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Constructing Uncertainty Sets for Robust Linear Optimization

Author

Listed:
  • Dimitris Bertsimas

    (Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • David B. Brown

    (The Fuqua School of Business, Duke University, Durham, North Carolina 27708)

Abstract
In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. (1999) [Artzner, P., F. Delbaen, J. Eber, D. Heath. 1999. Coherent measures of risk. Math. Finance 9 203--228.], and show that such risk measures, in conjunction with the support of the uncertain parameters, are equivalent to explicit uncertainty sets for robust optimization. We explore the structure of these sets in detail. In particular, we study a class of coherent risk measures, called distortion risk measures , which give rise to polyhedral uncertainty sets of a special structure that is tractable in the context of robust optimization. In the case of discrete distributions with rational probabilities, which is useful in practical settings when we are sampling from data, we show that the class of all distortion risk measures (and their corresponding polyhedral sets) are generated by a finite number of conditional value-at-risk (CVaR) measures. A subclass of the distortion risk measures corresponds to polyhedral uncertainty sets symmetric through the sample mean. We show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary, polyhedral uncertainty sets.

Suggested Citation

  • Dimitris Bertsimas & David B. Brown, 2009. "Constructing Uncertainty Sets for Robust Linear Optimization," Operations Research, INFORMS, vol. 57(6), pages 1483-1495, December.
    • Handle: RePEc:inm:oropre:v:57:y:2009:i:6:p:1483-1495
    DOI: 10.1287/opre.1080.0646
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    References listed on IDEAS

    as
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