Finite Convergence of the Moment-SOS Hierarchy on the 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.

• The paper establishes that for generic multihomogeneous polynomials, the Moment-SOS hierarchy achieves finite convergence with flat truncation certification.
• The methodology utilizes Morse theory and transversality arguments to prove that problematic instances form a Lebesgue measure zero subset.
• These results enable certifiable tensor approximations, such as best rank-one decompositions, and extend to broader semialgebraic feasible sets.

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

Problem Context and Main Results

This work investigates the polynomial optimization problem (POP) of minimizing a real multihomogeneous polynomial over the product of spheres, a setting relevant for best rank-one approximation of arbitrary real tensors. Such problems naturally arise in tensor decomposition, multilinear algebra, and quantum information theory. Despite the expressive power of the moment-sum-of-squares (SOS) hierarchy—a sequence of semidefinite programming (SDP) relaxations for polynomial optimization—guarantees for finite convergence are crucial for both theoretical tractability and practical deployment. This paper establishes that for generic multihomogeneous polynomials, the moment-SOS hierarchy certifiably achieves finite convergence on the product of spheres.

In detail, the main result proves: For a generic objective in the space of real multihomogeneous polynomials of any multidegree, the moment-SOS hierarchy converges to the true minimum in finitely many steps, with flat truncation providing a constructive certificate at a certain relaxation order.

This generalizes prior results that were previously established only for homogeneous polynomials over a single sphere.

Technical Framework

Multihomogeneous Polynomial Optimization

Given positive integers $m \geq 1$ and $n_i \geq 2$ for each $i\in[m]$, consider variables $\bm{x} = (\bm{x}_1, \ldots, \bm{x}_m)$ where $\bm{x}_i \in \mathbb{R}^{n_i}$, and let the product of spheres $$S = \mathbb{S}^{n_1-1} \times \cdots \times \mathbb{S}^{n_m-1}$$ define the feasible set. The objective function is a multihomogeneous polynomial $f \in \mathbb{R}[\bm{x}]_{=(d_1,\ldots,d_m)}$, i.e., it is homogeneous of degree $d_i$ in each block $\bm{x}_i$. The optimization task is

$\min_{\bm{x} \in S} f(\bm{x}).$

Such multihomogeneous POPs encompass, for instance, best rank-one tensor approximation, where $f(\bm{x})$ represents the contraction of a tensor with $m$ unit vectors.

The Moment-SOS Hierarchy

Lasserre's moment-SOS hierarchy produces a sequence of SDP relaxations, generating non-decreasing lower bounds to the global minimum. The hierarchy is built upon real algebraic geometry—specifically Positivstellensatz representations and moment duality. Although asymptotic convergence to the true minimum holds under Archimedean assumptions, finite convergence is generally not guaranteed and is heavily dependent on subtle geometric and algebraic properties of the feasible set and the objective.

The paper links finite convergence to satisfaction of strict local optimality conditions (linear independence constraint qualification (LICQ), strict complementarity, and second-order sufficiency) at all global minimizers, as established in [huangnie_newflatness_2024].

Main Theorem and Methods

The central theorem asserts finite convergence for generic multihomogeneous polynomials on $S$:

• The set of "bad" polynomials—those for which the hierarchy does not converge finitely—forms a Lebesgue measure zero subset in coefficient space.
• At a high enough order, solutions of the moment hierarchy satisfy flat truncation, enabling extraction of the optimizer measure and hence the minimizers.

Differential Topology and Morse Theory

To establish genericity of the relevant local optimality conditions, the authors invoke Morse-theoretic and transversality machinery. Specifically:

• Critical points of a smooth function restricted to a manifold are characterized using the tangent/cotangent structure and Lagrange multipliers.
• Nondegeneracy (the second-order sufficiency condition) is shown to hold generically via a parametric family argument: for a parametric family of polynomials, the set of coefficients for which critical points degenerate is of measure zero.
• The surjectivity of the parameter-to-cotangent map at each feasible point is established, ensuring the existence of sufficiently many parameters (monomials) to control criticality generically.
• The union bound over all active-constraint patterns is handled via parametric transversality, extending the results to settings with additional polynomial inequality constraints.

The proof leverages the block structure of the product of spheres; explicit calculations show the linear independence of active constraint gradients everywhere on $S$.

Application to Tensor Optimization

The key application is best rank-one tensor approximation. For a given tensor $$\mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m}$$, finding the rank-one tensor closest to $\mathcal{A}$ in Hilbert-Schmidt norm reduces to

$$\max_{\bm{x}_i \in \mathbb{S}^{n_i-1}} |\langle \mathcal{A}, \bm{x}_1 \otimes \cdots \otimes \bm{x}_m \rangle|,$$

which exactly fits the framework above. Thus,

• For generic tensors, the hierarchy computes the global best rank-one approximation in finitely many steps, with explicit certification of minimizers via flat extension.
• For symmetric tensors, the setup reduces to homogeneous polynomials over a single sphere, recovering prior results.

These guarantees enable efficient and certifiable algorithms for tensor approximation via general-purpose SDP relaxations.

Extension to Inequality Constraints and Broader Feasible Sets

The paper extends its finite convergence result to broader families of constraints, notably including semialgebraic feasible sets defined by equality and inequality constraints. By adapting the Morse-theoretic and transversality arguments, and under linear independence assumptions on active constraint gradients, the hierarchy achieves finite convergence generically for arbitrary feasible manifolds (e.g., products of simplices, other quadratic varieties), provided the polynomial space is sufficiently expressive relative to the feasible set's dimension.

This encompasses, for example, quantum separability testing, where moment approaches are applied to higher-order bi-spherical sets [doherty2004complete, dressler2022separability].

Implications and Future Directions

Practically, the results offer constructive, certified SDP-based solvers for generic multihomogeneous POPs on products of spheres (and, more generally, on semialgebraic manifolds), with applicability to polynomial approaches for tensor analysis and quantum separability.

Theoretically, these results:

• Generalize classical Morse theory and algebraic geometric arguments to the moment-SOS setting for broad multihomogeneous contexts.
• Elucidate the strong interplay between real algebraic geometry, differential topology, and polynomial optimization.
• Show that the obstacles to finite convergence are non-generic, and that the hierarchy can systematically certify optimality for almost all instances in high-dimensional polynomial spaces.

Future research may address explicit upper bounds on the required relaxation order—a major open problem—either through refined measure estimates, algebraic degree bounds, or exploiting further structure in tensorial/algebraic problems. Additionally, extending non-generic (structured tensor) results, or strengthening certificates for complex or symmetric settings, remains an important avenue. Finally, connections to hierarchies for quantum separability and extensions to generalized moment problems (including the optimization over products of higher-dimensional varieties or noncompact sets) provide abundant opportunities for cross-fertilization.

Conclusion

This work rigorously establishes that the moment-SOS hierarchy possesses the finite convergence property for generic multihomogeneous polynomial optimization over products of spheres and, by implication, for a wide range of tensor optimization problems. The main approach synthesizes deep tools from real algebraic geometry, SDP hierarchies, and Morse theory, yielding constructive and certifiable algorithms with robust theoretical guarantees for almost all instances of these challenging polynomial optimization problems.