Chance constrained problems: reformulation using penalty functions and sample approximation technique Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics 25th European Conference on Operational Research 8-11 July 2012, Vilnius M.Branda (Charles University) EURO 2012 1 / 31
Contents 1 Stochastic programming formulations 2 Relations between formulations 3 Sample approximations using Monte-Carlo techniques M.Branda (Charles University) EURO 2012 2 / 31
Stochastic programming formulations Contents 1 Stochastic programming formulations 2 Relations between formulations 3 Sample approximations using Monte-Carlo techniques M.Branda (Charles University) EURO 2012 3 / 31
Stochastic programming formulations Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇔ chance constrained problem objective problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check M.Branda (Charles University) EURO 2012 4 / 31
Stochastic programming formulations Optimization problem with a random factor Program with a random factor ξ min { f ( x ) : x ∈ X , g i ( x , ξ ) ≤ 0 , i = 1 , . . . , k } , where g i , i = 0 , . . . , k , are real functions on R n × R n ′ , X ⊆ R n and ξ ∈ R n ′ is a realization of a n ′ -dimensional random vector defined on the probability space (Ω , F , P ). If P is known , we can use chance constraints to deal with the random constraints... M.Branda (Charles University) EURO 2012 5 / 31
Stochastic programming formulations Chance constrained problem (CCP) Chance constrained problem ψ ǫ = min x ∈ X f ( x ) , s . t . � � P g 11 ( x , ξ ) ≤ 0 , . . . , g 1 k 1 ( x , ξ ) ≤ 0 ≥ 1 − ε 1 , . . . � � g m 1 ( x , ξ ) ≤ 0 , . . . , g mk m ( x , ξ ) ≤ 0 ≥ 1 − ε m , P with optimal solution x ǫ , where we denoted ǫ = ( ε 1 , . . . , ε m ) with levels ε j ∈ (0 , 1). The formulation covers the joint ( k 1 > 1 and m = 1) as well as the individual ( k j = 1 and m > 1) chance constrained problems as special cases. M.Branda (Charles University) EURO 2012 6 / 31
Stochastic programming formulations Solving chance constrained problems In general, the feasible region is not convex even if the functions are convex, it is even not easy to check feasibility because it leads to computations of multivariate integrals. Hence, we will try to reformulate the chance constrained problem using penalty functions. M.Branda (Charles University) EURO 2012 7 / 31
Stochastic programming formulations Penalty functions Penalty functions ϑ j : R m → R + , j = 1 , . . . , m , are continuous nondecreasing, equal to 0 on R m − and positive otherwise, e.g. k [ u i ] + � p , p ∈ N ϑ 1 , p ( u ) � � = i =1 ϑ 2 ( u ) 1 ≤ i ≤ k [ u i ] + , = max � = min t ≥ 0 : u i − t ≤ 0 , i = 1 , . . . , k } where u ∈ R m . Let p j denote the penalized constraints p j ( x , ξ ) = ϑ j ( g j 1 ( x , ξ ) , . . . , g jk j ( x , ξ )) , ∀ j . M.Branda (Charles University) EURO 2012 8 / 31
Stochastic programming formulations Penalty function problems Problem with penalties in the objective function m � � � ϕ N = min f ( x ) + N · E [ p j ( x , ξ )] x ∈ X j =1 with an optimal solution x N . In Ermoliev et al. (2000) for ϑ 1 , 1 and m = 1. Problem with generalized integrated chance constraints � � ϕ ICC = min f ( x ) : s . t . E [ p j ( x , ξ )] ≤ L j , j = 1 , . . . , m L x ∈ X for some prescribed bounds L j ≥ 0, L = ( L 1 , . . . , L m ) ′ , with an optimal (originally defined using u 2 , cf. Klein Haneveld (1986)). solution x ICC L M.Branda (Charles University) EURO 2012 9 / 31
Stochastic programming formulations Penalty function problems Problem with penalties in the objective function m � � � ϕ N = min f ( x ) + N · E [ p j ( x , ξ )] x ∈ X j =1 with an optimal solution x N . In Ermoliev et al. (2000) for ϑ 1 , 1 and m = 1. Problem with generalized integrated chance constraints � � ϕ ICC = min f ( x ) : s . t . E [ p j ( x , ξ )] ≤ L j , j = 1 , . . . , m L x ∈ X for some prescribed bounds L j ≥ 0, L = ( L 1 , . . . , L m ) ′ , with an optimal (originally defined using u 2 , cf. Klein Haneveld (1986)). solution x ICC L M.Branda (Charles University) EURO 2012 9 / 31
