Evaluating approximations of the semidefinite cone with trace normalized distance.

We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely DD n ∗ (resp., SDD n ∗ ), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. [5], between a set S and the semidefinite cone has the same value whenever SDD n ∗ ⊆ S ⊆ DD n ∗ . This implies that the norm normalized distance is not a sufficient measure to evaluate these approximations. As a new measure to compensate for the weakness of that distance, we propose a new distance, called the trace normalized distance. We prove that the trace normalized distance between DD n ∗ and S + n has a different value from the one between SDD n ∗ and S + n and give the exact values of these distances. [ABSTRACT FROM AUTHOR]