Dentcheva and Ruszczynski recently proposed using a stochastic dominance constraint to specify risk preferences in a stochastic program. Such a constraint requires the random outcome resulting from one’s decision to stochastically dominate a given random comparator. These ideas have been extended to problems with multiple random outcomes, using the notion of positive linear stochastic dominance. We propose a constraint using a different version of multivariate stochastic dominance. This version is natural due to its connection to expected utility maximization theory and is relatively tractable. In particular, we show that such a constraint can be formulated with linear constraints for the second-order dominance relation, and with mixed- integer constraints for the first-order relation. This is in contrast to a constraint on second-order positive linear dominance, for which no efficient algorithms are known. We tested these formulations in the context of two applications: budget allocation in a setting with multiple objectives and finding radiation treatment plans in the presence of organ motion.
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University of Wisconsin-Madison, May, 2010.
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View Models and Formulations for Multivariate Dominance Constrained Stochastic Programs