On semidefinite representations of non-closed sets

TL;DR

This paper introduces a new method to establish semidefinite representability for non-closed sets, expanding the understanding of spectrahedra and their projections in polynomial optimization.

Contribution

It develops a novel approach to prove semidefinite representability of non-closed sets, including interiors and face removals, which were not addressed in prior work.

Findings

Interior of semidefinite representable sets is semidefinite representable

Removing faces parametrized suitably preserves semidefinite representability

New method extends semidefinite representation theory to non-closed sets

Abstract

Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinite representable. So far, all results focus on the case of closed sets. In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinite representable set is shown to be semidefinite representable. More general, one can remove faces of a semidefinite representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way.