Convex Chance-Constrained Programs with Wasserstein Ambiguity

Chance constraints yield nonconvex feasible regions in general. In particular, when the uncertain parameters are modeled by a Wasserstein ball, existing studies showed that the distributionally robust (pessimistic) chance constraint admits a mixed-integer conic representation. This paper identifies sufficient conditions that lead to convex feasible regions of chance constraints with Wasserstein ambiguity. First, when uncertainty arises from the right-hand side of a pessimistic joint chance constraint, we show that the ensuing feasible region is convex if the Wasserstein ball is centered around a log-concave distribution (or, more generally, an α-concave distribution with $\alpha \ge -1$). In addition, we propose a block coordinate ascent algorithm and prove its convergence to global optimum, as well as the rate of convergence. Second, when uncertainty arises from the left-hand side of a pessimistic two-sided chance constraint, we show the convexity if the Wasserstein ball is centered around an elliptical and star unimodal distribution. In addition, we propose a family of second-order conic inner approximations, and we bound their approximation error and prove their asymptotic exactness. Furthermore, we extend the convexity results to optimistic chance constraints.

Funding: This work was supported by the National Science Foundation [Grants ECCS-1845980, OIA-2119691, and OIA-1946391] and the Air Force Office of Scientific Research [Grant FA9550-23-1-0323].

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2021.0709.