Perturbation Duality for Robust and Distributionally Robust Optimization: Short and General Proofs

Abstract

Duality is a foundational tool in robust and distributionally robust optimization (RO/DRO), underpinning both analytical insights and tractable reformulations. While most RO/DRO duality results are derived through saddle-point, Lagrangian, or conic arguments, this paper leverages perturbation duality. We show that this perspective provides a natural and unifying framework for deriving RO/DRO dual formulations, proving the associated duality results, and diagnosing the regularity assumptions on which they depend. First, guided by perturbation duality, we establish new duality theorems for a recent DRO framework that unifies several canonical models, including $ϕ$-divergence and Wasserstein models, through optimal transport subject to conditional moment constraints. Our results resolve an open conjecture on this DRO duality by clarifying the role of compactness: compactness itself is not necessary, but can be replaced by perturbation-based regularity conditions. Second, we revisit \emph{robust duality}, commonly described as \emph{primal-worst equals dual-best.} Using bifunctions, we unify dual-best formulations appearing in the literature and derive concise perturbation-based proofs that streamline recent results. Overall, the paper positions perturbation duality as a versatile and underutilized tool for RO and DRO, offering both conceptual unification and technical generality across a broad class of models.