Global convergence of two spectral conjugate gradient methods

Abstract

Two new nonlinear spectral conjugate gradient methods for solving unconstrained optimization problems are proposed. One is based on the Hestenes and Stiefel (HS) method and the spectral conjugate gradient method. The other is based on a mixed spectral HS-CD conjugate gradient method, which combines the advantages of the spectral conjugate gradient method, the HS method, and the CD method. The directions generated by the methods are descent directions for the objective function. Under mild conditions, we prove that the spectral conjugate gradient methods with an Armijo-type line search are globally convergent. Numerical results show the proposed methods are promising.