Distributed Optimization and Scaling Design for Solving Sylvester Equations

This paper develops distributed algorithms for solving Sylvester equations. The authors transform solving Sylvester equations into a distributed optimization problem, unifying all eight standard distributed matrix structures. Then the authors propose a distributed algorithm to find the least squares...

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Bibliographic Details
Published in:Journal of systems science and complexity 2024-12, Vol.37 (6), p.2487-2510
Main Authors: Cheng, Songsong, Yu, Xin, Zeng, Xianlin, Liang, Shu, Hong, Yiguang
Format: Article
Language:English
Subjects:
Algorithms
Complex Systems
Control
Design optimization
Design standards
Mathematics
Mathematics and Statistics
Mathematics of Computing
Multiagent systems
Operations Research/Decision Theory
Scaling
Statistics
Systems Theory
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Summary:This paper develops distributed algorithms for solving Sylvester equations. The authors transform solving Sylvester equations into a distributed optimization problem, unifying all eight standard distributed matrix structures. Then the authors propose a distributed algorithm to find the least squares solution and achieve an explicit linear convergence rate. These results are obtained by carefully choosing the step-size of the algorithm, which requires particular information of data and Laplacian matrices. To avoid these centralized quantities, the authors further develop a distributed scaling technique by using local information only. As a result, the proposed distributed algorithm along with the distributed scaling design yields a universal method for solving Sylvester equations over a multi-agent network with the constant step-size freely chosen from configurable intervals. Finally, the authors provide three examples to illustrate the effectiveness of the proposed algorithms.
ISSN:1009-6124
1559-7067
DOI:10.1007/s11424-024-3407-6