Positive Polynomials and Projections of Spectrahedra

TL;DR

This paper investigates the structure and limitations of spectrahedra projections in semidefinite optimization, proving new results on their convexity, facial structure, and Positivstellensatz, advancing understanding of semialgebraic sets.

Contribution

It introduces novel results on the closure properties, facial structure, and Positivstellensatz for projections of spectrahedra, addressing open problems and extending existing theories.

Findings

Closure of a projection of a spectrahedron is again such a projection.

Limitations of Lasserre and theta body relaxation methods are established.

A new Positivstellensatz for projections of spectrahedra is proved.

Abstract

This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain the main result of the paper "Exposed faces of semidefinite representable sets" of Netzer, Plaumann and Schweighofer. We also solve the open problems from that work. We further prove some helpful facts which can not be found in the existing literature, for example that the closure of a projection of a spectrahedron is again such a projection. We give a unified account of several results on convex hulls of curves and images of polynomial maps. We finally prove a Positivstellensatz for projections of spectrahedra, which exceeds the known results that only work for basic closed semialgebraic sets.