Solving Linear Equations with Separable Problem Data over Directed Networks

TL;DR

This paper introduces distributed algorithms for solving linear equations over directed networks, leveraging reformulation as an unconstrained optimization and dynamic average consensus, applicable to both time-varying and general directed graphs.

Contribution

It presents novel distributed algorithms that do not require invertible matrices and work on unbalanced directed networks, expanding the applicability of linear equation solutions in networked systems.

Findings

Algorithms are exponentially stable and distributed.

Effective on time-varying and unbalanced directed networks.

Numerical simulations confirm theoretical results.

Abstract

This paper deals with linear algebraic equations where the global coefficient matrix and constant vector are given respectively, by the summation of the coefficient matrices and constant vectors of the individual agents. Our approach is based on reformulating the original problem as an unconstrained optimization. Based on this exact reformulation, we first provide a gradient-based, centralized algorithm which serves as a reference for the ensuing design of distributed algorithms. We propose two sets of exponentially stable continuous-time distributed algorithms that do not require the individual agent matrices to be invertible, and are based on estimating non-distributed terms in the centralized algorithm using dynamic average consensus. The first algorithm works for time-varying weight-balanced directed networks, and the second algorithm works for general directed networks for which the communication graphs might not be balanced. Numerical simulations illustrate our results.