For a strongly convex unconstrained minimization problem, Newton’s method with a
line search offers rapid convergence to a minimizer by incorporating second-order infor-
mation. However, for a general unconstrained problem the Hessian need not be positive
definite, which can lead to the inability to find a search direction or to search directions
that fail to produce descent. Modified Newton methods are designed to handle this sit-
uation. This paper considers modifications to a Newton-conjugate-gradient approach
based on Lanczos tridiagonalization. Existing methods are extended using dynamic
modification schemes that provide a descent direction while keeping the modification
bounded. A Lanczos-based scheme and a conjugate-gradient (CG)-based scheme are
formulated and analyzed. These methods are mathematically equivalent in exact arith-
metic, but their practical behavior differs in finite precision. Finally, it is shown how the
convergence and computational cost are affected by varying the amount of additional
curvature introduced and by allowing the early termination of the inner iterations.