On the convergence of conjugate gradient methods invariant to nonlinear scaling

In order to construct more efficient methods for unconstrained optimization problems, several authors have considered more general functions than quadratic functions as a basis for conjugate gradient method in recent years. Although interesting numerical experiments have been obtained by using the new methods, their convergence has remained an open problem even when line searches are exact. Under some assumptions, two global convergence theorems for the extended Fletcher–Reeves and Polay–Ribiere methods proposed in Boland et al. are given in this paper.