A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization

Abstract

An accelerated hybrid Conjugate Gradient (CG) algorithm represents the subject of this paper. The parameter is computed as a convex combination of  Fletcher and Reeves, [22] and Wu-Chen,  [3], i.e. . The parameter  in the convex combination is computed in such a way that the direction corresponding to the CG algorithm is the best direction we know, i.e. the Newton direction, while the pair (,) satisfies the classical secant condition , where,  and . It is shown that both for uniformly convex functions and for general nonlinear functions the new proposed algorithm with strong Wolfe line search is globally convergent. This algorithm uses an acceleration scheme modifying the step-length  for improving the reduction of the function values along the iterations. The technique was given by (Andrei [15]). Numerical comparisons with some similar CG algorithms show that the new proposed hybrid computational scheme outperforms the CG algorithms given by Wu-Chen and FR. A set of 35 unconstrained optimization problems with  several different dimensions are used in this paper.