Second-Order Conic Programming Approach for Wasserstein Distributionally Robust Two-Stage Linear Programs

This article proposes a second-order conic programming (SOCP) approach to solve distributionally robust two-stage linear programs over 1-Wasserstein balls. We start from the case with distribution uncertainty only in the objective function and then explore the case with distribution uncertainty only in constraints. The former program is exactly reformulated as a tractable SOCP problem, whereas the latter one is proved to be generally NP-hard as it involves a norm maximization problem over a polyhedron. However, it reduces to an SOCP problem if the extreme points of the polyhedron are given as a prior. This motivates the design of a constraint generation algorithm with provable convergence to approximately solve the NP-hard problem. Moreover, the least favorable distribution achieving the worst case cost is given as an “empirical” distribution by simply perturbing each original sample for both cases. Finally, experiments illustrate the advantages of the proposed model in terms of the out-of-sample performance and computational complexity.
Note to Practitioners
—The two-stage program with distribution uncertainty is an important decision problem in broad applications, e.g., two-stage schedule problems, facility location problems, and recourse allocation problems. To deal with the uncertainty, this work proposes a novel data-driven model over the 1-Wasserstein ball and develops an efficient second-order conic programming (SOCP)-based solution approach, where the sample data set can be easily exploited to reduce the distribution uncertainty. The good out-of-sample performance and computational complexity of the proposed model are validated by the experiments on the two-stage portfolio programs and material order programs.