Exploiting effective negative curvature directions via SYMMBK algorithm, in 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. 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 (Bunch and Kaufman in Math Comput 31:163–179, 2003; Chandra in Conjugate gradient methods for partial differential equations, Yale University, New Haven, 1978; Conn et al. Trust-region methods. MPS-SIAM Series on Optimization, Philadelphia, 2000; HSL 2013: A collection of Fortran codes for large scale scientific computation. http://www.hsl.rl.ac.uk/), 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 curvatures is basically carried out as by–product of SYMMBK procedure, without storing no more than two additional vectors. Hence, no explicit matrix factorization or matrix storage is required. An extensive numerical experimentation has been performed on CUTEst problems; the obtained results have been analyzed also through novel profiles (Quality Profiles) which highlighted the good capability of the algorithms which use negative curvature directions to determine better local minimizers.