Derivative-free line search and DFP method for symmetric equations with global and superlinear convergence

Numerical Functional Analysis and Optimization

Volumn 20, Issue 1, 1999, Pages 59-77

  Donghui, Li ^a,b   Fukushima, Masao ^a  

^a KYOTO UNIVERSITY   (Japan)

^b HUNAN UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; FUNCTIONS; ITERATIVE METHODS; MATRIX ALGEBRA; PROBLEM SOLVING;

DERIVATIVE-FREE LINE SEARCH METHOD; GAUSS-NEWTON METHOD; GLOBAL CONVERGENCE METHOD; SUPERLINEAR CONVERGENCE METHOD; SYMMETRIC EQUATIONS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 0344211045     PISSN: 01630563     EISSN: None     Source Type: Journal    
DOI: 10.1080/01630569908816881     Document Type: Article

References (19)

1. C.G. Broyden, J.E. Dennis, Jr. and J.J. Moré, On the local and superlinear convergence of quasi-Newton methods, J. Inst. Math. Appl., 12 (1973) 223-246.
2. R. Byrd and J. Nocedal, A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. Numer. Anal., 26 (1989) 727-739.
3. R. Byrd, J. Nocedal and Y. Yuan, Global convergence of a class of quasi-Newton methods on convex problems, SIAM J. Numer. Anal., 24 (1987) 1171-1189.
4. J.E. Dennis, Jr. and J.J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comput., 28 (1974) 549-560.
5. J.E. Dennis and J.J. Moré, Quasi-Newton methods, motivation and theory, SIAM Rev., 19 (1977) 46-89.
6. L.C.W. Dixon, Variable metric algorithms: Necessary and sufficient conditions for identical behavior on nonquadratic functions, J. Optimization Theory Appl., 10 (1972) 34-40.
7. R. Fletcher, Practical Methods of Optimization, Second Edition, John Wiley & Sons, Chichester, 1987.
8. R. Fletcher, An overview of unconstrained optimization, in Algorithms for Continuous Optimization: The State of the Art, E. Spedicato, ed., Kluwer Academic Publishers, Boston, 1994, pp. 109-143.
9. A. Griewank and Ph.L. Toint, Local convergence analysis for partitioned quasi-Newton updates, Numer. Math., 39 (1982) 429-448.
10. A. Griewank, The 'global' convergence of Broyden-like methods with a suitable line search, J. Aust. Math. Soc., Ser. B, 28 (1986) 75-92.
11. A. Griewank, The global convergence of partitioned BFGS on problems with convex decompositions and Lipschitzian gradients, Math. Program., 50 (1991) 141-175.
12. D. Li, On the global convergence of DFP method, J. Hunan Univ., Nat. Sci., 20 (1993) 16-20.
13. D. Li and M. Fukushima, A globally and superlinearly convergent Gauss-Newton based BFGS method for symmetric equations, Technical Report 98006, Department of Applied Mathematics and Physics, Kyoto University, March 1998.
14. J. Nocedal, Theory of algorithms for unconstrained optimization, Acta Numer., 1 (1992) 199-242.
15. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
16. M.J.D. Powell, On the convergence of the variable metric algorithm, J. Inst. Math. Appl., 7 (1971) 21-36.
17. MJ.D. Powell, Some global convergence properties of a variable metric algorithm for minimization without exact line searches, in Nonlinear Programming, SIAM-AMS Proceedings, Vol. IX, R.W. Cottle, and C.E. Lemke, eds., SIAM, 1976, pp. 53-72.
18. Ph.L. Toint, Global convergence of the partitioned BFGS algorithm for convex partially separable optimization, Math. Program., 36 (1986) 290-306.
19. Y. Zhang and R.P. Tewarson, Quasi-Newton algorithms with updates from the pre-convex part of Broyden's family, IMA J. Numer. Anal., 8 (1988) 487-509.