Two-Stage Distributionally Robust Conic Linear Programming over 1-Wasserstein Balls

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Abstract

This paper studies two-stage distributionally robust conic linear programming under constraint uncertainty over type-1 Wasserstein balls. We present optimality conditions for the dual of the worst-case expectation problem, which characterizes worst-case uncertain parameters for its inner maximization problem. This condition offers an alternative proof, a counterexample, and an extension to previous works. Additionally, the condition highlights the potential advantage of a specific distance metric for out-of-sample performance, as exemplified in a numerical study on a facility location problem with demand uncertainty. Furthermore, cutting-plane-based algorithms, equipped with a unified scenario generation framework, are proposed for addressing both unbounded support and second-stage dual feasible regions, with a finite convergence proof under less stringent assumptions.