Use of directions of negative curvature in a modified Newton method

Abstract

A modified Newton method for the unconstrained minimization problem is presented. The modification occurs in nonconvex regions, where the information contained in the negative eigenvalues of the Hessian is taken into account by performing a line search along a path which is initially tangent to a direction of negative curvature. Termination criteria for the line search are given, and it is proved that the resulting iterates are guaranteed to converge, under reasonable conditions, to a critical point at which the Hessian is positive semidefinite. It is also shown how the Bunch and Parlett decomposition of a symmetric indefinite matrix can be used to give entirely adequate directions of negative curvature.