Limited-memory BFGS Systems with Diagonal Updates
Numerical Analysis
2012-02-02 v2 Optimization and Control
Abstract
In this paper, we investigate a formula to solve systems of the form (B + {\sigma}I)x = y, where B is a limited-memory BFGS quasi-Newton matrix and {\sigma} is a positive constant. These types of systems arise naturally in large-scale optimization such as trust-region methods as well as doubly-augmented Lagrangian methods. We show that provided a simple condition holds on B_0 and \sigma, the system (B + \sigma I)x = y can be solved via a recursion formula that requies only vector inner products. This formula has complexity M^2n, where M is the number of L-BFGS updates and n >> M is the dimension of x.
Cite
@article{arxiv.1112.6060,
title = {Limited-memory BFGS Systems with Diagonal Updates},
author = {Jennifer B. Erway and Roummel F. Marcia},
journal= {arXiv preprint arXiv:1112.6060},
year = {2012}
}