Contextual Distributionally Robust Chance-Constrained Programs: Exact Reformulations and Valid Inequalities

Abstract

We study a contextual distributionally robust chance-constrained programming (C-DRCCP) model that controls violation probabilities conditional on a neighborhood of a target context. The ambiguity set is defined over the joint distribution of contexts and outcomes using the Wasserstein distance. For affine safety systems with right-hand-side uncertainty, we derive an exact mixed-integer programming (MIP) reformulation. The reformulation reveals that worst-case conditional risk is determined by two coupled uses of the transportation budget: moving empirical mass into the conditioning neighborhood and moving conditioned mass into the failure region. Equivalently, robust feasibility holds when the minimum transportation cost needed to reach the conditional risk boundary is at least the available budget. Calibrating the ambiguity set gives a finite-sample guarantee for the true conditional violation probability. This characterization leads to strong valid inequalities that exploit the contextual transport. Each contextual mass allocation determines both the conditional scenario weights and the budget available to move outcomes into failure, whereas conventional quantile cuts utilize only the induced weights. Accounting for the remaining budget generates strengthened quantile cuts that strictly dominate standard quantile bounds and yield tighter MIP coefficients. We also derive rank inequalities from an exchange property of contextual transportation costs, linking feasible failure patterns across contextual allocations. Extensive numerical experiments demonstrate that these inequalities close much of the linear-relaxation gap, speed up the solution process, and enable large-sample instances that are intractable for the baseline formulation.