Relaxing LMI domination matricially

Given linear matrix inequalities (LMIs) L1 and L2 in the same number of variables it is natural to ask: (Q1) does one dominate the other, that is, does L1(X) 0 imply L2(X) 0? (Q2) 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 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 L2(x) = V 1 L1(x)V1 +

• Type: Journal
• Year: 2010
• Where: CDC
• Authors: J. William Helton, Igor Klep, Scott A. McCullough
• Keywords: CDC 2010, Control Systems, Linear Matrix Inequalities, Matrix, Relaxed Lmi Domination