Unifying Distributionally Robust Optimization via Optimal Transport Theory

In recent years, two prominent paradigms have shaped distributionally robust optimization (DRO), modeling distributional ambiguity through $ϕ$-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.

💡 Research Summary

The paper presents a unified framework that bridges the two dominant paradigms in distributionally robust optimization (DRO): φ‑divergence–based ambiguity sets and Wasserstein‑distance–based ambiguity sets. While φ‑divergences model uncertainty in likelihood ratios of a nominal distribution, Wasserstein distances capture geometric perturbations of outcomes. The authors propose to lift the original outcome space Z to a higher‑dimensional product space V × W, where V represents the original risk factors and W is a non‑negative weight variable that can adjust probability mass.

The central construct is an optimal transport (OT) discrepancy M(ν, ν̂) defined on probability measures over V × W, equipped with a conditional moment constraint Eπ