Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

  • Authors:

  • Aharon Ben-Tal;Dick den Hertog;Anja De Waegenaere;Bertrand Melenberg;Gijs Rennen

  • Affiliations:

  • Department of Industrial Engineering and Management, Technion--Israel Institute of Technology, Haifa 32000, Israel/ and CentER, Tilburg University, 5000 LE Tilburg, The Netherlands;Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, The Netherlands;Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, The Netherlands;Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, The Netherlands;Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, The Netherlands

  • Venue:
  • Management Science
  • Year:
  • 2013

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Abstract

In this paper we focus on robust linear optimization problems with uncertainty regions defined by φ-divergences for example, chi-squared, Hellinger, Kullback--Leibler. We show how uncertainty regions based on φ-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 φ-divergence uncertainty is tractable for most of the choices of φ 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. This paper was accepted by Gérard P. Cachon, optimization.