Modeling Hessian-vector products in nonlinear optimization: new Hessian-free methods

Abstract In this paper we suggest two ways of calculating interpolation models for unconstrained smooth nonlinear optimization when Hessian-vector products are available. The main idea is to interpolate the objective function using a quadratic on a set of points around the current one, and concurrently using the curvature information from products of the Hessian times appropriate vectors, possibly defined by the interpolating points. These enriched interpolating conditions then form an affine space of model Hessians or model Newton directions, from which a particular one can be computed once an equilibrium or least secant principle is defined. A first approach consists of recovering the Hessian matrix satisfying the enriched interpolating conditions, from which then a Newton direction model can be computed. In a second approach we pose the recovery problem directly in the Newton direction. These techniques can lead to a significant reduction in the overall number of Hessian-vector products when compared to the inexact or truncated Newton method, although simple implementations may pay a cost in the number of function evaluations and the dense linear algebra involved poses a scalability challenge.