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Huang, Hai; Wei, Zengxin; Shengwei, Yao

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

Appl. Math. Comput. 189, No. 2, 1241-1245 (2007).
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.
Reviewer: Efstratios Rappos (Athens)

Cited in 23 Documents

MSC:

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

Keywords:

inexact line search; sufficient descent condition; global convergence; unconstrained optimization; conjugate gradient method

Citations:

Zbl 1116.65073
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\textit{H. Huang} et al., Appl. Math. Comput. 189, No. 2, 1241--1245 (2007; Zbl 1131.65049)
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References:

[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)
[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
[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: 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: 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, (Adams, L.; Nazareth, J. L., Linear and Nonlinear Conjugate Gradient Related Methods (1995), SIAM: SIAM Philadelphia), 9-23 · Zbl 0866.65037
[17] Polak, E., Optimization: Algorithms and Consistent Approximations (1997), Springer: 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, (Lecture Notes in Mathematics, vol. 1066 (1984), Springer: Springer Berlin), 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, (Abadie, J., Integer and Nonlinear Programming (1970), North-Holland: North-Holland Amsterdam), 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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