On Distributionally Robust Chance Constrained Program with Wasserstein Distance

Transcription

1 On Distributionally Robust Chance Constrained Program with Wasserstein Distance Weijun Xie 1 1 Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA 4061 June 15, 018 Abstract This paper studies a distributionally robust chance constrained program DRCCP) with Wasserstein ambiguity set, where the uncertain constraints should satisfy with a probability at least a given threshold for all the probability distributions of the uncertain parameters within a chosen Wasserstein distance from an empirical distribution. In this work, we investigate equivalent reformulations and approximations of such problems. We first show that a DRCCP can be reformulated as a conditionalvalue-at-risk constrained optimization problem, and thus admits tight inner and outer approximations. When the metric space of uncertain parameters is a normed vector space, we show that a DR- CCP of bounded feasible region is mixed integer representable by introducing big-m coefficients and additional binary variables. For a DRCCP with pure binary decision variables, by exploring submodular structure, we show that it admits a big-m free formulation and can be solved by branch and cut algorithm. This result can be generalized to mixed integer DRCCPs. Finally, we present a numerical study to illustrate effectiveness of the proposed methods. wxie@vt.edu. 1

2 1 Introduction 1.1 Setting A distributionally robust chance constrained program DRCCP) is of the form: min c x, s.t. x S, inf P ξ : ax) ξi b i x), i [I] P P 1 ɛ. 1a) 1b) 1c) In 1), the vector x R n denotes the decision variables; the vector c R n denotes the objective function coefficients; the set S R n denotes deterministic constraints on x; and the constraint 1c) is a chance constraint involving I uncertain inequalities specified by the random vectors ξ i supported on set Ξ i R n for each i [I] with a joint probability distribution P from a family P, termed ambiguity set. We let [R] := 1,,..., R for any positive integer R, and for each uncertain constraint i [I], ax) R n and b i x) R denote affine mappings of x such that ax) = ηx + 1 η)e and b i x) = Bi x + bi with η 0, 1, all-one vector e R n, B i R n, and b i R, respectively. For notational convenience, we let Ξ = i [I] Ξ i and ξ = ξ 1,..., ξ I ). ote that i) for any i, j [I] and i j, random vectors ξ i and ξ j can be correlated; and ii) we use η 0, 1 to differentiate whether 1c) involves left-hand uncertainty i.e., η = 1) or right-hand uncertainty i.e., η = 0). The chance constraint 1c) requires that all I uncertain constraints are simultaneously satisfied for all the probability distributions from ambiguity set P with a probability at least 1 ɛ), where ɛ 0, 1) is a specified risk tolerance. We call 1) a single DRCCP if I = 1 and a joint DRCCP if I. Also, 1) is termed a DRCCP with right-hand uncertainty if η = 0 and a DRCCP with left-hand uncertainty, otherwise. For a joint DRCCP, if I =, ξ 1 = ξ, we call 1) as a two-sided DRCCP. We denote the feasible region induced by 1c) as Z := x R n : inf P ξ : ax) ξi b i x), i [I] 1 ɛ. ) P P In this paper, we consider Wassertein ambiguity set P. A1) The Wasserstein ambiguity set P is defined as P = P P 0 Ξ) : E ξ, ζ)] P P ζ[d δ, 3) where P ζ denotes the discrete empirical distribution of ζ on the countable support Z = ζ j Ξ with point mass function p j, d : Ξ Ξ R + denotes distance metric and δ > 0 denotes the Wasserstein radius. We assume that Ξ, d) is a totally bounded Polish separable complete metric) space with distance metric d, i.e., for every ɛ > 0, there exists a finite covering of Ξ by balls with radius at most ɛ. ote that The Wasserstein metric measures the distance between true distribution and empirical distribution and is able to recover the true distribution when the number of sampled data goes to infinity [14]. 1. Related Literature There are significant work on reformulation, convexity and approximations of set Z under various ambiguity sets [7, 18, 19, 1, 37, 39]). For a single DRCCP, when P consists of all probability distributions with given first and second moments, the set Z is second-order conic representable [7, 13]. Similar

3 convexity results hold for single DRCCP when P also incorporates other distributional information such as the support of ξ [10], the unimodality of P [18, 4], or arbitrary convex mapping of ξ [37]. For a joint DRCCP, [19] provided the first convex reformulation of Z in the absence of coefficient uncertainty, i.e. η = 0, when P is characterized by the mean, a positively homogeneous dispersion measure, and a conic support of ξ. For the more general coefficient uncertainty setting, [37] identified several sufficient conditions for Z to be convex e.g., when P is specified by one moment constraint), and [36] showed that Z is convex for two-sided DRCCP when P is characterized by the first two moments. When DRCCP set Z is not convex, many inner convex approximations have been proposed. In [9], the authors proposed to aggregate the multiple uncertain constraints with positive scalars in to a single constraint, and then use conditional-value-at-risk CVaR) approximation scheme [8] to develop an inner approximation of Z. This approximation is shown to be exact for single DRCCP when P is specified by first and second moments in [43] or, more generally, by convex moment constraints in [37]. In [38], the authors provided several sufficient conditions under which the well-known Bonferroni approximation of joint DRCCP is exact and yields a convex reformulation. Recently, there are many successful developments on data driven distributionally robust programs with Wasserstein ambiguity set 3) [16, 7, 41]. For instance, [16, 7] studied its reformulation under different settings. Later on, [4, 15, 3, 3] applied it to the optimization problems related with machine learning. Other relevant works can be found [3, 17,, 6]. However, there is very limited literature on DRCCP with Wasserstein ambiguity set. In [35], the authors proved that it is strongly P-hard to optimize over the DRCCP set Z with Wasserstein ambiguity set and proposed a bicriteria approximation for a class of DRCCP with covering uncertain constraints i.e., S is a closed convex cone and Ξ i R n, B i R n +, b i R for each i [I]). In [11], the authors considered two-sided DRCCP with right-hand uncertainty and proposed its tractable reformulation, while in [0], the authors studied CVaR approximation of DRCCP. As far as the author is concerned, there is no work on developing tight approximations and exact reformulations of general DRCCP with Wasserstein ambiguity set. 1.3 Contributions In this paper, we study approximations and exact reformulations of DRCCP under Wasserstein ambiguity set. In particular, our main contributions are summarized as below. 1. We derive a deterministic equivalent reformulation for set Z and show that this reformulation admits a conditional-value-at-risk CVaR) interpretation. Based upon this fact, we are able to derive tight inner and outer approximations.. When the support Ξ is an n I- dimensional vectlor space and the distance metric is a norm i.e, dξ, ζ) = ξ ζ ), we show that the feasible region S Z of a DRCCP, once bounded, is mixed integer representable with big-m coefficients and I additional binary variables. We also derive compact formulations for the proposed inner and outer approximations and compare their strengths. 3. When the decision variables are pure binary i.e., S 0, 1 n ), we first show that the nonlinear constraints in the reformulation can be recast as submodular knapsack constraints. Then, by exploring the polyhedral properties of submodular functions, we propose a new big-m free mixed integer linear reformulation. In a numerical study, we further show that the proposed formulation can be effectively solved by branch and cut algorithm. The remainder of the paper is organized as follows. Section presents exact reformulation of DR- CCP set Z as well as its inner and outer approximations under a general setting. Section 3 provides mixed integer) convex reformulations of feasible region S Z and its inner and outer approximations when the metric space of the random variables is a normed vector space. Section 4 studies binary DR- CCP i.e., S 0, 1 n ), develops a big-m free formulation, and numerically illustrates the proposed 3

