{"ID":2826149,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.19141","arxiv_id":"2512.19141","title":"Solving Stengle's Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations","abstract":"We revisit Stengle's classical univariate polynomial optimization example $min 1 - x^2 s.t. (1 - x^2)^3 \\geq 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r \\geq 3$ has the exact value $-1/r(r - 2)$. For this we construct in rational arithmetic a dual polynomial sum-of-squares (SOS) certificate and a primal moment sequence representing a finitely atomic measure. The key ingredients are elementary trigonometric properties of Chebyshev and Gegenbauer polynomial, and a Christoffel-Darboux kernel argument.","short_abstract":"We revisit Stengle's classical univariate polynomial optimization example $min 1 - x^2 s.t. (1 - x^2)^3 \\geq 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r \\geq 3$ has the exact value $-1/r(r - 2)$. For this we construct in rational arithme...","url_abs":"https://arxiv.org/abs/2512.19141","url_pdf":"https://arxiv.org/pdf/2512.19141v1","authors":"[\"Didier Henrion\"]","published":"2025-12-22T08:39:57Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}