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A new trust region method for nonsmooth equations

Published online by Cambridge University Press:  17 February 2009

Y. F. Yang
Y. F. Yang
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha 410082, China; e-mail: yyangf@hotmail.com.
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Abstract

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We propose a new trust region algorithm for solving the system of nonsmooth equations F(x) = 0 by using a smooth function satisfying the Jacobian consistency property to approximate the nonsmooth function F(x). Compared with existing trust region methods for systems of nonsmooth equations, the proposed algorithm possesses some nice convergence properties. Global convergence is established and, in particular, locally superlinear or quadratical convergence is obtained if F is semismooth or strongly semismooth at the solution.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 4 , April 2003 , pp. 595 - 607
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Chen, X., Qi, L. and Sun, D., “Global and superlinear convergence of the smoothing Newton method and its applications to general box constrained variational inequalities”, Math. Comp. 222 (1998) 519–540.CrossRefGoogle Scholar
[2]Dennis, J. E., Li, S. B. and Tapia, R. A., “A unified approach to global convergence of trust region methods for nonsmooth optimization”, Math. Program. 68 (1995) 319–346.CrossRefGoogle Scholar
[3]Friedlander, A., Martínez, J. M. and Santos, S. A., “A new trust region algorithm for bound constrained minimization”, Appl. Math. Optim. 30 (1994) 235–266.CrossRefGoogle Scholar
[4]Gabriel, S. A. and Moré, J. J., “Smoothing of mixed complementarity problems”, in Complementarity and variational problems: State of the Art (eds. Ferris, M. C. and Pang, J. S.), (SIAM, Philadelphia, 1997) 105–116.Google Scholar
[5]Gabriel, A. and Pang, J. S., “A trust region method for constrained nonsmooth equations”, in Large scale optimization: State of the Art (eds. Hager, W. W., Hearn, D. W. and Pardalos, P. M.), (Kluwer Academic Publishers, Boston, MA, 1994) 159–186.Google Scholar
[6]Han, S. P., Pang, J. S. and Rangaraj, N., “Globally convergent Newton methods for nonsmooth equations”, Math. Oper. Res. 17 (1992) 586–607.CrossRefGoogle Scholar
[7]Jiang, H., Fukushima, M., Qi, L. and Sun, D., “A trust region method for solving generalized complementarity problems”, SIAM J. Optim. 8 (1998) 140–157.CrossRefGoogle Scholar
[8]Kanzow, C. and Pieper, H., “Jacobian smoothing methods for nonlinear complementarity problems”, SIAM J. Optim. 9 (1999) 342–373.CrossRefGoogle Scholar
[9]Kanzow, C. and Zupke, M., “Inexact trust region methods for nonlinear complementarity problems”, in Reformulation: Nonsmooth, piecewise smooth, semismooth and smoothing methods (eds. Fukushima, M. and Qi, L.), (Kluwer Academic Publishers, Norwell, MA, 1998) 211–233.CrossRefGoogle Scholar
[10]Martínez, J. M. and Moretti, A. C., “A trust region method for minimization of nonsmooth functions with linear constraints”, Math. Program. 76 (1997) 431–449.CrossRefGoogle Scholar
[11]Martínez, J. M. and Santos, S. A., “A trust region strategy for minimization on arbitrary domains”, Math. Program. 68 (1995) 267–302.CrossRefGoogle Scholar
[12]Moré, J. J., “Recent developments in algorithms and software for trust region methods”, in Mathematical programming: State of the Art (eds. Bachem, A., Grotschel, M. and Korte, B.), (Springer, Berlin, 1983) 258–287.CrossRefGoogle Scholar
[13]Pang, J. S. and Qi, L., “Nonsmooth equations: motivation and algorithms”, SIAM J. Optim. 3 (1993) 443–465.CrossRefGoogle Scholar
[14]Qi, L., “Convergence analysis of some algorithms for solving nonsmooth equations”, Math. Oper. Res. 18 (1993) 227–244.CrossRefGoogle Scholar
[15]Qi, L., “Trustregion algorithms for solving nonsmooth equation”, SIAM J. Optim. 5 (1995) 219–230.CrossRefGoogle Scholar
[16]Qi, L. and Chen, X., “A globally convergent successive approximation method for nonsmooth equations”, SIAM J. Control Optim. 38 (1995) 402–418.CrossRefGoogle Scholar
[17]Qi, L. and Sun, J., “A nonsmooth version of Newton's method”, Math. Program. 58 (1993) 353–367.CrossRefGoogle Scholar
[18]Qi, L. and Sun, J., “A trust region algorithm for minimization of locally Lipschitzian functions”, Math. Program. 66 (1994) 25–43.CrossRefGoogle Scholar
[19]Yuan, Y., “An example of only linearly convergence of trust region algorithms for nonsmooth optimization”, IMA J. Numer. Anal. 4 (1984) 327–335.CrossRefGoogle Scholar
[20]Yuan, Y., “Conditions for convergence of trust region algorithms for nonsmooth optimization”, Math. Program. 31 (1985) 220–228.CrossRefGoogle Scholar
[21]Yuan, Y., “On the superlinear convergence of a trust region algorithm for nonsmooth optimization”, Math. Program. 31 (1985) 269–285.CrossRefGoogle Scholar