A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle

Jianzhe Zhen (), Daniel Kuhn () and Wolfram Wiesemann ()
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Jianzhe Zhen: School of Economics and Management, University of Chinese Academy of Sciences, Beijing 100864, China
Daniel Kuhn: College of Management of Technology, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Wolfram Wiesemann: Imperial College Business School, Imperial College London, London SW7 2AZ, United Kingdom

Operations Research, 2025, vol. 73, issue 2, 862-878

Abstract: Robust optimization and distributionally robust optimization are modeling paradigms for decision making under uncertainty where the uncertain parameters are only known to reside in an uncertainty set or are governed by any probability distribution from within an ambiguity set, respectively, and a decision is sought that minimizes a cost function under the most adverse outcome of the uncertainty. In this paper, we develop a rigorous and general theory of robust and distributionally robust nonlinear optimization using the language of convex analysis. Our framework is based on a generalized “primal-worst-equals-dual-best” principle that establishes strong duality between a semi-infinite primal worst and a nonconvex dual best formulation, both of which admit finite convex reformulations. This principle offers an alternative formulation for robust optimization problems that obviates the need to mobilize the machinery of abstract semi-infinite duality theory to prove strong duality in distributionally robust optimization. We illustrate the modeling power of our approach through convex reformulations for distributionally robust optimization problems whose ambiguity sets are defined through general optimal transport distances, which generalize earlier results for Wasserstein ambiguity sets.

Keywords: Optimization; (distributionally) robust optimization; convex analysis; optimal transport (search for similar items in EconPapers)
Date: 2025
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