Distributed computation of linear matrix equations: An optimization perspective

This paper investigates the distributed computation of the well-known linear matrix equation in the form of AXB = F, with the matrices A, B, X, and F of appropriate dimensions, over multiagent networks from an optimization perspective. In this paper, we consider the standard distributed matrix-information structures, where each agent of the considered multiagent network has access to one of the subblock matrices of A, B, and F To be specific, we first propose different decomposition methods to reformulate the matrix equations in standard structures as distributed constrained optimization problems by introducing substitutional variables; we show that the solutions of the reformulated distributed optimization problems are equivalent to least squares solutions to original matrix equations; and we design distributed continuous-time algorithms for the constrained optimization problems, even by using augmented matrices and a derivative feedback technique. Moreover, we prove the exponential convergence of the algorithms to a least squares solution to the matrix equation for any initial condition.