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Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

Author

Listed:
  • Aharon Ben-Tal

    (Department of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa 32000, Israel; and CentER, Tilburg University, 5000 LE Tilburg, The Netherlands)

  • Dick den Hertog

    (Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, The Netherlands)

  • Anja De Waegenaere

    (Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, The Netherlands)

  • Bertrand Melenberg

    (Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, The Netherlands)

  • Gijs Rennen

    (Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, The Netherlands)

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. This paper was accepted by Gérard P. Cachon, optimization.

Suggested Citation

  • 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.
  • Handle: RePEc:inm:ormnsc:v:59:y:2013:i:2:p:341-357
    DOI: 10.1287/mnsc.1120.1641
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    References listed on IDEAS

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    1. Georgia Perakis & Guillaume Roels, 2008. "Regret in the Newsvendor Model with Partial Information," Operations Research, INFORMS, vol. 56(1), pages 188-203, February.
    2. Dimitris Bertsimas & David B. Brown, 2009. "Constructing Uncertainty Sets for Robust Linear Optimization," Operations Research, INFORMS, vol. 57(6), pages 1483-1495, December.
    3. John Y. Campbell, 2000. "Asset Pricing at the Millennium," Journal of Finance, American Finance Association, vol. 55(4), pages 1515-1567, August.
    4. Louis Eeckhoudt & Christian Gollier & Harris Schlesinger, 1995. "The Risk-Averse (and Prudent) Newsboy," Management Science, INFORMS, vol. 41(5), pages 786-794, May.
    5. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    6. Lorenzo Garlappi & Raman Uppal & Tan Wang, 2007. "Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach," The Review of Financial Studies, Society for Financial Studies, vol. 20(1), pages 41-81, January.
    7. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    8. D. Goldfarb & G. Iyengar, 2003. "Robust Portfolio Selection Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 1-38, February.
    9. Broniatowski, Michel & Keziou, Amor, 2009. "Parametric estimation and tests through divergences and the duality technique," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 16-36, January.
    10. Frank Fabozzi & Dashan Huang & Guofu Zhou, 2010. "Robust portfolios: contributions from operations research and finance," Annals of Operations Research, Springer, vol. 176(1), pages 191-220, April.
    11. Michael R. Wagner, 2010. "Fully Distribution-Free Profit Maximization: The Inventory Management Case," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 728-741, November.
    12. I Moon & E A Silver, 2000. "The multi-item newsvendor problem with a budget constraint and fixed ordering costs," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 51(5), pages 602-608, May.
    13. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    14. Erick Delage & Yinyu Ye, 2010. "Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems," Operations Research, INFORMS, vol. 58(3), pages 595-612, June.
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    More about this item

    Keywords

    robust optimization; [phi]-divergence; goodness-of-fit statistics;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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