Nonconvex optimization methods for ground states in disordered continuous-spin models

TL;DR

This paper applies Riemannian optimization techniques to find ground states in disordered continuous-spin models, demonstrating MBH's superior performance over MS in complex energy landscapes.

Contribution

It introduces a novel reformulation of the energy landscape problem and compares advanced global optimization methods for better ground state identification.

Findings

MBH outperforms MS in finding lower-energy configurations

Riemannian optimization improves computational efficiency

Establishes a link between spin models and global optimization

Abstract

This work explores the global optimization problem of finding lowest-energy configurations (ground states) in disordered continuous spins models from statistical physics, with a particular focus on the random field XY model. Due to an extremely non-convex nature of the associated energy landscape, this problem remains highly challenging. From an optimization perspective, we reformulate the traditional angular Hamiltonian as a constrained problem on the Cartesian product of spheres, allowing the application of Riemannian optimization techniques, which show better computational performances. We compare a MultiStart (MS) strategy against a Monotonic Basin Hopping (MBH) framework, with the aim of highlighting the limitations of standard approaches and the resulting need to resort to more advanced global optimization techniques. Our results demonstrate that MBH consistently outperforms MS in identifying lower-energy configurations, offering superior computational efficiency and numerical stability. This approach establishes a robust link between continuous-spin systems and continuous global optimization, providing a high-performance benchmark for exploring complex energy landscapes.