The matricial relaxation of a linear matrix inequality

AbstractGiven linear matrix inequalities (LMIs) L1 and L2 it is natural to ask: (Q1) when does one dominate the other, that is, does $${L_1(X) \succeq 0}$$ imply $${L_2(X) \succeq 0}$$? (Q2) when are they mutually dominant, that is, when do they have the same solution set? The matrix cube problem of Ben-Tal and Nemirovski (SIAM J Optim 12:811–833, 2002) is an example of LMI domination. Hence such problems 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. As an example, to test the strength of this relaxation we specialize it to the matrix cube problem and obtain essentially the relaxation given in Ben-Tal and Nemirovski (SIAM J Optim 12:811–833, 2002). Thus our relaxation could be viewed as generalizing it. Assume there is an X such that L1(X) and L2(X) are both positive definite, 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$$\begin{array}{ll}L_2(x) = V_1^{*} L_1(x) V_1 + \cdots + V_\mu^{*} L_1(x) V_{\mu}. \quad \quad \quad ({\rm A}_1)\end{array}$$As for (Q2) we show that L1 and L2 are mutually dominant if and only if, up to certain redundancies described in the paper, L1 and L2 are unitarily equivalent. Algebraic certificates for positivity, such as (A1) for linear polynomials, are typically called Positivstellensätze. The paper goes on to derive a Putinar-type Positivstellensatz for polynomials with a cleaner and more powerful conclusion under the stronger hypothesis of positivity on an underlying bounded domain of the form $${ \{X \mid L(X)\succeq0\}. }$$ 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 τ from a subspace of matrices to a matrix algebra is “completely positive”. Complete positivity is one of the main techniques of modern operator theory and the theory of operator algebras. On one hand it provides tools for studying LMIs and on the other hand, since completely positive maps are not so far from representations and generally are more tractable than their merely positive counterparts, the theory of completely positive maps provides perspective on the difficulties in solving LMI domination problems.