Local and superlinear convergence of quasi-Newton methods based on modified secant conditions

Journal of Computational and Applied Mathematics

Volumn 205, Issue 1, 2007, Pages 617-632

  Yabe, Hiroshi ^a   Ogasawara, Hideho ^a   Yoshino, Masayuki ^b  

^a TOKYO UNIVERSITY OF SCIENCE   (Japan)

^b HITACHI LTD   (Japan)

Author keywords

Broyden family; Local and q superlinear convergence; Modified secant condition; Quasi Newton method; Unconstrained minimization

Indexed keywords

APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; FUNCTION EVALUATION; ITERATIVE METHODS; MATRIX ALGEBRA;

BROYDEN FAMILY; MODIFIED SECANT CONDITION; Q-SUPERLINEAR CONVERGENCE; QUASI-NEWTON METHODS; UNCONSTRAINED MINIMIZATION;

LINEAR SYSTEMS;

EID: 34247398380     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2006.05.018     Document Type: Article

References (15)

1. Broyden C.G., Dennis Jr. J.E., and Moré J.J. On the local and superlinear convergence of quasi-Newton methods. J. Inst. Math. Appl. 12 (1973) 223-245
2. Byrd R.H., Nocedal J., and Yuan Y. Global convergence of a class of quasi-Newton methods on convex problems. SIAM J. Numer. Anal. 24 (1987) 1171-1190
3. Dennis Jr. J.E., and Moré J.J. A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput. 28 (1974) 549-560
4. J.E. Dennis Jr., R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1983.
5. Ford J.A., and Moghrabi I.A. Multi-step quasi-Newton methods for optimization. J. Comput. Appl. Math. 50 (1994) 305-323
6. Ford J.A., and Moghrabi I.A. Using function-values multi-step quasi-Newton methods. J. Comput. Appl. Math. 66 (1996) 201-211
7. Huang H.Y. Unified approach to quadratically convergent algorithms for function minimization. J. Optim. Theory Appl. 5 (1970) 405-423
8. Powell M.J.D. How bad are the BFGS and DFP methods when the objective function is quadratic?. Math. Programming 34 (1986) 34-47
9. Stachurski A. Superlinear convergence of Broyden's bounded θ-class of methods. Math. Programming 20 (1981) 196-212
10. Yabe H., and Yamaki N. Some properties of Oren's SSVM type algorithm for unconstrained minimization. TRU Mathematics 16 (1980) 103-111
11. Yabe H., and Yamaki N. A bounded deterioration property of a symmetric positive definite class of Newton-like methods and its application. J. Oper. Res. Soc. of Japan 42 (1999) 373-388
12. Yamaki N., and Yabe H. A family of the quasi-Newton methods. TRU Math. 16 (1980) 49-54
13. Zhang J.Z., Deng N.Y., and Chen L.H. New quasi-Newton equation and related methods for unconstrained optimization. J. Optim. Theory Appl. 102 (1999) 147-167
14. Zhang Y., and Tewarson R.P. Quasi-Newton algorithms with updates from the preconvex part of Broyden's family. IMA J. Numer. Anal. 8 (1988) 487-509
15. Zhang J.Z., and Xu C.X. Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations. J. Comput. Appl. Math. 137 (2001) 269-278