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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.

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