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

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.