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Solving Stengle’s Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations

Year: 2026

Type: article

Abstract: We revisit Stengle’s classical univariate polynomial optimization example min 1 - x 2 s.t. ( 1 - x 2 ) 3 ≥ 0 whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order r ≥ 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.

Source: Open Journal of Mathematical Optimization

DOI: 10.5802/ojmo.53

Author: Didier Henrion

Institutions: Centre National de la Recherche Scientifique, Laboratoire d'Analyse et d'Architecture des Systèmes, Université de Toulouse, Czech Technical University in Prague

Language: English

Topic: Advanced Optimization Algorithms Research

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

OpenAlex ID: W7117131232

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