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.

Automated summary

This article introduces a second-order conic programming approach to solve distributionally robust two-stage linear programs over 1-Wasserstein balls, addressing distribution uncertainty in both the objective function and constraints. By formulating the problems as tractable SOCP problems and developing a constraint generation algorithm, it provides a solution to NP-hard problems. Additionally, it introduces the concept of the least favorable distribution and demonstrates the model's advantages in terms of out-of-sample performance and computational complexity through experiments.