Mathematical Programming

, Volume 117, Issue 1–2, pp 111–127 | Cite as

Optimization with multivariate stochastic dominance constraints

  • Darinka Dentcheva
  • Andrzej Ruszczyński


We consider stochastic optimization problems where risk-aversion is expressed by a stochastic ordering constraint. The constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector. We identify a suitable multivariate stochastic order and describe its generator in terms of von Neumann–Morgenstern utility functions. We develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint. Assuming convexity we show that the Lagrange multipliers corresponding to dominance constraints are elements of the generator of this order, thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints. Furthermore, we obtain necessary conditions of optimality for non-convex problems under additional smoothness assumptions.


Optimality Duality Utility Stochastic order Risk 

Mathematics Subject Classification (2000)

90C15 90C46 90C48 46N10 60E15 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesStevens Institute of TechnologyHobokenUSA
  2. 2.Department of Management Science and Information SystemsRutgers UniversityPiscatawayUSA

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