On distributionally robust chance constrained programs with Wasserstein distance

Abstract

In the optimization under uncertainty, decision-makers first select a wait-and-see policy before any realization of uncertainty and then place a here-and-now decision after the uncertainty has been observed. Two-stage stochastic programming is a popular modeling paradigm for the optimization under uncertainty that the decision-makers first specifies a probability distribution, and then seek the best decisions to jointly optimize the deterministic wait-and-see and expected here-and-now costs. In practice, such a probability distribution may not be fully available but is probably observable through an empirical dataset. Therefore, this paper studies distributionally robust two-stage stochastic program (DRTSP) which jointly optimizes the deterministic wait-and-see and worst-case expected here-and-now costs, and the probability distribution comes from a family of distributions which are centered at the empirical distribution using ∞−Wasserstein metric. There have been successful developments on deriving tractable approximations of the worst-case expected here-and-now cost in DRTSP. Unfortunately, limited results on exact tractable reformulations of DRTSP. This paper fills this gap by providing sufficient conditions under which the worst-case expected here-and-now cost in DRTSP can be efficiently computed via a tractable convex program. By exploring the properties of binary variables, the developed reformulation techniques are extended to DRTSP with binary random parameters. The main tractable reformulations in this paper are projected into the original decision space and thus can be interpreted as conventional two-stage stochastic programs under discrete support with extra penalty terms enforcing the robustness. These tractable results are further demonstrated to be sharp through complexity analysis.

Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning

“…If the undesirable event can be influenced so as drive its worst-case probability below a prescribed tolerance, we face a distributionally robust chance constraint. Even though distributionally robust chance constrained programs with Wasserstein ambiguity sets around the empirical distribution are intractable in general, they are sometimes equivalent to mixed-integer linear programs that can be solved with off-the-shelf software [20,112]. In contrast, distributionally robust chance constrained programs with moment ambiguity sets can often be reformulated as (or tightly approximated by) tractable conic programs [16,21,48,115].…”

Optimization under uncertainty in the era of big data and deep learning: When machine learning meets mathematical programming

“…Recently, data-driven chance constraints over Wasserstein balls were exactly reformulated as mixed-integer conic constraints [145,146]. Leveraging the strong duality result [147], distributionally robust chance constrained programs with Wasserstein ambiguity set were studied for linear constraints with both right and left hand uncertainty [148], as well as for general nonlinear constraints [149].…”

Data-Driven Chance Constrained Optimization under Wasserstein Ambiguity Sets

“…Remark V.2. (Comparison with literature and exactness of CVaR approximation): In [27], [28], authors derive the reformulation given in Proposition V.1 for the case when Ξ = R m . In addition, they show that when Ξ = R m and N α ≤ 1, the CVaR approximation is exact, i.e., X DCP = X CDCP .…”

“…The authors in [26] first showed that it is strongly NP-Hard to solve a DRCCP with Wasserstein ambiguity sets and proposed a bi-criteria approximation scheme for covering constraints. While preparing this paper, we became aware of two recent working papers that presented reformulations and approximations of DRCCPs under Wasserstein ambiguity sets [27], [28] and for constraint functions that are affine in both the decision variable and the uncertainty. Both [27], [28] show that the exact feasibility set of DRCCPs with affine constraints can be reformulated as mixed integer conic programs.…”

“…While preparing this paper, we became aware of two recent working papers that presented reformulations and approximations of DRCCPs under Wasserstein ambiguity sets [27], [28] and for constraint functions that are affine in both the decision variable and the uncertainty. Both [27], [28] show that the exact feasibility set of DRCCPs with affine constraints can be reformulated as mixed integer conic programs. Specifically, Xie [27] studies individual chance constraints and joint chance constraints with right hand side uncertainty, while Chen et.…”