Finite Convergence of the Moment-SOS Hierarchy on the Product of Spheres
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914201500385280 |
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| author | Halaseh, Sami Magron, Victor Skomra, Mateusz |
| author_facet | Halaseh, Sami Magron, Victor Skomra, Mateusz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11119 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite Convergence of the Moment-SOS Hierarchy on the Product of Spheres Halaseh, Sami Magron, Victor Skomra, Mateusz Optimization and Control Algebraic Geometry Differential Geometry 90C23, 15A69, 53Z99, 14P10 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. |
| title | Finite Convergence of the Moment-SOS Hierarchy on the Product of Spheres |
| topic | Optimization and Control Algebraic Geometry Differential Geometry 90C23, 15A69, 53Z99, 14P10 |
| url | https://arxiv.org/abs/2512.11119 |