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Convergence analysis of nonmonotone Levenberg-Marquardt algorithms for complementarity problem. (English) Zbl 1202.65073
The convergence of two nonmonotone Levenberg-Marquardt algorithms for nonlinear complementarity problem is studied and proved. Under some mild assumptions, and requiring only the solution of a linear system at each iteration, the proposed nonmonotone Levenberg-Marquardt algorithms are shown to be globally convergent.

MSC:
65K05 Numerical mathematical programming methods
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
Software:
levmar
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References:
[1] Fischer, A., Solution of monotone complementarity problems with locally Lipschitzian functions, Math. program., 76, 513-532, (1997) · Zbl 0871.90097
[2] Fischer, A.; Jeyakumar, V.; Luc, D.T., Solution point characterizations and convergence analysis of a descent algorithm for nonsmooth continuous complementarity problems, J. optim. theory appl., 110, 493-513, (2001) · Zbl 1064.90048
[3] Facchinei, F.; Kanzow, C., A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems, Math. program., 76, 493-512, (1997) · Zbl 0871.90096
[4] Fasano, G.; Lampariello, F.; Sciandrone, M., A truncated nonmonotone gauss – newton method for large-scale nonlinear least-squares problems, Comput. optim. appl., 34, 343-358, (2006) · Zbl 1122.90094
[5] Gao, D.Y., Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications, Nonlinear anal., 42, 1161-1193, (2000) · Zbl 0983.49024
[6] Gao, Y., Newton methods for solving two classes of nonsmooth equations, Appl. math., 46, 215-229, (2001) · Zbl 1068.65063
[7] Geiger, C.; Kanzow, C., On the resolution of monotone complementarity problems, Comput. optim. appl., 5, 155-173, (1996) · Zbl 0859.90113
[8] Huang, Z.; Ma, G., On the computation of an element of Clarke generalized Jacobian for a vector-valued MAX function, Nonlinear anal. TMA, (2009)
[9] Jiang, H., Unconstrained minimization approaches to nonlinear complementarity problems, J. global optim., 9, 169-181, (1996) · Zbl 0868.90122
[10] Kanzow, C., Nonlinear complementarity as unconstrained optimization, J. optim. theory appl., 88, 139-155, (1996) · Zbl 0845.90120
[11] Kanzow, C.; Yamashita, N.; Fukushima, M., Levenberg – marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, J. comput. appl. math., 173, 321-343, (2005) · Zbl 1065.65070
[12] Ma, C.F.; Jiang, L.; Wang, D., The convergence of a smoothing damped gauss – newton method for nonlinear complementarity problem, Nonlinear anal. RWA, 10, 2072-2087, (2009) · Zbl 1163.90752
[13] Ma, C.F.; Tang, J.; Chen, X.H., A globally convergent levenberg – marquardt method for solving nonlinear complementarity problem, Appl. math. comput., 192, 370-381, (2007) · Zbl 1193.90205
[14] Ma, C.F.; Tang, J., The quadratic convergence of a smoothing levenberg – marquardt method for nonlinear complementarity problem, Appl. math. comput., 197, 566-581, (2008) · Zbl 1141.65044
[15] Ma, C.F.; Jiang, L.H., Some research on levenberg – marquardt method for the nonlinear equations, Appl. math. comput., 184, 1032-1040, (2007) · Zbl 1114.65054
[16] Qi, L.; Tseng, P., On almost smooth functions and piecewise smooth functions, Nonlinear anal. TMA, 67, 773-794, (2007) · Zbl 1125.26019
[17] Sun, D.; Qi, L., On NCP-functions, Comput. optim. appl., 13, 201-220, (1999) · Zbl 1040.90544
[18] Tseng, P., Growth behavior of a class of merit functions for the nonlinear complementarity problem, J. optim. theory appl., 1, 17-37, (1996) · Zbl 0866.90127
[19] Yamashita, N.; Fukushima, M., Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Math. program., 76, 469-491, (1997) · Zbl 0872.90102
[20] Zhang, J.; Chen, L., Nonmonotone levenberg – marquardt algorithms and their convergence analysis, J. optim. theory appl., 2, 393-418, (1997) · Zbl 0886.90129
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