# On Linear Optimization over Wasserstein Balls

arXiv: https://arxiv.org/abs/2004.07162 · HTML: https://mlsanity.com/paper/2004.07162/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2020.

## Built on by (most cited first, 24 of 24)

- [The Many Faces of Adversarial Risk](https://mlsanity.com/paper/2201.08956/) (2022)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Mixed Nash Equilibria in the Adversarial Examples Game](https://mlsanity.com/paper/2102.06905/) (2021)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Unifying Distributionally Robust Optimization via Optimal Transport Theory](https://mlsanity.com/paper/2308.05414/) (2023)
- [Distributionally Robust Statistical Verification with Imprecise Neural Networks](https://mlsanity.com/paper/2308.14815/) (2023)
- [Distributionally Robust Fair Principal Components via Geodesic Descents](https://mlsanity.com/paper/2202.03071/) (2022)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow](https://mlsanity.com/paper/2410.00229/) (2024)
- [Distributionally Robust Optimization via Iterative Algorithms in Continuous Probability Spaces](https://mlsanity.com/paper/2412.20556/) (2024)
- [Sampling from the Mean-Field Stationary Distribution](https://mlsanity.com/paper/2402.07355/) (2024)
- [Worst-case generation via minimax optimization in Wasserstein space](https://mlsanity.com/paper/2512.08176/) (2025)
- [Learning Against Distributional Uncertainty: On the Trade-off Between Robustness and Specificity](https://mlsanity.com/paper/2301.13565/) (2023)
- [Markov Decision Processes under Model Uncertainty](https://mlsanity.com/paper/2206.06109/) (2022)
- [Distributional Robustness Bounds Generalization Errors](https://mlsanity.com/paper/2212.09962/) (2022)
- [Wasserstein Distributionally Robust Shallow Convex Neural Networks](https://mlsanity.com/paper/2407.16800/) (2024)
- [Automatic Outlier Rectification via Optimal Transport](https://mlsanity.com/paper/2403.14067/) (2024)
- [Global Optimisation of Black-Box Functions with Generative Models in the Wasserstein Space](https://mlsanity.com/paper/2407.11917/) (2024)
- [A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimization](https://mlsanity.com/paper/2502.17602/) (2025)
- [Distributionally Robust Deep Q-Learning](https://mlsanity.com/paper/2505.19058/) (2025)
- [Distributionally and Adversarially Robust Logistic Regression via Intersecting Wasserstein Balls](https://mlsanity.com/paper/2407.13625/) (2024)
- [Provable Robust Overfitting Mitigation in Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2503.04315/) (2025)
- [Data-driven robust Markov decision processes on Borel spaces: performance guarantees via an axiomatic approach](https://mlsanity.com/paper/2603.08979/) (2026)
- [Policy Gradient Learning for Distributionally Robust Markov Decision Processes under Wasserstein Ambiguity](https://mlsanity.com/paper/2606.27610/) (2026)

## Abstract (arXiv)

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.