Papers › Moment Relaxations for Data-Driven Wasserstein Distributionally Robust Optimization
Moment Relaxations for Data-Driven Wasserstein Distributionally Robust Optimization
Shixuan Zhang, Suhan Zhong
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We propose moment relaxations for data-driven p-Wasserstein distributionally robust optimization (p-WDRO) problems that are defined by polynomials. The proposed moment relaxations admit Benders-type decomposition with parallel evaluation of the subgradients using each sample subproblem, which enables efficient solution for larger training sets. We then identify conditions on p and the defining polynomial degrees such that the proposed k-th order moment relaxations preserve the asymptotic consistency of the original p-WDRO (i.e., the relaxation gap is bounded at most linearly by the Wasserstein radius). In particular, these conditions translate to effective bounds on k, which lead to polynomially sized semidefinite optimization formulations that are compatible with existing solvers. Numerical experiments on a box-constrained regression problem and a two-stage production problem are included to demonstrate the scalability and the effectiveness of the proposed moment relaxations.
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