4 methods. Section 5 concludes the paper. otation: The following notation is used throughout the paper. We use bold-letters e.g., x, A) to denote vectors or matrices, and use corresponding non-bold letters to denote their components. We let e be the all-ones vector, and let e i be the ith standard basis vector. Given an integer n, we let [n] := 1,,..., n, and use R n + := x R n : x l 0, l [n] and R n := x R n : x l 0, l [n]. Given a real number t, we let t) + := maxt, 0. Given a finite set I, we let I denote its cardinality. We let ξ denote a random vector with support Ξ and denote one of its realization by ξ. Given a set R, the characteristic function χ R x) = 0 if x R, and, otherwise, while the indicator function Ix R) =1 if x R, and 0, otherwise. For a matrix A, we let A i denote ith row of A and A j denote jth column of A. Additional notation will be introduced as needed. Given a subset T [n], we define an n-dimensional 1, if τ T binary vector e T as e T ) τ = 0, if τ [n] \ T. General Case: Reformulations and Approximations In this section, we will study an equivalent reformulation of set Z under Assumption A1). This reformulation has conditional-value-at-risk CVaR) interpretation, therefore allows us to derive tight inner and outer approximations..1 Exact Reformulation In this part, we will reformulate set Z into its deterministic counterpart. The main idea of this reformulation is that we first use the strong duality result from [16] to formulate the worst-case chance constraint into its dual form, and then break down the indicator function according to its definition. Theorem 1. Under Assumption A1), set Z is equivalent to Z = x Rn : δλ ɛ 1 min λfx, ζ j ) 1, 0, λ 0, 4) where Proof. ote that fx, ζ) := min inf i [I] ξ Ξ,ax) ξ i>b ix) inf P ξ : ax) ξi b i x), i [I] 1 ɛ P P is equivalent to sup P ξ : ax) ξi > b i x), i [I] ɛ. P P By Theorem 1 in [16], sup P P P ξ : ax) ξi > b i x), i [I] is equivalent to min λ 0 λδ 1 ξ Ξ dξ, ζ). 5) [ inf λdξ, ζ j ) I ax) ξ i > b i x), i [I] )]. 6a) Thus set Z is equivalent to Z := x Rn : λδ 1 [ inf λdξ, ζ j ) I ax) ξ i > b i x), i [I] )] ɛ, λ 0. ξ Ξ 6b) 4

5 We now break down the indicator function in the infimum of 6b) and reformulate it as below. Claim 1. for given λ 0 and ζ Z, we have [ inf λdξ, ζ) I ax) ξ i > b i x), i [I] )] = min min ξ Ξ inf i [I] ξ Ξ,ax) ξ i>b ix) [λdξ, ζ) 1], 0. 6c) Proof. We first note that I ax) ξ i > b i x), i [I] ) = max i [I] I ax) ξ i > b i x) ). Thus, [ inf λdξ, ζ) I ax) ξ i > b i x), i [I] )] = min inf [ λdξ, ζ) I ax) ξ i > b i x) )]. ξ Ξ i [I] ξ Ξ Therefore, we only need to show that for any i [I], [ inf λdξ, ζ) I ax) ξ i > b i x) )] = min inf [λdξ, ζ) 1], 0. 6d) ξ Ξ ξ Ξ,ax) ξ i>b ix) There are two cases: Case 1. If ax) ζ i > b i x), then in the left-hand side of 6d), the infimum is equal to 1 by letting ξ := ζ, which equals to the right-hand side since the infimum is also achieved by ξ := ζ. Case. If ax) ζ i b i x), then for any ξ Ξ, we either have ax) ξ i > b i x) or ax) ξ i b i x). Hence, the left-hand side of 6d) is equivalent to [ λdξ, ζ) I ax) ξ i > b i x) )] inf ξ Ξ = min = min inf ξ Ξ,ax) ξ i>b ix) [λdξ, ζ) 1], inf [λdξ, ζ) 1], 0 ξ Ξ,ax) ξ i>b ix) where inf ξ Ξ,ax) ξ i b ix) [dξ, ζ)] = 0 by letting ξ := ζ. Thus, By Claim 1, set Z is equivalent to 4). inf ξ Ξ,ax) ξ i b ix), [λdξ, ζ)] In Theorem 1, we must have λ > 0, thus can define a new variable γ = 1 λ in 4) and reformulate set Z into the following equivalent form. Theorem. Under Assumption A1), set Z is equivalent to Z = x Rn : δ ɛγ 1 min fx, ζ j ) γ, 0, γ 0, 7) where function f, ) is defined in 5). Proof. ext, let Z denote the set in the right-hand side of 7), we only need to show that sets Z, Z are equivalent, i.e., Z = Z. Z Z ) Given x Z, there exists λ 0 such that x, λ) satisfies 4). If λ > 0, then let γ = 1 λ. Then it is easy to see that x, γ) satisfies 7). Hence, x Z. ow suppose that λ = 0, then in 4), we have a contradiction that ɛ 0, 1). ɛ 1 5

