A second-order gradient method for convex minimization

This work addresses the strictly convex unconstrained minimization problem via a modified version of the gradient method. The proposal is a line search method that uses a search direction based on the gradient method. This new direction is constructed by a mixture of the negative direction of the gradient with another particular direction that uses second-order information. Unlike Newton-type methods, our algorithm does not need to compute the inverse of the Hessian of the objective function. We analyze the global convergence under an exact line search. A numerical study is carried out, to illustrate the numerical effectiveness of the method by comparing it with some conjugate gradient methods and also with the Barzilai-Borwein gradient method both in quadratic and non-linear problems.