On Linear Optimization over Wasserstein Balls

2020/04/15 by Man–Chung Yue, Daniel Kühn, Yue, Man-Chung +3 · 3 citations
Decision Sciences · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Optimization and Control (math.OC) #Point processes and geometric inequalities #Risk and Portfolio Optimization #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.2004.07162

openalex publication_date 2020/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities to formulate and solve data-driven optimization problems with rigorous statistical guarantees. In this technical note we prove that the Wasserstein ball is weakly compact under mild conditions, and we offer necessary and sufficient conditions for the existence of optimal solutions. We also characterize the sparsity of solutions if the Wasserstein ball is centred at a discrete reference measure. In comparison with the existing literature, which has proved similar results under different conditions, our proofs are self-contained and shorter, yet mathematically rigorous, and our necessary and sufficient conditions for the existence of optimal solutions are easily verifiable in practice.

Citations

• Learning with a Wasserstein Loss
• Distributionally Robust Stochastic Optimization with Wasserstein Distance
• Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations
• Calculating Optimistic Likelihoods Using (Geodesically) Convex Optimization

Cited by

• Worst-case generation via minimax optimization in Wasserstein space
• Scaling limits of multi-period distributionally robust optimization problems
• Loss-aware distributionally robust optimization via trainable optimal transport ambiguity sets

Related

• Distributionally Robust Optimization over Wasserstein Balls with i.i.d. Structure
• Comparative Analysis of Two-Stage Distributionally Robust Optimization over 1-Wasserstein and 2-Wasserstein Balls
• Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality
• Data-driven Distributionally Robust Optimization Using the Wasserstein\n Metric: Performance Guarantees and Tractable Reformulations
• Wasserstein Distributionally Robust Optimization: Theory and\n Applications in Machine Learning