# On Solving L-SR1 Trust-Region Subproblems

arXiv: https://arxiv.org/abs/1506.07222 · HTML: https://mlsanity.com/paper/1506.07222/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2015.

## Built on by (most cited first, 6 of 6)

- [Globally Convergent Multilevel Training of Deep Residual Networks](https://mlsanity.com/paper/2107.07572/) (2021)
- [Parallel Trust-Region Approaches in Neural Network Training: Beyond Traditional Methods](https://mlsanity.com/paper/2312.13677/) (2023)
- [A machine learning analysis of the relationship between some underlying medical conditions and COVID-19 susceptibility](https://mlsanity.com/paper/2112.12901/) (2021)
- [Artificial Intelligence-Based Analytics for Impacts of COVID-19 and Online Learning on College Students' Mental Health](https://mlsanity.com/paper/2202.07441/) (2022)
- [A Novel Fast Exact Subproblem Solver for Stochastic Quasi-Newton Cubic Regularized Optimization](https://mlsanity.com/paper/2204.09116/) (2022)
- [On the efficiency of Stochastic Quasi-Newton Methods for Deep Learning](https://mlsanity.com/paper/2205.09121/) (2022)

## Abstract (arXiv)

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.