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 isto 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 form then an affine space of model Hessians or model Newton directions, from which a particular one can be computed once an equilibrium or least secant principleis 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 techniquescan lead to a significant reduction in the overall number of Hessian-vector products whencompared to the inexact or truncated Newton method, although simple implementationsmay pay a cost in linear algebra or number of function evaluations.