Globally convergent Polak-Ribière-Polyak conjugate gradient methods under a modified Wolfe line search

It is well known that global convergence has not been established for the Polak-Ribière-Polyak (PRP) conjugate gradient method using the standard Wolfe conditions. In the convergence analysis of PRP method with Wolfe line search, the (sufficient) descent condition and the restriction βk⩾0 are indispensable (see [4,7]). This paper shows that these restrictions could be relaxed. Under some suitable conditions, by using a modified Wolfe line search, global convergence results were established for the PRP method. Some special choices for βk which can ensure the search direction's descent property were also discussed in this paper. Preliminary numerical results on a set of large-scale problems were reported to show that the PRP method's computational efficiency is encouraging.