Optimal degree bounds for sum-of-squares representations

Abstract

The moment-sum-of-squares (SOS) hierarchy provides a sequence of bounds on the minimum of a polynomial over a basic closed semialgebraic set, i.e., a subset of $\mathbb{R}^n$ defined by finitely many polynomial constraints. We establish optimal bounds on the error of the SOS hierarchy in the following settings: 1. If the constraints define an Archimedean quadratic module: the error of the Putinar-based hierarchy at level $k$ is $Θ((\log k)^4/k^2)$ in the worst case. 2. If the constraints define a compact semialgebraic set: the error of the Schmüdgen-based hierarchy at level $k$ is $Θ((\log k)^2/k^2)$ in the worst case. 3. If the constraints define an Archimedean quadratic module and satisfy linear independence constraint qualification conditions (LICQ): the error of the hierarchy at level $k$ is $Θ(1/k^{4})$ in the worst case. Here, the upper bound holds already for the Putinar-based hierarchy, while the lower bound holds even against the Schmüdgen-based hierarchy. Equivalently, these give best-possible effective versions of Putinar's and Schmüdgen's Positivstellensätze for these settings (in terms of their dependence on $k$).