On the Strong Convergence of a Sufficient Descent Polak-Ribière-Polyak Conjugate Gradient Method

Abstract

Recently, Zhang et al. 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 ${\text{lim\hspace{0.17em}inf}}_{k\to \infty }\parallel \nabla f(x_k)\parallel =0$ when an Armijo-type line search is used. In this paper, motivated by the line searches proposed by Shi et al. and Zhang et al., we propose two new Armijo-type line searches and show that the SDPRP method has strong convergence in the sense that ${\text{lim}}_{k\to \infty }\parallel \nabla f(x_k)\parallel =0$ 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.