×

A descent nonlinear conjugate gradient method for large-scale unconstrained optimization. (English) Zbl 1117.65097

Author’s summary: A new nonlinear conjugate gradient method is proposed for large-scale unconstrained optimization which possesses the following three properties: (i) the sufficient descent property holds without any line searches; (ii) employing some steplength technique which ensures the Zoutendijk condition to be held, this method is globally convergent; (iii) this method inherits an important property of the Polak-Ribière-Polyak (PRP) method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. Preliminary numerical results show that this method is very promising.

MSC:

65K05 Numerical mathematical programming methods
90C06 Large-scale problems in mathematical programming
90C30 Nonlinear programming

Software:

minpack
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahmed, T.; Storey, D., Efficient hybrid conjugate gradient techniques, J. Optimiz. Theory Appl., 64, 379-394 (1990) · Zbl 0666.90063
[2] 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
[3] Dai, Y.; Han, J.; Liu, G.; Sun, D.; Yin, H.; Yan, Y., Convergence properties of nonlinear conjugate methods, SIAM J. Optimiz., 2, 345-358 (1999) · Zbl 0957.65062
[4] Dai, Y.; Yuan, Y., A nonlinear conjugate gradient with a strong global convergence properties, SIAM J. Optimiz., 10, 177-182 (2000) · Zbl 0957.65061
[5] Dai, Y.; Yuan, Y., Convergence properties of the Fletch-Reeves method, IMA J. Numer. Anal., 16, 2, 155-164 (1996) · Zbl 0851.65049
[6] Dai, Y.; Yuan, Y., Nonlinear Conjugate Gradient Methods (2000), Science Press of Shanghai: Science Press of Shanghai Shanghai, pp. 24-26 (in Chinese)
[7] Fletcher, R., Practical Method of Optimization. Practical Method of Optimization, Unconstrained Optimization, vol. I (1997), Wiley: Wiley New York · Zbl 0905.65002
[8] Fletcher, R.; Reeves, C., Function minimization by conjugate gradients, Comput. J., 7, 149-154 (1964) · Zbl 0132.11701
[9] Gibert, J. C.; Nocedal, J., Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optimiz., 2, 21-42 (1992) · Zbl 0767.90082
[10] Hu, Y. F.; Storey, C., Global convergence result for conjugate gradient method, J. Optimiz. Theory Appl., 71, 399-405 (1991) · Zbl 0794.90063
[11] 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
[12] Liu, G.; Han, J.; Yin, H., Global convergence of the Fletcher-Reeves algorithm with inexact line search, Appl. Math. JCU, 10B, 75-82 (1995) · Zbl 0834.90122
[13] Liu, Y.; Storey, C., Efficient generalized conjugate gradient algorithms. Part 1: Theory, J. Optimiz. Theory Appl., 69, 129-137 (1992) · Zbl 0702.90077
[14] Morè, J. J.; Garbow, B. S.; Hillstrome, K. E., Testing unconstrained optimization software, ACM Trans. Math. Software, 7, 17-41 (1981) · Zbl 0454.65049
[15] Nocedal, J.; Wright, S. J., Numerical optimization, (Springer Series in Operations Research (1999), Springer: Springer New York) · Zbl 1104.65059
[16] Polak, E.; Ribiére, G., Note Sur la convergence de directions conjugèes, Rev. Francaise Informat Recherche Operationelle 3e Annèe, 16, 35-43 (1969) · Zbl 0174.48001
[17] Polyak, B. T., The conjugate gradient method in extreme problems, USSR Comp. Math. Math. Phys., 9, 94-112 (1969) · Zbl 0229.49023
[18] Powell, M. J.D., Restart procedures for the conjugate gradient method, Math. Program., 12, 241-254 (1977) · Zbl 0396.90072
[19] Powell, M. J.D., Nonconvex minimization calculations and the conjugate gradient method, (Lecture Notes in Mathematics, 1066 (1984), Springer-Verlag: Springer-Verlag Berlin), 122-141 · Zbl 0531.65035
[20] Sun, J.; Zhang, J., Convergence of conjugate gradient methods without line search, Annals of Operations Research, 103, 161-173 (2001) · Zbl 1014.90071
[21] Wei, Z.; Li, G.; Qi, L., New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems, Appl. Math. Comput., 179, 407-430 (2006) · Zbl 1106.65055
[22] Wolfe, P., Convergence conditions for ascent methods, SIAM Rev., 11, 226-235 (1969) · Zbl 0177.20603
[23] Zoutendijk, G., Nonlinear programming computational methods, (Abadie, J., Integer and Nonlinear Programming (1970), North-Holland: North-Holland Amsterdam), 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.