Nonlinear distributionally robust optimization

• 成果类型: Article
• 署名作者: Sheriff, Mohammed Rayyan; Esfahani, Peyman Mohajerin
• 署名单位: Delft University of Technology
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-024-02151-7
• 发表日期: 2025
• 页码: 639-698
• 关键词: minimization; RISK

摘要

This article focuses on a class of distributionally robust optimization (DRO) problems where, unlike the growing body of the literature, the objective function is potentially nonlinear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present both theoretical and computational challenges. Motivated by this, we propose an alternative notion for the derivative and corresponding smoothness based on Gateaux (G)-derivative for generic risk measures. These concepts are explained via three running risk measure examples of variance, entropic risk, and risk on finite support sets. We then propose a G-derivative based Frank-Wolfe (FW) algorithm for generic nonlinear optimization problems in probability spaces and establish its convergence under the proposed notion of smoothness in a completely norm-independent manner. We use the set-up of the FW algorithm to devise a methodology to compute a saddle point of the nonlinear DRO problem. Finally, we validate our theoretical results on two cases of the entropic and variance risk measures in the context of portfolio selection problems. In particular, we analyze their regularity conditions and sufficient statistic, compute the respective FW-oracle in various settings, and confirm the theoretical outcomes through numerical validation.