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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.

MSC:

65K05 Numerical mathematical programming methods
90C53 Methods of quasi-Newton type
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References:

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