 / 55

Contents 1 Exact penalty method and calmness 2 Stochastic programming formulations 3 Relations between formulations 4 Sample approximations using Monte-Carlo techniques 5 References M.Branda (Charles University) SPaA 2013 2 / 55

Exact penalty method and calmness Contents 1 Exact penalty method and calmness 2 Stochastic programming formulations 3 Relations between formulations 4 Sample approximations using Monte-Carlo techniques 5 References M.Branda (Charles University) SPaA 2013 3 / 55

Exact penalty method and calmness Basis of the exact penalty method I.I. Eremin (1966). Penalty method in convex programming . Soviet Math. Dokl. 8, 459–462 W.I. Zangwill (1967). Nonlinear programming via penalty functions . Management Sci. 13, 344–358 M.Branda (Charles University) SPaA 2013 4 / 55

Exact penalty method and calmness Nonlinear programming problems and penalty problem Nonlinear programming problem min x ∈ X f ( x ) s . t . g j ( x ) ≤ 0 , j = 1 , . . . , m , where f , g j : R n → R , X ⊆ R n . Corresponding penalty problem min x ∈ X f ( x ) + N · α ( x ) , where m � | g j ( x ) | p + , p ∈ N . α ( x ) = j =1 M.Branda (Charles University) SPaA 2013 5 / 55

Exact penalty method and calmness Nonlinear programming problems and penalty problem Nonlinear programming problem min x ∈ X f ( x ) s . t . g j ( x ) ≤ 0 , j = 1 , . . . , m , where f , g j : R n → R , X ⊆ R n . Corresponding penalty problem min x ∈ X f ( x ) + N · α ( x ) , where m � | g j ( x ) | p + , p ∈ N . α ( x ) = j =1 M.Branda (Charles University) SPaA 2013 5 / 55

Exact penalty method and calmness Exterior penalty method Bazaraa et al. (2006), Theorem 9.2.2 (also for equality constraints): Proposition Let f , g j be continuous, X � = ∅ , the underlying problem have a feasible solution. Assume that for each N there is a solution x N ∈ X of the penalty problem and { x N } is contained in a compact subset of X. Then x ∈ X { f ( x ) : g j ( x ) ≤ 0 , j = 1 , . . . , m } = sup min θ ( N ) = lim N →∞ θ ( N ) , N ≥ 0 where x ∈ X f ( x ) + N · α ( x ) . θ ( N ) = min Furthermore, the limit of any convergent subsequence of { x N } is an optimal solution to the original problem, and N · α ( x N ) → 0 as N → ∞ . Employed by Ermoliev et al. (2001), Branda (2012a, 2012b, 2013), Branda and Dupaˇ cov´ a (2012). M.Branda (Charles University) SPaA 2013 6 / 55

Exact penalty method and calmness Exact absolute value penalty method Remark If for some N > 0 it holds α ( x N ) = 0 , then x N is the optimal solution of the NLP, see Bazaraa et al. (2006), Corollary 9.2.2. It is a question, how to ensure this situations known as “exact penalization” in general. Bazaraa et al. (2006), Theorem 9.3.1 (also for equality constraints): Proposition Let ( x ∗ , v ∗ ) ∈ R n × R m + be a KKT point. Moreover, suppose that f , g j are j , x ∗ minimizes also the penalized convex functions. Then for N ≥ max j v ∗ objective with p = 1 (L 1 penalty). M.Branda (Charles University) SPaA 2013 7 / 55

Exact penalty method and calmness Exact absolute value penalty method Remark If for some N > 0 it holds α ( x N ) = 0 , then x N is the optimal solution of the NLP, see Bazaraa et al. (2006), Corollary 9.2.2. It is a question, how to ensure this situations known as “exact penalization” in general. Bazaraa et al. (2006), Theorem 9.3.1 (also for equality constraints): Proposition Let ( x ∗ , v ∗ ) ∈ R n × R m + be a KKT point. Moreover, suppose that f , g j are j , x ∗ minimizes also the penalized convex functions. Then for N ≥ max j v ∗ objective with p = 1 (L 1 penalty). M.Branda (Charles University) SPaA 2013 7 / 55

