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Hidden conic quadratic representation of some nonconvex quadratic optimization problems

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Abstract

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.

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Authors and Affiliations

  1. Department of Industrial Engineering and Management, Technion-Israel Institute of Technology, 32000 , Haifa, Israel

    Aharon Ben-Tal

  2. CentER, Extramural Fellow, Tilburg University, Tilburg, The Netherlands

    Aharon Ben-Tal

  3. Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE , Tilburg, The Netherlands

    Dick den Hertog

Authors
  1. Aharon Ben-Tal
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  2. Dick den Hertog
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Correspondence to Aharon Ben-Tal.

Additional information

This paper was written when the first author was visiting Centrum Wiskunde & Informatica in Amsterdam, The Netherlands, as a CWI Distinguished Scientist. He is also supported by Israel-USA Science Foundation (DSF) Grant number 2008302.

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Ben-Tal, A., den Hertog, D. Hidden conic quadratic representation of some nonconvex quadratic optimization problems. Math. Program. 143, 1–29 (2014). https://doi.org/10.1007/s10107-013-0710-8

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  • DOI: https://doi.org/10.1007/s10107-013-0710-8

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Mathematics Subject Classification (2000)