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 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.