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Extending the Scope of Robust Quadratic Optimization

Author

Listed:
  • Ahmadreza Marandi

    (Department of Industrial Engineering and Innovation Sciences, Eindhoven University of Technology, Eindhoven 5600 MB, Netherlands)

  • Aharon Ben-Tal

    (CentER, Tilburg University, Tilburg 5037 AB, Netherlands)

  • Dick den Hertog

    (Amsterdam Business School, University of Amsterdam, Amsterdam 1012 WX, Netherlands)

  • Bertrand Melenberg

    (Tilburg School of Economics and Management, Tilburg University, Tilburg 5037 AB, Netherlands)

Abstract

We derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We do this for a broad range of uncertainty sets. Our results provide extensions to known results from the literature. We also consider hard quadratic constraints: those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints, we derive inner and outer tractable approximations. As an application, we show how to construct a natural uncertainty set based on a statistical confidence set around a sample mean vector and covariance matrix and use this to provide a tractable reformulation of the robust counterpart of an uncertain portfolio optimization problem. We also apply the results of this paper to norm approximation problems. Summary of Contribution: This paper develops new theoretical results and algorithms that extend the scope of a robust quadratic optimization problem. More specifically, we derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We also consider hard quadratic constraints: those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints, we derive inner and outer tractable approximations.

Suggested Citation

  • Ahmadreza Marandi & Aharon Ben-Tal & Dick den Hertog & Bertrand Melenberg, 2022. "Extending the Scope of Robust Quadratic Optimization," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 211-226, January.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:1:p:211-226
    DOI: 10.1287/ijoc.2021.1059
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    References listed on IDEAS

    as
    1. Trapani, Lorenzo, 2016. "Testing for (in)finite moments," Journal of Econometrics, Elsevier, vol. 191(1), pages 57-68.
    2. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    3. 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.
    4. 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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