A new filled function method for an unconstrained nonlinear equation. (English) Zbl 1213.65077

The authors transform an unconstrained system of nonlinear equations into a special optimization problem. For this purpose, a new filled function is constructed by using the properties of the transformed optimization problem. The authors investigate theoretical and numerical properties of the proposed filled function, and is proposed an algorithm for a solution. In addition, under some conditions, a solution or an approximate solution to the system of nonlinear equations in finite iterations is found.


65H10 Numerical computation of solutions to systems of equations
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
Full Text: DOI


[1] Soleimani-damaneh, M.; Nieto, J., Nonsmooth multiple-objective optimization in separable Hilbert spaces, Nonlinear Anal. TMA, 71, 10, 4553-4558 (2009) · Zbl 1167.49308
[2] Martinez, J. M.; Qi, L. Q., Inexact Newton methods for solving nonsmooth equations, J. Comput. Appl. Math., 60, 1-2, 127-145 (1999) · Zbl 0833.65045
[3] Soleimani-damaneh, M., Infinite (semi-infinite) problems to characterize the optimality of nonlinear optimization problems, Eur. J. Ope. Res., 188, 1, 49-56 (2008) · Zbl 1142.90032
[4] Soleimani-damaneh, M., Multiple-objective programs in Banach spaces: sufficiency for (proper) optimality, Nonlinear Anal. TMA, 67, 3, 958-962 (2007) · Zbl 1121.90113
[5] Kanzow, C., Global optimization techniques for mixed complementarity problems, J. Global Optim., 16, 1-21 (2000) · Zbl 1009.90119
[6] Nazareth, J. L.; Qi, L., Globalization of Newton’s methods for solving nonlinear equations, Numer. Linear Algebra Appl., 3, 239-249 (1996) · Zbl 0858.65047
[7] Qi, L.; Yang, Y. F., NCP functions applied to Lagrangian globalization for the nonlinear complementarity problem, J. Global Optim., 24, 261-283 (2002) · Zbl 1047.90072
[8] Tong, X. J.; Qi, L.; Yang, Y. F., The Lagrangian globalization method for nonsmooth constrained equations, Comput. Optim. Appl., 33, 89-109 (2006) · Zbl 1103.90077
[9] Levy, A. V.; Montalvo, A., The tunneling algorithm for the global minimization of functions, SIAM J. Sci. Stat. Comput., 6, 15-29 (1985) · Zbl 0601.65050
[10] Ge, R. P., A filled function for finding global minimizer of a function of several variables, Math. Program., 46, 191-204 (1990) · Zbl 0694.90083
[11] Wu, Z. Y.; Mammodov, M.; Bai, F. S.; Yang, Y. J., A filled function method for nonlinear equations, Appl. Math. Comput., 189, 1196-1204 (2007) · Zbl 1122.65355
[12] Wang, C. J.; Yang, Y. J.; Li, J., A new filled function method for unconstrained global optimization, J. Comput. Appl. Math., 225, 68-79 (2009) · Zbl 1162.65033
[13] Yang, Y. J.; Shang, Y. L., A new filled function method for constrained global optimization, Appl. Math. Comput., 173, 501-512 (2006) · Zbl 1094.65063
[14] Zhang, L. S.; NG, C. K.; Li, D.; Tian, W. W., A new filled function method for global optimization, J. Global Optim., 28, 17-43 (2004) · Zbl 1061.90109
[15] Floudas, C. A.; Pardalos, P. M.; Adjiman, C. S.; Esposito, W. R.; Gumus, Z. H.; Harding, S. T.; Klepeis, J. L.; Meyer, C. A.; Schweiger, C. A., Handbook of Test Problems in Local and Global Optimization (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, the Netherlands · Zbl 0943.90001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.