Distributed optimisation approach to least-squares solution of Sylvester equations

In this study, the authors design distributed algorithms for solving the Sylvester equation AX+XB=C in the sense of least squares over a multi-agent network. In the problem setup, every agent in the interconnected system only has local information of some columns or rows of data matrices A, B and C, and exchanges information among neighbour agents. They propose algorithms with mainly focusing on a specific partition case, whose designs can be easily generalised to other partitions. Three distributed continuous-time algorithms aim at two cases for seeking a least-squares/regularisation solution from the viewpoint of optimisation. Due to the equivalence between an equilibrium point of each system under discussion and an optimal solution to the corresponding optimisation problem, the authors make use of semi-stability theory and methods in convex optimisation to prove convergence theorems of proposed algorithms that arrive at a least-squares/regularisation solution.