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A nonmonotonic trust region method for constrained optimization problems

Published online by Cambridge University Press:  17 February 2009

Jianzhong Zhang  and
Detong Zhu
Jianzhong Zhang
Affiliation:
Department of Mathematics, City University of Hong Kong, Hong Kong.
Detong Zhu
Affiliation:
Department of Mathematics, City University of Hong Kong, Hong Kong.
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Abstract

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In this paper we propose an easy-to-implement algorithm for solving general nonlinear optimization problems with nonlinear equality constraints. A nonmonotonic trust region strategy is suggested which does not require the merit function to reduce its value in every iteration. In order to deal with large problems, a reduced Hessian is used to replace a full Hessian matrix. To avoid solving quadratic trust region subproblems exactly which usually takes substantial computation, we only require an approximate solution which requires less computation. The calculation of correction steps, necessary from a theoretical view point to overcome the Maratos effect but which often brings in negative results in practice, is avoided in most cases by setting a criterion to judge its necessity. Global convergence and a local superlinear rate are then proved. This algorithm has a good performance.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 4 , April 1999 , pp. 542 - 567
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Boggs, P. T. and Tolle, J. W., “A family of descent functions for constrained optimization”, SIAM J. Numer.Anal. 21 (1984) 1146–1161.CrossRefGoogle Scholar
[2]Byrd, R. H. and Nocedal, J., “An analysis of reduced Hessian methods for constrained optimization”, Mathematical Programming 49 (1991) 285–323.Google Scholar
[3]Byrd, R. H. and Schnable, R. B., “Continuity of the null space basis and constrained optimization”, Mathematical Programming 35 (1986) 32–41.Google Scholar
[4]Byrd, R. H., Schnable, R. B. and Shultz, G. A., “A trust region algorithm for nonlinearly constrained optimization”, SIAM J. Numer. Anal. 24 (1987) 1152–1170.Google Scholar
[5]Byrd, R. H., Schnable, R. B. and Shultz, G. A., “Approximate solution of the trust region problem by minimization over two-dimensional subspaces”, Mathematical Programming 40 (1988) 247–263.Google Scholar
[6]Celis, M. R., Dennis, J. E. and Tapia, R. A., “A trust region strategy for nonlinear equality constrained optimization”, in Numerical Optimization (ed. Boggs, P. et al. ), (SIAM, Philadelphia, 1985) 71–82.Google Scholar
[7]Coleman, T. F. and Conn, A. R., “On the local convergence of a quasi-Newton method for the nonlinear programming problem”, SIAM J. Numer. Anal 21 (1984) 755–769.CrossRefGoogle Scholar
[8]Coleman, T. F. and Sorensen, D. C., “A note on the computation of an orthonormal basis for the null space of a matrix”, Mathematical Programming 29 (1984) 234–242.Google Scholar
[9]Deng, N. Y., Xiao, Y. and Zhou, F. J., “A nonmonotone dogleg method for unconstrained optimization”, J. Optim. Theory and Applications 76 (1993) 259–285.Google Scholar
[10]Dennis, J. E. Jr and Mei, H., “Two new unconstrained optimization algorithms which use function and gradient values”, J. Optim. Theory and Applications 28 (1979) 453–482.CrossRefGoogle Scholar
[11]Dennis, J. E. and Schnable, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, New Jersey, 1983).Google Scholar
[12]Ferris, M. C. and Lucidi, S., “Nonmonotone stabilization methods for nonlinear equations”, J. Optim. Theory and Applications 81 (1994) 53–71.CrossRefGoogle Scholar
[13]Gill, P. E., Murray, W., Saunders, M. A., Stewart, G. W. and Wright, M. H., “Properties of a representation of a basis for the null space”, Mathematical Programming 33 (1985) 172–186.Google Scholar
[14]Grippo, L., Lampariell, F. and Lucidi, S., “A nonmonotone line search technique for Newton's methods”, SIAM J Numer. Anal. 33 (1986) 707–716.CrossRefGoogle Scholar
[15]Hock, W. and Schittkowski, K., Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems 187 (Springer, Berlin, 1981).Google Scholar
[16]Nocedal, J. and Overton, M. L., “Projected Hessian updating algorithms for nonlinearly constrained optimization”, SIAM J. Numer. Anal 19 (1985) 821–850.Google Scholar
[17]Powell, M. J. D., “A hybrid method for nonlinear equations”, in Numerical Methods for Nonlinear Algebraic Equations (ed. Rabinowitz, P.), (Gordon and Breach, London, 1970) 87–114.Google Scholar
[18]Powell, M. J. D., “A Fortran subroutine for solving systems of nonlinear algebraic equations”, in Numerical Methods for Nonlinear Algebraic Equations (ed. Rabinowitz, P.), (Gordon and Breach, London, 1970) 115–161.Google Scholar
[19]Powell, M. J. D. and Yuan, Y., “A trust region algorithm for equality constrained optimization”, Mathematical Programming 49 (1991) 189–211.Google Scholar
[20]Schittkowski, K., More Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems 282 (Springer, Berlin, 1987).Google Scholar
[21]Stewart, G. W., Introduction to Matrix Computations (Academic Press, Orlando, Florida, 1973).Google Scholar
[22]Zhang, J. Z. and Chen, L. H., “Some nonmonotone Levenberg-Marquardt algorithms and their convergence analysis”, J. Optim. Theory and Applications, to appear.Google Scholar
[23]Zhang, J. Z., Xu, X. J. and Zhu, D. T., “A nonmonotonic dogleg method for unconstrained optimization”, Central European J. Oper. Res. and Economics 3 (1994) 123–138.Google Scholar
[24]Zhang, J. and Zhu, D., “A trust region typed dogleg method for nonlinear optimization”, Optimization 21 (1990) 543–557.CrossRefGoogle Scholar
[25]Zhang, J. and Zhu, D., “A projective quasi-Newton method for nonlinear optimization”, J. Comp. Appl. Math. 53 (1994) 291–307.CrossRefGoogle Scholar
[26]Zhang, J., Zhu, D. and Hou, S., “Some improved two-sided projected quasi-Newton algorithms and their convergence, Part 1: Method and global convergence”, Acta Mathematicae Applicatae Sinica 5 (1989) 33–45.CrossRefGoogle Scholar
[27]Zhang, J., Zhu, D. and Hou, S., “Some improved two-sided projected quasi-Newton algorithms and their convergence, Part 2: Local convergence rate and numerical results”, Acta Mathematicae Applicatae Sinica 5 (1989) 46–59.CrossRefGoogle Scholar