Polynomial Matrix Inequality and Semidefinite Representation

TL;DR

This paper investigates conditions under which convex sets defined by polynomial or rational matrix inequalities can be represented as semidefinite programs, providing explicit constructions and examples for various cases.

Contribution

It establishes sufficient conditions for semidefinite representability of sets defined by polynomial or rational matrix inequalities, including explicit SDP constructions.

Findings

SDP representability when the domain is the whole space and the polynomial is matrix sos-concave.

SDP representability for compact convex domains with strictly matrix concave polynomials.

Explicit SDP representations for rational matrix functions that are q-module matrix concave.

Abstract

Consider a convex set S defined by a matrix inequality of polynomials or rational functions over a domain. The set S is called semidefinite programming (SDP) representable or just semidefinite representable if it equals the projection of a higher dimensional set which is defined by a linear matrix inequality (LMI). This paper studies sufficient conditions guaranteeing semidefinite representability of S. We prove that S is semidefinite representable in the following cases: (i) the domain is the whole space and the matrix polynomial is matrix sos-concave; (ii) the domain is compact convex and the matrix polynomial is strictly matrix concave; (iii) the rational matrix function is q-module matrix concave on the domain. Explicit constructions of SDP representations are given. Some examples are illustrated.