Stochastic programming formulations History and applications of the penalty approach in SP Pr´ ekopa (1973): CPP and penalization Dupaˇ cov´ a et al (1991): Water management (empirical study) Ermoliev et al (2000): Managing exposure to catastrophic risks (asymptotic equivalence with particular penalty) M.B. and Dupaˇ cov´ a (2008, 2012): Contamination technique for CCP (asymptotic equivalence using general penalty functions) ˇ Zampachov´ a (2009): Beam design (reliability problem with partial differential equations - nonlinear - significant reduction of computational time) M.B. (2009, 2012A): Value at Risk optimization with transaction costs and integer allocations (general penalty functions and several CC) M.B (2011): Blending problem (asymptotic equivalence with generalized integrated chance constraints) M.Branda (Charles University) EURO 2012 10 / 31
Relations between formulations Contents 1 Stochastic programming formulations 2 Relations between formulations 3 Sample approximations using Monte-Carlo techniques M.Branda (Charles University) EURO 2012 11 / 31
Relations between formulations Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇐ ⇒ with penalty ⇔ chance constrained problem objective problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check M.Branda (Charles University) EURO 2012 12 / 31
Relations between formulations Asymptotic equivalence and bounds M.B. (2012A): Under the following assumptions, the asymptotic equivalence of the CCP and the PPO problems can be shown: Continuity of the constraints and the probabilistic functions. Compactness of the fixed set of feasible solutions. Existence of integrable majorants . Existence of a permanently feasible solution . THEN for any prescribed ǫ ∈ (0 , 1) m there always exists N large enough so that minimization of the penalty objective generates optimal solutions x N which also satisfy the chance constraints with the given ǫ . M.Branda (Charles University) EURO 2012 13 / 31
Relations between formulations Denote η = κ/ (2(1 + κ )), and for arbitrary N > 0 and ǫ ∈ (0 , 1) m put � � ε j ( x ) = P p j ( x , ξ ) > 0 , j = 1 , . . . , m , m m � � E [ p j ( x , ξ )] , β ǫ ( x ) = ε − η α N ( x ) = N · E [ p j ( x , ξ )] , max j =1 j =1 where ε max = max j ε j and [1 / N 1 /η ] = (1 / N 1 /η , . . . , 1 / N 1 /η ). THEN bounds on the optimal values can be constructed: ϕ 1 /ε η max ( x N ) − β ǫ ( x N ) ( x ǫ ( x N ) ) ≤ ψ ǫ ( x N ) ≤ ϕ N − α N ( x N ) , ψ ǫ ( x N ) + α N ( x N ) ≤ ϕ N ≤ ψ [1 / N 1 /η ] + β [1 / N 1 /η ] ( x [1 / N 1 /η ] ) , with N → + ∞ α N ( x N ) = lim N → + ∞ ε j ( x N ) = lim ε max → 0 + β ǫ ( x ǫ ) = 0 lim for any sequences of optimal solutions x N and x ǫ . M.Branda (Charles University) EURO 2012 14 / 31
Relations between formulations Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇐ ⇒ chance constrained problem objective problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check M.Branda (Charles University) EURO 2012 15 / 31
Relations between formulations Asymptotic equivalence and bounds M.B. (2011): Under the following assumptions, the asymptotic equivalence of the ICC and the PPO problems can be shown: Continuity of the constraints. Compactness of the fixed set of feasible solutions. Existence of integrable majorants . Existence of a permanently feasible solution . THEN for any prescribed L j ≥ 0 there always exists N large enough so that minimization of the penalty function problem generates the optimal solutions x N which also satisfy the integrated chance constraints with given L = ( L 1 , . . . , L m ) ′ . M.Branda (Charles University) EURO 2012 16 / 31
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