Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion
Author(s)Bertsimas, Dimitris J.; Doan, Xuan Vinh; Natarajan, Karthik; Teo, Chung-Piaw
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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.
DepartmentMassachusetts Institute of Technology. Operations Research Center; Sloan School of Management
Mathematics of Operations Research
Institute for Operations Research and the Management Sciences
Bertsimas, D. et al. “Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion.” Mathematics of Operations Research 35.3 (2010): 580–602.
Author's final manuscript