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Scenario Grouping and Decomposition Algorithms for Chance-constrained Programs Siqian Shen Dept. of Industrial and Operations Engineering University of Michigan Joint work with Yan Deng (UMich, Google) Shabbir Ahmed (Georgia Tech) Jon Lee (UMich) (supported by NSF grants CMMI-1433066, 1633196 & ONR N00014-14-0315) INFORMS Optimization Soceity Conference March 24, 2018 1/33

Outline Introduction Optimization-based Scenario Grouping Base Model MILP Reformulation Branch-and-Cut Heuristic-based Scenario Grouping Numerical Studies Experimental Design Results of Group-based Bounds Results of Scenario Decomposition with Grouping 2/33

Chance-constrained Program c ⊤ x min (1a) s.t. P { x ∈ F ( ξ ) } ≥ 1 − ǫ (1b) x ∈ X ⊆ R d , (1c) ◮ x : a d -dimensional decision vector; c ∈ R d : cost parameter ◮ ξ : a multivariate random vector. (W.l.o.g., we consider a finite support Ξ = { ξ 1 , . . . , ξ K } , and each scenario is realized with equal probability.) ◮ F ( ξ ) ⊆ R d : a region parameterized by ξ . Let F k = F ( ξ k ). ◮ X : a deterministic feasible region; either continuous or discrete. 3/33

Literature Review Chance-constrained programs have wide applications in energy, healthcare, transportation problems, but in general nonconvex and intractable to solve. We present some main literature below. ◮ Convex approximations : Pr´ ekopa (1970), Nemirovski and Shapiro (2006) ◮ SAA and MILP-based algorithms : Luedtke and Ahmed (2008),Pagnoncelli et al. (2009), Luedtke et al. (2010), K¨ u¸ c¨ ukyavuz (2012), Luedtke (2014), Song et al. (2014), Ahmed et al. (2016) ◮ Scenario decomposition : Watson et al. (2010), Ahmed (2013), Carøe and Schultz (1999), Dentcheva and R¨ omisch (2004), Miller and Ruszczy´ nski (2011), Collado et al. (2012), Deng et al. (2016) 4/33

Quantile Bounds I An equiavlent formulation of model (1) is: v ∗ := min c ⊤ x CCP : K � ∈ F k ) ≤ K ′ s.t. I ( x / k =1 x ∈ X , where I ( · ) is an indicator function and K ′ := ⌊ ǫ K ⌋ . Using binary variables to model outcomes of the indicator function in all the scenarios, we can further reformulate CCP as an MILP with a knapsack constraint (see Ahmed et al. (2016)) and solve it by branch-and-cut (see Luedtke (2014)) 5/33

Quantile Bounds II ◮ To satisfy the probabilistic constraint, x must fall in sufficiently many subregions F k ’s. 6/33

Quantile Bounds II ◮ To satisfy the probabilistic constraint, x must fall in sufficiently many subregions F k ’s. ◮ The optimal objective values of the K scenario subproblems: � � c ⊤ x : x ∈ F k , x ∈ X ψ k := min , ∀ k = 1 , . . . , K . (2) 6/33

Quantile Bounds II ◮ To satisfy the probabilistic constraint, x must fall in sufficiently many subregions F k ’s. ◮ The optimal objective values of the K scenario subproblems: � � c ⊤ x : x ∈ F k , x ∈ X ψ k := min , ∀ k = 1 , . . . , K . (2) ◮ Then order them to obtain a permutation σ of the set { 1 , . . . , K } such that ψ σ 1 ≥ · · · ≥ ψ σ K . 6/33

Quantile Bounds II ◮ To satisfy the probabilistic constraint, x must fall in sufficiently many subregions F k ’s. ◮ The optimal objective values of the K scenario subproblems: � � c ⊤ x : x ∈ F k , x ∈ X ψ k := min , ∀ k = 1 , . . . , K . (2) ◮ Then order them to obtain a permutation σ of the set { 1 , . . . , K } such that ψ σ 1 ≥ · · · ≥ ψ σ K . ◮ Given K ′ = ⌊ ǫ K ⌋ , the ( K ′ + 1) th quantile value, ψ σ K ′ +1 , is a valid lower bound for CCP, due to that x will fall in at least one F k with k ∈ { σ 1 , . . . , σ K ′ +1 } . 6/33

Scenario Grouping I We partition { 1 , . . . , K } , into N disjoint subsets G 1 , . . . , G N , and obtain a relaxation of CCP as: c ⊤ x SGM : v SGM := min   N � �  ≤ K ′ s.t.  x / ∈ F k I n =1 k ∈ G n x ∈ X . ◮ SGM is not a relaxation, if { G 1 , . . . , G N } does not form a partition of scenarios { 1 , . . . , K } . 7/33

Scenario Grouping II We consider the quantile bound of SGM as: � N � I ( ρ ≤ φ n ) ≥ K ′ + 1 v Q � SGM := max ρ : , where we solve n =1 (3) � � � c ⊤ x : x ∈ φ n := min F k , x ∈ X , n = 1 , . . . , N . (4) k ∈ G n ◮ We call Model (4) group subproblems . 8/33

Scenario Grouping II We consider the quantile bound of SGM as: � N � I ( ρ ≤ φ n ) ≥ K ′ + 1 v Q � SGM := max ρ : , where we solve n =1 (3) � � � c ⊤ x : x ∈ φ n := min F k , x ∈ X , n = 1 , . . . , N . (4) k ∈ G n ◮ We call Model (4) group subproblems . ◮ The grouping-based quantile bound v Q SGM is a valid lower bound for CCP (i.e., v Q SGM ≤ v SGM ≤ v ∗ ). 8/33

