Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion

• 成果类型: Article
• 署名作者: Bertsimas, Dimitris; Doan, Xuan Vinh; Natarajan, Karthik; Teo, Chung-Piaw
• 署名单位: Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT); City University of Hong Kong; National University of Singapore
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.1100.0445
• 发表日期: 2010
• 页码: 580-602
• 关键词: expectation; bounds

摘要

We propose a semidefinite optimization (SDP) model for the class of minimax two-stage stochastic linear optimization problems with risk aversion. The distribution of second-stage random variables belongs to a set of multivariate distributions with known first and second moments. For the minimax stochastic problem with random objective, we provide a tight SDP formulation. The problem with random right-hand side is NP-hard in general. In a special case, the problem can be solved in polynomial time. Explicit constructions of the worst-case distributions are provided. Applications in a production-transportation problem and a single facility minimax distance problem are provided to demonstrate our approach. In our experiments, the performance of minimax solutions is close to that of data-driven solutions under the multivariate normal distribution and better under extremal distributions. The minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity.