On the complexity of matrix Putinar's Positivstellensatz

TL;DR

This paper establishes polynomial degree bounds for matrix Putinar's Positivstellensatz on semialgebraic sets, providing convergence rate estimates for matrix sum-of-squares relaxations and addressing open questions in the field.

Contribution

It proves polynomial bounds on degrees in matrix Positivstellensatz representations and convergence rates, extending results to unbounded sets via homogenization.

Findings

Polynomial degree bounds for matrix Positivstellensatz representations.

Polynomial bounds on convergence rates of matrix sum-of-squares relaxations.

Extension of bounds to unbounded sets using homogenization techniques.

Abstract

This paper studies the complexity of matrix Putinar's Positivstellens{\"a}tz on the semialgebraic set that is given by the polynomial matrix inequality. \rev{When the quadratic module generated by the constrained polynomial matrix is Archimedean}, we prove a polynomial bound on the degrees of terms appearing in the representation of matrix Putinar's Positivstellens{\"a}tz. Estimates on the exponent and constant are given. As a byproduct, a polynomial bound on the convergence rate of matrix sum-of-squares relaxations is obtained, which resolves an open question raised by Dinh and Pham. When the constraining set is unbounded, we also prove a similar bound for the matrix version of Putinar--Vasilescu's Positivstellens{\"a}tz by exploiting homogenization techniques.