Solving Stengle's Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914214160891904 |
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| author | Henrion, Didier |
| author_facet | Henrion, Didier |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solving Stengle's Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations Henrion, Didier Optimization and Control 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. |
| title | Solving Stengle's Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2512.19141 |