Network flows as least squares solvers for linear equations

This paper presents a first-order continuous-time distributed step-size algorithm for computing the least squares solution to a linear equation over networks. Given the uniqueness of the solution and nonintegrable step size, the convergence results are provided for fixed graphs. For the nonunique solution and square integrable step size, the convergence is shown for constantly connected switching graphs. We also validate the results and illustrate possible impacts on the convergence speed using a few numerical examples.

[1]  Bahman Gharesifard,et al.  Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs , 2012, IEEE Transactions on Automatic Control.

[2]  Brian D. O. Anderson,et al.  Network Flows That Solve Linear Equations , 2015, IEEE Transactions on Automatic Control.

[3]  Jing Wang,et al.  Control approach to distributed optimization , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[4]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[5]  Jing Wang,et al.  Solving Systems of Linear Equations by Distributed Convex Optimization in the Presence of Stochastic Uncertainty , 2014 .

[6]  P. Barooah,et al.  Graph Effective Resistance and Distributed Control: Spectral Properties and Applications , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[7]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Multi-Agent Optimization , 2009, IEEE Transactions on Automatic Control.

[8]  Angelia Nedic,et al.  Distributed Optimization Over Time-Varying Directed Graphs , 2015, IEEE Trans. Autom. Control..

[9]  Jing Wang,et al.  Distributed Least Square with intermittent communications , 2012, 2012 American Control Conference (ACC).

[10]  Cong Wang,et al.  Harnessing the Cloud for Securely Outsourcing Large-Scale Systems of Linear Equations , 2013, IEEE Transactions on Parallel and Distributed Systems.

[11]  Ali H. Sayed,et al.  Diffusion recursive least-squares for distributed estimation over adaptive networks , 2008, IEEE Transactions on Signal Processing.

[12]  Shaoshuai Mou,et al.  An asynchronous distributed algorithm for solving a linear algebraic equation , 2013, 52nd IEEE Conference on Decision and Control.

[13]  B. Anderson,et al.  Exponential Least Squares Solvers for Linear Equations over Networks , 2017 .

[14]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[15]  Jing Wang,et al.  Distributed solution of linear equations over unreliable networks , 2016, 2016 American Control Conference (ACC).

[16]  Shaoshuai Mou,et al.  A fixed-neighbor, distributed algorithm for solving a linear algebraic equation , 2013, 2013 European Control Conference (ECC).

[17]  Shaoshuai Mou,et al.  Decentralized gradient algorithm for solution of a linear equation , 2015, ArXiv.

[18]  Shaoshuai Mou,et al.  A Distributed Algorithm for Solving a Linear Algebraic Equation , 2013, IEEE Transactions on Automatic Control.