Consensus and balancing on the three-sphere

We study consensus and anti-consensus on the 3-sphere as the global optimization problems. The corresponding gradient descent algorithm is a dynamical systems on S 3 , that is known in Physics as non-Abelian Kuramoto model . This observation opens a slightly different insight into some previous resu...

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Bibliographic Details
Published in:Journal of global optimization 2020-03, Vol.76 (3), p.575-586
Main Authors: Crnkic, Aladin, Jacimovic, Vladimir
Format: Article
Language:English
Subjects:
Algorithms
Analysis
Computer Science
Dynamical systems
Global optimization
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Real Functions
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Summary:We study consensus and anti-consensus on the 3-sphere as the global optimization problems. The corresponding gradient descent algorithm is a dynamical systems on S 3 , that is known in Physics as non-Abelian Kuramoto model . This observation opens a slightly different insight into some previous results and also enables us to prove some novel results concerning consensus and balancing over the complete graph. In this way we fill some gaps in the existing theory. In particular, we prove that the anti-consensus algorithm over the complete graph on S 3 converges towards a balanced configuration if a certain mild condition on initial positions of agents is satisfied. The form of this condition indicates an unexpected relation with some important constructions from Complex Analysis.
ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-018-0723-1