Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

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Abstract

In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-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 ø-divergence uncertainty is tractable for most of the choices of ø 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.
Original languageEnglish
Place of PublicationTilburg
PublisherOperations research
Volume2011-061
Publication statusPublished - 2011

Publication series

NameCentER Discussion Paper
Volume2011-061

Fingerprint

Divergence
Linear Optimization
Optimization Problem
Uncertainty
Confidence Set
Asset Pricing
Chi-squared
Inventory Control
Robust Optimization
Uncertain Parameters
Expected Utility
Finance
Random variable
Moment
Optimization
Term
Relevance

Keywords

  • robust optimization
  • ø-divergence
  • goodness-of-fit statistics

Cite this

Ben-Tal, A., den Hertog, D., De Waegenaere, A. M. B., Melenberg, B., & Rennen, G. (2011). Robust Solutions of Optimization Problems Affected by Uncertain Probabilities. (CentER Discussion Paper; Vol. 2011-061). Tilburg: Operations research.
Ben-Tal, A. ; den Hertog, D. ; De Waegenaere, A.M.B. ; Melenberg, B. ; Rennen, G. / Robust Solutions of Optimization Problems Affected by Uncertain Probabilities. Tilburg : Operations research, 2011. (CentER Discussion Paper).
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Ben-Tal, A, den Hertog, D, De Waegenaere, AMB, Melenberg, B & Rennen, G 2011 'Robust Solutions of Optimization Problems Affected by Uncertain Probabilities' CentER Discussion Paper, vol. 2011-061, Operations research, Tilburg.

Robust Solutions of Optimization Problems Affected by Uncertain Probabilities. / Ben-Tal, A.; den Hertog, D.; De Waegenaere, A.M.B.; Melenberg, B.; Rennen, G.

Tilburg : Operations research, 2011. (CentER Discussion Paper; Vol. 2011-061).

Research output: Working paperDiscussion paperOther research output

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T1 - Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

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N2 - In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-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 ø-divergence uncertainty is tractable for most of the choices of ø 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.

AB - In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-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 ø-divergence uncertainty is tractable for most of the choices of ø 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.

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KW - goodness-of-fit statistics

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BT - Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

PB - Operations research

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Ben-Tal A, den Hertog D, De Waegenaere AMB, Melenberg B, Rennen G. Robust Solutions of Optimization Problems Affected by Uncertain Probabilities. Tilburg: Operations research. 2011. (CentER Discussion Paper).