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

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 set points around current one and concurrently curvature information from Hessian times appropriate vectors, possibly defined by interpolating points. These enriched conditions form then an affine space model Hessians or Newton directions, which particular can be computed once equilibrium least secant principle defined. A first approach consists recovering matrix satisfying conditions, direction computed. second pose recovery problem directly in direction. techniques lead significant reduction overall number compared inexact truncated method, although simple implementations may pay cost linear algebra evaluations.