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

Abstract

We revisit Stengle's classical univariate polynomial optimization example $min1-x^{2}s.t.(1-x^2{)}^3\ge 0$ 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 polynomial, and a Christoffel-Darboux kernel argument.