Exact penalty method and calmness Calmness Calm problems Calm set-valued mappings (graphs) Calm functions M.Branda (Charles University) SPaA 2013 8 / 55

Exact penalty method and calmness A general mathematical program A general relaxed mathematical program min f ( x ) s . t . F ( x ) + u ∈ Λ f : R n → R , F : R n → R m , Λ ⊆ R m closed, u ∈ R m . Underlying problem for u = 0. We denote d Λ ( x ) = min x ′ ∈ Λ � x − x ′ � . M.Branda (Charles University) SPaA 2013 9 / 55

Exact penalty method and calmness Calm problems Burke (1991a), Definition 1.1: Definition Let x ∗ be feasible for the unperturbed problem. Then the problem is said to be calm at x ∗ if there exist constant ˜ N ≥ 0 (modulus) and ǫ > 0 (radius) such that for all ( x , u ) ∈ R n × R m satisfying x ∈ B ǫ ( x ∗ ) and F ( x ) + u ∈ Λ, one has f ( x ) + ˜ N � u � ≥ f ( x ∗ ) . Note that then x ∗ is necessarily a local solution to the unperturbed problem. M.Branda (Charles University) SPaA 2013 10 / 55

Exact penalty method and calmness Exact penalization Burke (1991a), Theorem 1.1: Proposition Let x ∗ be feasible for the unperturbed problem, i.e. with u = 0 . Then the unperturbed problem is calm at x ∗ with modulus ˜ N and radius ǫ > 0 if and only if x ∗ is a local minimum of the function f ( x ) + N · d Λ ( F ( x )) over the neighbourhood B ǫ ( x ∗ ) for all N ≥ ˜ N. Penalty function d Λ ( x ) = min x ′ ∈ Λ � x − x ′ � . M.Branda (Charles University) SPaA 2013 11 / 55

Exact penalty method and calmness Lipschitz-like properties of set-valued mappings Set valued mapping (multifunction) Z : Y ⇒ X between metric spaces X and Y . Local Lipschitz property at y : ∃ L , ε > 0 d Z ( y 1 ) ( x ) ≤ L · d ( y 1 , y 2 ) , ∀ x ∈ Z ( y 2 ) , ∀ y 1 , y 2 ∈ B ε ( y ) . Aubin property at y : ∃ L , ε > 0 d Z ( y 1 ) ( x ) ≤ L · d ( y 1 , y 2 ) , ∀ x ∈ Z ( y 2 ) ∩ B ε ( x ) , ∀ y 1 , y 2 ∈ B ε ( y ) , where x ∈ Z ( y ). Local upper Lipschitz property at y : ∃ L , ε > 0 d Z ( y ) ( x ) ≤ L · d ( y , y ) , ∀ x ∈ Z ( y ) , ∀ y ∈ B ε ( y ) . Calmness at ( y , x ) ∈ Gph Z : ∃ L , ε > 0 d Z ( y ) ( x ) ≤ L · d ( y , y ) , ∀ x ∈ Z ( y ) ∩ B ε ( x ) , ∀ y ∈ B ε ( y ) . M.Branda (Charles University) SPaA 2013 12 / 55