6 Z Z ) Similarly, given x Z, there exists γ 0 such that x, γ) satisfies 7). If γ > 0, then let λ = 1 γ. Then it is easy to see that x, λ) satisfies 4). Hence, x Z. ow suppose that γ = 0, then in 7), we have min fx, ζ j ) γ, 0 := 0 for each j []. Thus, 7) reduces to δ 0 contradicting that δ > 0. Before showing a conditional-value-at-risk CVaR) interpretation of set Z, let us begin with the following two definitions. Given a random variable X, let P and F X ) be its probability distribution and cumulative distribution function, respectively. Then 1 ɛ)-value at risk VaR) of X is VaR 1 ɛ X) := min s : F Xs) 1 ɛ, while its 1 ɛ)-conditional-value-at-risk CVaR) [31] is defined as CVaR 1 ɛ X) := min β β + 1 [ ] ɛ E P X β + With the definitions above, we observe that set Z in 7) has a CVaR interpretation. Corollary 1. Under Assumption A1), set Z is equivalent to Z = x R n : δ [ ɛ + CVaR 1 ɛ fx, ζ) ] 0, 8) where f, ) is defined in 5), and CVaR 1 ɛ [ fx, ζ) ] [ = min γ γ + 1 ɛ E P ζ fx, ζ) ] γ. + Proof. First, we observe that the constraint in 7) directly implies γ > 0, thus the nonnegativity constraint of γ can be dropped, i.e., equivalently, we have Z = x Rn : δ ɛ γ + 1 max fx, ζ j ) + γ, 0 0 ɛ. Z = ext, in the above formulation, let γ := γ and replace the existence of γ by finding the best γ such that the constraint still holds, we arrive at which is equivalent to 8). x Rn : δ ɛ + min γ γ + 1 ɛ. max fx, ζ j ) γ, 0 0, In the following sections, our derivations of exact reformulations is based upon Theorem while the approximations are mainly according to CVaR interpretation in Corollary 1. 6

7 . Outer and Inner Approximations In this subsection, we will introduce one outer approximation and three different inner approximations by exploring the exact reformulations in the previous section. VaR Outer Approximation: ote from [31] that for any random variable X, we have ) ) CVaR 1 ɛ X = VaR 1 ɛ X + 1 [ )] ) ɛ E X VaR1 ɛ X VaR 1 ɛ X. Therefore, in Corollary 1, if we replace CVaR 1 ɛ ) by VaR 1 ɛ ), then we have the following outer approximation of set Z. Theorem 3. Under Assumption A1), set Z is outer approximated by Z VaR = x R n : P ζ fx, ζ) δ ɛ + 1 ɛ. 9) Proof. Due to the well known result in [31] that CVaR 1 ɛ [ fx, ζ) ] [ VaR 1 ɛ fx, ζ) ]. Therefore, set Z is outer approximated by Z VaR = x R n : δ [ ɛ + VaR 1 ɛ fx, ζ) ] 0, which is equivalent to Z VaR = x R n : P ζ fx, ζ) δ 1 ɛ. ɛ Inner Approximation I- Robust Scenario Approximation: On the other hand, we notice that for any random variable X, we have ) ) CVaR 1 ɛ X CVaR 1 X := ess. sup X). Thus, in Corollary 1, if we replace CVaR 1 ɛ ) by ess. sup ), then we have the following inner approximation of set Z. Theorem 4. Under Assumption A1), set Z is inner approximated by Z S = x R n : fx, ζ j ) δɛ, j []. 10) Proof. Since CVaR 1 ɛ [ fx, ζ) ] [ ess. sup fx, ζ) ], and ζ is a discrete random vector, therefore, set Z can inner approximated by which is equivalent to 10). Z S = x R n : P ζ fx, ζ) δ = 1, ɛ ote that set Z S has a similar structure as scenario approach to chance constrained problem [6, 8, 9], and indeed can be viewed as a robust scenario approach to chance constrained problem. We will discuss more on this fact in Section 3.. Inner Approximation II- Inner Chance Constrained Approximation: ext we propose a chance constrained inner approximation of DRCCP set Z by constructing a feasible γ in 7). 7

8 Theorem 5. Under Assumption A1) and ɛ 0, 1), ɛ / Z +, set Z is inner approximated by Z I = x R n : P ζ fx, ζ) δ 1 α, 0 α ɛ. 11) ɛ α Proof. For any x Z I, we would like to show that x Z. Since x Z I, there exists an α such that x, α) satisfies constraints in 11). ow let us define γ =. Let define a set Since δ ɛ α 1 δ ɛ α C = j [] : fx, ζ j ) γ. δ ɛ α := γ, thus by 11), C α. Hence, min fx, ζ j ) γ, 0 = 1 fx, ζ j ) γ ) C γ α γ = δ ɛγ, j C where the first inequality is due to fx, ζ j ) 0 and the second inequality is due to C α. We remark that this result together with set Z VaR shows that DRCCP set Z can be inner and outer approximated by regular chance constraints with empirical probability distribution P ζ. We also note that i) set Z S is a special case of set Z I by letting α = 0, thus, we must have Z S Z I ; ii) there are ɛ + 1 non-dominant α values, that is, we must have α 0, 1,..., ɛ. Indeed, suppose that α i 1, i ) for an i [ ɛ ], then the feasible region expands if we decrease the value of α to i 1. Therefore, to optimize over set Z I, we can enumerate these ɛ + 1 values of α and choose the one which yields the smallest objective value. These two results are summarized below. Corollary. Suppose that Assumption A1) holds and ɛ 0, 1), ɛ / Z + and set Z I is defined in 11), then i) Z S Z I ; and ii) set Z I is equivalent to Z I = x R n : P ζ fx, ζ) δ 1 α, α 0, 1 ɛ α,..., ɛ. 1) Inner Approximation III- CVaR Approximation: Finally, we close this section by studying a well known convex approximation of a chance constraint, which is to replace the nonconvex chance constraint by a convex constraint defined by CVaR cf., [8]). For a DRCCP, the resulting formulation is Z CVaR = x R n : sup inf P P β [ ɛβ + E P max i [I] ax) ξ b i x) ) ) ] + β 0. + Set Z CVaR 13) is convex and is an inner approximation of set Z. The following results show a reformulation of set Z CVaR. We would like to acknowledge that this result has been independently observed by a recent work in [0]. For the completeness of this paper, we present a proof with our notation as below. 13) Theorem 6. Set Z CVaR Z and is equivalent to [ Z CVaR = x : ɛβ + λδ 1 inf λdξ, ζ j ) ξ Ξ λ, β 0. max i [I] ax) ξ i b i x) ) ) ] + β + 0, 14a) 14b) 8

