The Nonlinear Conjugate Gradient Method

The conjugate gradient method has played an efficient role in solving unconstrained optimization,especially in solving large-scale nonlinear optimization due to the simplicity of their iterations and their very low memory requirements. Some well-known conjugate gradient methods are FR, PRP, CD and HS so on. Recently, using a new conjugacy condition proposed by Dai and Liao, Wei zengxin, Zhang jianzhong, Hiroshi Yabe et al. respectively proposed a class of new nonlinear conjugate gradient methods, which use both available gradient and function value information and have good numerical experiments.In this paper, we establish two classes of conjugate gradient methods .The first class, based on the conjugate gradient method of Hiroshi Yabe and Masahiro Takano, given the choice ofμ_k-1 (because u_k-1 -s_k-1 orμ_k-1= y_k-1 has good numerical experiments),μ_k-1 is modified by arithmetic mean method and the convex combination of s_k-1 and y_k-1, otherwiseθ_k is modified, therefore three new methods are obtained. The modified three new nonlinear conjugate gradient methods make the width ofβ_k widen, which are proved globally convergence with the strong Wolfe line search. Finally, the numerical results show this class of conjugate gradient methods is very efficient.In 1987, Fletcher proposed conjugate descent method (CD method), which ensured every search direction descend, but global convergence property is not good . Combined CD method and a new conjugate gradient method given by Liu.Y, Storey. C, the second class of nonlinear conjugate gradient method is proposed, which not only has descent property, but also is proved global convergence with the general Wolfe line search. Finally, the numerical results show this class of conjugate gradient methods is very efficient.