A Continuous Nonlinear Optimization Perspective on the Spin Glass Problem
Phil Duxbury, Carlile Lavor, Luiz Leduino de Salles-Neto
TL;DR
The paper tackles the Spin Glass Problem (SGP), whose energy is represented by $F(x)=-\sum_{(i,j)\in E} J_{i,j} x_i x_j$ with spins $x_i\in\{-1,1\}$ and which is closely related to Max-Cut. It proposes a Continuous Spin Glass (CSG) model, a box-constrained continuous relaxation $F_2(x)=-\sum_{i<j} J_{i,j} x_i x_j$ with $x_i\in[-1,1]$, and leverages Rosenberg's theorem to show that a global optimum of the continuous problem corresponds to a discrete optimum, enabling a constructive mapping back to a ground-state configuration. Through extensive computational experiments on Biq Mac benchmarks, the CSG model, solved with a modern global optimizer like GUROBI on general-purpose CPU hardware, often matches or surpasses the best linearization-based approaches in both solution quality and runtime, and remains robust on large-scale Gset instances. The study positions continuous optimization as a practical and certifiable alternative to Ising-machine heuristics for hard binary quadratic problems, with potential extensions to other unconstrained binary optimization problems and hybrid strategies.
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.