On the Use of the SYMMBK Algorithm for Computing Negative Curvature Directions Within Newton–Krylov Methods

In this paper, we consider the issue of computing negative curvature directions, for nonconvex functions, within Newton–Krylov methods for large scale unconstrained optimization. This issue has been widely investigated in the literature, and different approaches have been proposed. We focus on the well known SYMMBK method proposed for solving large scale symmetric possibly indefinite linear systems [3, 5, 7, 20], and show how to exploit it to yield an effective negative curvature direction. The distinguishing feature of our proposal is that the computation of such negative curvature direction is iteratively carried out, without storing no more than a couple of additional vectors. The results of a preliminary numerical experience are reported showing the reliability of the novel approach we propose.