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The quadratic convergence of a smoothing Levenberg-Marquardt method for nonlinear complementarity problem. (English) Zbl 1141.65044
The authors approximate the problem of the least \({l}_2\)-norm solution of the equivalent nonsmooth equations of a nonlinear complementarity problem with a family of parameterized twice smooth optimization problem by making use of a new smoothing function. A smoothing Levenberg-Marquardt method is presented to solve the parameterized smooth optimization problem. The proposed method is shown to be globally convergent under an assumption that the level set of the problem is compact. By making use of the smooth and semismooth technique, the local quadratic convergence of the proposed method under some assumptions is established.

MSC:
65K05 Numerical mathematical programming methods
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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[1] Pang, J.S., Complementarity problems, (), 271-338 · Zbl 0833.90114
[2] ()
[3] Harker, P.T.; Pang, J.S., Finite-dimensional variational inequality and nonlinear complementarity problems, a survey review of theory, algorithms and applications, Math. prog., 48, 161-220, (1990) · Zbl 0734.90098
[4] Dannis, J.E.; Martinez, H.J.; Tapia, R.A., Convergence theory for the structured BFGS secant method with an application to nonlinear least squares, J. optim. theory appl., 61, 161-178, (1989) · Zbl 0645.65026
[5] Yabe, H.; Takahashi, T., Factorized quasi-Newton method for nonlinear least squares problems, Math. program., 51, 75-100, (1991) · Zbl 0737.90064
[6] Zhang, J.; Chen, L., Nonmonotone levenberg – marquardt algorithms and their convergence analysis, J. optim. theory appl., 92, 393-418, (1997) · Zbl 0886.90129
[7] Zhang, J.; Chen, L.; Deng, N., A family of scaled factorized Broyden-like methods for nonlinear least squares problems, SIAM J. optim., 10, 1163-1179, (2000) · Zbl 0994.65072
[8] Chen, B.; Harker, P.T., A noninterior continuation method for quadratic and linear programming, SIAM J. optim., 3, 503-515, (1993) · Zbl 0795.90040
[9] Burke, J.; Xu, S., The global linear convergence of a non-interior path-following algorithm for linear complementarity problems, Math. oper. res., 23, 719-734, (1998) · Zbl 0977.90056
[10] Kanzow, C., Some noninterior continuation methods for linear complementarity problems, SIAM J. matrix anal. appl., 17, 851-868, (1996) · Zbl 0868.90123
[11] Chen, C.; Mangasarian, O.L., A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. optim. appl., 5, 97-138, (1996) · Zbl 0859.90112
[12] Chen, X.J.; Qi, L.; Sun, D., Global and superlinear convergence of the smoothing Newton method and its applications general box constrained variational inequalities, Math. comput., 67, 519-540, (1998) · Zbl 0894.90143
[13] Qi, L.; Sun, D., Improving the convergence of non-interior point algorithm for nonlinear complementarity problems, Math. comput., 69, 301-325, (2000)
[14] Chen, X.J.; Ye, Y., On homotopy-smoothing methods for variational inequalities, SIAM J. control optim., 37, 589-616, (1999) · Zbl 0973.65051
[15] Qi, L.; Sun, D.; Zhou, G.L., A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems, Math. program., 87, 1-35, (2000) · Zbl 0989.90124
[16] Sun, D.; Qi, L., Solving variational inequality problems via smoothing-nonsmooth reformulations, J. comput. appl. math., 129, 37-62, (2001) · Zbl 0987.65059
[17] Chen, X.; Tseng, P., Non-interior continuation methods for solving semidefinite complementarity problems, Math. program., 95, 431-474, (2003) · Zbl 1023.90046
[18] Huang, Z.; Han, J., Non-interior continuation methods for solving the monotone semidefinite complementarity problems, Appl. math. optim., 47, 195-211, (2003) · Zbl 1030.65069
[19] Tseng, P., Merit functions semi-definite complementarity problems, Math. program., 83, 159-185, (1998) · Zbl 0920.90135
[20] Clarke, F.H., Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0727.90045
[21] Changfeng Ma, Jia Tang, Xiaohong Chen, A globally convergent Levenberg-Marquardt method for solving nonlinear complementarity problem, Appl. Math. Comput., in press, doi:10.1016/j.amc.2007.03.014. · Zbl 1193.90205
[22] Fischer, A., Solution of monotone complementarity problems with locally Lipschitzian functions, Math. program., 76, 513-532, (1997) · Zbl 0871.90097
[23] Bhatia, R., Matrix analysis, (1997), Springer-Verlag New York
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