A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization

TL;DR

A novel elementary proof is presented that demonstrates that strong duality is contingent on a certain interchangeability principle and extended to encompass risk-averse optimization and globalized distributionally robust counterparts.

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.

Theorems & Definitions (29)

• Lemma 1
• Theorem 1
• Remark 1: $p$-Wasserstein distance
• Remark 2: Necessity of \ref{['IP']}
• Remark 3: Continuity at $\rho=0$
• Definition 1
• Proposition 1
• Remark 4
• Proposition 2
• Example 1: Empirical distribution