Journal of Optimization Theory and Applications

, Volume 171, Issue 3, pp 998–1007 | Cite as

Copositivity and Sparsity Relations Using Spectral Properties



A standard quadratic optimization problem consists in minimizing a quadratic form over the standard simplex. This problem has a closer connection with copositive optimization, where the copositivity appears in local optimality conditions. In this paper, we establish a relationship between the sparsity of a solution of the standard quadratic optimization problem, from one hand, and the copositivity of the Hessian matrix of the Lagrangian from other hand. More precisely, if the Hessian matrix is copositive we prove that the number of zero components, of a local minimizer associated with a standard quadratic optimization problem, is greater or equal to the number of negative eigenvalues counting multiplicities. Moreover, we show that if the number of zero components of a local minimizer is equal to the number of negative eigenvalues of the Hessian matrix, then the strict complementarity condition is satisfied and the critical cone is a linear subspace.


Standard quadratic optimization problem Copositive matrices Local minimum Eigenvalues 

Mathematics Subject Classification

90C30 90C20 49K10 15A18 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.University of Tunis, ENSITTunisTunisia
  2. 2.University of Tunis El Manar, FSTEl Manar, TunisTunisia

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