Relaxing LMI domination matricially

J. William Helton, Igor Klep, Scott A. McCullough
2010 49th IEEE Conference on Decision and Control (CDC)  
Given linear matrix inequalities (LMIs) L1 and L2 in the same number of variables it is natural to ask: (Q1) 1) does one dominate the other, that is, does L1(X) 0 imply L2(X) 0? 2) are they mutually dominant, that is, do they have the same solution set? Such problems can be NP-hard. We describe 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 semidefinite
more » ... semidefinite programs (SDP) which we show how to construct. For our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the existence of matrices Vj such that 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. 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".
doi:10.1109/cdc.2010.5717737 dblp:conf/cdc/HeltonKM10 fatcat:whig4wogh5aa3dv22owifgmspa