Semidefinite programming in matrix unknowns which are dimension free

TL;DR

This paper surveys the development of dimension-free semidefinite programming in non-commutative algebra, highlighting its applications in control theory and the structural advantages over scalar-variable problems.

Contribution

It provides a comprehensive overview of convexity, non-commutative real algebraic geometry, and their integration into dimension-free semidefinite programming.

Findings

Non-commutative problems exhibit cleaner qualitative properties.

Relaxation from scalar to matrix variables reveals elegant structures.

Dimension-free SDP is closely linked to non-commutative algebraic geometry.

Abstract

One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain non-commutative polynomials in matrices. These polynomials depend only on the system layout and do not change with the size of the matrices involved, hence such problems are called "dimension-free". Analyzing dimension-free problems has led to the development recently of a non-commutative (nc) real algebraic geometry (RAG) which, when combined with convexity, produces dimension-free Semidefinite Programming. This article surveys what is known about convexity in the non-commutative setting and nc SDP and includes a brief survey of nc RAG. Typically, the qualitative properties of the non-commutative case are much cleaner than those of their scalar counterparts - variables in R^g. Indeed we describe how relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure.