Universality of the quadratic convergence exponent for bivariate POPs

Establish that, under the usual Archimedean assumptions and for every fixed bivariate polynomial optimization problem, the moment-SOS hierarchy satisfies an error bound e_r=O(1/r^2), thereby determining whether the quadratic convergence exponent remains universal in bivariate polynomial optimization.

Background

The paper proves a universal O(1/r^2) upper bound for the moment-SOS relaxation error of every fixed univariate polynomial optimization problem on a bounded subset of the real line, and gives a degree-four example attaining the exact rate 1/(2r(r-1)). It also shows that a particular bivariate cusp problem inherits the same asymptotic quadratic rate through an exact univariate reduction. However, these results do not establish a universal rate for general bivariate polynomial optimization problems. The authors explicitly identify as unresolved whether any bivariate problem can converge more slowly than O(1/r^2), and propose proving that no such example exists under the usual Archimedean assumptions and fixed problem data.