The nonlinear conjugate gradient (CG) algorithm is a very effective method for optimization, especially for large-scale problems, because of its low memory requirement and simplicity. Zhang et al. (IMA J. Numer. Anal. 26:629-649, 2006) firstly propose a three-term CG algorithm based on the well known Polak-Ribière-Polyak (PRP) formula for unconstrained optimization, where their method has the s...

Recently, important contributions on convergence studies of conjugate gradient methods have been made by Gilbert and Nocedal [6]. They introduce a “sufficient descent condition” to establish global convergence results, whereas this condition is not needed in the convergence analyses of Newton and quasi-Newton methods, [6] hints that the sufficient descent condition, which was enforced by their ...

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To achieve a conjugate gradient method which is strong in theory and efficient practice for solving unconstrained optimization problem, we propose hybridization of the Hager Zhang (HZ) Polak-Ribière Polyak (PRP) methods possesses an important property well known PRP method: tendency to turn towards steepest descent direction if small step generated away from solutio...