Finite Convergence of the Moment-SOS Hierarchy on the Product of Spheres

TL;DR

This paper proves that the moment-SOS hierarchy finitely converges for polynomial optimization problems on the product of spheres, extending previous results to multihomogeneous cases using differential geometry and Morse theory.

Contribution

It establishes finite convergence of the moment-SOS hierarchy for multihomogeneous polynomial optimization on product of spheres, generalizing prior single sphere results.

Findings

Finite convergence holds generically for multihomogeneous polynomials.

Uses differential geometry and Morse theory to verify local optimality conditions.

Extends previous results from single sphere to product of spheres.

Abstract

We study the polynomial optimization problem of minimizing a multihomogeneous polynomial over the product of spheres. This polynomial optimization problem models the tensor optimization problem of finding the best rank one approximation of an arbitrary tensor. We show that the moment-SOS hierarchy has finite convergence in this case, for a generic multihomogeneous objective function. To show finite convergence of the hierarchy, we use a result of Huang et al. [SIAM J. Optim. 34(4) (2024), pp 3399-3428], which relies on local optimality conditions. To prove that the local optimality conditions hold generically, we use techniques from differential geometry and Morse theory. This work generalizes the main result of Huang [Optim. Lett. 17(5) (2023), pp 1263-1270], which shows finite convergence for the case of a homogeneous polynomial over a single sphere.