Abstract
In this article we consider the complementarity eigenvalue problem, which is of great interest to many researchers due to its numerous applications in Engineering and Physics. We approach its solution as a nonlinear complementarity problem using a quasi-Newton method, a type of method that, as far as we know, has not been used for this purpose. We verify that the problem satisfies certain hypotheses that allow the use of a global quasi-Newton algorithm and we analyze its numerical performance. Numerical tests show the efficiency of the algorithm used and make it a good alternative to solve complementary eigenvalue problems.
References
Arenas, F., Martínez, H. J., & Perez, R. (2014). Redefinicion de la función de ´complementariedad de kanzow. , 18(2), 111-122. doi:10.25100/rc.v18i2.6105
Arenas, F., Martínez, H. J., & Perez, R. (2015). Least change secant update methods for nonlinear complementarity problem. , 11(21), 11-36. doi: 10.17230/ingciencia.11.21.1
Arenas, F. E., Martínez, H. J., & Perez, R. (2020, May). A local jacobian smoothing method for solving nonlinear complementarity problems. Universitas Scientiarum, 25(1), 149–174. doi: 10.11144/Javeriana.SC25-1.aljs
Arias, C., Martínez, H., & Perez, R. ´ (2017). A nonsmooth global cuasi-newton method for nonlinear complementarity problems. Computer Methods in Applied Mechanics and Engineering, 13(1), 1 - 15.
Brown, P. N., & Saad, Y. (1994). Convergence theory of nonlinear newton–krylov algorithms. SIAM Journal on Optimization, 4(2), 297-330.
Broyden, C. G. (1965). A Class of Methods for Solving Nonlinear Simultaneous Equations. Mathematics of Computation, 19(92), 577–593.
Clarke, F. H. (1975). Generalized gradients and applications. Transactions of the american society, 205(), 247-262.
Dennis, J. E., & Schnabel, R. B. (1996). Numerical methods for unconstrained optimization and nonlinear equations. Society for Industrial and Applied Mathematics.
Facchinei, F., & Pang, J. (2003). Finite-dimensional variational inequalities and complementarity problems (Vol. II; Springer, Ed.). Springer-Verlag. doi:10.1007/b97544
Ferris, M. C., & Pang, J. S. (1997). Engineering and economic applications of complementarity problems. SIAM Review, 39(), 669-713.
Iusem, A. N., Júdice, J. J., Sessa, V., & Sarabando, P. (2019). Splitting methods for the eigenvalue complementarity problem. Optimization Methods and Software, 34(6), 1184-1212. doi: 10.1080/10556788.2018.1479408
Júdice, J. J., Sherali, H. D., & Ribeiro, I. (2007). The eigenvalue complementarity problem. Computational Optimization and Applications, 37(2), 139-156. doi: 10.1007/s10589-007-9017-0
Kanzow, C., & Kleinmichel, H. (1998). A new class of semismooth newtontype methods for nonlinear complementarity problems. , 11(3), 227–251. doi:10.1023/A:1026424918464
Mart´ınez, J. M. (2000). Practical quasi-newton methods for solving nonlinear systems. Journal of Computational and Applied Mathematics, 124(1), 97-121. (Numerical Analysis 2000. Vol. IV: Optimization and Nonlinear Equations) doi:10.1016/S03770427(00)00434-9
Martínez, J. M., & Zambaldi, M. C. (1992). An inverse column-updating method for solving large–scale nonlinear systems of equations. Optimization Methods and Software, 1(2), 129-140. doi: 10.1080/10556789208805512
Perez, R., Arenas, F., Martínez, H., & Arias, C. (2019). El problema de complementariedad no lineal (Vol. 1). Programa editorial Universidad del Valle.
Pinto da Costa, A., Martins, J., Figueiredo, I. N., & Júdice, J. (2004). The directional instability problem in systems with frictional contacts. Computer Methods in Applied Mechanics and Engineering, 193(3), 357 - 384. doi: 10.1016/j.cma.2003.09.013
Queiroz, M., Júdice, J., & Humes, C. J. (2004). The symmetric eigenvalue complementarity problem. Mathematics of Computation, 73(248), 1849–1863.
Schubert, L. K. (1970). Modification of a quasi-newton method for nonlinear equations with a sparse jacobian. Mathematics of Computation, 24(), 27-30.
Seeger, A. (1999). Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions. Linear Algebra and its Applications, 292(1), 1-14. doi:10.1016/S0024-3795(99)00004-X
Sherali, H. D., & Tuncbilek, C. H. (1992, 03). A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. Journal of Global Optimization, 2(), 101-112. doi: 10.1007/BF00121304
Sánchez, W., Pérez, R., & Martínez, H. (2021, oct.). Un algoritmo global con jacobiano suavizado para problemas de complementariedad no lineal. Revista integracion, temas de matemáticas, 39(2), 191–215. doi: 10.18273/revint.v39n2-20210004
Yong, L. (2010). Nonlinear complementarity problem and solution methods. In Proceedings of the 2010 international conference on artificial intelligence and computational intelligence: Part i (p. 461-469). Springer-Verlag.

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