Comparative Analysis of Two-Stage Distributionally Robust Optimization over 1-Wasserstein and 2-Wasserstein Balls

TL;DR

This paper compares 1- and 2-Wasserstein ambiguity sets in two-stage distributionally robust optimization, showing 2-Wasserstein sets often provide better decision-making and avoid pathological issues present in 1-Wasserstein sets.

Contribution

It demonstrates the advantages of 2-Wasserstein ambiguity sets over 1-Wasserstein sets, including closed-form solutions and broader applicability in robust optimization.

Findings

2-Wasserstein sets avoid pathological behaviors of 1-Wasserstein sets.

Closed-form solutions for newsvendor problem with 2-Wasserstein ambiguity.

2-Wasserstein balls outperform 1-Wasserstein balls across various radii.

Abstract

This paper investigates advantages of using 2-Wasserstein ambiguity sets over 1-Wasserstein sets in two-stage distributionally robust optimization with right-hand side uncertainty. We examine the worst-case distributions within 1- and 2-Wasserstein balls under both unrestricted and nonnegative orthant supports, highlighting a pathological behavior arising in 1-Wasserstein balls. Closed-form solutions for a single-scenario newsvendor problem illustrate that 2-Wasserstein balls enable more informed decisions. Additionally, a penalty-based dual interpretation suggests that 2-Wasserstein balls may outperform 1-Wasserstein balls across a broader range of Wasserstein radii, even with general support sets.