The matricial relaxation of a linear matrix inequality

Abstract

Given linear matrix inequalities (LMIs) L1 and L2, it is natural to ask: (Q1) when does one dominate the other, that is, does L1(X) PsD imply L2(X) PsD? (Q2) when do they have the same solution set? Such questions can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables xj. With this relaxation, the domination questions (Q1) and (Q2) have elegant answers, indeed reduce to constructible semidefinite programs. Assume there is an X such that L1(X) and L2(X) are both PD, and suppose the positivity domain of L1 is bounded. For our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the existence of matrices Vj such that L2(x)=V1* L1(x) V1 + ... + Vk* L1(x) Vk. As for (Q2) we show that, up to redundancy, L1 and L2 are unitarily equivalent. Such algebraic certificates are typically called Positivstellensaetze and the above are examples of such for linear polynomials. The paper goes on to derive a cleaner and more powerful Putinar-type Positivstellensatz for polynomials positive on a bounded set of the form X | L(X) PsD. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map from a subspace of matrices to a matrix algebra is "completely positive".