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Optimization with multivariate stochastic dominance constraints

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ, 07030, USA

    Darinka Dentcheva

  2. Department of Management Science and Information Systems, Rutgers University, Piscataway, NJ, 08854, USA

    Andrzej Ruszczyński

Authors
  1. Darinka Dentcheva
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  2. Andrzej Ruszczyński
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Corresponding author

Correspondence to Andrzej Ruszczyński.

Additional information

This paper is dedicated to Stephen M. Robinson, with gratitude for his help and in appreciation of his fundamental contributions to nonlinear optimization and set-valued analysis.

This research was supported by the NSF awards DMS-0603728, DMS-0604060, DMI-0354500 and DMI-0354678.

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Dentcheva, D., Ruszczyński, A. Optimization with multivariate stochastic dominance constraints. Math. Program. 117, 111–127 (2009). https://doi.org/10.1007/s10107-007-0165-x

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  • DOI: https://doi.org/10.1007/s10107-007-0165-x

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