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On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets

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An Erratum to this article was published on 03 July 2013

Abstract

In the paper we prove that any nonconvex quadratic problem over some set \({K\subset \mathbb {R}^n}\) with additional linear and binary constraints can be rewritten as a linear problem over the cone, dual to the cone of K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actually a semidefinite programming problem. This generalizes results obtained by Sturm and Zhang (Math Oper Res 28:246–267, 2003). Our result also generalizes the well-known completely positive representation result from Burer (Math Program 120:479–495, 2009), which is actually a special instance of our result with \({K=\mathbb{R}^n_{+}}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Erlangen-Nuremberg, Martensstr. 3, 91058, Erlangen, Germany

    Gabriele Eichfelder

  2. Faculty of information studies in Novo mesto, Novi trg 5, 8000, Novo mesto, Slovenia

    Janez Povh

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  1. Gabriele Eichfelder
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  2. Janez Povh
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Correspondence to Gabriele Eichfelder.

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Eichfelder, G., Povh, J. On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets. Optim Lett 7, 1373–1386 (2013). https://doi.org/10.1007/s11590-012-0450-3

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  • DOI: https://doi.org/10.1007/s11590-012-0450-3

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