Distributed Computation of Linear Matrix Equations: An Optimization Perspective

TL;DR

This paper presents distributed algorithms for solving linear matrix equations over multi-agent networks by formulating the problem as a constrained optimization task and proving exponential convergence to least squares solutions.

Contribution

It introduces novel decomposition methods and continuous-time algorithms for distributed matrix equation solving with proven exponential convergence.

Findings

Algorithms converge exponentially to least squares solutions.

Distributed methods work with partial matrix information in multi-agent networks.

Solutions are equivalent to least squares solutions of the original matrix equations.

Abstract

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 multi-agent networks from an optimization perspective. In this paper, we consider the standard distributed matrix-information structures, where each agent of the considered multi-agent network has access to one of the sub-block 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.