New conjugacy condition and related new conjugate gradient methods for unconstrained optimization

The conjugate gradient (CG) method has played a special role in solving large-scale nonlinear optimization due to the simplicity of their iterations and their very low memory requirements. Based on a new quasi-Newton equation proposed in [Z. Wei, G. Li, L. Qi, New quasi-newton methods for unconstrain optimization, preprint, Z. Wei, G. Yu, G. Yuan, Z. Lian, The superlinear convergence of a modified BFGS-type method for unconstrained optimization, Comput. Optim. Appl. 29(3) (2004) 315–332], we establish a new conjugacy condition for CG methods and propose several new CG methods. It is a interesting feature that these new CG methods take both the gradient and function value information. Under some suitable conditions, the global convergence is achieved for these methods. The numerical results show that one of our new CG methods is very encouraging.