Exploiting SYMMBK method for the full computation of negative curvature directions

In this paper we consider the issue of computing negative curvature directions, for non-convex functions, within Newton–Krylov methods for large scale unconstrained optimization. In the last decades this issue has been widely investigated in the literature, and different approaches have been proposed. We focus on the well known SYMMBK method introduced for solving large scale symmetric possibly indefinite linear systems, and show how to exploit it to yield an effective negative curvature direction in optimization frameworks. The distinguishing feature of our proposal is that the computation of negative curvature directions is basically carried out as by–product of SYMMBK procedure, without storing no more than one additional vector. Hence, no explicit matrix factorization or matrix storage is required. A numerical experience is reported, showing the reliability of the novel approach we propose. In addition, we also propose a novel general tool, to profile the quality of the solutions found by different solvers in optimization frameworks. The new proposed tool, namely Quality Profile, draws its inspiration from both performance and data profiles, sharing with them a number of basic properties but also showing several strong differences on fruitful use cases.