Nonlinear Rayleigh quotient optimization
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911222048227328 |
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| author | Salizzoni, Flavio Sodomaco, Luca Weigert, Julian |
| author_facet | Salizzoni, Flavio Sodomaco, Luca Weigert, Julian |
| contents | Rayleigh quotient minimization deals with optimizing a quadratic homogeneous function over a sphere. Its critical points correspond to the normalized eigenvectors of the symmetric matrix associated with the quadratic form. In this paper, we consider a homogeneous polynomial objective function $f$ over a sphere, a projective algebraic variety $X$, and we study the $X$-eigenpoints of $f$, which are classes of critical points of $f$ constrained to the sphere and the affine cone over $X$. The number of $X$-eigenpoints of a generic polynomial $f$ is the Rayleigh-Ritz degree of $X$. This invariant is a version of the Euclidean distance degree of a Veronese embedding of $X$. We provide concrete formulas in various scenarios, including those involving varieties of rank-one tensors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17760 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Rayleigh quotient optimization Salizzoni, Flavio Sodomaco, Luca Weigert, Julian Algebraic Geometry Optimization and Control 13P25, 14N05, 14N10, 14Q20, 15A18, 58K05, 90C26 Rayleigh quotient minimization deals with optimizing a quadratic homogeneous function over a sphere. Its critical points correspond to the normalized eigenvectors of the symmetric matrix associated with the quadratic form. In this paper, we consider a homogeneous polynomial objective function $f$ over a sphere, a projective algebraic variety $X$, and we study the $X$-eigenpoints of $f$, which are classes of critical points of $f$ constrained to the sphere and the affine cone over $X$. The number of $X$-eigenpoints of a generic polynomial $f$ is the Rayleigh-Ritz degree of $X$. This invariant is a version of the Euclidean distance degree of a Veronese embedding of $X$. We provide concrete formulas in various scenarios, including those involving varieties of rank-one tensors. |
| title | Nonlinear Rayleigh quotient optimization |
| topic | Algebraic Geometry Optimization and Control 13P25, 14N05, 14N10, 14Q20, 15A18, 58K05, 90C26 |
| url | https://arxiv.org/abs/2510.17760 |