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A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem

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  • Brás, Carmo P.
  • Fischer, Andreas
  • Júdice, Joaquim J.
  • Schönefeld, Klaus
  • Seifert, Sarah

Abstract

In this paper, we address the solution of the symmetric eigenvalue complementarity problem (EiCP) by treating an equivalent reformulation of finding a stationary point of a fractional quadratic program on the unit simplex. The spectral projected-gradient (SPG) method has been recommended to this optimization problem when the dimension of the symmetric EiCP is large and the accuracy of the solution is not a very important issue. We suggest a new algorithm which combines elements from the SPG method and the block active set method, where the latter was originally designed for box constrained quadratic programs. In the new algorithm the projection onto the unit simplex in the SPG method is replaced by the much cheaper projection onto a box. This can be of particular advantage for large and sparse symmetric EiCPs. Global convergence to a solution of the symmetric EiCP is established. Computational experience with medium and large symmetric EiCPs is reported to illustrate the efficacy and efficiency of the new algorithm.

Suggested Citation

  • Brás, Carmo P. & Fischer, Andreas & Júdice, Joaquim J. & Schönefeld, Klaus & Seifert, Sarah, 2017. "A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 36-48.
    • Handle: RePEc:eee:apmaco:v:294:y:2017:i:c:p:36-48
    DOI: 10.1016/j.amc.2016.09.005
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    References listed on IDEAS

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    1. Andrei Patrascu & Ion Necoara, 2015. "Efficient random coordinate descent algorithms for large-scale structured nonconvex optimization," Journal of Global Optimization, Springer, vol. 61(1), pages 19-46, January.
    2. Luís Fernandes & Joaquim Júdice & Hanif Sherali & Masao Fukushima, 2014. "On the computation of all eigenvalues for the eigenvalue complementarity problem," Journal of Global Optimization, Springer, vol. 59(2), pages 307-326, July.
    3. Julia Sponsel & Stefan Bundfuss & Mirjam Dür, 2012. "An improved algorithm to test copositivity," Journal of Global Optimization, Springer, vol. 52(3), pages 537-551, March.
    4. L. Fernandes & A. Fischer & J. Júdice & C. Requejo & J. Soares, 1998. "A block active set algorithm for large-scalequadratic programming with box constraints," Annals of Operations Research, Springer, vol. 81(0), pages 75-96, June.
    5. Samir Adly & Hadia Rammal, 2015. "A New Method for Solving Second-Order Cone Eigenvalue Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 563-585, May.
    6. A. Pinto da Costa & A. Seeger, 2010. "Cone-constrained eigenvalue problems: theory and algorithms," Computational Optimization and Applications, Springer, vol. 45(1), pages 25-57, January.
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    Cited by:

    1. Joaquim Júdice & Valentina Sessa & Masao Fukushima, 2022. "Solution of Fractional Quadratic Programs on the Simplex and Application to the Eigenvalue Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 545-573, June.
    2. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2022. "Minimization over the $$\ell _1$$ ℓ 1 -ball using an active-set non-monotone projected gradient," Computational Optimization and Applications, Springer, vol. 83(2), pages 693-721, November.
    3. Masao Fukushima & Joaquim Júdice & Welington Oliveira & Valentina Sessa, 2020. "A sequential partial linearization algorithm for the symmetric eigenvalue complementarity problem," Computational Optimization and Applications, Springer, vol. 77(3), pages 711-728, December.
    4. Niu, Yi-Shuai & Júdice, Joaquim & Le Thi, Hoai An & Pham, Dinh Tao, 2019. "Improved dc programming approaches for solving the quadratic eigenvalue complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 95-113.
    5. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2020. "An active-set algorithmic framework for non-convex optimization problems over the simplex," Computational Optimization and Applications, Springer, vol. 77(1), pages 57-89, September.

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