{"ID":2852907,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.17760","arxiv_id":"2510.17760","title":"Nonlinear Rayleigh quotient optimization","abstract":"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.","short_abstract":"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 proje...","url_abs":"https://arxiv.org/abs/2510.17760","url_pdf":"https://arxiv.org/pdf/2510.17760v1","authors":"[\"Flavio Salizzoni\",\"Luca Sodomaco\",\"Julian Weigert\"]","published":"2025-10-20T17:12:15Z","proceeding":"math.AG","tasks":"[\"math.AG\",\"math.OC\"]","methods":"[]","has_code":false}