Second-order Conic Programming Approach for Wasserstein Distributionally Robust Two-stage Linear Programs

TL;DR

This paper develops a second-order conic programming approach to solve distributionally robust two-stage linear programs over Wasserstein balls, offering efficient reformulations and algorithms for problems with distribution uncertainty.

Contribution

It introduces SOCP reformulations for distributionally robust problems and proposes a constraint generation algorithm with convergence guarantees.

Findings

The SOCP approach effectively handles distribution uncertainty in the objective.

The constraint generation algorithm approximates NP-hard problems with provable convergence.

Experiments show improved out-of-sample performance and reduced computational complexity.

Abstract

This paper proposes a second-order conic programming (SOCP) approach to solve distributionally robust two-stage stochastic linear programs over 1-Wasserstein balls. We start from the case with distribution uncertainty only in the objective function and exactly reformulate it as an SOCP problem. Then, we study the case with distribution uncertainty only in constraints, and show that such a robust program is generally NP-hard as it involves a norm maximization problem over a polyhedron. However, it is reduced to an SOCP problem if the extreme points of the polyhedron are given as a prior. This motivates to design a constraint generation algorithm with provable convergence to approximately solve the NP-hard problem. In sharp contrast to the exiting literature, the distribution achieving the worst-case cost is given as an "empirical" distribution by simply perturbing each sample for both cases. Finally, experiments illustrate the advantages of the proposed model in terms of the out-of-sample performance and the computational complexity.