The proof of the sufficient descent condition of the Wei-Yao-Liu conjugate gradient method under the strong Wolfe-Powell line search. (English) Zbl 1131.65049

This short article investigates the sufficient descent condition of a new conjugate gradient method. In the first section an overview of conjugate gradient methods is presented and, in particular, the new approach by Z. Wei, S. Yao and L. Liu [ibid. 183, No. 2, 1341–1350 (2006; Zbl 1116.65073)]. In the second section the main result of this article is presented, namely, that for the case of the parameter \(\sigma<1/4\) the Wei-Yai-Liu conjugate gradient method possesses the sufficient decent condition.


65K05 Numerical mathematical programming methods
90C52 Methods of reduced gradient type
90C30 Nonlinear programming


Zbl 1116.65073
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] Chen, X.; Sun, J., Global convergence of a two-parameter family of conjugate gradient methods without line search, J. comput. appl. math., 146, 37-45, (2002) · Zbl 1018.65081
[3] Dai, Y., Convergence of nonlinear conjugate methods, J. comput. math., 19, 539-549, (2001)
[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] 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.
[7] Dai, Y.; Yuan, Y., A nonlinear conjugate gradient with a strong global convergence properties, SIAM J. optim., 10, 177-182, (2000)
[8] Dai, Y.; Yuan, Y., Nonlinear conjugate gradient methods, (2000), Science Press of Shanghai Shanghai · Zbl 1030.90141
[9] Dai, Y.; Yuan, Y., An efficient hybrid conjugate gradient method for unconstrained optimization, Ann. oper. res., 103, 33-47, (2001) · Zbl 1007.90065
[10] Fletcher, R., Practical method of optimization, Unconstrained optimization, vol. I, (1997), Wiley New York
[11] Fletcher, R.; Reeves, C., Function minimization by conjugate gradients, Compute. J., 7, 149-154, (1964) · Zbl 0132.11701
[12] Gilbert, J.C.; Nocedal, J., Global convergence properties of conjugate gradient methods for optimization, SIAM J. optim., 2, 21-42, (1992) · Zbl 0767.90082
[13] Grippo, L.; Lucidi, S., A globally convergent version of the polak – ribière gradient method, Math. prog., 78, 375-391, (1997) · Zbl 0887.90157
[14] Hestenes, M.R.; Stiefel, E., Method of conjugate gradient for solving linear equations, J. res. nat. bur. stand., 49, 409-436, (1952) · Zbl 0048.09901
[15] Liu, Y.; Storey, C., Efficient generalized conjugate gradient algorithms, part 1: theory, J. optim. theory appl., 69, 129-137, (1992) · Zbl 0702.90077
[16] Nocedal, J., Conjugate gradient methods and nonlinear optimization, (), 9-23 · Zbl 0866.65037
[17] Polak, E., Optimization: algorithms and consistent approximations, (1997), Springer New York · Zbl 0899.90148
[18] Polak, E.; Ribière, G., Note sur la convergence de directions conjugèes, Rev. francaise informat recherche operationelle, 3e annèe, 16, 35-43, (1969) · Zbl 0174.48001
[19] Polyak, B.T., The conjugate gradient method in extreme problems, USSR comp. math. math. phys., 9, 94-112, (1969) · Zbl 0229.49023
[20] Powell, M.J.D., Nonconvex minimization calculations and the conjugate gradient method, (), 122-141 · Zbl 0531.65035
[21] Sun, J.; Zhang, J., Convergence of conjugate gradient methods without line search, Ann. oper. res., 103, 161-173, (2001) · Zbl 1014.90071
[22] Zoutendijk, G., Nonlinear programming computational methods, (), 37-86 · Zbl 0336.90057
[23] Wei, Z., The convergence properties of some conjugate gradient methods, Appl. math. comput., 183, 1341-1350, (2006) · Zbl 1116.65073
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.