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 ≥ 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.
Fields of Study & Subjects
Polynomial and algebraic computation
Advanced Optimization Algorithms Research
Mathematical functions and polynomials
Publication details
- Journal / source
- Open Journal of Mathematical Optimization
- Volume
- 7
- Pages
- 1–18
- Publication date
- 23 September 2026
- Language
- EN