Some sufficient descent conjugate gradient methods and their global convergence

Modified Hestense–Stiefel, Polak–Ribière–Polyak and Liu–Storey conjugate gradient methods are developed using some new techniques. The proposed methods can generate sufficient descent directions without any line search. Under some conditions, global convergence results of the methods are established when the Wolfe or Armijo line search is used. Moreover, the r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document}-linear convergence rate of the methods are analyzed. Numerical comparisons are given with some existing conjugate gradient methods using the unconstrained optimization problems in the CUTEr library.