Abstract

Linear matrix inequalities LMIs are common in many areas: control systems, combinatorial optimization, statistics, etc. They often have unknowns x= ( x_1, ... ,x_n) with x_j scalars, but in many problems of control, certainly the classical ones, the unknowns enter naturally as matrices.

The talk treats several topics involving LMIs with matrix unknowns:

A basic question in light of the fact that convexity, a seemingly much weaker condition than being an LMI , guarantees numerical success is: How much more restricted are LMIs than Convex MIs? It turns out that scalar unknowns vs matrix unknowns makes a huge difference in the answer.

Can we transform a problem to being convex?

LMI domination: L dominates \L means L(x) is positive definite implies \L(x) is positive definite. Checking for domination numerically can be NP hard. However, we relax the problem by insisting on domination whenever the unknowns x_j are matrices. There is an elegant algebraic characterization of relaxed LMI domination and numerical solution is no longer NP hard. Roughly what we observed is that this matrix relaxation corresponds exactly to a very natural procedure in modern in modern functional analysis.