×

CDT like approaches for the system of nonlinear equations. (English) Zbl 1088.65049

Summary: The Celis-Dennis-Tapia ( CDT) subproblem approach gives a new use and is employed to tackle systems of nonlinear equations [cf. M. R. Celis, J. E. Dennis, and R. A. Topia, Proc. SIAM Conf., Boulder/Colo. 1984, 71–82 (1985; Zbl 0566.65048)]. In the new method, the system of nonlinear equations is transferred into a constrained nonlinear programming problem, which is then solved by CDT subproblem algorithms. We handle a CDT subproblem to obtain a trial step. Some criterion, similar to that in the trust region technique, is then employed to determine whether to accept this trial point or not. In the new algorithm, an \(\varepsilon\) approximate solution or an locally infeasible point is obtained. In essence, constrained optimization methods are utilized to cope with the system of nonlinear equations.

MSC:

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

Citations:

Zbl 0566.65048
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Celis, M. R.; Dennis, J. E.; Tapia, R. A., A trust region strategy for nonlinear equality constrained optimization, (Boggs, P. T.; Schnable, R. B., Numerical Optimization 1984 (1985), SIAM: SIAM Philadelphia, USA), 71-82 · Zbl 0566.65048
[2] Dennis, J. E.; Schnable, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1983), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ
[3] Fan, J. Y., A modified Levenberg-Marquardt algorithm for singular system of nonlinear equations, Journal of Computational Mathematics, 21, 625-636 (2003) · Zbl 1032.65053
[4] Fletcher, R., Practical Methods of Optimization, vol. 2, Constrained Optimization, Vol. 2 (1981), John Wiley and Sons: John Wiley and Sons New York and Toronto · Zbl 0474.65043
[5] Nie, P. Y., A null space method for solving system of nonlinear equations, Applied Mathematics and Computation., 149, 215-226 (2004) · Zbl 1044.65041
[6] Nie, P. Y.; Fan, J. Y., A derivative-free filter method for solving nonlinear complementarity problems, Applied Mathematics and Computation, 161, 787-797 (2005) · Zbl 1122.90080
[7] Powell, M. J.D., A hybrid method for nonlinear equations, (Rabinowitz, P., Numerical Methods for Nonlinear Algebric Equations (1970), Gordon and Breach: Gordon and Breach London) · Zbl 0277.65028
[8] Powell, M. J.D.; Yuan, Y., A recursive quadratic programming algorithm for equality constrained optimization, Mathematical Programming, 35, 265-278 (1986) · Zbl 0598.90079
[9] Powell, M. J.D.; Yuan, Y., A trust region algorithm for equality constrained optimization, Mathematical Programming, 49, 189-211 (1990) · Zbl 0816.90121
[10] Yuan, Y., A dual algorithm for minimizing a quadratic funcyion with two quadratic constraints, Journal of Computational Mathematics, 9, 348-359 (1991) · Zbl 0758.65050
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.