Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911790889172992 |
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| author | Miller, Jared Meroni, Chiara Tacchi, Matteo Velasco, Mauricio |
| author_facet | Miller, Jared Meroni, Chiara Tacchi, Matteo Velasco, Mauricio |
| contents | This paper presents an algorithm to maximize the volume of an affine slice through a given semialgebraic set. This slice-volume task is formulated as an infinite-dimensional linear program in continuous functions, inspired by prior work in volume computation of semialgebraic sets. A convergent sequence of upper-bounds to the maximal slice volume are computed using the moment-Sum-of-Squares hierarchy of semidefinite programs in increasing size. The computational complexity of this scheme can be reduced by utilizing topological structure (in dimensions 2, 3, 4, 8) and symmetry. This numerical convergence can be accelerated through the introduction of redundant Stokes-based constraints. Demonstrations of slice-volume calculation are performed on example sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04438 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming Miller, Jared Meroni, Chiara Tacchi, Matteo Velasco, Mauricio Optimization and Control Algebraic Geometry This paper presents an algorithm to maximize the volume of an affine slice through a given semialgebraic set. This slice-volume task is formulated as an infinite-dimensional linear program in continuous functions, inspired by prior work in volume computation of semialgebraic sets. A convergent sequence of upper-bounds to the maximal slice volume are computed using the moment-Sum-of-Squares hierarchy of semidefinite programs in increasing size. The computational complexity of this scheme can be reduced by utilizing topological structure (in dimensions 2, 3, 4, 8) and symmetry. This numerical convergence can be accelerated through the introduction of redundant Stokes-based constraints. Demonstrations of slice-volume calculation are performed on example sets. |
| title | Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming |
| topic | Optimization and Control Algebraic Geometry |
| url | https://arxiv.org/abs/2403.04438 |