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A new method for solving Pareto eigenvalue complementarity problems. (English) Zbl 1296.90124
Summary: 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 $$\mathrm{SNM}_{\min}$$ and $$\mathrm{SNM}_{\mathrm{FB}}$$, two methods widely discussed in the literature for solving nonlinear complementarity problems, by using the performance profiles as a comparing tool [E. D. Dolan and J. J. Moré, Math. Program. 91, No. 2 (A), 201–213 (2002; Zbl 1049.90004)]. The performance measures, used to analyze the three solvers on a set of matrices mostly taken from the Matrix Market [R. F. Boisvert, R. Pozo, K. Remington, K. Barrett and J. J. Dongarra, “Matrix market: a web resource for test matrix collections. In: The Quality of Numerical Software: Assessment and Enhancement, London: Chapman and Hall, 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.

##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
##### Software:
MatrixMarket; PATH Solver
Full Text:
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