Scenario Grouping II We consider the quantile bound of SGM as: � N � I ( ρ ≤ φ n ) ≥ K ′ + 1 v Q � SGM := max ρ : , where we solve n =1 (3) � � � c ⊤ x : x ∈ φ n := min F k , x ∈ X , n = 1 , . . . , N . (4) k ∈ G n ◮ We call Model (4) group subproblems . ◮ The grouping-based quantile bound v Q SGM is a valid lower bound for CCP (i.e., v Q SGM ≤ v SGM ≤ v ∗ ). ◮ Related work: Escudero et al. (2013), Crainic et al. (2014), Ryan et al. (2016) (all for expectation-based stochastic programs) 8/33

Outline Introduction Optimization-based Scenario Grouping Base Model MILP Reformulation Branch-and-Cut Heuristic-based Scenario Grouping Numerical Studies Experimental Design Results of Group-based Bounds Results of Scenario Decomposition with Grouping 9/33

Notation and Parameter Recall the following parameter: o K : the number of scenarios, N : the number of groups o K ′ = ⌊ ǫ K ⌋ where ǫ is the risk tolerance level in Model (1). Decision variables: o y kn ∈ { 0 , 1 } , k = 1 , . . . , K , n = 1 , . . . , N : whether scenario k is assigned to group G n , such that y kn = 1 if yes, and = 0 o.w. Procedures: o solve the ordered objective values φ 1 , . . . , φ N of group subproblems (4) and maximize φ K ′ +1 for K ′ = ⌊ ǫ K ⌋ . 10/33

Optimal Scenario Grouping Model To obtain the tightest quantile bound v Q SGM that is equal to φ K ′ +1 : QGP : max (5a) φ K ′ +1 φ n ≤ min { c ⊤ x : I ( x ∈ F k ) ≥ y kn , ∀ k , x ∈ X} s.t. ∀ n = 1 , . . . , N (5b) φ n − φ n +1 ≥ 0 ∀ n = 1 , . . . , N − 1 (5c) N � y kn = 1 ∀ k = 1 , . . . , K (5d) n =1 K � y kn ≤ P ∀ n = 1 , . . . , N (5e) k =1 y kn ∈ { 0 , 1 } ∀ n = 1 , . . . , N , k = 1 , . . . , K . (5f) (5c) are to avoid symmetric solutions. We restrict each group size by an integer parameter P in (5e) with P ≥ K / N . Without this, i.e., if P = K , the model will allocate scenarios densely into K ′ + 1 groups and make some subproblems hard to solve. 11/33

Outline Introduction Optimization-based Scenario Grouping Base Model MILP Reformulation Branch-and-Cut Heuristic-based Scenario Grouping Numerical Studies Experimental Design Results of Group-based Bounds Results of Scenario Decomposition with Grouping 12/33

Special Cases and MILP Reformulation Consider chance-constrained linear programs with X = R d + and F k = { x : A k x ≥ r k } . We can reformulate (5b) as � � c ⊤ x : A k x ≥ r k − M k (1 − y kn ) , ∀ k = 1 , . . . , K , x ∈ R d φ n ≤ min , + (6) + be the dual of the k th set of constraints in the Let λ kn ∈ R m k minimization problem in (6). The dual problem is: K � � � r T k λ kn − M T D n ( y ) := max k λ kn (1 − y kn ) (7a) k =1 K � A T s.t. k λ kn ≤ c (7b) k =1 λ kn ∈ R m k + . (7c) 13/33

Linearize Dual Formulation Further define w kn ≡ λ kn y kn , ∀ k = 1 , . . . , K , n = 1 , . . . , N and use the McCormick inequalities, we can linearize the dual and derive an equivalent MILP to replace the right-hand side of (6) in QGP: K � � � r T k λ kn − M T k λ kn + M T max (8a) k w kn y ,λ, w k =1 s.t. (7b) w kn ≤ λ kn , w kn ≤ λ kn y kn , ∀ k = 1 , . . . , K (8b) w kn ≥ λ kn − λ kn (1 − y kn ) , ∀ k = 1 , . . . , K (8c) λ kn , w kn ∈ R m k + , y kn ∈ { 0 , 1 } , ∀ k = 1 , . . . , K . (8d) The overall MILP for optimal scenario grouping is: � K � � � r T k λ kn − M T k λ kn + M T max φ K ′ +1 : φ n ≤ k w kn , n = 1 , . . . , N , φ,λ, w , y k =1 � (5c)–(5e), (7b), (8b), (8d) . 14/33

Outline Introduction Optimization-based Scenario Grouping Base Model MILP Reformulation Branch-and-Cut Heuristic-based Scenario Grouping Numerical Studies Experimental Design Results of Group-based Bounds Results of Scenario Decomposition with Grouping 15/33

General Case and Branch-and-Cut Consider a master problem of QGP as: max { φ K ′ +1 : (5c)–(5e) , ( y , φ ) ∈ A , φ n ∈ R , y kn ∈ { 0 , 1 } , ∀ k , n } . y , ˆ For any (ˆ φ ) (where ˆ y could be fractional), consider and define a group set G ∗ n := { k ∈ { 1 , . . . , K } : ˆ y kn > 0 } for each n = 1 , . . . , N .     � φ ∗  c ⊤ x : x ∈ n = min F k , x ∈ X  . k ∈ G ∗ n n < ˆ If φ ∗ φ n , following integer L-shaped method, we add a cut    � φ n ≤ ( U − φ ∗  + U n ) y kn − 1 (9) k :ˆ y kn =0 where U = max { c ⊤ x : x ∈ X} (an upper bound). 16/33