A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks

Seyyed Shaho Alaviani, Nicola Elia

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

This paper considers the problem of solving linear algebraic equations of the form Ax=b among multi agents which seek a solution by using local information in presence of random communication topologies. The equation is solved by m agents where each agent only knows a subset of rows of the partitioned matrix [A,b]. The problem is formulated such that this formulation does not need the distribution of random interconnection graphs. Therefore, this framework includes asynchronous updates or unreliable communication protocols. The random Krasnoselskii-Mann iterative algorithm is applied that converges almost surely and in mean square to a solution of the problem for any matrices A and b and any initial conditions of agents' states. The algorithm is a totally asynchronous algorithm without requiring a priori B-connectivity and distribution dependency assumptions. The algorithm is able to solve the problem even if the weighted matrix of the graph is periodic and irreducible for synchronous protocol. It is demonstrated that the limit point to which the agents' states converge is determined by the unique solution of a convex optimization problem regardless of the distribution of random co

Original languageEnglish (US)
JournalIEEE Transactions on Automatic Control
DOIs
StateAccepted/In press - 2020

Bibliographical note

Funding Information:
Manuscript received May 29, 2019; revised January 19, 2020 and June 16, 2020; accepted June 29, 2020. Date of publication July 20, 2020; date of current version April 26, 2021. This work was supported by the National Science Foundation under Grant CCF-1320643, under Grant CNS-1239319, and AFOSR Grant FA 9550-15-1-0119. This paper was presented in part at 57th IEEE Conference on Decision and Control, Miami Beach, FL, USA, December 2018. [36]. Recommended by Associate Editor D. Hristu-Varsakelis. (Corresponding author: Seyyed Shaho Alaviani.) Seyyed Shaho Alaviani was with the Iowa State University, Ames, IA 50011, USA, and now with the Department of Mechanical Engineering, Clemson University, Clemson, SC 29634 USA (e-mail: salavia@clemson.edu).

Publisher Copyright:
IEEE

Keywords

  • Convex functions
  • Distributed algorithms
  • Hilbert space
  • Mathematical model
  • Network topology
  • Protocols
  • Topology
  • asynchronous
  • distributed algorithm
  • linear algebraic equations
  • random graphs

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