On distributionally robust chance constrained programs with Wasserstein distance

• Published in: Mathematical programming Vol. 186; no. 1-2; pp. 115 - 155
• Main Author: Xie, Weijun
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2021; Springer Nature B.V
• Subjects: Algorithms; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Constraints; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Mixed integer; Numerical Analysis; Optimization; Parameter uncertainty; Robustness (mathematics); Theoretical; 90C47; 90C11; 90C15
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-019-01445-5

This paper studies a distributionally robust chance constrained program (DRCCP) with Wasserstein ambiguity set, where the uncertain constraints should be satisfied 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 conditional value-at-risk constrained optimization problem, and thus admits tight inner and outer approximations. We also show that a DRCCP 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 the submodular structure, we show that it admits a big-M free formulation, which can be solved by a branch and cut algorithm. Finally, we present a numerical study to illustrate the effectiveness of the proposed formulations.