Shifted L-BFGS Systems

TL;DR

This paper introduces an efficient recursive method for solving shifted L-BFGS systems, which are common in optimization algorithms, enabling faster and more stable computations in relevant applications.

Contribution

The paper presents a novel recursion-based approach for solving systems involving limited-memory BFGS matrices plus a shift, improving efficiency and stability.

Findings

Recursion method requires only vector inner products.

Effective for various shift matrices G.

Numerical experiments confirm efficiency and stability.

Abstract

We investigate fast direct methods for solving systems of the form (B + G)x = y, where B is a limited-memory BFGS matrix and G is a symmetric positive-definite matrix. These systems, which we refer to as shifted L-BFGS systems, arise in several settings, including trust-region methods and preconditioning techniques for interior-point methods. We show that under mild assumptions, the system (B + G)x = y can be solved in an efficient and stable manner via a recursion that requies only vector inner products. We consider various shift matrices G and demonstrate the effectiveness of the recursion methods in numerical experiments.