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

TL;DR

This paper introduces two novel methods for calculating interpolation models in nonlinear optimization using Hessian-vector products, aiming to reduce the number of such products needed compared to traditional methods.

Contribution

The paper proposes two new approaches for modeling Hessian-vector products in nonlinear optimization, enhancing efficiency over existing inexact Newton methods.

Findings

Significant reduction in Hessian-vector product computations.

New interpolation models improve optimization efficiency.

Methods balance computational cost and accuracy.

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 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.