- Evaluating approximations of the semidefinite cone with trace normalized distance Yuzhu Wang (wangyuzhu820yahoo.co.jp) Akiko Yoshise (yoshisesk.tsukuba.ac.jp) Abstract: We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely ${\cal DD}_n^*$ (resp., ${\cal SDD}_n^*$), as an approximation of the semidefinite cone. Using the measure proposed by Blekherman et al. (2020) called norm normalized distance, we prove that the norm normalized distance between a set ${\cal S}$ and the semidefinite cone has the same value whenever ${\cal SDD}_n^* \subseteq {\cal S} \subseteq {\cal DD}_n^*$. 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 trace normalized distance. We prove that the trace normalized distance between ${\cal DD}_n^*$ and ${\cal S}^n_+$ has a different value from the one between ${\cal SDD}_n^*$ and ${\cal S}^n_+$, and give the exact values of these distances. Keywords: Semidefinite optimization problem; Diagonally dominant matrix; Scaled diagonally dominant matrix. Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Citation: Discussion Paper Series, No.1376, Department of Policy and Planning Sciences, University of Tsukuba Download: [PDF]Entry Submitted: 05/28/2021Entry Accepted: 05/29/2021Entry Last Modified: 05/30/2021Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.