A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization

TL;DR

This paper introduces a unified duality proof for Wasserstein distributionally robust optimization, applicable to various costs, loss functions, and distributions, using convex analysis and measurable selection conditions.

Contribution

It provides a general duality framework for Wasserstein DRO that simplifies existing proofs and extends to multiple complex stochastic optimization problems.

Findings

Duality holds under broad conditions for Wasserstein DRO.

The proof relies on one-dimensional convex analysis and measurable selection.

Applications include robust Markov decision processes and multistage stochastic programming.

Abstract

We present a general duality result for Wasserstein distributionally robust optimization that holds for any Kantorovich transport cost, measurable loss function, and nominal probability distribution. Assuming an interchangeability principle inherent in existing duality results, our proof only uses one-dimensional convex analysis. Furthermore, we demonstrate that the interchangeability principle holds if and only if certain measurable projection and weak measurable selection conditions are satisfied. To illustrate the broader applicability of our approach, we provide a rigorous treatment of duality results in distributionally robust Markov decision processes and distributionally robust multistage stochastic programming. Additionally, we extend our analysis to other problems such as infinity-Wasserstein distributionally robust optimization, risk-averse optimization, and globalized distributionally robust counterpart.