We consider a stochastic program with expected value constraints. We analyze this problem in a general context via Distributionally Robust Optimization (DRO) approach using 1 or 2-Wasserstein metrics where the ambiguity set depends on the decision. We show that this approach can be reformulated as a finite-dimensional optimization problem, and, in some cases, this can be convex. Additionally, we establish criteria to determine the feasibility of the problem in terms of the Wasserstein radius and the level of the constraint. Finally, we present numerical results in the context of inventory management and portfolio optimization. In the portfolio optimization context, we present the advantages that our approach has over some existing non-robust methods using real financial market data.
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