Anwa Zhou, Zhibin Zhu, Hao Fan, Qian Qing
.
DOI: 10.4236/am.2011.23035   PDF    HTML     6,274 Downloads   13,407 Views   Citations

Abstract

In this paper, three new hybrid nonlinear conjugate gradient methods are presented, which produce suf?cient descent search direction at every iteration. This property is independent of any line search or the convexity of the objective function used. Under suitable conditions, we prove that the proposed methods converge globally for general nonconvex functions. The numerical results show that all these three new hybrid methods are efficient for the given test problems.

Keywords

Conjugate Gradient Method, Descent Direction, Global Convergence

Share and Cite:

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. Fletcher and C. Reeves, “Function Minimization by Conjugate Gradients,” The Computer Journal, Vol. 7, No. 2, 1964, pp. 149-154. doi:10.1093/comjnl/7.2.149
[2]	R. Fletcher, “Practical Methods of Optimization, Unconstrained Optimization,” Wiley, New York, 1987.
[3]	Y. H. Dai and Y. Yuan, “A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property,” SIAM Journal on Optimization, Vol. 10, No. 1, 1999, pp. 177-182. doi:10.1137/S1052623497318992
[4]	M. R. Hestenes and E. L. Stiefel, “Methods of Conjugate Gradients for Solving Linear Systems,” Journal of Research of the National Bureau of Standards, Vol. 49, No. 6, 1952, pp. 409-432.
[5]	B. Polak and G. Ribiere, “Note Surla Convergence des Méthodes de Directions Conjuguées,” Revue Francaise d’Informatique et de Recherche Opérationnelle, Vol. 16, No. 1, 1969, pp. 35-43.
[6]	B. T. Polyak, “The Conjugate Gradient Method in Extreme Problems,” USSR Computational Mathematics and Mathematical Physics, Vol. 9, No. 4, 1969, pp. 94-112. doi:10.1016/0041-5553(69)90035-4
[7]	Y. L. Liu and C. S. Storey, “Efficient Generalized Conjugate Gradient Algorithms, Part 1: Theory,” Journal of Optimization Theory and Applications, Vol. 69, No. 1, 1991, pp. 129-137. doi:10.1007/BF00940464
[8]	Y. Yuan and W. Sun, “Theory and Methods of Optimization,” Science Press of China, Beijing, 1999.
[9]	Y. H. Dai and Y. Yuan, “Nonlinear Conjugate Gradient Methods,” Shanghai Scientific and Technical Publishers, Shanghai, 1998.
[10]	G. Zoutendijk, “Nonlinear Programming, Computational Methods,” In: J. Abadie Ed., Integer and Nonlinear Programming, North-Holland, Amsterdam, 1970, pp. 37-86.
[11]	M. Al-Baali, “Descent Property and Global Convergence of the Fletcher–Reeves Method with Inexact Line Search,” IMA Journal of Numerical Analysis, Vol. 5, No. 1, 1985, pp.121-124. doi:10.1093/imanum/5.1.121
[12]	M. J. D. Powell, “Nonconvex Minimization Calculations and the Conjugate Gradient Method,” Lecture Notes in Mathematics, Vol. 1066, No. 122, 1984, pp. 121-141.
[13]	D. Touati-Ahmed and C. Storey, “Efficient Hybrid Conjugate Gradient Techniques,” Journal of Optimization Theory and Applications, Vol. 64, No. 2, 1990, pp. 379–397. doi:10.1007/BF00939455
[14]	Y. H. Dai and Y. Yuan, “An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimization,” Annals of Operations Research, Vol. 103, No. 1-4, 2001, pp. 33-47. doi:10.1023/A:1012930416777
[15]	J. C. Gilbert and J. Nocedal, “Global Convergence Properties of Conjugate Gradient Methods for Optimization,” SIAM Journal Optimization, Vol. 2, No. 1, 1992, pp. 21-42. doi:10.1137/0802003
[16]	W. W. Hager and H. Zhang, “A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search,” SIAM Journal Optimization, Vol. 16, No. 1, 2005, pp. 170-192. doi:10.1137/030601880
[17]	L. Zhang, W. J. Zhou and D. H. Li, “A Descent Modified Polak-Ribière-Polyak Conjugate Gradient Method and Its Global Convergence,” IMA Journal of Numerical Analysis, Vol. 26, No. 4, 2006, pp. 629-640. doi:10.1093/imanum/drl016
[18]	L. Zhang, W. J. Zhou and D. H. Li, “Global Convergence of a Modified Fletcher-Reeves Conjugate Method with Armijo-Type Line Search,” Numerische Mathematik, Vol. 104, No. 4, 2006, pp. 561-572. doi:10.1007/s00211-006-0028-z
[19]	L. Zhang, “Nonlinear Conjugate Gradient Methods for Optimization Problems,” Ph.D. Thesis, Hunan University, 2006.
[20]	L. Zhang and W. J. Zhou, “Two Descent Hybrid Conjugate Gradient Methods for Optimization,” Journal of Computational and Applied Mathematics, Vol. 216, No. 1, 2008, pp. 251-264. doi:10.1016/j.cam.2007.04.028
[21]	P. Wolfe, “Convergence Conditions for Ascent Methods,” SIAM Review, Vol. 11, No. 2, 1969, pp. 226-235. doi:10.1137/1011036
[22]	W. Hock and K. Schittkowski, “Test Examples for Nonlinear Programming Codes,” Journal of Optimization Theory and Applications, Vol. 30, No. 1, 1981, pp. 127-129. doi:10.1007/BF00934594