This paper proposes a tractable stochastic approach based on an entropic regularization of the distributionally robust value function, which makes it possible to compute stochastic gradient estimators, and the combination of these estimators with a stochastic Frank-Wolfe algorithm, allowing us to optimize the regularized robust objective while naturally handling constraints.
Abstract
We consider constrained optimization problems in which input data are affected by estimation errors. In such settings, Wasserstein distributionally robust optimization provides a principled framework to mitigate model risk by optimizing against worst-case distributions within Wasserstein ambiguity sets. However, the numerical resolution of the resulting problems remains challenging, especially in constrained and combinatorial settings. In this paper, we propose a tractable stochastic approach based on two key ingredients: (i) an entropic regularization of the distributionally robust value function, which makes it possible to compute stochastic gradient estimators, and (ii) the combination of these estimators with a stochastic Frank-Wolfe algorithm, allowing us to optimize the regularized robust objective while naturally handling constraints. We illustrate the method, and its interests against empirical risk minimization, on two classical optimization problems, the minimum quadratic spanning tree and the traffic assignment problems. Our approach provides a general, practical way to address Wasserstein distributionally robust formulations in the presence of constraints.
A novel formulation of a Wasserstein-2 metric that uses the Bures-Wasserstein (BW) metric over probability measures with finite second moments is developed, which allows the worst-case distribution to endogenously determine both how many mixture components receive mass and where their means and covariances lie within a continuous support.
Wasserstein distributionally robust optimization (DRO) is a versatile and widely adopted framework for decision-making under uncertainty, yet its standard deterministic reformulations generally contain non-convex inner subproblems that are challenging to solve. To address this issue, we propose a shrinkage path heuristic that reduces the solution of a DRO problem to a one-dimensional search over the line segment connecting the (typically benign) sample average approximation (SAA) and the (more demanding but practically solvable) classical robust optimization solution. We derive a priori suboptimality bounds in stylized settings and, for the general case, a posteriori bounds obtained by applying a similar heuristic to a dual formulation. Numerical experiments on a multi-item newsvendor and an appointment scheduling problem show that the shrinkage path heuristic attains 85-110% (resp. 45-70%) of the out-of-sample performance improvements of Wasserstein DRO over SAA, at a fraction of the computational cost.
Ling-Jun Meng, Ryan Cory-Wright, W. Wiesemann· 0 citations
Distributionally robust optimization (DRO) with optimal transport ambiguity sets is traditionally solved by reformulating the minimax problem into a single-level convex program. While theoretically tractable, these reformulations introduce numerous auxiliary variables and demanding conic constraints that scale poorly in practice. In this paper, we address this challenge by reducing the inner worst-case expectation problem exactly to a scalar budget allocation task. This structural insight yields an efficient algorithm that bypasses large lifted reformulations, alongside a fast post-processing scheme to recover an optimal worst-case distribution supported on at most $N+1$ points, where $N$ denotes the sample size. We embed this procedure within an oracle-based distributional best-response framework to directly compute an approximate primal-dual solution to the overall DRO problem. Furthermore, we extend our analysis to the dual DRO formulation, proving the existence of a least-favorable distribution supported on at most $\min\{N+n+1, KN\}$ atoms, where $n$ and $K$ denote the decision dimension and number of loss components, respectively, and provide an efficient convex programming reduction to extract it from the solution of the primal DRO. Numerical experiments demonstrate that the proposed approach significantly outperforms state-of-the-art reformulation-based solvers.
Guixian Chen, S. Fattahi, Soroosh Shafiee· 1 citation
As a popular optimization scheme, distributionally robust optimization (DRO) protects decisions against ambiguity in probability distributions. For (single-stage) DRO, prevailing dual reformulations can become difficult when model or ambiguity-set structures are complex. We study DRO from a primal perspective, working directly with distributions in ambiguity sets on closed, potentially unbounded sample spaces. This perspective leads to an algorithmic framework, referred to as BiCS, that constructs and leverages distribution cuts to achieve strong performance. We show that BiCS is applicable to standard DRO, almost-sure DRO, DRO with various chance constraints, and DRO with ambiguity sets strengthened by local information. Numerical experiments with moment and Wasserstein ambiguity sets show that this framework demonstrates superior performance, including solving cases where the examined compact reformulations are unavailable or computationally difficult. The local-information study also makes changes in worst-case distributions directly visible.
We consider solving an optimization instance in which the objective is quadratic and where the decision variable is a probability measure. Our class of problems are motivated by applications arising from optimal transport (with the Gromov-Wasserstein problem being a prominent example) as well as energy landscape minimization. Because the objective depends quadratically on the decision variable, our class of problems fall outside the standard modeling framework of the Generalized Moment Problems (which requires the objective to be linear). To this end, we propose a hierarchy of convex relaxations based on searching over probability measures over products of the base space. These have a natural interpretation with the moment Sum-of-squares hierarchy-a prominent framework for solving polynomial optimization instances, which we adapt to accommodate probability measures. A key conceptual contribution is to introduce a notion of positive-semidefiniteness that extends the usual notion over matrices. Under the assumption that the decision variables satisfy certain marginal constraints (as in the Kantorovich formulation of the optimal transport problem), we establish convergence of our hierarchy towards the globally optimal solution. Under the additional assumption that the objective is a polynomial, we propose a moment-SOS type hierarchy of finite dimensional semidefinite programs whose optimal solution converges to that of the original quadratic optimization over measures. We demonstrate our framework with numerical experiments. More generally, optimization over measures where the objective and/or constraint depends on the decision in a polynomial way is a fundamental problem. It is hoped that our work provides a road-map as to how the ideas of the SOS-ordinarily developed for polynomial optimization-may be applied to a broader class of non-linear problems involving measures.
Given a dataset where a portion of the samples are contaminated, our goal is to recover the underlying clean population distribution. To this end, we propose Wasserstein Filtering (WF), a novel sample selection framework that discards a fraction of suspicious samples and estimates the target distribution using the empirical measure of the remaining data. The core insight is to select a subset of samples whose empirical distribution maximizes its Wasserstein distance to the fully contaminated empirical distribution, thereby preferentially isolating and removing geometrically influential outliers. To render this optimization computationally tractable, we introduce three algorithms: a marginal screening scheme, SinkMarg, and two joint optimization algorithms, SinkWF and SlicedWF, leveraging entropic optimal transport and sliced Wasserstein approximations, respectively. On the theoretical front, we introduce the Far Exclusion and Local Projection (FELP) contamination model, which characterizes corruptions consisting of well-separated outliers and locally indistinguishable perturbations. Under this model, we prove that the WF estimator achieves minimax optimality over distribution families with bounded covariance. Extensive numerical experiments on synthetic datasets, benchmark anomaly detection suites, and robust generative learning with diffusion models demonstrate that WF serves as a highly practical, model-agnostic preprocessing tool. It delivers competitive outlier detection performance and provides substantial downstream benefits for generative modeling under heavy contamination.