New step lengths in conjugate gradient methods

Consider any conjugate gradient method for finding a zero point of a given gradient whose function is implicit. We propose two different types of conditions for selecting the step length using the gradient information only. One is used for re-proving known convergence results under the same gradient-Lipschitz assumption. Moreover, if the gradient is merely continuous then we are still able to get some interesting convergence results. The other also allows for convergence of the resulting conjugate gradient methods, with an application to convergence analysis of the Fletcher–Reeves conjugate gradient method. Preliminary numerical experiments show the efficiency of our proposed step length rules in practice.