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Deriving Robust Counterparts of Nonlinear Uncertain Inequalities

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  • Ben-Tal, A.
  • Hertog, D. den
  • Vial, J.P.

    (Tilburg University, Center for Economic Research)

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    Abstract

    Abstract: In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.

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    Bibliographic Info

    Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2012-053.

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    Date of creation: 2012
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    Handle: RePEc:dgr:kubcen:2012053

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    Web page: http://center.uvt.nl

    Related research

    Keywords: Fenchel duality; robust counterpart; nonlinear inequality; robust optimization; support functions;

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    References

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    1. 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.
    2. Ben-Tal, A. & Hertog, D. den, 2011. "Immunizing Conic Quadratic Optimization Problems Against Implementation Errors," Discussion Paper 2011-060, Tilburg University, Center for Economic Research.
    3. 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.
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    Cited by:
    1. Ruiter, F.J.C.T. de & Ben-Tal, A. & Brekelmans, R.C.M. & Hertog, D. den, 2014. "Adjustable Robust Optimizations with Decision Rules Based on Inexact Revealed Data," Discussion Paper 2014-003, Tilburg University, Center for Economic Research.
    2. 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.
    3. Soleimanian, Azam & Salmani Jajaei, Ghasemali, 2013. "Robust nonlinear optimization with conic representable uncertainty set," European Journal of Operational Research, Elsevier, vol. 228(2), pages 337-344.
    4. 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.
    5. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    6. Gorissen, Bram L. & den Hertog, Dick, 2013. "Robust counterparts of inequalities containing sums of maxima of linear functions," European Journal of Operational Research, Elsevier, vol. 227(1), pages 30-43.

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