A Matrix Positivstellensatz with lifting polynomials

TL;DR

This paper introduces a new matrix Positivstellensatz with lifting polynomials to certify set containment in polynomial matrix inequalities, enabling semidefinite programming solutions and applications to spectrahedrop containment.

Contribution

It proposes a novel matrix Positivstellensatz with lifting polynomials for set containment certification, extending existing methods under classical conditions.

Findings

Containment can be certified via semidefinite programming.

Applicable to spectrahedrop containment problems.

Provides a new algebraic certificate for polynomial matrix inequalities.

Abstract

Given the projections of two semialgebraic sets defined by polynomial matrix inequalities, it is in general difficult to determine whether one is contained in the other. To address this issue we propose a new matrix Positivstellensatz that uses lifting polynomials. Under the classical archimedean condition and some mild natural assumptions, we prove that such a containment holds if and only if the proposed matrix Positivstellensatz is satisfied. The corresponding certificate can be searched for by solving a semidefinite program. An important application is to certify when a spectrahedrop (i.e., the projection of a spectrahedron) is contained in another one.