Upper bound on the finite convergence order of the moment-SOS hierarchy on the product of spheres

Determine an explicit upper bound on the relaxation order k at which the moment-SOS hierarchy attains the true minimum for the polynomial optimization problem of minimizing a multihomogeneous polynomial f over the feasible set S = S^{n_1−1} × ⋯ × S^{n_m−1}. The bound should be expressed in terms of the dimensions n_i and the multidegrees d_i of f, and specify how large k must be to guarantee finite convergence for this problem.

Background

The paper proves that, for a generic multihomogeneous polynomial objective, the moment-SOS hierarchy has finite convergence on the product of spheres, and that flat truncation certificates are available at sufficiently high order. While asymptotic convergence is guaranteed by Putinar’s Positivstellensatz, finite convergence is established generically using local optimality conditions and Morse-theoretic arguments.

Despite the finite convergence guarantee, the authors note that the exact relaxation order at which convergence occurs is unknown. Prior works have studied convergence rates or provided degree bounds in other settings; adapting such techniques to the product of spheres could yield explicit bounds on the order required for finite convergence in this setting.

References

While Theorem \ref{main_result} guarantees convergence at some finite order of the moment-SOS hierarchy, we do not know how high this order may be.

— Finite Convergence of the Moment-SOS Hierarchy on the Product of Spheres  (2512.11119 - Halaseh et al., 11 Dec 2025) in Section 6, Further Work