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In this paper, we suggest two ways of calculating interpolation models forunconstrained smooth nonlinear optimization when Hessian-vector products areavailable. The main idea is to interpolate the objective function using aquadratic on a set of points around the current one and concurrently using thecurvature information from products of the Hessian times appropriate vectors,possibly defined by the interpolating points. These enriched interpolatingconditions form then an affine space of model Hessians or model Newtondirections, from which a particular one can be computed once an equilibrium orleast secant principle is defined.  A first approach consists of recovering the Hessian matrix satisfying theenriched interpolating conditions, from which then a Newton direction model canbe computed. In a second approach we pose the recovery problem directly in theNewton direction. These techniques can lead to a significant reduction in theoverall number of Hessian-vector products when compared to the inexact ortruncated Newton method, although simple implementations may pay a cost inlinear algebra or number of function evaluations.

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