A criterion for the global convergence of conjugate gradient methods under strong Wolfe line search

Abstract

From 1952 until now, the sufficient descent property and the global convergence of conjugate gradient (CG) methods have been studied extensively. However, the sufficient descent property and the global convergence of some CG methods such as the method of Polak, Ribière, and Polyak (PRP) and the method of Hestenes and Stiefel (HS) have not been established yet under the strong Wolfe line search. In this paper, based on Yousif (Yousif, 2020) we present a criterion that guarantees the generation of descent search directions property and the global convergence of CG methods when they are applied under the strong Wolfe line search. Moreover, the PRP and the HS methods are restricted in order to satisfy the presented criterion, so new modified versions of PRP and HS are proposed. Finally, to support the theoretical proofs, a numerical experiment is done.