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Robust solutions of optimization problems affected by uncertain probabilities

Aharon Ben-Tal(abental***at***ie.technion.ac.il)
Dick Den Hertog(D.denHertog***at***uvt.nl)
Anja De Waegenaere(A.M.B.DeWaegenaere***at***uvt.nl)
Bertrand Melenberg(B.Melenberg***at***uvt.nl)
Gijs Rennen(G.Rennen***at***gmail.com)

Abstract: In this paper we focus on robust linear optimization problems with uncertainty regions defined by phi-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on phi-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with phi-divergence uncertainty is tractable for most of the choices of phi typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.

Keywords: robust optimization, phi-divergence, goodness-of-fit statistics.

Category 1: Robust Optimization

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Category 3: Linear, Cone and Semidefinite Programming

Citation: CentER Discussion Paper CDP 2011-061, May 2011, CentER, Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands

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Entry Submitted: 06/27/2011
Entry Accepted: 06/27/2011
Entry Last Modified: 06/27/2011

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