A New Hessian Approximation for Non-Convex Unconstrained Minimization Methods

In this work, we present a technique for a step-length selection in the frame of gradient descent methods. This method selects the step-length according to a modified backtracking procedure, along the negative gradient, using a new scalar approximation of the Hessian of the minimization function based on the function values and its gradients in two successive points along the iterations. The resulting method belongs to the same class of gradient descent with linear convergence property. Our numerical results show that the new method compares favorable with the Barzilai and Borwein (1988) and Andrei (2005) approaches. The main advantage of this new method is its implementations in both convex and non-convex functions