  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 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 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
linear algebra or number of function evaluations.
