Finite Newton-Step Generation via Subspace Quasi-Newton Methods for Quadratic Problems

Abstract

We study the curvature information sufficient for finite Newton-step generation by quasi-Newton methods on strictly convex quadratic problems. In exact arithmetic, the memoryless BFGS quasi-Newton method terminates finitely on such a problem when exact line search is used, although its quasi-Newton approximation incorporates curvature information associated with only one direction. We show that exact line search can be replaced by curvature information associated with one additional direction. Information associated with at most two directions suffices to generate the Newton step after finitely many iterations, independently of the preceding step lengths. More precisely, knowledge of the action of the Hessian on a suitably chosen subspace of dimension at most two is sufficient to generate a step that decomposes into a subspace Newton step and a component parallel to the next conjugate direction. Applying this construction recursively generates the full Newton step after finitely many iterations, after which a unit step reaches the minimizer. The required Hessian actions can also be recovered from gradient differences, yielding a first-order formulation. The resulting algorithm is intended as a constructive device for identifying sufficient curvature information, rather than as a practical alternative to the method of conjugate gradients.