On Solving L-SR1 Trust-Region Subproblems

TL;DR

This paper introduces an efficient solver for large-scale trust-region subproblems using L-SR1 matrices, leveraging eigenspace decomposition and the Sherman-Morrison-Woodbury formula to achieve high accuracy and computational efficiency.

Contribution

The paper presents a novel solver that exploits the compact representation of L-SR1 matrices, enabling direct, high-accuracy solutions for trust-region subproblems, including the hard case.

Findings

Effective in solving large-scale problems

High-accuracy solutions in the hard case

Numerical experiments confirm efficiency

Abstract

In this article, we consider solvers for large-scale trust-region subproblems when the quadratic model is defined by a limited-memory symmetric rank-one (L-SR1) quasi-Newton matrix. We propose a solver that exploits the compact representation of L-SR1 matrices. Our approach makes use of both an orthonormal basis for the eigenspace of the L-SR1 matrix and the Sherman-Morrison-Woodbury formula to compute global solutions to trust-region subproblems. To compute the optimal Lagrange multiplier for the trust-region constraint, we use Newton's method with a judicious initial guess that does not require safeguarding. A crucial property of this solver is that it is able to compute high-accuracy solutions even in the so-called hard case. Additionally, the optimal solution is determined directly by formula, not iteratively. Numerical experiments demonstrate the effectiveness of this solver.