Abstract
This paper proposes a filter secant method with nonmonotone line search for non- linear equality constrained optimization. The Hessian of the Lagrangian is approximated using the BFGS secant update. This new method has more flexibility for the acceptance of the trial step and requires less computational costs compared with themonotone one. The global and local convergence of the proposed method are given under some reasonable conditions. Further, two-step Q-superlinear convergence rate is established by introducing second order correction step. The numerical experiments are reported to show the effectiveness of the proposed algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Mathematical Programming, 2002, 91(2): 239–269.
R. Fletcher, S. Leyffer, and P. L. Toint, On the global convergence of a filter-SQP algorithm, SIAM J. Optim., 2002, 13: 44–59.
R. Fletcher, N. I. M. Gould, S. Leyffer, P. L. Toint, and A. Wächter, Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming, SIAM J. Optim., 2002, 13: 635–659.
C. C. Gonzaga, E. Karas, and M. Vanti, A globally convergent filter method for nonlinear programming, Technical Report, Department of Mathematics, Federal University of Santa Catarina, Brazil, 2001.
M. Ulbrich and S. Ulbrich, Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function, Math. Program, 2003, 95: 103–135.
C. Gu and D. T. Zhu, A filter interior-point algorithm with projected Hessian updating for nonlinear optimization, J. Appl. Math. Comput, 2009, 29: 67–80.
A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Motivation and Global convergence, SIAM J. Comput, 2005, 16: 1–31.
A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Local convergence, SIAM J. Optim., 2005, 6: 32–48.
L. Grippo, F. Lampariello, and S. Ludidi, A nonmonotone line search technique for Newtons method, SIAM J. Numer. Anal., 1986, 23: 707–716.
K. Su and Z. Yu, A modified SQP method with nonmonotone technique and its global convergence, Comput. Math. Appl., 2009, 57: 240–247.
J. Zhang and D. T. Zhu, A projective quasi-Newton method for nonlinear optimization, J. Comput. Appl. Math., 1994, 53: 291–307.
R. Fontecilla, Local convergence of secant methods for nonlinear constrained optimization, SIAM J. Numer. Anal., 1988, 25: 692–712.
J. Nocedal and S. Wright, Numerical Optimization, Springer-Verlag, New York, NY, 1999.
W. Hock and K. Schittkowski, Test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematics System, Springer-Verlag, Berlin, Heidelberg, 1981.
K. Schittkowski, More test examples for nonlinear mathematical programming codes, Lecture Notes in Economics and Mathematics System, Springer-Verlag, Berlin, Heidelberg, 1987.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the National Science Foundation of China under Grant No. 10871130, the Ph.D. Foundation under Grant No. 20093127110005, the Shanghai Leading Academic Discipline Project under Grant No. S30405, and the Shanghai Finance Budget Project under Grant Nos. 1139IA0013 and 1130IA15.
Rights and permissions
About this article
Cite this article
Gu, C., Zhu, D. A filter secant method with nonmonotone line search for equality constrained optimization. J Syst Sci Complex 23, 846–860 (2010). https://doi.org/10.1007/s11424-010-9051-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-010-9051-3