First-Order Methods for Wasserstein Distributionally Robust Constrained Optimization

We consider constrained problems in which input data are affected by errors. In such settings, Wasserstein distributionally robust optimization provides a principled framework to mitigate model risk by optimizing against worst-case data distributions within Wasserstein ambiguity sets. However, the numerical resolution of the resulting problems remains challenging, especially in constrained settings. In this paper, we provide a general, practical way to solve Wasserstein distributionally robust formulations in the presence of constraints. Our approach only requires a linear minimization oracle for the feasible set, and combines two key ingredients: (i) an entropic regularization of the distributionally robust value function, which makes it possible to compute stochastic gradient estimators, and (ii) a stochastic Frank-Wolfe algorithm, which minimizes the regularized robust objective while naturally handling constraints. We illustrate the method, its tractability, and its benefits over empirical risk minimization, on two operations research problems: the traffic assignment and the minimum quadratic spanning tree.

Recommended citation: Hubert Villuendas, Mathieu Besançon and Jérôme Malick. (2026). "First-Order Methods for Wasserstein Distributionally Robust Constrained Optimization." preprint arXiv:2607.11460.
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