A Data-Driven Distributionally Robust Bound on the Expected Optimal Value of Uncertain Mixed 0-1 Linear Programming

TL;DR

This paper develops a data-driven, distributionally robust method to bound the expected optimal value of uncertain mixed 0-1 linear programs using Wasserstein ambiguity sets, reformulated as tractable semidefinite programs.

Contribution

It introduces a novel Wasserstein-based ambiguity set approach for mixed 0-1 programming, providing a tractable reformulation and improved bounds compared to existing methods.

Findings

The approach provides tight bounds on the expected optimal value.

Numerical experiments demonstrate the method's effectiveness across applications.

The reformulation into semidefinite programs enables practical computation.

Abstract

This paper studies the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from the observed samples and containing the true distribution with a high statistical guarantee. The problem of interest is to investigate the bound on the expected optimal value over the Wasserstein ambiguity set. Under standard assumptions, we reformulate the problem into a copositive program, which naturally leads to a tractable semidefinite-based approximation. We compare our approach with a moment-based approach from the literature on three applications. Numerical results illustrate the effectiveness of our approach.