Shape-Changing Trust-Region Methods Using Multipoint Symmetric Secant Matrices

TL;DR

This paper introduces a novel trust-region optimization method that employs shape-changing norms and densely-initialized multipoint symmetric secant matrices, demonstrating improved performance in large-scale nonconvex problems.

Contribution

It is the first to integrate shape-changing norms with MSS matrices in trust-region methods, enhancing approximation and convergence.

Findings

Outperforms existing MSS trust-region methods in numerical tests

Effective for large-scale nonconvex unconstrained optimization

Shows improved convergence and robustness

Abstract

In this work, we consider methods for large-scale and nonconvex unconstrained optimization. We propose a new trust-region method whose subproblem is defined using a so-called "shape-changing" norm together with densely-initialized multipoint symmetric secant (MSS) matrices to approximate the Hessian. Shape-changing norms and dense initializations have been successfully used in the context of traditional quasi-Newton methods, but have yet to be explored in the case of MSS methods. Numerical results suggest that trust-region methods that use densely-initialized MSS matrices together with shape-changing norms outperform MSS with other trust-region methods.