Two fundamental convergence theorems for nonlinear conjugate gradient methods and their applications

Two fundamental convergence theorems are given for nonlinear conjugate gradient methods only under the descent condition. As a result, methods related to the Fletcher-Reeves algorithm still converge for parameters in a slightly wider range, in particular, for a parameter in its upper bound. For methods related to the Polak-Ribiére algorithm, it is shown that some negative values of the conjugate parameters do not prevent convergence. If the objective function is convex, some convergence results hold for the Hestenes-Stiefel algorithm.