9 Proof. ote that Wassersetein ambiguity set P is weakly compact [5], thus according to Theorem.1 in [33], set Z CVaR is equivalent to Z CVaR = x : inf β [ ɛβ + sup P P E P max ax) ξ b i x) ) ) ] + β 0. i [I] + ote that in the 15), the infimum must be achieved. Indeed, we first note that for any β < 0, the inequality in 15) will not be satisfied. Thus, we must have β 0. On the other hand, we note that ɛβ + sup P P E P max ax) ξ b i x) ) ) + β ɛβ + E P i [I] ζ + max i [I] ) ) ax) ζ bi x) + β + where the inequality is due to P ζ P. The right-hand side of the above inequality will be equal to ) ) 1 ɛ)β > 0 for any β > max i [I], b i x) ax) ζ j i, 0. Thus, the best β in 15) is bounded, i.e., Z CVaR is equivalent to Z CVaR = x : ɛβ + sup E P max ax) ξ i b i x) ) ) + β 0, β 0. 16) P P i [I] + By Theorem 1 in [16], the above formulation is further equal to [ Z CVaR = x : min ɛβ + λδ 1 inf λdξ, ζ j ) max ax) ξ b i x) ) ) ] + β 0, λ 0 ξ Ξ i [I] + β 0, which is equivalent to 14). 15) 3 DRCCP with ormed Vector Space Ξ, d) ote that the results in the previous section are quite general. In this section, we will show that these results can be significantly simplified given that Ξ, d) is a normed vector space. In particular, we make the following assumption. A) The support Ξ is an n I-dimensional vector space and distance metric dξ, ζ) = ξ ζ. 3.1 Exact Mixed Integer Program Reformulation In this subsection, we show that set Z is mixed integer representable under Assumptions A1)-A). To begin with, we observe that under additional Assumption A), function f, ) in Theorem can be explicitly calculated. Thus, Theorem 7. Under Assumptions A1)-A), set Z is equivalent to δ ɛγ 1 Z = x Rn : γ 0, min fx, ζ j ) γ, 0, 17a) 17b) 9

10 where fx, ζ) = min min i [I]\Ix) max b i x) ax) ζ i, 0 ax), min i Ix) χ x:b ix)<0x), 18) Ix) = [I] if ax) 0 and Ix) =, otherwise, and characteristic function χ R x) = if x / R and 0, otherwise. ote that the result in Theorem 7 can be further simplified by reformulating set Z as a disjunction of a nonconvex set and a convex set. Theorem 8. Suppose Assumptions A1)-A) hold. Then Z = Z 1 Z, where δν ɛγ 1 z j, Z 1 = x R n z : j + γ max b i x) ax) ζ j i, 0, i [I], j [], z j 0, j [], ax) ν, ν > 0, γ 0, 19a) 19b) 19c) 19d) 19e) and Z = x R n :ax) = 0, b i x) 0, i [I]. 0) Proof. We need to show that Z 1 Z Z and Z Z 1 Z. Z 1 Z Z. Given x Z, we have Ix) = [I], thus fx, ζ) defined in 18)) is. Thus, let γ = δ ɛ. Clearly, γ, x) satisfies all the constraints in 17), i.e., x Z. Hence, Z Z. Given x Z 1, there exists γ, ν, z, x) which satisfies constraints in 19). Suppose that Ix) = [I], then we have ax) = 0. Hence, for each i Ix), we have 19a) and 19b) imply that which is equivalent to δν ɛγ 1 That is, b i x) > 0. Thus, x Z Z. z j 1 max b i x), 0 γ), max b i x), 0 δν + 1 ɛ)γ > 0. ow we suppose that Ix) =. For each i [I], 19a) and 19b) along with ν > 0 imply that z j 1 ν min ν min max b i x) ax) ζ j i, 0 γ i [I] ν, 0 max b i x) ax) ζ j min min i, 0 γ i [I] ax) ν, 0 = min fx, ζ j ) γ ν, 0 where the second inequality due to 19d). Then according to 19a), we have δ ɛ γ ν 1 z j ν 1 min fx, ζ j ) γ ν, 0 i.e., γ/ν, x) satisfies the constraints in 17), i.e., x Z. Thus, Z 1 Z. 10

11 Z Z 1 Z Similarly, given x Z, there exists γ, x) which satisfies constraints in 17). Suppose that ax) = 0, then we must have b i x) 0 for all i [I], otherwise, we have fx, ζ j ) = 0 for all j [I]. Then 17a) is equivalent to 0 < δ ɛ 1)γ a contradiction that γ 0, ɛ 0, 1). Hence, we must x Z. From now on, we assume that ax) 0. Let us define γ = γ ax), ν = ax), and z j = min i [I] maxb i x) ax) ζ j i, 0 γ, 0) for each j []. Clearly, γ, ν, z, x) satisfies constraints in 19), i.e., x Z 1. We remark that set Z is usually trivial. Remark 1. i) If η = 1, then ii) if η = 0, then Z =. Z = x R n : x = 0, b i 0, i [I] ; On the other hand, set Z 1 can be formulated as a mixed integer set when it is bounded, i.e., we can introduce binary variables to represent constraints 19b). Theorem 9. Suppose there exists an M R I + such that max x Z 1 b i x) ax) ζ j i M ij for all i [I], j []. Then Z 1 is mixed integer representable as below: δν ɛγ 1 z j, z j + γ s ij, i [I], j [], Z 1 = x R n : s ij b i x) ax) ζ j i, i [I], j [], s ij M ij y ij, s ij b i x) ax) ζ j i + M ij1 y ij ), i [I], j [], ax) ν, ν > 0, γ 0, s ij 0, z j 0, y ij 0, 1, i [I], j []. 1a) 1b) 1c) 1d) 1e) 1f) Proof. To prove that Z 1 is equivalent to the right-hand side of 1), it is sufficient to show that for each i [I], j [], max b i x) ax) ζ j i, 0 = s ij. There are three cases: Case 1. if b i x) ax) ζ j i < 0, then we must have y ij = 0 otherwise, we have s ij b i x) ax) ζ j i < 0, a contradiction that s ij 0). Hence, we have s ij = 0 = max b i x) ax) ζ j i, 0. Case. if b i x) ax) ζ j i = 0, then for any y ij 0, 1, we have s ij = 0 = max b i x) ax) ζ j i, 0. Case 3. if b i x) ax) ζ j i > 0, then we must have y ij = 1 otherwise, we have b i x) ax) ζ j i s ij M j y j = 0, a contradiction that s ij b i x) ax) ζ j i > 0). Thus, we has s ij = b i x) ax) ζ j i = max b i x) ax) ζ j i, 0. 11

