Sample average approximation of stochastic dominance constrained programs

Jian Hu*, Tito Homem-De-Mello, Sanjay Mehrotra

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

47 Scopus citations

Abstract

In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with the case of multi-variate stochastic dominance under general distributions and nonlinear functions. We introduce the concept of C -dominance, which generalizes some notions of multi-variate dominance found in the literature. We apply the Sample Average Approximation (SAA) method to this problem, which results in a semi-infinite program, and study asymptotic convergence of optimal values and optimal solutions, as well as the rate of convergence of the feasibility set of the resulting semi-infinite program as the sample size goes to infinity. We develop a finitely convergent method to find an ε -optimal solution of the SAA problem. An important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value. We also show that the bounds are asymptotically tight as the sample size goes to infinity.

Original languageEnglish (US)
Pages (from-to)171-201
Number of pages31
JournalMathematical Programming
Volume133
Issue number1-2
DOIs
StatePublished - Jun 2012

Keywords

  • Convex programming
  • Cutting plane algorithms
  • Sample average approximation
  • Semi-infinite programming
  • Stochastic dominance
  • Stochastic programming

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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