  We consider minimization of a smooth nonconvex objective function using an
iterative algorithm based on Newton's method and the linear conjugate gradient
algorithm, with explicit detection and use of negative curvature directions for
the Hessian of the objective function. The algorithm tracks Newton-conjugate
gradient procedures developed in the 1980s closely, but includes enhancements
that allow worst-case complexity results to be proved for convergence to points
that satisfy approximate first-order and second-order optimality conditions.
The complexity results match the best known results in the literature for
second-order methods.
