Another improved Wei–Yao–Liu nonlinear conjugate gradient method with sufficient descent property

Recently, Zhang [13] take a little modification to the Wei–Yao–Liu nonlinear conjugate gradient method proposed by Wei et al. [10] such that the modified method (called NPRP method in Zhang [13]) satisfies sufficient descent condition with greater parameter σ∈0,12 in the strong Wolfe line search and converges globally for nonconvex minimization with the strong Wolfe line search. In this paper, we take a little modification to the NPRP method such that the modified NPRP method possesses better properties than the NPRP method in Zhang [13]. Firstly, the modified NPRP method possesses the sufficient descent property without any line searches. Secondly, the modified NPRP method converges globally for nonconvex minimization with Wolfe line search or Armijo line search. Moreover, we extend these results to the Hestenes–Stiefel (HS) method and prove that the modified HS method also possesses sufficient descent property and global convergence with the standard Wolfe conditions. Numerical results are reported by utilizing some test problems in the CUTE library.