Extending the Scope of Robust Quadratic Optimization

Ahmadreza Marandi, A. Ben-Tal, Dick den Hertog, Bertrand Melenberg

Research output: Working paperDiscussion paperOther research output

Abstract

In this paper, we derive tractable reformulations of the robust counterparts of convex quadratic and conic quadratic constraints with concave uncertainties for a broad range of uncertainty sets. For quadratic constraints with convex uncertainty, it is well-known that the robust counterpart is, in general, intractable. Hence, we derive tractable inner and outer approximations of the robust counterparts of such constraints. The approximations are made by replacing the quadratic terms in the uncertain parameters with suitable linear upper and lower bounds. Furthermore, when the uncertain parameters consist of a mean vector and covariance matrix, we construct a natural uncertainty set using an asymptotic confidence level and show that its support function is semi-definite representable. Finally, we apply our results to a portfolio choice, a norm approximation, and a regression line problem.
Original languageEnglish
PublisherOptimization Online
Publication statusSubmitted - Jun 2017

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Covariance matrix
Uncertainty

Keywords

  • Robust optimization
  • Quadratic programming
  • Safe approximation
  • Mean-variance uncertainty

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Marandi, Ahmadreza ; Ben-Tal, A. ; den Hertog, Dick ; Melenberg, Bertrand. / Extending the Scope of Robust Quadratic Optimization. Optimization Online, 2017.
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Extending the Scope of Robust Quadratic Optimization. / Marandi, Ahmadreza; Ben-Tal, A.; den Hertog, Dick; Melenberg, Bertrand.

Optimization Online, 2017.

Research output: Working paperDiscussion paperOther research output

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T1 - Extending the Scope of Robust Quadratic Optimization

AU - Marandi, Ahmadreza

AU - Ben-Tal, A.

AU - den Hertog, Dick

AU - Melenberg, Bertrand

PY - 2017/6

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N2 - In this paper, we derive tractable reformulations of the robust counterparts of convex quadratic and conic quadratic constraints with concave uncertainties for a broad range of uncertainty sets. For quadratic constraints with convex uncertainty, it is well-known that the robust counterpart is, in general, intractable. Hence, we derive tractable inner and outer approximations of the robust counterparts of such constraints. The approximations are made by replacing the quadratic terms in the uncertain parameters with suitable linear upper and lower bounds. Furthermore, when the uncertain parameters consist of a mean vector and covariance matrix, we construct a natural uncertainty set using an asymptotic confidence level and show that its support function is semi-definite representable. Finally, we apply our results to a portfolio choice, a norm approximation, and a regression line problem.

AB - In this paper, we derive tractable reformulations of the robust counterparts of convex quadratic and conic quadratic constraints with concave uncertainties for a broad range of uncertainty sets. For quadratic constraints with convex uncertainty, it is well-known that the robust counterpart is, in general, intractable. Hence, we derive tractable inner and outer approximations of the robust counterparts of such constraints. The approximations are made by replacing the quadratic terms in the uncertain parameters with suitable linear upper and lower bounds. Furthermore, when the uncertain parameters consist of a mean vector and covariance matrix, we construct a natural uncertainty set using an asymptotic confidence level and show that its support function is semi-definite representable. Finally, we apply our results to a portfolio choice, a norm approximation, and a regression line problem.

KW - Robust optimization

KW - Quadratic programming

KW - Safe approximation

KW - Mean-variance uncertainty

M3 - Discussion paper

BT - Extending the Scope of Robust Quadratic Optimization

PB - Optimization Online

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Marandi A, Ben-Tal A, den Hertog D, Melenberg B. Extending the Scope of Robust Quadratic Optimization. Optimization Online. 2017 Jun.