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Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints

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Abstract

Stochastic dominance relations are well studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance constraints can be solved by linear programming. However, problems involving first order stochastic dominance constraints are potentially hard due to the non-convexity of the associated feasible regions. In this paper we consider a mixed 0–1 linear programming formulation of a discrete first order constrained optimization model and present a relaxation based on second order constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional second order constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a real world portfolio optimization problem.

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Authors and Affiliations

  1. Manufacturing Systems and Industrial Engineering Program, Faculty of Engineering and Natural Sciences, SABANCI University, 34956, Orhanli, Tuzla, Istanbul, Turkey

    Nilay Noyan

  2. Department of Management Science and Information Systems, Rutgers University, 94 Rockefeller Road, Piscataway, NJ, 08854, USA

    Andrzej Ruszczyński

Authors
  1. Nilay Noyan
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  2. Andrzej Ruszczyński
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Correspondence to Andrzej Ruszczyński.

Additional information

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

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Noyan, N., Ruszczyński, A. Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints. Math. Program. 114, 249–275 (2008). https://doi.org/10.1007/s10107-007-0100-1

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