A new class of Conjugate Gradient Methods with extended Nonmonotone Line Search

In this paper, we propose a new nonlinear conjugate gradient method for large-scale unconstrain optimization which possesses the following properties:(i)the sufficient descent condition -g(k)(T)d(k) >= 7/8 parallel to gk parallel to(2) holds without any line searchcs;(ii)With exact line search, this method reduces to a nonlinear version of the Liu-Storey conjugate gradient scheme.(iii)Under some assumption, global convergence of this method is proved with a new nonmonotone line search.Preliminary numerical results show that this method is very efficient.