Spectral method and its application to the conjugate gradient method

A new method used to prove the global convergence of the nonlinear conjugate gradient methods, the spectral method, is presented in this paper, and it is applied to a new conjugate gradient algorithm with sufficiently descent property. By analyzing the descent property, several concrete forms of this algorithm are suggested. Under standard Wolfe line searches, the global convergence of the new algorithm is proven for nonconvex functions. Preliminary numerical results for a set of 720 unconstrained optimization test problems verify the performance of the algorithm and show that the new algorithm is competitive with CG_DESCENT algorithm.