Exact penalty method and calmness Lipschitz-like properties of set-valued mappings Set valued mapping (multifunction) Z : Y ⇒ X between metric spaces X and Y . Local Lipschitz property at y : ∃ L , ε > 0 d Z ( y 1 ) ( x ) ≤ L · d ( y 1 , y 2 ) , ∀ x ∈ Z ( y 2 ) , ∀ y 1 , y 2 ∈ B ε ( y ) . Aubin property at y : ∃ L , ε > 0 d Z ( y 1 ) ( x ) ≤ L · d ( y 1 , y 2 ) , ∀ x ∈ Z ( y 2 ) ∩ B ε ( x ) , ∀ y 1 , y 2 ∈ B ε ( y ) , where x ∈ Z ( y ). Local upper Lipschitz property at y : ∃ L , ε > 0 d Z ( y ) ( x ) ≤ L · d ( y , y ) , ∀ x ∈ Z ( y ) , ∀ y ∈ B ε ( y ) . Calmness at ( y , x ) ∈ Gph Z : ∃ L , ε > 0 d Z ( y ) ( x ) ≤ L · d ( y , y ) , ∀ x ∈ Z ( y ) ∩ B ε ( x ) , ∀ y ∈ B ε ( y ) . M.Branda (Charles University) SPaA 2013 12 / 55

Exact penalty method and calmness Lipschitz-like properties of set-valued mappings Set valued mapping (multifunction) Z : Y ⇒ X between metric spaces X and Y . Local Lipschitz property at y : ∃ L , ε > 0 d Z ( y 1 ) ( x ) ≤ L · d ( y 1 , y 2 ) , ∀ x ∈ Z ( y 2 ) , ∀ y 1 , y 2 ∈ B ε ( y ) . Aubin property at y : ∃ L , ε > 0 d Z ( y 1 ) ( x ) ≤ L · d ( y 1 , y 2 ) , ∀ x ∈ Z ( y 2 ) ∩ B ε ( x ) , ∀ y 1 , y 2 ∈ B ε ( y ) , where x ∈ Z ( y ). Local upper Lipschitz property at y : ∃ L , ε > 0 d Z ( y ) ( x ) ≤ L · d ( y , y ) , ∀ x ∈ Z ( y ) , ∀ y ∈ B ε ( y ) . Calmness at ( y , x ) ∈ Gph Z : ∃ L , ε > 0 d Z ( y ) ( x ) ≤ L · d ( y , y ) , ∀ x ∈ Z ( y ) ∩ B ε ( x ) , ∀ y ∈ B ε ( y ) . M.Branda (Charles University) SPaA 2013 12 / 55

Exact penalty method and calmness Lipschitz-like properties of set-valued mappings Set valued mapping (multifunction) Z : Y ⇒ X between metric spaces X and Y . Local Lipschitz property at y : ∃ L , ε > 0 d Z ( y 1 ) ( x ) ≤ L · d ( y 1 , y 2 ) , ∀ x ∈ Z ( y 2 ) , ∀ y 1 , y 2 ∈ B ε ( y ) . Aubin property at y : ∃ L , ε > 0 d Z ( y 1 ) ( x ) ≤ L · d ( y 1 , y 2 ) , ∀ x ∈ Z ( y 2 ) ∩ B ε ( x ) , ∀ y 1 , y 2 ∈ B ε ( y ) , where x ∈ Z ( y ). Local upper Lipschitz property at y : ∃ L , ε > 0 d Z ( y ) ( x ) ≤ L · d ( y , y ) , ∀ x ∈ Z ( y ) , ∀ y ∈ B ε ( y ) . Calmness at ( y , x ) ∈ Gph Z : ∃ L , ε > 0 d Z ( y ) ( x ) ≤ L · d ( y , y ) , ∀ x ∈ Z ( y ) ∩ B ε ( x ) , ∀ y ∈ B ε ( y ) . M.Branda (Charles University) SPaA 2013 12 / 55

Exact penalty method and calmness Perturbed constraint set Let F : X → Y be a continuous mapping, Λ ⊆ Y be a closed set. Denote by M ( u ) = { x ∈ X : F ( x ) + u ∈ Λ } the perturbation of the (original) constraint set M (0) = F − 1 (Λ). M.Branda (Charles University) SPaA 2013 13 / 55

Exact penalty method and calmness Set-valued mapping calmness implies problem calmness Hoheisel et al. (2010), Proposition 3.5: Proposition Let x ∗ be a local minimizer such that M is calm at (0 , x ∗ ) . Then the original problem is calm at x ∗ . M.Branda (Charles University) SPaA 2013 14 / 55