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

TL;DR

This paper analyzes Stengle's classical polynomial optimization problem, demonstrating that the moment-SOS hierarchy yields exact relaxation values at order r ≥ 3 using rational arithmetic and classical polynomial properties.

Contribution

It provides the first explicit rational arithmetic proof of the exactness of the moment-SOS hierarchy for Stengle's example at relaxation order r ≥ 3.

Findings

Moment-SOS relaxation order r ≥ 3 yields exact value -1/r(r - 2)

Constructs dual SOS certificate and primal measure in rational arithmetic

Uses Chebyshev, Gegenbauer polynomials, and Christoffel-Darboux kernel

Abstract

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