On the strong convergence of a sufficient descent Polak-Ribière-Polyak conjugate gradient method
Summary: Recently, \textit{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 \textit{Z.-J. Shi} and \textit{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.
scientific article; zbMATH DE number 7022086