An accelerated conjugate gradient algorithm with guaranteed descent and conjugacy conditions for unconstrained optimization

In this paper we suggest a new conjugate gradient algorithm that for all both the descent and the conjugacy conditions are guaranteed. The search direction is selected as a linear combination of 0 k ≥ 1 k g + − and where , k s 1 1 ( ) k k g f x + + = ∇ , and the coefficients in this linear combination are selected in such a way that both the descent and the conjugacy condition are satisfied at every iteration. It is shown that for general nonlinear functions with bounded Hessian the algorithm with strong Wolfe line search generates directions bounded away from infinity. The algorithm uses an acceleration scheme that modify the steplength 1 k k s x x + = − k k α in such a manner as to improve the reduction of the function values along the iterations. Numerical comparisons with some conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the CUTE library, show that the computational scheme outperform the known conjugate gradient algorithms like Hestenes and Stiefel, Polak, Ribière and Polyak, Dai and Yuan or hybrid Dai and Yuan as well as CG_DESCENT by Hager and Zhang with Wolfe line search.