On the strong convergence of a sufficient descent Polak-Ribière-Polyak conjugate gradient method. (English) Zbl 1469.65108

Summary: Recently, L. Zhang et al. [IMA J. Numer. Anal. 26, No. 4, 629–640 (2006; Zbl 1106.65056)] proposed a sufficient descent Polak-Ribière-Polyak (SDPRP) conjugate gradient method for large-scale unconstrained optimization problems and proved its global convergence in the sense that \(\lim\inf_{k \rightarrow \infty} \| \nabla f(x_k) \| = 0\) when an Armijo-type line search is used. In this paper, motivated by the line searches proposed by Z.-J. Shi and J. Shen [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 6, 1428–1441 (2007; Zbl 1120.49027)] and Zhang et al. [loc. cit.], we propose two new Armijo-type line searches and show that the SDPRP method has strong convergence in the sense that \(\lim_{k \rightarrow \infty} \| \nabla f(x_k) \| = 0\) under the two new line searches. Numerical results are reported to show the efficiency of the SDPRP with the new Armijo-type line searches in practical computation.


65K05 Numerical mathematical programming methods
90C53 Methods of quasi-Newton type
Full Text: DOI


[1] Polak, B.; Ribière, G., Note surla convergence des méthodes de directions conjuguées, Revue Francaise Imformat Recherche Opertionelle, 16, 35-43 (1969) · Zbl 0174.48001
[2] Polyak, B. T., The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9, 4, 94-112 (1969) · Zbl 0229.49023
[3] Fletcher, R.; Reeves, C., Function minimization by conjugate gradients, The Computer Journal, 7, 149-154 (1964) · Zbl 0132.11701
[4] Hestenes, M. R.; Stiefel, E. L., Method of conjugate gradient for solving linear systems, Journal of Research of the National Bureau of Standards, 49, 409-432 (1952)
[5] Zhang, L.; Zhou, W. J.; Li, D. H., Some descent three-term conjugate gradient methods and their global convergence, Optimization Methods and Software, 22, 4, 697-711 (2007) · Zbl 1220.90094
[6] Dai, Y. H.; Yuan, Y., A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal on Optimization, 10, 1, 177-182 (1999) · Zbl 0957.65061
[7] Zhang, L.; Zhou, W.; Li, D. H., A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26, 4, 629-640 (2006) · Zbl 1106.65056
[8] Shi, Z. J.; Shen, J., Convergence of the Polak-Ribière-Polyak conjugate gradient method, Nonlinear Analysis, Theory, Methods and Applications, 66, 6, 1428-1441 (2007) · Zbl 1120.49027
[9] Shi, Z.-J.; Shen, J., Convergence of Liu-Storey conjugate gradient method, European Journal of Operational Research, 182, 2, 552-560 (2007) · Zbl 1121.90125
[10] Zhou, W. J.; Zhou, Y. H., On the strong convergence of a modified Hestenes-Stiefel method for nonconvex optimization, Journal of Industrial and Management Optimization, 9, 4, 893-899 (2013) · Zbl 1281.90062
[11] Neculai, A.
[12] Dolan, E. D.; Moré, J. J., Benchmarking optimization software with performance profiles, Mathematical Programming B, 91, 2, 201-213 (2002) · Zbl 1049.90004
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.