A continuous nonlinear optimization perspective on the Spin Glass Problem

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. • The paper builds on Rosenberg’s classical result for multilinear optimization over box constraints and applies it to the Spin Glass Problem (SGP), showing that the continuous relaxation can be used (together with a simple problem-specific argument) to recover optimal discrete spin configurations. • The proposed continuous formulation, solved with a modern global optimization solver, attains high-quality solutions on standard benchmark instances and often matches or surpasses the best results obtained by recent integer programming linearization techniques.