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Copositivity and Sparsity Relations Using Spectral Properties

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Abstract

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.

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Authors and Affiliations

  1. University of Tunis, ENSIT, 5 Avenue Taha Hussein, 1008, Tunis, Tunisia

    Abdeljelil Baccari

  2. University of Tunis El Manar, FST, Campus Universitaire, 2092, El Manar, Tunis, Tunisia

    Mourad Naffouti

Authors
  1. Abdeljelil Baccari
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  2. Mourad Naffouti
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Correspondence to Mourad Naffouti.

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Baccari, A., Naffouti, M. Copositivity and Sparsity Relations Using Spectral Properties. J Optim Theory Appl 171, 998–1007 (2016). https://doi.org/10.1007/s10957-016-0997-8

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  • DOI: https://doi.org/10.1007/s10957-016-0997-8

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