How Grossone Can Be Helpful to Iteratively Compute Negative Curvature Directions

Abstract

We consider an iterative computation of negative curvature directions, in large scale optimization frameworks. We show that to the latter purpose, borrowing the ideas in [1], [3] and [4], we can fruitfully pair the Conjugate Gradient (CG) method with an algebraic approach involving the use of grossone [5]. In particular, though in principle the CG method is well-posed only on positive definite linear systems, the use of grossone can enhance the performance of the CG, allowing the computation of negative curvature directions, too. The overall method in our proposal significantly generalizes the theory proposed for [1] and [3], and straightforwardly allows the use of a CG-based method on indefinite Newton's equations.