Convergence rate of the moment-SOS hierarchy for univariate polynomial optimization

Abstract

We study the convergence rate of the moment-SOS (sum-of-squares) hierarchy for polynomial optimization problems (POPs) on a bounded subset of the real line described by arbitrary polynomial inequalities. We prove that, for every fixed univariate POP, the relaxation error is bounded by $O(1/r^2)$, where $r$ is the relaxation order. In particular, boundary degeneracies in the polynomial description of the feasible set affect the constant but not the convergence exponent. The proof combines the structure of finitely generated univariate quadratic modules with a Chebyshev polynomial construction that approximately recovers the natural generators of the feasible set while controlling the degree of the certificate. We also give an elementary degree-four example for which the relaxation error is exactly $1/(2r(r-1))$, showing that the quadratic rate is optimal. Equivalent reformulations connect this example to a cubic univariate problem and to a bivariate POP whose feasible set has a cusp singularity at the minimizer.