12 ote that there are various methods introduced by literature [30, 34] to obtain big-m coefficients and in the numerical study section, we will derive the big-m coefficients by inspection. In formulation 1), there are I binary variables and big-m coefficients, causing it very challenging to solve. In the next section, we will show that set Z 1 can be approximated to arbitrary accuracy by a big-m free formulation, and a branch and cut algorithm can be used to solve the approximated formulation. DRCCP with Right-hand Uncertainty: In this special case, we consider DRCCP with right-hand uncertainty, i.e., η = 0, ax) = e. We first note that by Theorem 7, set Z with ax) = e yields a more compact formulation. Theorem 10. If η = 0, ax) = e, then under Assumptions A1)-A), set Z is equivalent to the following mathematical program: δ ɛγ 1 z j, a) Z = x R n : z j + γ) n max b i x) e ζ j i, 0, j [], i [I], b) z j 0, j [], γ 0. c) Proof. The result directly follows from Theorem 7. To reformulate set Z in ) as a mixed integer program, we observe that without loss of generality, e ζ j i is nonnegative for all j [], i [I]. Indeed, suppose that L := min,i [I] e ζ j i < 0, then we can redefine e ζ j i := e ζ j i L and b ix) := b i x) L for all j [], i [I]. Theorem 11. Suppose that η = 0, ax) = e and e ζ j i that for all j [], i [I]. Then set Z is max x Z b ix) e ζ j i M ij z j + γ) n s ij, j [], i [I], Z = x R n : s ij b i x) e ζ j i y ij, j [], i [I], δ ɛγ 1 0 for all i [I] and there exists an M RI + such z j, s ij M ij y ij, j [], i [I], s ij b i x) e ζ j i, j [], i [I], γ 0, z j 0, y ij 0, 1, j [], i [I]. Proof. Let Ẑ denote the set on right-hand side of 3), we would like to show that Ẑ = Z. Z Ẑ. Given x Z, there exists γ, z, x) which satisfies constraints ). ow let s ij = maxb i x) e ζ j i, 0 and y ij = 1 if b i x) e ζ j i, 0, otherwise for each j [], i [I]. We only need to show that γ, s, z, y, x) satisfies the constraints in 3). Clearly, constraints 3a), 3b), 3d), 3e) and 3f) are satisfied, and for each j [], i [I] such that y ij = 1, constraints 3c) are also satisfied since s ij = max b i x) e ζ j i, 0 = b i x) e ζ j i. It remains to show that for each j [], i [I] such that y ij = 0, constraints 3c) will be also satisfied, i.e., s ij = 0 b i x). 3a) 3b) 3c) 3d) 3e) 3f) 1

13 Z Suppose that there exists a i 0 [I] such that b i0 x) < 0. By assumption, we know that e ζ j i 0 0 for all j []. Thus, we have max b i0 x) e ζ j i 0, 0 = 0 for all j []. According to constraints b), we have z j +γ 0, i.e., z j γ for each j []. Substituting this inequality into constraint a), we finally have δ ɛγ 1 z j γ which implies that γ δ 1 ɛ < 0, a contradiction that γ 0. Ẑ. Given x Z, there exists γ, s, z, y, x) which satisfies the constraints in 3). To prove that γ, z, x) satisfies constraints ), we only need to show that s ij max b i x) e ζ j i, 0 for each j [], i [I]. Indeed, if y ij = 1, we have s ij b i x) e ζ j i maxb ix) e ζ j i, 0, otherwise, we have s ij 0 maxb i x) e ζ j i, 0. We finally remark that formulation 3) can be stronger than 1) since we only need to compute the largest upper bound of b i x) e ζ j i and define it as M ij rather than the largest absolute value of b i x) e ζ j i. 3. Inner and Outer Approximations In the previous subsection, we develop exact mixed integer reformulations of set Z under various setting. However, these reformulations might be difficult to solve, especially when the number of empirical data points becomes large i.e., is large), there are a large number i.e., I ) of binary variables in the reformulations. In this subsection, we will investigate compact formulations of the inner and outer approximations of set Z proposed in Section, which involve fewer or even zero binary variables. VaR Outer Approximation: We first study the reformulation of the outer approximation Z VaR. Theorem 1. Under Assumptions A1)-A), set Z is outer approximated by δ Z VaR = x R n : P ζ ɛ ax) + ax) ζi b i x), i [I] 1 ɛ. 4) Proof. By Theorem 3 with fx, ζ) = min min i [I]\Ix) max b i x) ax) ζ, 0 ax), min i Ix) χ b ix)<0x), and Ix) = [I] if ax) 0, otherwise, Ix) =, we have maxbix) ax) ζ,0 Z VaR = x R n : P ax) ζ δ ɛ, i [I] \ Ix), χ bix)<0x) δ ɛ, i Ix) 1 ɛ, which is equivalent to 4). 13

