Moment Relaxations for Data-Driven Wasserstein Distributionally Robust Optimization

TL;DR

This work develops moment relaxations for data-driven Wasserstein DRO to yield tractable surrogates for single- and two-stage problems while preserving asymptotic consistency as the Wasserstein radius vanishes. By representing inner max problems via truncated moment sequences and SOS-based semidefinite constraints, the authors obtain SDP-friendly relaxations that can be solved in parallel per data sample and provide subgradients for outer optimization. They establish conditions under which these relaxations achieve $\mathcal{O}(r)$-consistency with the original ESO problem, including degree and boundedness/compactness requirements; they also propose strengthened two-stage relaxations to handle unbounded recourse. Numerical experiments on a two-stage production problem illustrate the trade-off between conservatism and out-of-sample performance and demonstrate the potential of moment relaxations to yield tractable, data-driven decision-making under distributional uncertainty. The work identifies practical directions for improving scalability (e.g., SOC relaxations, low-rank methods) and extending applicability to non-polynomial costs via polynomial approximations.

Abstract

We propose moment relaxations for data-driven Wasserstein distributionally robust optimization problems. Conditions are identified to ensure asymptotic consistency of such relaxations for both single-stage and two-stage problems, together with examples that illustrate their necessity. Numerical experiments are also included to illustrate the proposed relaxations.

Figures (2)

• Figure 1: Comparison of Mean Costs for the Two-Stage Production Problem
• Figure 2: Out-of-sample Performance for the Two-Stage Production Problem

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Example 2.1
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof