A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization
This work provides a foundational theoretical framework for distributionally robust optimization, impacting researchers and practitioners in optimization and machine learning, though it is incremental in extending existing duality results.
The paper presents a general duality proof for Wasserstein distributionally robust optimization that applies to any transport cost, loss function, and nominal distribution, using one-dimensional convex analysis and establishing conditions for interchangeability. It extends this approach to problems like distributionally robust Markov decision processes and multistage stochastic programming.
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.