14 ote that in 4), we arrive at a regular chance constrained program with discrete random vector ζ, which can be reformulated as mixed integer program with big-m coefficients cf., [1, 5]). A particular interpretation of formulation 4) is that in order to enforce the robustness, we further penalize the lefthand side of uncertain constraints by the dual norm ax). Inner Approximation I- Robust Scenario Approximation: We next consider robust scenario approximation Z S. Theorem 13. Under Assumptions A1)-A), set Z is inner approximated by Z S = x R n : δ ɛ ax) + ax) ζ j b i x), j [], i [I]. 5) Proof. The proof is similar to Theorem 1, thus is omitted. We remark that set Z S in 5) is very similar to scenario approach to chance constrained program [6], i.e., generate i.i.d. samples ζ j and enforce the constraints corresponding to each sample. The difference is that in set Z S, we also add a penalty δ ɛ ax) to each of the sampled constraints. Inner Approximation II- Inner Chance Constraint Approximation: The second inner approximation set Z I is nonconvex and according to Theorem 5, we can formulate it as below. Theorem 14. Suppose that Assumptions A1)-A) hold and ɛ 0, 1), ɛ / Z +, then set Z is inner approximated by δ Z I = x R n : P ζ ɛ α ax) + ax) ζi b i x), i [I] 1 α, 0 α ɛ. 6) Proof. The proof is similar to Theorem 1, hence is omitted. ote that for any given α, set Z I is mixed integer representable with big-m coefficients. Since from Corollary, there are only ɛ + 1 effective values of α that we can choose from, thus Z I can be formulated as a disjunction of ɛ + 1 mixed integer sets. Inner Approximation III- CVaR Approximation: ext, we study the CVaR approximation. Theorem 15. Under Assumptions A1)-A), set Z is inner approximated by δν ɛγ 1 z j, Z CVaR = x R n z : j + γ b i x) ax) ζ j i, j [], i [I], z j 0, j [], ax) ν,, ν 0, γ 0, 7a) 7b) 7c) 7d) 7e) Proof. By Theorem 6, we have set Z CVaR is equal to [ Z CVaR = x R n : ɛβ + λδ 1 λ, β 0. inf ξ λ ξ ζ j max i [I] ax) ξ i b i x) ) ) ] + β + 0, 14

15 which is further equivalent to ɛβ + λδ 1 Z CVaR = x R n : ax) λ, i [I], λ, β 0. [ min min i [I] In the above formulation, let ν = λ, γ = β and also let z j = min linearize it for each j []. Thus, we arrive at 7). ) ] b i x) ax) ζ j i β, 0 0, ) ] [min i [I] b i x) ax) ζ j i β, 0 and We remark that we can directly derive the reformulation of set Z CVaR in 7) based upon formulation 17). Indeed, since maxb i x) ax) ζ i, 0 b i x) ax) ζ i, by replacing maxb i x) ax) ζ i, 0 with b i x) ax) ζ, then function fx, ζ) is lower bounded by fx, ζ) fx, ζ) = min min i [I]\Ix) b i x) ax) ζ i, min ax) χ b i Ix) ix)<0x). Thus, set Z can be inner approximated by the following set δ ɛγ 1 min fx, ζ j ) γ, 0, x Rn : γ 0, which is exactly equivalent to Z CVaR by introducing additional variables to linearize the nonlinear function min fx, ζ) γ, 0. From this observation, we note that Z CVaR = Z if ɛ 1. Corollary 3. Suppose that Assumptions A1)-A) hold and ɛ 0, 1/], then set Z = Z CVaR. Proof. We note that set Z CVaR Z = Z 1 Z, where Z 1 and Z are defined in 19) and 0), respectively. We note that set Z Z CVaR. Indeed, suppose that x Z, i.e., ax) = 0, b i x) 0 for each i [I], then let ν = 0, γ = 0 and z j = 0 for each j []. Clearly, ν, γ, z, x) satisfies the constraints in 7). Hence, x Z CVaR. Thus, it is sufficient to show that Z 1 Z CVaR. Indeed, given x Z 1, there exists ν, γ, z) such that ν, γ, z, x) satisfies the constraints in 19). We only need to show that z j + γ > 0 for each j []. Suppose that there exists a j 0 [] such that z j0 + γ 0. Then according to 19a), we have δ 1 \j 0 z j + 1 ɛγ + z j 0 ) 0 where the second inequality is due to ɛ 1 and z j0 + γ 0, a contradiction that δ > 0. Therefore, in 19b), we must have max b i x) ax) ζ j i, 0 = b i x) ax) ζ j i for each i [I], j []. Hence, ν, γ, z, x) satisfies the constraints in 7), i.e., x Z CVaR. Formulation Comparisons: Finally, we would like to compare sets Z S, Z CVaR. Indeed, we can show that Z S Z CVaR, i.e., set Z S is at least as conservative as set Z CVaR. Theorem 16. Let Z S, Z CVaR be defined in 5), 7), respectively. Then Z S Z CVaR. 15

16 Proof. Given x Z S, we only need to show that x Z CVaR. Indeed, let us consider ν = ax), γ = ax) ɛ, z j = 0 for all j [], then we see that ν, γ, z, x) satisfies the constraints in 7), i.e., x Z CVaR. We illustrate sets Z, Z VaR, Z CVaR, Z S, Z I with the following example. Example 1. Suppose = 3, n =, I =, δ = 1/6, ɛ = /3 and ζ1 1 = 1, 0), ζ 1 = 0, 3), ζ1 = 3, 0), ζ = 0, 1), ζ1 3 =, 0), ζ 3 = 0, ), b 1 x) = x 1, ax) = 1, b x) = x. Then, we have Z = x 1, x ) : + x 1, 3 + x x 1, x ) : 3 + x 1, + x x 1, x ) : 3 x 1, 3 x 3, 6 + x 1 + x Z VaR = x 1, x ) : + 4 x 1, x x 1, x ) : x 1, + 4 x Z CVaR = x 1, x ) : 3 x 1, 3 x, 6 + x 1 + x Z S = x 1, x ) : x 1, x Z I = x 1, x ) : + x 1, 3 + x x 1, x ) : 3 + x 1, + x Clearly, we have Z S x 1, x ) : x 1, 3 + ZCVaR Z I 4 x Z Z VaR see Figure 1 for an illustration). Finally, the inclusive relationships among sets Z, Z VaR, Z S, Z I Z CVaR are illustrated in Figure and their reformulations are summarized in Table 1. Table 1: Summary of formulation results in Section 3. Set Z Set Z VaR Set Z S Set Z I Set Z CVaR Mixed-integer Mixed-integer Convex Mixed-integer Convex Theorem 9 Theorem 1 Theorem 13 Theorem 14 Theorem 15 4 DRCCP with Pure Binary Decision Variables In this section, we will study DRCCP with pure binary decision variables x 0, 1 n, i.e., in addition to Assumptions A1)-A), we further assume that A3) the set S is binary, that is, S 0, 1 n. Indeed, we remark that if S is a bounded mixed integer set, we can introduce On logn)) additional binary variables to approximate set S with arbitrary accuracy via binary approximation of continuous variables c.f., [4]). For binary DRCCP, we will show that the reformulations in the previous section can be improved. 16

