Distributionally Robust Optimization over Wasserstein Balls with i.i.d. Structure

TL;DR

This work advances distributionally robust optimization by focusing on i.i.d. uncertainty through a structured Wasserstein ambiguity set consisting of product measures $P^{\otimes N}$ with $P$ in a Wasserstein ball around $\widehat{P}$. To overcome non-convexity, the authors develop a sequence of convex relaxations based on lifting, symmetrization, and inclusion of symmetric marginals, proving strong duality for each relaxation and convergence to the original problem value under suitable conditions. They establish finiteness, existence, and tighter bounds for the inner UQ problem, characterize the relaxation gap (including cases with zero gap when the loss is concave), and relate the relaxed problems to de Finetti-type mixtures of product measures. The framework yields tractable reformulations via Lagrange duality, including explicit semi-infinite programs for concave polyhedral losses, and numerical experiments demonstrate the practical advantages of structured ambiguity over unstructured DRO in capturing uncertainty with reduced conservatism. Overall, the paper provides a rigorous, scalable approach to structured DRO with i.i.d. components and offers insights, exactness results, and computational tools relevant to control, optimization, and data-driven decision-making.

Abstract

We consider distributionally robust optimization problems where the uncertainty is modeled via a structured Wasserstein ambiguity set. Specifically, the ambiguity is restricted to product measures $P^{\otimes N}$, where $P$ lies within a Wasserstein ball centered at an empirical distribution $\widehat{P}$. This structure reflects the assumption of independent and identically distributed (i.i.d.) uncertainty components and yields a non-convex ambiguity set that is strictly contained in its unstructured counterpart, thereby reducing conservatism. The resulting optimization problem is generally intractable due to the loss of convexity. We address this by introducing a sequence of tractable convex relaxations, each admitting strong duality, and prove that this sequence converges to the original problem value under suitable conditions. Numerical examples are provided to illustrate the effectiveness of the proposed approach. As a byproduct of our proofs, we establish a novel formula, of independent interest, relating the Wasserstein distance of a mixture of product distributions to the Wasserstein distance between its constituent measures.

Figures (3)

• Figure 1: Objective values for Examples \ref{['ex:symmetrization']} and \ref{['ex:liftedRelaxation']}
• Figure 2: Relaxation gap of the structured DRO problem for different values of $M$ and $\rho$.
• Figure 3: Objective values and optimal decision vectors for the example from Section \ref{['sec:structuredDRONumerical']}.

Theorems & Definitions (82)

• Theorem 1
• Theorem 2: zhang2024short, blanchet2019quantifying
• Lemma 1
• Theorem 3
• Example 1
• Example 2
• Lemma 2
• Lemma 3
• Example 3
• Definition 1