Unifying Distributionally Robust Optimization via Optimal Transport Theory

TL;DR

This paper introduces a unified framework for distributionally robust optimization by combining $\,phi$-divergences and Wasserstein distances through optimal transport theory, enabling more flexible modeling of distributional ambiguity.

Contribution

It proposes a novel OT-based formulation that unifies existing DRO paradigms and provides duality results and tractable reformulations for practical applications.

Findings

Unified framework bridging $\,phi$-divergences and Wasserstein distances.

Duality results for the proposed OT-based DRO.

Tractable reformulations demonstrating practical utility.

Abstract

In recent years, two prominent paradigms have shaped distributionally robust optimization (DRO), modeling distributional ambiguity through $\phi$-divergences and Wasserstein distances, respectively. While the former focuses on ambiguity in likelihood ratios, the latter emphasizes ambiguity in outcomes and uses a transportation cost function to capture geometric structure in the outcome space. This paper proposes a unified framework that bridges these approaches by leveraging optimal transport (OT) with conditional moment constraints. Our formulation enables adversarial distributions to jointly perturb likelihood ratios and outcomes, yielding a generalized OT coupling between the nominal and perturbed distributions. We further establish key duality results and develop tractable reformulations that highlight the practical power of our unified approach.