ALSO-X#: Better Convex Approximations for Distributionally Robust Chance Constrained Programs

Abstract

This paper studies distributionally robust chance constrained programs (DRCCPs), where the uncertain constraints must be satisfied with at least a probability of a prespecified threshold for all probability distributions from the Wasserstein ambiguity set. As DRCCPs are often nonconvex and challenging to solve optimally, researchers have been developing various convex inner approximations. Recently, ALSO-X has been proven to outperform the conditional value-at-risk (CVaR) approximation of a regular chance constrained program when the deterministic set is convex. In this work, we relax this assumption by introducing a new ALSO-X\# method for solving DRCCPs. Namely, in the bilevel reformulations of ALSO-X and CVaR approximation, we observe that the lower-level ALSO-X is a special case of the lower-level CVaR approximation and the upper-level CVaR approximation is more restricted than the one in ALSO-X. This observation motivates us to propose the ALSO-X\#, which still resembles a bilevel formulation -- in the lower-level problem, we adopt the more general CVaR approximation, and for the upper-level one, we choose the less restricted ALSO-X. We show that ALSO-X\# can always be better than the CVaR approximation and can outperform ALSO-X under regular chance constrained programs and type $\infty-$Wasserstein ambiguity set. We also provide new sufficient conditions under which ALSO-X\# outputs an optimal solution to a DRCCP. We apply the proposed ALSO-X\# to a wireless communication problem and numerically demonstrate that the solution quality can be even better than the exact method.