Robust Solutions of Optimization Problems Affected by Uncertain Probabilities
AbstractIn 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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Bibliographic InfoPaper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2011-061.
Date of creation: 2011
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Web page: http://center.uvt.nl
robust optimization; ø-divergence; goodness-of-fit statistics;
Other versions of this item:
- Aharon Ben-Tal & Dick den Hertog & Anja De Waegenaere & Bertrand Melenberg & Gijs Rennen, 2013. "Robust Solutions of Optimization Problems Affected by Uncertain Probabilities," Management Science, INFORMS, vol. 59(2), pages 341-357, April.
- C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
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- Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
- Louis Eeckhoudt & Christian Gollier & Harris Schlesinger, 1995. "The Risk-Averse (and Prudent) Newsboy," Management Science, INFORMS, vol. 41(5), pages 786-794, May.
- Gorissen, B.L. & Ben-Tal, A. & Blanc, J.P.C. & Hertog, D. den, 2012. "A New Method for Deriving Robust and Globalized Robust Solutions of Uncertain Linear Conic Optimization Problems Having General Convex Uncertainty Sets," Discussion Paper 2012-076, Tilburg University, Center for Economic Research.
- Ben-Tal, A. & Hertog, D. den & Laurent, M., 2011. "Hidden Convexity in Partially Separable Optimization," Discussion Paper 2011-070, Tilburg University, Center for Economic Research.
- Ben-Tal, A. & Hertog, D. den & Vial, J.P., 2012. "Deriving Robust Counterparts of Nonlinear Uncertain Inequalities," Discussion Paper 2012-053, Tilburg University, Center for Economic Research.
- Yanikoglu, I. & Hertog, D. den & Kleijnen, Jack P.C., 2013. "Adjustable Robust Parameter Design with Unknown Distributions," Discussion Paper 2013-022, Tilburg University, Center for Economic Research.
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