The superlinear convergence analysis of a nonmonotone BFGS algorithm on convex objective functions

Acta Mathematica Sinica, English Series

Volumn 24, Issue 1, 2008, Pages 35-42

  Yuan, Gong Lin ^a   Wei, Zeng Xin ^a,b  

^a EAST CHINA UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^b GUANGXI UNIVERSITY   (China)

Author keywords

BFGS method; Nonmonotone linesearch; Superlinear convergence

Indexed keywords

EID: 40449112915     PISSN: 14398516     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10114-007-1012-y     Document Type: Article

References (22)

1. Byrd, R., Nocedal, J., Yuan, Y.: Global convergence of a class of quasi-Newton methods on convex problems. SIAM Journal on Numerical Analysis, 24, 1171-1189 (1987)
2. Powell, M. J. D.: Some properties of the variable metric algorithm, In F. A. Lootsma, (ed.), Numerical Methods for Nonlinear Optimization, Academia Press, London, 1972
3. Powell, M. J. D.: Some global convergence properties of a variable Metric algorithm for minimization without exact linesearches, In Nonlinear Programming, SIAM-AMS Proceedings, R. W. Cottle and C. E. Lemke(eds.), Vol. IX., American Mathematrical Society, Providence, RI, 1976
4. Byrd, R., Nocedal, J.: A tool for the analysis of quasi-Newton methods with application to unconstrained minimization. SIAM Journal on Numerical Analysis, 26, 727-739 (1989)
5. Powell, M. J. D.: On the Convergence of the variable metric agorithm. J. of the Institute of Mathematics and its Applications, 7, 21-36 (1971)
6. Werner, J.: Uber die Globale Konvergenze von variable-metric verfahen mit nichtexakter schrittweitenbestimmung. Numer. Math., 31, 321-334 (1978)
7. Han, J. Y., Liu, G. H.: Notes on the general form of stepsize selection. OR and Decision Making, I, 619-624 (1992)
8. Liu, G. H., Han, J. Y.: Global convergence Analysis of the variable metric algorithm with a generalized Wolf linesearch, Technical Report, Institute of Applied Mathematics, Academia Sinica, Beijing, China, 029, 1993
9. Liu, G. H., Han, J. Y., Sun, D. F.: Global convergence Analysis of the BFGS Algorithm with Nonmonotone linesearch. Optimization, 34, 147-159 (1995)
10. Liu, G. H., Peng, J. M.: The convergence properties of a nonmonotonic algorithm. J. of Computational Mathematics, 1, 65-71 (1992)
11. Yin, H. X., Du, D. L.: The global convergence of a self-scaling BFGS algorithm with nonmonotone line search for unconstrained nonconvex optimizaation problems. Acta Mathematica Sinica, English Series, 23(8), (2007)
12. Xu, D. C.: Global convergence of the Broyden's Class of Quasi-Newton Methods with Nonomonotone Linesearch. Acta Mathematicae Applicatae Sinica, English Series, 19(1), 19-24 (2003)
13. Davidon, W. C.: Variable metric methods for minimization, Argonne National Lab Report, Argonne, IL., 1959
14. Li, D., Fukushima, M.: A modified BFGS method and its global convergence in nonconvex minimization. Journal of Computational and Applied Mathematics, 129, 15-35 (2001)
15. Powell, M. J. D.: A new algorithm for unconstrained optimation, In: Nonlinear Programming, J. B. Rosen, O. L. Mangasarian and K. Ritter, eds. Academic Press, New York, 1970
16. Wei, Z., Li, G., Qi, L.: New quasi-Newton methods for unconstrained optimization problems. Applied Mathematics and Computation, 175, 1156-1188 (2006)
17. Wei, Z., Qi, L., Chen, X.: An SQP-type method and its application in stochastic programming. Journal of Optimization Theory and Applications, 116, 205-228 (2003)
18. Wei, Z., Yu, G., Yuan, G., Lian, Z.: The superlinear convergence of a modified BFGS-type method for unconstrained optimization. Computational Optimization and Applications, 29, 315-332 (2004)
19. Bürmenei, Á., Bratkovič, F., Puhan, J., Fajfar, I., Tuma, T.: Extended global convergence framework for unconstrained optimization. Acta Mathematica Sinca, English Series, 20(3), 433-440 (2004)
20. Han, J. Y., Liu, G. H.: Global convergence analysis of a new nonmonotone BFGS algorithm on convex objective Functions. Computational Optimization and Applications, 7, 277-289 (1997)
21. Griewank, A., Toint, Ph. L.: Local convergence analysis for partitioned quasi-Newton updates. Numer. Math., 39, 429-448 (1982)
22. Broyden, C. G., Dennis, J. E., Moré, J. J.: On the local and supelinear convergence of quasi-Newton methods. J. Inst. Math. Appl., 12, 223-246 (1973)