Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

Abstract

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 counterpartsvariables in R g . Indeed we describe how relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure.

The convex Positivstellensatz in a free algebra

“…[BCR98,Las10,Lau09,Mar08,PD01,Sce09]. Positivstellensätze for polynomials in noncommuting variables are creatures of this century -see [HKM12,HM04a,KS07,PNA10,DLTW08]; for software equipped to dealing with positive noncommutative polynomials we refer to [HOSM+,CKP11]. Often in the noncommutative setting such theorems have cleaner statements than their commutative counterparts.…”

Aspects of non-commutative function theory

“…This has been studied, for example, in [15,16,18,19]. Let us consider a simple example, the Riccati matrix inequality: AX C XA XBB X C C C > 0:…”

Chapter 8: Free Convex Algebraic Geometry

“…For further engineering motivation we recommend the paper [SI95] or the longer version [SIG97] for related new directions. Descriptions of Positivstellensätze are in the surveys [HKM12b,dOHMP09,HP07,Smü09] with the first three also briskly touring free convexity. The survey article [HMPV09] is aimed at engineers.…”