An Extended Spectral Conjugate Gradient Method for Unconstrained Optimization Problems

In this paper, an extended spectral conjugate gradient method is proposed for solving unconstrained optimization problems, where the search direction is a linear combination of the gradient vector at current iteration and the search direction at the previous iteration. Instead of specifying a fixed expression to compute each combination coefficient in the existent methods, only suitable conditions are presented for the combination coefficients such that the values of coefficients are chosen freely in a range. Under some mild assumptions, with step lengths satisfying the Armijo condition, global convergence is established for the developed algorithm. It is shown that some existent methods are the special cases of the presented method in this paper.