New conjugacy condition and related new conjugate gradient methods for unconstrained optimization. (English) Zbl 1116.65069

The authors study the conjugate gradient method for solving large-scale nonlinear optimization problems. In the first two sections, the authors present the necessary background relating to conjugate methods in general and choices for the conjugacy condition. The third section contains the main contribution of this paper which is a new conjugacy condition derived by the authors using a new quasi-Newton equation. This equation uses not only the gradient value information but also the information relating to the function value.
Several theorems are then presented, with proof, which include the properties of the proposed conjugacy condition and a study of the properties (e.g., convergence) of the derived algorithm. The paper concludes with a section containing the results of the performed numerical experimentation and a list of relevant references.


65K05 Numerical mathematical programming methods
90C26 Nonconvex programming, global optimization
90C30 Nonlinear programming
90C53 Methods of quasi-Newton type


Full Text: DOI


[1] Al-Baali, A., Descent property and global convergence of the fletcher – reeves method with inexact line search, IMA J. numer. anal., 5, 121-124, (1985) · Zbl 0578.65063
[2] Armijo, L., Minimization of functions having Lipschitz conditions partial derivatives, Pacific J. math., 16, 1-3, (1966) · Zbl 0202.46105
[3] X. Chen, J. Sun, Global convergence of two-parameter family of conjugate gradient methods without line search, J. Comput. Appl. Math. 146 (2002) 37-45. · Zbl 1018.65081
[4] Y. Dai, Convergence of Polak-Ribière-Polyak conjugate gradient method with constant stepsizes, Manuscript, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, 2001.
[5] Dai, Y.; Han, J.; Liu, G.; Sun, D.; Yin, H.; Yan, Y., Convergence properties of nonlinear conjugate methods, SIAM J. optim., 2, 345-358, (1999) · Zbl 0957.65062
[6] Dai, Y.; Liao, L.Z., New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. math. optim., 43, 87-101, (2001) · Zbl 0973.65050
[7] Y. Dai, Y. Yuan, Further studies on the Polak-Ribière-Polyak method, Research Report ICM-95-040, Institute of Computational Mathematics and Scientific/ Engineering Computing, Chinese Academy of Sciences, 1995.
[8] Dai, Y.; Yuan, Y., Nonlinear conjugate gradient methods, (2000), Science Press of Shanghai Shanghai · Zbl 1030.90141
[9] Gibert, J.C.; Nocedal, J., Global convergence properties of conjugate gradient methods for optimization, SIAM J. optim., 2, 21-42, (1992) · Zbl 0767.90082
[10] Moreau, J.J., Proximite et dualite dans un espace hilbertien, Bull. soc. math. France, 93, 273-299, (1965) · Zbl 0136.12101
[11] Morè, J.J.; Garbow, B.S.; Hillstrome, K.E., Testing unconstrained optimization software, AVM trans. math. software, 7, 17-41, (1981) · Zbl 0454.65049
[12] Nocedal, J., Conjugate gradient methods and nonlinear optimization, (), 9-23 · Zbl 0866.65037
[13] Powell, M.J.D., Restart procedures for the conjugate gradient method, Math. programming, 12, 241-254, (1977) · Zbl 0396.90072
[14] M.J.D. Powell, Nonconvex Minimization Calculations and the Conjugate Gradient Method, in: Lecture Notes in Mathematics, vol. 1066, Springer, Berlin, 1984, pp. 122-141. · Zbl 0531.65035
[15] Z. Wei, G. Li, L. Qi, New quasi-newton methods for unconstrain optimization, preprint.
[16] Wei, Z.; Qi, L., Convergence analysis of a proximal Newton method, Numer. funct. anal. optim., 17, 463-472, (1996) · Zbl 0884.90123
[17] Wei, Z.; Qi, L.; Birge, J.R., A new method for nonsmooth convex optimization, J. inequalities appl., 2, 157-179, (1998) · Zbl 0903.90131
[18] Wei, Z.; Yu, G.; Yuan, G.; Lian, Z., The superlinear convergence of a modified BFGS-type method for unconstrained optimization, Comput. optim. appl., 29, 3, 315-332, (2004) · Zbl 1070.90089
[19] Wolfe, P., Convergence conditions for ascent methods, SIAM rev., 11, 226-235, (1969) · Zbl 0177.20603
[20] Wolfe, P., Convergence conditions for ascent methods II: some corrections, SIAM rev., 11, 185-188, (1971) · Zbl 0216.26901
[21] Zoutendijk, G., Nonlinear programming computational methods, (), 37-86 · Zbl 0336.90057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.