Abstract: Hager and Zhang in [4] proposed a new nonlinear conjugate gradient method (HZ method) and proved that this method is globally convergent when the line search fulfills the Wolfe conditions or the Goldstein’s conditions for strongly convex functions. But no global convergence results were obtained for nonconvex objective functions with Armijo line search. In this paper, the authors introduce a cautious HZ method and prove that the proposed method with Armijo line search converges globally even if the minimization function is
nonconvex. The authors also present some numerical results to show the efficiency of the proposed method.

Key words: HZ method, Armijo line search, Global convergence