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

Gabriele Eichfelder (Gabriele.Eichfelder***at***am.uni-erlangen.de)
Janez Povh (janez.povh***at***fis.unm.si)

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 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 a results obtained by Sturm and Zhang ([J.F. Sturm and S. Zhang, On cones of nonnegative quadratic functions. Math. Oper. Res. 28 (2003)]), since we can handle problems with many linear and binary constraints. Our result also generalizes the well-known completely positive representation result from Burer ([S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120 (Ser. A) (2009), pp. 479–-495]), which is actually a special instance of our result with $K=\mathbb{R}^n$.

Keywords: set-positivity, semidefinite programming, copositive programming, mixed integer programming.

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Nonlinear Optimization (Quadratic Programming )

Category 3: Convex and Nonsmooth Optimization

Citation: Preprint No. 349, Preprint-Series of the Institute of Applied Mathematics, Univ. Erlangen-Nürnberg, Germany, 2011. See also published version in Optimization Letters DOI: 10.1007/s11590-012-0450-3

Download: [PDF]

Entry Submitted: 06/30/2011
Entry Accepted: 07/03/2011
Entry Last Modified: 02/20/2012

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