A Continuous Nonlinear Optimization Perspective on the Spin Glass Problem

TL;DR

This paper introduces a continuous nonlinear optimization model for the Spin Glass Problem, enabling conversion of continuous solutions into optimal discrete configurations, and demonstrates its effectiveness compared to existing methods.

Contribution

It provides a novel continuous formulation for the Spin Glass Problem that aligns with discrete solutions and is compatible with modern optimization software.

Findings

The continuous relaxation can be converted into optimal discrete solutions.

The approach matches or surpasses existing integer programming techniques.

It offers a practical tool for combining physics and combinatorial optimization.

Abstract

We present a continuous nonlinear optimization model for the Spin Glass Problem (SGP), building on a classical result by Rosenberg (1972), which shows that for a class of multilinear polynomial problems the optimal values of the continuous relaxation and the corresponding discrete model coincide. Using the SGP as a case study, we provide a simple, problem-specific argument showing how any optimal solution returned by a continuous solver can be converted into an optimal discrete spin configuration, even when the solver outputs non-integer values. The relaxed model remains nonconvex and does not alter the inherent computational hardness of the problem, but it offers a direct and conceptually transparent continuous formulation that can be handled by modern global optimization software. Computational experiments on standard benchmark instances indicate that this approach can match, and in several cases surpass, recent integer programming linearization techniques, making it a practical and complementary tool for researchers working at the interface between statistical physics and combinatorial optimization.