Extended Barzilai-Borwein method for unconstrained minimization problems

In 1988, Barzilai and Borwein presented a new choice of step size for the gradient method for solving unconstrained minimization problems. Their method aimed to accelerate the convergence of the steepest descent method. The Barzilai-Borwein method has a low storage requirement and inexpensive computations. Therefore, many authors have paid attention to the Barzilai-Borwein method and have proposed some variants to solve large-scale unconstrained minimization problems. In this paper, we extend the Barzilai-Borwein-type methods of Friedlander et al. to more general class and establish global and Q-superlinear convergence properties of the proposed method for minimizing a strictly convex quadratic function. Furthermore, we apply our method to general objective functions. Finally, some numerical experiments are given.