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A new method for solving Pareto eigenvalue complementarity problems

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  • Samir Adly
  • Hadia Rammal

Abstract

In this paper, we introduce a new method, called the Lattice Projection Method (LPM), for solving eigenvalue complementarity problems. The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues and eigenvectors of the complementarity problems. The LPM is compared to SNM min and SNM FB , two methods widely discussed in the literature for solving nonlinear complementarity problems, by using the performance profiles as a comparing tool (Dolan, Moré in Math. Program. 91:201–213, 2002 ). The performance measures, used to analyze the three solvers on a set of matrices mostly taken from the Matrix Market (Boisvert et al. in The quality of numerical software: assessment and enhancement, pp. 125–137, 1997 ), are computing time, number of iterations, number of failures and maximum number of solutions found by each solver. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving eigenvalue complementarity problems. Finally, Pareto bi-eigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method. Copyright Springer Science+Business Media New York 2013

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  • Samir Adly & Hadia Rammal, 2013. "A new method for solving Pareto eigenvalue complementarity problems," Computational Optimization and Applications, Springer, vol. 55(3), pages 703-731, July.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:3:p:703-731
    DOI: 10.1007/s10589-013-9534-y
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    References listed on IDEAS

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    1. Mohamed Hanafi & Jos Berge, 2003. "Global optimality of the successive Maxbet algorithm," Psychometrika, Springer;The Psychometric Society, vol. 68(1), pages 97-103, March.
    2. C. Kanzow & N. Yamashita & M. Fukushima, 1997. "New NCP-Functions and Their Properties," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 115-135, July.
    3. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
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    Cited by:

    1. Chen Ling & Hongjin He & Liqun Qi, 2016. "Higher-degree eigenvalue complementarity problems for tensors," Computational Optimization and Applications, Springer, vol. 64(1), pages 149-176, May.
    2. 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.
    3. Pinto da Costa, A. & Seeger, A. & Simões, F.M.F., 2017. "Complementarity eigenvalue problems for nonlinear matrix pencils," Applied Mathematics and Computation, Elsevier, vol. 312(C), pages 134-148.
    4. Chen Ling & Hongjin He & Liqun Qi, 2016. "On the cone eigenvalue complementarity problem for higher-order tensors," Computational Optimization and Applications, Springer, vol. 63(1), pages 143-168, January.
    5. Fatemeh Abdi & Fatemeh Shakeri, 2017. "A New Descent Method for Symmetric Non-monotone Variational Inequalities with Application to Eigenvalue Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 923-940, June.

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