### Computing modified Newton directions using a partial Cholesky
factorization

The effectiveness of Newton's method for finding an unconstrained
minimizer of a strictly convex twice continuously differentiable
function has prompted the proposal of various *modified* Newton
methods for the nonconvex case.
*Line search* modified Newton methods utilize a linear
combination of a descent direction and a direction of negative
curvature. If these directions are *sufficient* in a certain
sense, and a suitable line search is used, the resulting method will
generate limit points that satisfy the second-order necessary
conditions for optimality.

We propose an efficient method for computing a descent direction and a
direction of negative curvature that is based on a partial Cholesky
factorization of the Hessian. This factorization not only gives
theoretically satisfactory directions, but also requires only a
*partial* pivoting strategy, i.e., the equivalent of only two
rows of the Schur complement need be examined at each step.

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