Reduced-Hessian Quasi-Newton Methods for Unconstrained Optimization

Quasi-Newton methods are reliable and efficient on a wide range of problems, but they can require many iterations if the problem is ill-conditioned or if a poor initial estimate of the Hessian is used. In this paper, we discuss methods designed to be more efficient in these situations. All the methods to be considered exploit the fact that quasi-Newton methods accumulate approximate second-derivative information in a sequence of expanding subspaces. Associated with each of these subspaces is a certain reduced approximate Hessian that provides a complete but compact representation of the second derivative information approximated up to that point. Algorithms that compute an explicit reduced-Hessian approximation have two important advantages over conventional quasi-Newton methods. First, the amount of computation for each iteration is significantly less during the early stages. This advantage is increased by forcing the iterates to linger on a manifold whose dimension can be significantly smaller than the subspace in which curvature has been accumulated. Second, approximate curvature along directions that lie off the manifold can be reinitialized as the iterations proceed, thereby reducing the influence of a poor initial estimate of the Hessian. These advantages are illustrated by extensive numerical results from problems in the CUTE test set. Our experiments provide strong evidence that reduced-Hessian quasi-Newton methods are more efficient and robust than conventional BFGS methods and some recently proposed extensions.