On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solutions. (English) Zbl 1069.65055

Tong, X. J.; Qi, L.

J. Optimization Theory Appl. 123, No. 1, 187-211 (2004).

The authors discuss a so called trust-region method for the numerical solving of a system of constrained nonlinear equations. The new method can be regarded as an extension of the work of the authors [ibid. 120, No. 3, 601–625 (2004; Zbl 1140.65331)] and of the method of C. Kanzow [Complementarity: Applications, Algorithms and Extensions (Kluwer Academic Publishers, Dordrecht), 179–200 (2001; Zbl 0983.90060)]. The proposed method is globally convergent. Local superlinear and quadratic convergence of the algorithm under the condition of a local error bound, are proved. Eight numerical examples taken from the literature are performed by the proposed method, and the results are promising. In most of the examples, the iterative sequence converges to the solution quickly.

Reviewer: Iulian Coroian (Baia Mare)

MSC:

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

Keywords:

constrained nonlinear equations; trust-region methods; global convergence; superlinear/quadratic convergence; error bounds; numerical examples

Citations:

Zbl 0983.90060; Zbl 1140.65331

References:

[1]	Yamashita, N., and Fukushima, M., On the Rate of Convergence of the Levenberg-Marquardt Method, Computing (Supplement), Vol. 15, pp. 239–249, 2001. · Zbl 1001.65047
[2]	Fan, J., and Yuan, Y., On the Convergence of a New Levenberg-Marquardt Method, Report, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Science, Beijing, China, 2001.
[3]	El-Hawary, M. E., Optimal Power Flow: Solution Techniques, Requirements, and Challenges, IEEE Service Center, Piscataway, New Jersey, 1996.
[4]	Wood, A. J., and Wollenberg, B. F., Power Generation, Operation, and Control, John Wiley and Sons, New York, NY, 1996.
[5]	Chiang, H. D., and Fekih-Ahmed, L., Quasi-Stability Regions of Nonlinear Dynamical Systems:Optimal Estimations, IEEE Transactions on Circuits and Systems, Vol. 43, pp. 636–643, 1996.
[6]	Chiang, H. D., Hirsch, M. W., and Wu, F. F., Stability Regions of Nonlinear Autonomous Dynamical Systems, IEEE Transactions on Automatic Control, Vol. 33, pp. 16–27, 1988. · Zbl 0639.93043
[7]	Bellavia, S., Macconi, M., and Morini, B., An Af ne Scaling Trust-Region Approach to Bound-Constrained Nonlinear Systems, Technical Report, Dipartmento di Energetica, University of Florence, Florence, Italy, 2001. · Zbl 1018.65067
[8]	Gabriel, S. A., and Pang, J. S., A Trust-Region Method for Constrained Nonsmooth Equations, Large-Scale Optimization:State of the Art, Edited by W. W. Hager, D. W. Hearn, and P. M. Pardalos, Kluwer Academic Publishers, Boston, Massachusetts, pp. 159–186, 1994.
[9]	Kanzow, C., An Active-Set Type Newton Method for Constrained Nonlinear Equations, Complementarity: Applications, Algorithms, and Extensions, Edited by M. C. Ferris, O. L. Mangasarian, and J. S. Pang, Kluwer Academic Publishers, Dordrecht, Holland, pp. 179–200, 2001. · Zbl 0983.90060
[10]	Kanzow, C., Strictly Feasible Equation-Based Method for Mixed Complementarity Problems, Numerische Mathematik, Vol. 89, pp. 135–160, 2001. · Zbl 0992.65070
[11]	Kanzow, C., Yamashita, N., and Fukushima, M., Leveberg-Marquardt Methods for Constrained Nonlinear Equations with Strong Local Convergence Properties, Technical Report 2002–2007, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, Japan, 2002.
[12]	Kozakevich, D. N., Martinez, J. M., and Santos, S. A., Solving Nonlinear Systems of Equations with Simple Bounds, Journal of Computational and Applied Mathematics, Vol. 16, pp. 215–235, 1997. · Zbl 0896.65041
[13]	Monteiro, R. D. C., and Pang, J. S., A Potential Reduction Newton Method for Constrained Equations, SIAM Journal on Optimization, Vol. 9, pp. 729–754, 1999. · Zbl 0958.65069
[14]	Qi, L., Tong, X. J., and Li, D. H., An Active-Set Projected Trust-Region Algorithm for Box-Constrained Nonsmooth Equations, Journal of Optimization Theory and Applications, Vol. 120, 2004. · Zbl 1140.65331
[15]	Ulbrich, M., Nonmonotone Trust-Region Method for Bound-Constrained Semi-smooth Equations with Applications to Nonlinear Mixed Complementarity Problems, SIAM Journal on Optimization, Vol. 11, pp. 889–917, 2001. · Zbl 1010.90085
[16]	Dan, H., Yamashita, N., and Fukushima, M., Convergence Properties of the Inexact Levenberg-Marquardt Method under Local Error Bound Conditions, Optimization Methods and Software, Vol. 17, pp. 605–626, 2002. · Zbl 1030.65049
[17]	Tseng, P., Error Bounds and Superlinear Convergence Analysis of Some Newton-Type Methods in Optimization, Nonlinear Optimization and Related Topics, Edited by G. Di Pillo and F. Giannessi, Kluwer Academic Publishers, Boston, Massachusetts, pp. 445–462, 2000. · Zbl 0965.65091
[18]	Steihaug, T., The Conjugate Gradient Method and Trust Region in Large-Scale Optimization, SIAM Journal on Numerical Analysis, Vol. 20, pp. 626–637, 1983. · Zbl 0518.65042
[19]	Yuan, Y., On the Truncated Conjugate Gradient Method, Mathematical Programming, Vol. 87, pp. 561–573, 2000. · Zbl 0955.65039
[20]	H. Jiang, H., and Qi, L., ANew Nonsmooth Equations Approach to Nonlinear Complementarity Problems, SIAM Journal on Control and Optimization, Vol. 35, pp. 178–193, 1997. · Zbl 0872.90097
[21]	Hock, W., and Schittkowski, K., Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, Germany, Vol. 187, 1981. · Zbl 0452.90038