Every free basic convex semi-algebraic set has an LMI representation

TL;DR

This paper proves that every free basic convex semi-algebraic set can be represented by a Linear Matrix Inequality (LMI), establishing a fundamental link between convexity and LMI representations in non-commutative algebra.

Contribution

It establishes that convexity of certain non-commutative semi-algebraic sets implies they have an LMI representation, solving a core problem in non-commutative convex algebra.

Findings

Convexity implies LMI representation for bounded non-commutative semi-algebraic sets.

The main theorem confirms the converse of the known fact that LMI sets are convex.

The result has implications for semidefinite programming and systems engineering.

Abstract

The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex basic non-commutative semi-algebraic set. The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and systems engineering. For p(x) a non-commutative polynomial in free variables x= (x1, ... xg) we can substitute a tuple of symmetric matrices X= (X1, ... Xg) for x and obtain a matrix p(X). Assume p is symmetric with p(0) invertible, let Ip denote the set {X: p(X) is an invertible matrix}, and let Dp denote the component of Ip containing 0. THEOREM: If the set Dp is uniformly bounded independent of the size of the matrix tuples, then Dp has an LMI representation if and only if it is convex. Linear engineering systems problems are called "dimension free" if they can be stated purely in terms of a signal flow diagram with L2 performance measures, e.g., H-infinity control. Conjecture: A dimension free problem can be made convex if and only it can be made into an LMI. The theorem here settles the core case affirmatively.