Convergence rates of S.O.S hierarchies for polynomial semidefinite programs

TL;DR

This paper develops a sum-of-squares hierarchy for polynomial semidefinite programs, providing new convergence rate results and bounds on polynomial degrees needed for positivity certification within matrix polynomial constraints.

Contribution

It introduces a novel S.o.S hierarchy for matrix polynomial constraints and analyzes its convergence rates, extending Putinar's theorem to this setting.

Findings

Derived convergence rates for the S.o.S hierarchy.

Established bounds on degrees of polynomials for positivity certification.

Extended Putinar's theorem for matrix polynomial semidefinite sets.

Abstract

We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly address the matrix-based constraint, making it applicable to both discrete and continuous polynomial optimization problems. We investigate the convergence rates of these bounds in both types of problems. The proposed method yields a variant of Putinar's theorem, tailored for positive polynomials within a compact semidefinite set $\mathcal{X}$ defined by a matrix polynomial semidefinite constraint. More specifically, we derive novel insights into the convergence rates and bounds on the degree of the S.o.S polynomials required to certify positivity on $\mathcal{X}$, based on Jackson's theorem and a variant of the {\L}ojasiewicz inequality.