Regularization of Limited Memory Quasi-Newton Methods for Large-Scale Nonconvex Minimization

TL;DR

This paper introduces a regularized limited memory quasi-Newton method for large-scale nonconvex optimization, demonstrating its competitive performance and convergence properties through theoretical analysis and numerical experiments.

Contribution

It develops a novel regularized L-BFGS algorithm tailored for large-scale nonconvex problems, combining regularization with limited memory techniques.

Findings

Regularized L-BFGS is competitive with state-of-the-art methods.

The method shows promising results on large-scale test problems.

Regularization improves performance especially with nonmonotonic strategies.

Abstract

This paper deals with regularized Newton methods, a flexible class of unconstrained optimization algorithms that is competitive with line search and trust region methods and potentially combines attractive elements of both. The particular focus is on combining regularization with limited memory quasi-Newton methods by exploiting the special structure of limited memory algorithms. Global convergence of regularization methods is shown under mild assumptions and the details of regularized limited memory quasi-Newton updates are discussed including their compact representations. Numerical results using all large-scale test problems from the CUTEst collection indicate that our regularized version of L-BFGS is competitive with state-of-the-art line search and trust-region L-BFGS algorithms and previous attempts at combining L-BFGS with regularization, while potentially outperforming some of them, especially when nonmonotonicity is involved.