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

• 成果类型: Article
• 署名作者: Zhen, Jianzhe; Kuhn, Daniel; Wiesemann, Wolfram
• 署名单位: Chinese Academy of Sciences; University of Chinese Academy of Sciences, CAS; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Imperial College London
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2021.0268
• 发表日期: 2025
• 关键词: PORTFOLIO OPTIMIZATION; deriving robust; uncertainty; PERSPECTIVE; RISK

摘要

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.