COMPUTING MODIFIED NEWTON DIRECTIONS USING A PARTIAL CHOLESKY FACTORIZATION

The effectiveness of Newton's method for finding an unconstrained mini mizer of a strictly convex twice continuously differentiable function has prompted the proposal of various modified Newton methods for the n onconvex case. Linesearch modified Newton methods utilize a linear com bination of a descent direction and a direction of negative curvature. If these directions are sufficient in a certain sense, and a suitable linesearch is used, the resulting method will generate limit points t hat satisfy the second-order necessary conditions for optimality. The authors propose an efficient method for computing a descent direction and a direction of negative curvature that is based on a partial Chole sky factorization of the Hessian. This factorization not only gives th eoretically satisfactory directions, but also requires only a partial pivoting strategy; i.e., the equivalent of only two rows of the Schur complement needs to be examined at each step.