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 s...

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Bibliographic Details
Published in:Optimization letters 2023-05, Vol.17 (4), p.917-934
Main Authors: Wang, Yuzhu, Yoshise, Akiko
Format: Article
Language:English
Subjects:
Computational Intelligence
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Operations Research/Decision Theory
Optimization
Original Paper
Simulation
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Summary: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.
ISSN:1862-4472
1862-4480
DOI:10.1007/s11590-022-01908-3