We revisit Stengle’s classical univariate polynomial optimization example $\min 1-x^2$ s.t. $(1-x^2)^3\ge 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r\ge 3$ 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 polynomials, and a Christoffel–Darboux kernel argument.
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Didier Henrion 1
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Didier Henrion. Solving Stengle’s Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations. Open Journal of Mathematical Optimization, Volume 7 (2026), article no. 5, 18 p.. doi: 10.5802/ojmo.53
@article{OJMO_2026__7__A5_0,
author = {Didier Henrion},
title = {Solving {Stengle{\textquoteright}s} {Example} in {Rational} {Arithmetic:} {Exact} {Values} of the {Moment-SOS} {Relaxations}},
journal = {Open Journal of Mathematical Optimization},
eid = {5},
pages = {1--18},
year = {2026},
publisher = {Universit\'e de Montpellier},
volume = {7},
doi = {10.5802/ojmo.53},
language = {en},
url = {https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.53/}
}
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