17 x Z CVaR ZS, 3) Z VaR Z I Z, ) 3, ) x 1 Figure 1: Illustration of Example 1 Figure : Summary of formulation comparisons 17

18 4.1 Polyhedral Results of Submodular Functions: A Review Our main derivation of stronger formulations is based upon some polyhedral results of submodular functions, which will be briefly reviewed in this subsection. We first begin with the following lemmas on submodular functions. Lemma 1. Given d 1 R n +, d, d 3 R, function fx) = max d 1 x + d, d 3 ) is submodular over the binary hypercube. Proof. For simplicity, given a T [n], we define a binary vector e T 0, 1 n such that 1, if l T e T ) l = 0, if l [n] \ T. According to the definition of submodular function [1], we only need to show that fe T1 t) fe T1 ) fe T t) fe T ) for any T 1 T and t [n] \ T. There are three cases: Case 1. if i T 1 d 1i + d d 3, since d 1 R n +, then we must have fe T1 t) fe T1 ) = fe T t) fe T ) = d 1t. Case. if i T 1 d 1i + d < d 3 but i T d 1i + d d 3, then we must have where the inequality is due to d 1 R n +. fe T1 t) fe T1 ) = 0 fe T t) fe T ) = d 1t, Case 3. if i T d 1i + d < d 3, since d 1 R n +, then we must have fe T1 t) fe T1 ) = fe T t) fe T ) = 0. Lemma. Given q 1, function fx) = x q with q 1 is submodular over the binary hypercube. Proof. This is because fx) = x q = q l [n] x l, and ge x) is a submodular function if g ) is a concave function cf., [40]). ext, we will introduce polyhedral properties of submodular functions. For any given submodular function fx) with x 0, 1 n, let us denote Π f to be its epigraph, i.e., Π f = x, φ) : φ fx), x 0, 1 n. Then the convex hull of Π f is characterized by the system of extended polymatroid inequalities EPI) [, 40], i.e., conv Π f ) = x, φ) : f0) + ρ σl x σl φ, σ Ω, x [0, 1] n, 8) l [n] where Ω denotes a collection of all the permutations of set [n] and ρ σl = fe A σ l ) fe A σ l 1 ) for each l [n] with A σ 0 =, A σ 1, if τ T l = σ 1,..., σ l and e T ) τ = 0, if τ [n] \ T. In addition, although there are n! number of inequalities in 8), these inequalities can be easily separated by greedy procedure. 18

19 Lemma 3. [, 40]) Suppose x, φ) / conv Π f ), and σ Ω be a permutation of [n] such that x σ1... x σn. Then x, φ) must violate the constraint f0) + l [n] ρ σ l x σl φ. From Lemma 3, we see that to separate a point x, φ) from conv Π f ), we only need to sort the coordinates of x in a descending order, i.e., x σ1... x σn. Then x, φ) can be the separated by the constraint f0) + l [n] ρ σ l x σl φ from conv Π f ). The time complexity of this separating procedure is On log n). 4. Reformulating a Binary DRCCP by Submodular Knapsack Constraints: Big-M free In this section, we will replace the nonlinar constraints defining the feasible region of a binary DR- CCP i.e., set S Z) by submodular upper bound knapsack) constraints. These constraints can be equivalently described by the system of EPI in 8), therefore we obtain a big-m free mixed integer representation of set S Z. First, we introduce n complementing binary variables of x, denoted by w, i.e., w l + x l = 1 for each l [n]. With these n additional variables, we can reformulate function b i x) ax) ζ j i as b i x) ax) ζ j i = r ijx + t ijw + u ij 9) for each i [I], j [] such that r ij R n +, t ij R n +. Indeed, since ax) = ηx + 1 η)e and b i x) = Bi x + bi, in 9), we can choose r ijl = B il IB il > 0) ηζ j il Iζj il < 0), t ijl = B il IB il < 0) + ηζ j il Iζj il > 0), u ij = b i 1 η)e ζ j i + ) B iτ IB iτ < 0) ηζ j iτ Iζj iτ > 0), τ [n] for each l [n], i [I], j []. Thus, from above discussion, we can formulate S Z 1 note that set Z = Z 1 Z according to Theorem 8) as the following mixed integer set with submodular knapsack constraints. Theorem 17. Suppose that Assumptions A1)-A3) hold. Then S Z = S Z 1 ) S Z ), where δν ɛγ 1 z j, max rijx + t ijw + u ij, 0 z j γ, i [I], j [], S Z 1 = x S : z j 0, j [], η x + 1 η) e ν, w l + x l = 1, l [n], ν 1, γ 0, w 0, 1 n 30a) 30b) 30c) 30d) 30e) 30f) 30g) and S Z = x S :ax) = 0, b i x) 0, i [I] 31) Proof. From the discussion above and the fact that ax) = ηx + 1 η)e with η 0, 1, we have constraints 19b) and 19d) is equivalent to 30b) and 30d). We only need to show that ν 1 in set S Z 1. 19

