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 <inline-formula><tex-math notation="LaTeX">${{AXB}} = F$</tex-math></inline-formula>, with the matrices <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$B$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$X$</tex-math></inline-formula>, and <inline-formula><tex-math notation="LaTeX">$F$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$B$</tex-math></inline-formula>, and <inline-formula><tex-math notation="LaTeX">$F$</tex-math></inline-formula>. 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.