Improving directions of negative curvature in an efficient manner

In order to converge to second-order KKT points, second derivative information has to be taken into account. Therefore, methods for minimization satisfying convergence to second-order KKT points must, at least implicitly, compute a direction of negative curvature of an indefinite matrix. An important issue is to determine the quality of the negative curvature direction. This problem is closely related to the symmetric eigenvalue problem. More specifically we want to develop algorithms that improve directions of negative curvature with relatively little effort, extending the proposals by Boman and Murray. This paper presents some technical improvements with respect to their work. In particular, we study how to compute ‘good’ directions of negative curvature. In this regard, we propose a new method and we present numerical experiments that illustrate its practical efficiency compared to other proposals.