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We consider a general class of two-stage distributionally robust optimization(DRO) problems which includes prominent instances such as task scheduling, theassemble-to-order system, and supply chain network design. The ambiguity set isconstrained by fixed marginal distributions that are not necessarily discrete.We develop a numerical algorithm for computing approximately optimal solutionsof such problems. Through replacing the marginal constraints by a finitecollection of linear constraints, we derive a relaxation of the DRO problemwhich serves as its upper bound. We can control the relaxation error to bearbitrarily close to 0. We develop duality results and transform the inf-supproblem into an inf-inf problem. This leads to a numerical algorithm fortwo-stage DRO problems with marginal constraints which solves a linearsemi-infinite optimization problem. Besides an approximately optimal solution,the algorithm computes both an upper bound and a lower bound for the optimalvalue of the problem. The difference between the computed bounds provides adirect sub-optimality estimate of the computed solution. Most importantly, onecan choose the inputs of the algorithm such that the sub-optimality iscontrolled to be arbitrarily small. In our numerical examples, we apply theproposed algorithm to task scheduling, the assemble-to-order system, and supplychain network design. The ambiguity sets in these problems involve a largenumber of marginals, which include both discrete and continuous distributions.The numerical results showcase that the proposed algorithm computeshigh-quality robust decisions along with their corresponding sub-optimalityestimates with practically reasonable magnitudes that are notover-conservative.

Numerical method for approximately optimal solutions of two-stage distributionally robust optimization with marginal constraints