Journal of Global Optimization

, Volume 47, Issue 3, pp 437–456 | Cite as

Convergence rate for consensus with delays



We study the problem of reaching a consensus in the values of a distributed system of agents with time-varying connectivity in the presence of delays. We consider a widely studied consensus algorithm, in which at each time step, every agent forms a weighted average of its own value with values received from the neighboring agents. We study an asynchronous operation of this algorithm using delayed agent values. Our focus is on establishing convergence rate results for this algorithm. In particular, we first show convergence to consensus under a bounded delay condition and some connectivity and intercommunication conditions imposed on the multi-agent system. We then provide a bound on the time required to reach the consensus. Our bound is given as an explicit function of the system parameters including the delay bound and the bound on agents’ intercommunication intervals.


Distributed consensus Asynchronous Convergence Delays 


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  1. 1.
    Angeli, D., Bliman, P.-A.: Convergence speed of unsteady distributed consensus: decay estimate along the settling spanning-trees. (2006)
  2. 2.
    Bertsekas D.P., Tsitsiklis J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont, MA (1997)Google Scholar
  3. 3.
    Bliman, P.-A., Ferrari-Trecate, G.: Average consensus problems in networks of agents with delayed communications. (2005)
  4. 4.
    Blondel, V.D., Hendrickx, J.M., Olshevsky, A., Tsitsiklis, J.N.: Convergence in multiagent coordination, consensus, and flocking. Proceeding of 44th IEEE Conference on Decision and Control, pp. 2996–3000 (2005)Google Scholar
  5. 5.
    Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Gossip algorithms: design, analysis, and applications. Proceedings of IEEE INFOCOM, 24th Joint Conference of the IEEE Computer and Communications Societies, vol. 3, pp. 1653–1664 (2005)Google Scholar
  6. 6.
    Cao, M., Spielman, D.A., Morse, A.S.: A lower bound on convergence of a distributed network consensus algorithm. Proceedings of 44th IEEE Conference on Decision and Control, pp. 2356–2361 (2005)Google Scholar
  7. 7.
    Carli, R., Fagnani, F., Speranzon, A., Zampieri, S.: Communication constraints in coordinated consensus problems. Proceedings of IEEE American Control Conference, pp. 4189–4194 (2006)Google Scholar
  8. 8.
    Carli, R., Fagnani, F., Frasca, P., Taylor, T., Zampieri, S.: Average consensus on networks with transmission noise or quantization. Proceedings of European Control Congerence (2007)Google Scholar
  9. 9.
    Jadbabaie A., Lin J., Morse S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003)CrossRefGoogle Scholar
  10. 10.
    Kashyap, A., Basar, T., Srikant, R.: Consensus with quantized information updates. Proceedings of 45th IEEE Conference on Decision and Control, pp. 2728–2733 (2006)Google Scholar
  11. 11.
    Li S., Basar T.: Asymptotic agreement and convergence of asynchronous stochastic algorithms. IEEE Trans. on Autom. Control 32(7), 612–618 (1987)CrossRefGoogle Scholar
  12. 12.
    Moallemi, C., Van Roy, B.: Consensus propagation. Advances in neural information processing systems 18. In: Weiss, Y., Scholkopf, B., Platt, J. (eds.) Proceedings of Neural Information Processing Systems, pp. 899–906 (2005)Google Scholar
  13. 13.
    Nedić, A., Ozdaglar, A. : On the rate of convergence of distributed subgradient methods for multi-agent optimization. Proceedings of 46th IEEE Conference on Decision and Control, pp. 4711–4716 (2007)Google Scholar
  14. 14.
    Nedić A., Ozdaglar A.: Distributed subradient methods for multi-agent optimization. IEEE Trans. on Autom. Control (2009, to appear)Google Scholar
  15. 15.
    Olfati-Saber R., Murray R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004)CrossRefGoogle Scholar
  16. 16.
    Olshevsky, A., Tsitsiklis, J.N.: Convergence rates in distributed consensus and averaging, Proceedings of 46th IEEE Conference on Decision and Control, pp. 3387–3392 (2007)Google Scholar
  17. 17.
    Olshevsky, A., Tsitsiklis, J.N.:Convergence speed in distributed consensus and averaging, SIAM J. Control Optim. archive link: arXiv:math/0612682v1 (2006, forthcoming)
  18. 18.
    Tsitsiklis, J.N.: Problems in decentralized decision making and computation. Ph.D. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (1984)Google Scholar
  19. 19.
    Tsitsiklis J.N., Athans M.: Convergence and asymptotic agreement in distributed decision problems. IEEE Trans. Autom. Control 29(1), 42–50 (1984)CrossRefGoogle Scholar
  20. 20.
    Tsitsiklis J.N., Bertsekas D.P., Athans M.: Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Trans. Autom. Control 31(9), 803–812 (1986)CrossRefGoogle Scholar
  21. 21.
    Xiao L., Boyd S.: Fast linear iterations for distributed averaging. Syst. Control Lett. 53, 65–78 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of Industrial and Enterprise Systems EngineeringUniversity of IllinoisUrbanaChampaignUSA
  2. 2.Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA

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