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A new method for solving Pareto eigenvalue complementarity problems

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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 SNMmin and SNMFB, 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.

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Notes

  1. As stopping criteria, we use ∥Φ(z k)∥<10−8.

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Acknowledgements

The first author wishes to thank Professor Jonathan Borwein for some stimulating discussions about the new method LPM and particularly Lemma 3.

The authors wish to thank also Professor Alberto Seeger for his careful reading of the first version of the manuscript.

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Authors and Affiliations

  1. XLIM UMR-CNRS 7252, Université de Limoges, 87060, Limoges, France

    Samir Adly & Hadia Rammal

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  1. Samir Adly
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  2. Hadia Rammal
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Correspondence to Samir Adly.

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Dedicated to Jonathan Borwein in honor of his 60th birthday.

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Adly, S., Rammal, H. A new method for solving Pareto eigenvalue complementarity problems. Comput Optim Appl 55, 703–731 (2013). https://doi.org/10.1007/s10589-013-9534-y

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