20 If η = 0, then η x + 1 η) e = e 1, then we are done. ow suppose that η = 1. We note that if x = 0, then the constraints 19) imply that b i x) > 0 for each i [I]. Thus, if x = 0, then set Z 1 Z. Therefore, without loss of generality, we can assume that in set Z 1, x 0. ote that S Z 1 0, 1 n, therefore, x 0 implies that x 1, thus, v η x + 1 η) e = x 1. From the proof of Theorem 17, we note that if η = 1 and b i δ ɛ for each i [I], then we have Z Z 1 Corollary 4. Suppose that Assumptions A1)-A3) hold, η = 1 and b i δ ɛ for each i [I]. Then S Z = S Z 1. Proof. We only need to show that 0 S Z 1. Suppose x = 0, i.e., w = e. Let us set ν = 1, γ = 0, z = 0. Then it is easy to see that x, w, z, γ, ν) satisfies the constraints in 30), i.e., 0 S Z 1. We note that the left-hand sides of constraints 30b) and 30d) are submodular functions according to Lemma 1 and Lemma, thus, we can equivalently replace these constraints with the convex hulls of epigraphs of their associated submodular functions. Thus, Corollary 5. Suppose that Assumptions A1)-A3) hold. Then δν ɛγ 1 z j, x, w, z j γ) convπ ij ), i [I], j [], S Z 1 = x S : z j 0, j [], x, ν) convπ 0 ), w l + x l = 1, l [n], ν 1, γ 0, w 0, 1 n, 3a) 3b) 3c) 3d) 3e) 3f) where Π ij = x, w, φ) : max r ijx + t ijw + u ij, 0 φ, x, w 0, 1 n, i [I], j [], Π 0 = x, φ) : η x + 1 η) e φ, x 0, 1 n 33a) 33b) and convπ ij ) i [I],, convπ 0 ) can be described by the system of EPI in 8). ote that the optimization problem min x S Z1 c x can be solved by branch and cut algorithm. In particular, at each branch and bound node, denoted as x, ŵ, ẑ, γ, ν), there might be too many i.e., I +1) valid inequalities to add, since in 3b) and 3d), there are I +1 convex hulls of epigraphs i.e., convπ ij ) i [I],, convπ 0 )) to be separated from. Therefore, instead, we can first check and find the epigraphs of κ κ 1) most violated constraints in 30b) and 30d), i.e., find the epigraphs corresponding to the κ largest values in the following set max r ij x + t ijŵ + u ij, 0 + ẑ j + γ i [I], η x + 1 η) e ν. Finally, we can generate and add valid inequalities by separating x, ŵ, ẑ, γ, ν) from the convex hulls of these κ epigraphs according to Lemma umerical Study In this subsection, we present a numerical study to compare the big-m formulation in Theorem 9 with big-m free formulation in Theorem 17 and its corollaries on the distributionally robust multidimensional knapsack problem DRMKP) [10, 34, 37]. In DRMKP, there are n items and I knapsacks. Additionally, c j represents the value of item j for all j [n], ξ i := [ ξ i1,..., ξ in ] represents the vector of 0

21 random item weights in knapsack i, and b i > 0 represents the capacity limit of knapsack i, for all i [I]. The binary decision variable x j = 1 if the jth item is picked and 0 otherwise. We use the Wasserstein ambiguity set under Assumptions A1) and A) with L - norm as distance metric. With the notation above, DRMKP is formulated as v = max c x, x 0,1 n s.t. inf P ξ i x b i, i [I] P P 1 ɛ. 34) To test the proposed formulations, we generate 10 random instances with n = 0 and I = 10, indexed by 1,,..., 10. For each instance, we generate = 1000 empirical samples ζ j R+ I n from a uniform distribution over a box [1, 10] I n. For each l [n], we independently generate c l from the uniform distribution on the interval [1, 10], while for each i [I], we set b i := 100. We test these 10 random instances with risk parameter ɛ 0.05, 0.10 and Wasserstein radius δ 0.1, 0.. Our first approach is to solve the big-m reformulation of DRMKP in Theorem 9, which reads as follows: v = max c x, x 0,1 n s.t. δν ɛγ 1 z j, z j + γ s ij, i [I], j [], s ij b i x ζ j i, i [I], j [], 35) s ij M ij y ij, s ij b i x ζ j i + M ij1 y ij ), i [I], j [], x ν, ν 1, γ 0, s ij 0, z j 0, y ij 0, 1, i [I], j [], where M ij = maxb i, b i e ζ j i for each i [I], j []. We compare this formulation with another big-m free formulation in Theorem 17 and its corollaries, which reads as follows: v = max c x, x 0,1 n s.t. δν ɛγ 1 z j, w, z j γ) convπ ij ), i [I], j [], z j 0, j [], 36) x, ν) convπ 0 ), w l + x l = 1, l [n], ν 1, γ 0, w [0, 1] n, where Π ij = w, φ) : max ζ j i ) w + b i ζ j i ) e, 0 φ, w 0, 1 n, i [I], j [], Π 0 = x, φ) : x φ, x 0, 1 n 37a) 37b) and their convex hulls, convπ ij ) i [I],, convπ 0 ) can be described by the system of EPI 8). ote that the fact that 35) and 36) are exact reformulations of DRMKP follows from Corollary 4 since b i δ ɛ for all i [I]. 1

22 We use the commercial solver Gurobi version 7.5, with default settings) to solve the instances of formulation 35). We set the time limit of solving each instance to be 3600 seconds. The results are displayed in Table. We use UB, LB, GAP, Opt. Val. and Time to denote the best upper bound, the best lower bound, optimality gap, the optimal objective value and the total running time, respectively. All instances were executed on a MacBook Pro with a.80 GHz processor and 16GB RAM. From Table, we observe that the overall running time of DRMKP formulation 36) significantly outperforms that of 35), i.e., almost all of the instances of formulation 36) can be solved within 10 minutes, while the majority of the instances of formulation 35) reach the time limit. The main reasons are two-fold: i) formulation 35) involves O I + n) binary variables and O I) continuous variables, while formulation 35) only involves On) binary variables and O) continuous variables; and ii) formulation 35) contains big-m coefficients, while formulation 36) is big-m free. We also observe that, as the risk parameter ɛ increases or Wasserstein radius δ decreases, both formulations take longer to solve but formulation 36) still significantly outperforms formulation 35). These results demonstrate the effectiveness our proposed approaches. 5 Conclusion In this paper, we studied a distributionally robust chance constrained problem DRCCP) with Wasserstein ambiguity set. We showed that a DRCCP can be formulated as a conditional-value-at-risk constrained optimization, thus admits tight inner and outer approximations. When the metric space of random variables is normed vector space, we showed that a DRCCP is mixed integer representable with big-m coefficients and additional binary variables, i.e., a DRCCP can be formulated as a mixed integer conic program. We also compared various inner and outer approximations and proved their corresponding inclusive relations. We further proposed a big-m free formulation for a binary DRCCP. The numerical studies demonstrated that the developed big-m free formulation can significantly outperform the big-m one. Acknowledgments The author would like to thank Professor Shabbir Ahmed Georgia Tech) for his helpful comments on an earlier version of the paper.

23 Table : Performance comparison of formulation 35) and formulation 36) ɛ δ Instances n I Formulation 35) Formulation 36) UB LB Time GAP Opt. Val. Time % % % % % % % % % % Average % A A A A % % A A A A A A % A A % Average % % % % % % % % % % % Average % % % % % A A % % A A % % Average % The A represents that no feasible solution has been found within the time limit 3

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