Stochastic Decomposition Method for Two-Stage Distributionally Robust Optimization

TL;DR

This paper introduces a sequential sampling algorithm for two-stage distributionally robust linear programs, improving solution efficiency and convergence by using data-driven ambiguity set approximations and only two subproblems per iteration.

Contribution

It develops the DRSD method, a distributionally robust adaptation of stochastic decomposition, with proven convergence and computational advantages over existing methods.

Findings

The DRSD method converges to an optimal solution with probability one.

Numerical experiments show DRSD outperforms distributionally robust L-shaped method.

The approach effectively handles data-driven ambiguity sets in stochastic programming.

Abstract

In this paper, we present a sequential sampling-based algorithm for the two-stage distributionally robust linear programming (2-DRLP) models. The 2-DRLP models are defined over a general class of ambiguity sets with discrete or continuous probability distributions. The algorithm is a distributionally robust version of the well-known stochastic decomposition algorithm of Higle and Sen (Math. of OR 16(3), 650-669, 1991) for a two-stage stochastic linear program. We refer to the algorithm as the distributionally robust stochastic decomposition (DRSD) method. The key features of the algorithm include (1) it works with data-driven approximations of ambiguity sets that are constructed using samples of increasing size and (2) efficient construction of approximations of the worst-case expectation function that solves only two second-stage subproblems in every iteration. We identify conditions under which the ambiguity set approximations converge to the true ambiguity sets and show that the DRSD method asymptotically identifies an optimal solution, with probability one. We also computationally evaluate the performance of the DRSD method for solving distributionally robust versions of instances considered in stochastic programming literature. The numerical results corroborate the analytical behavior of the DRSD method and illustrate the computational advantage over an external sampling-based decomposition approach (distributionally robust L-shaped method).