Optimization Letters

, Volume 7, Issue 6, pp 1373–1386

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

Original Paper


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 \({K=\mathbb{R}^n_{+}}\).


Set-positivity Semidefinite programming Copositive programming Mixed integer programming 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Faculty of information studies in Novo mestoNovo mestoSlovenia

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