Article ID: | iaor20106292 |
Volume: | 35 |
Issue: | 3 |
Start Page Number: | 580 |
End Page Number: | 602 |
Publication Date: | Aug 2010 |
Journal: | Mathematics of Operations Research |
Authors: | Bertsimas Dimitris, Teo Chung-Piaw, Doan Xuan Vinh, Natarajan Karthik |
Keywords: | programming (semidefinite) |
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.