Two modified spectral conjugate gradient methods and their global convergence for unconstrained optimization

In this paper, two modified spectral conjugate gradient methods which satisfy sufficient descent property are developed for unconstrained optimization problems. For uniformly convex problems, the first modified spectral type of conjugate gradient algorithm is proposed under the Wolfe line search rule. Moreover, the search direction of the modified spectral conjugate gradient method is sufficiently descent for uniformly convex functions. Furthermore, according to the Dai-Liao's conjugate condition, the second spectral type of conjugate gradient algorithm can generate some sufficient decent direction at each iteration for general functions. Therefore, the second method could be considered as a modification version of the Dai-Liao's algorithm. Under the suitable conditions, the proposed algorithms are globally convergent for uniformly convex functions and general functions. The numerical results show that the approaches presented in this paper are feasible and efficient.