A modified three–term conjugate gradient method with sufficient descent property

A hybridization of the three-term conjugate gradient method proposed by Zhang et al. and the nonlinear conjugate gradient method proposed by Polak and Ribière, and Polyak is suggested. Based on an eigenvalue analysis, it is shown that search directions of the proposed method satisfy the sufficient descent condition, independent of the line search and the objective function convexity. Global convergence of the method is established under an Armijo-type line search condition. Numerical experiments show practical efficiency of the proposed method.