Computation of linear algebraic equations with solvability verification over multi-agent networks

Xianlin Zeng; Kai Cao

Kybernetika (2017)

  • Volume: 53, Issue: 5, page 803-819
  • ISSN: 0023-5954

Abstract

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In this paper, we consider the problem of solving a linear algebraic equation A x = b in a distributed way by a multi-agent system with a solvability verification requirement. In the problem formulation, each agent knows a few columns of A , different from the previous results with assuming that each agent knows a few rows of A and b . Then, a distributed continuous-time algorithm is proposed for solving the linear algebraic equation from a distributed constrained optimization viewpoint. The algorithm is proved to have two properties: firstly, the algorithm converges to a least squares solution of the linear algebraic equation with any initial condition; secondly, each agent in the algorithm knows the solvability property of the linear algebraic equation, that is, each agent knows whether the obtained least squares solution is an exact solution or not.

How to cite

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Zeng, Xianlin, and Cao, Kai. "Computation of linear algebraic equations with solvability verification over multi-agent networks." Kybernetika 53.5 (2017): 803-819. <http://eudml.org/doc/294712>.

@article{Zeng2017,
abstract = {In this paper, we consider the problem of solving a linear algebraic equation $Ax=b$ in a distributed way by a multi-agent system with a solvability verification requirement. In the problem formulation, each agent knows a few columns of $A$, different from the previous results with assuming that each agent knows a few rows of $A$ and $b$. Then, a distributed continuous-time algorithm is proposed for solving the linear algebraic equation from a distributed constrained optimization viewpoint. The algorithm is proved to have two properties: firstly, the algorithm converges to a least squares solution of the linear algebraic equation with any initial condition; secondly, each agent in the algorithm knows the solvability property of the linear algebraic equation, that is, each agent knows whether the obtained least squares solution is an exact solution or not.},
author = {Zeng, Xianlin, Cao, Kai},
journal = {Kybernetika},
keywords = {multi-agent network; distributed optimization; linear algebraic equation; least squares solution; solvability verification},
language = {eng},
number = {5},
pages = {803-819},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Computation of linear algebraic equations with solvability verification over multi-agent networks},
url = {http://eudml.org/doc/294712},
volume = {53},
year = {2017},
}

TY - JOUR
AU - Zeng, Xianlin
AU - Cao, Kai
TI - Computation of linear algebraic equations with solvability verification over multi-agent networks
JO - Kybernetika
PY - 2017
PB - Institute of Information Theory and Automation AS CR
VL - 53
IS - 5
SP - 803
EP - 819
AB - In this paper, we consider the problem of solving a linear algebraic equation $Ax=b$ in a distributed way by a multi-agent system with a solvability verification requirement. In the problem formulation, each agent knows a few columns of $A$, different from the previous results with assuming that each agent knows a few rows of $A$ and $b$. Then, a distributed continuous-time algorithm is proposed for solving the linear algebraic equation from a distributed constrained optimization viewpoint. The algorithm is proved to have two properties: firstly, the algorithm converges to a least squares solution of the linear algebraic equation with any initial condition; secondly, each agent in the algorithm knows the solvability property of the linear algebraic equation, that is, each agent knows whether the obtained least squares solution is an exact solution or not.
LA - eng
KW - multi-agent network; distributed optimization; linear algebraic equation; least squares solution; solvability verification
UR - http://eudml.org/doc/294712
ER -

References

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  1. Godsil, C., Royle, G. F., 10.1007/978-1-4613-0163-9, Springer-Verlag, New York 2001. Zbl0968.05002MR1829620DOI10.1007/978-1-4613-0163-9
  2. Haddad, W. M., Chellaboina, V., Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach., Princeton University Press, New Jersey 2008. Zbl1142.34001MR2381711
  3. Hui, Q., Haddad, W. M., Bhat, S. P., 10.1109/tac.2009.2029397, IEEE Trans. Automat. Control 54 (2009), 2465-2470. MR2562855DOI10.1109/tac.2009.2029397
  4. Liu, Y., Lageman, C., Anderson, B., Shi, G., An Arrow-Hurwicz-Uzawa type flow as least squares solver for network linear equations., arXiv:1701.03908v1. 
  5. Liu, J., Morse, A. S., Nedic, A., Basar, T., 10.1016/j.automatica.2017.05.004, Automatica 83 (2017), 37-46. MR3680412DOI10.1016/j.automatica.2017.05.004
  6. Liu, J., Mou, S., Morse, A. S., 10.1109/TAC.2017.2714645, IEEE Trans. Automat Control PP (2017), 99, 1-1. DOI10.1109/TAC.2017.2714645
  7. Mou, S., Liu, J., Morse, A. S., 10.1109/tac.2015.2414771, IEEE Trans. Automat. Control 60 (2015), 2863-2878. MR3419577DOI10.1109/tac.2015.2414771
  8. Nedic, A., Ozdaglar, A., Parrilo, P. A., 10.1109/tac.2010.2041686, IEEE Trans. Automat. Control 55 (2010), 922-938. MR2654432DOI10.1109/tac.2010.2041686
  9. Ni, W., Wang, X., 10.14736/kyb-2016-6-0898, Kybernetika 52 (2016), 898-913. MR3607853DOI10.14736/kyb-2016-6-0898
  10. Qiu, Z., Liu, S., Xie, L., 10.1016/j.automatica.2016.01.055, Automatica 68 (2016), 209-215. MR3483686DOI10.1016/j.automatica.2016.01.055
  11. Ruszczynski, A., Nonlinear Optimization., Princeton University Press, New Jersey 2006. MR2199043
  12. Shi, G., Anderson, B. D. O., 10.1109/acc.2016.7525353, In: American Control Conference, Boston 2016, pp. 2864-2869. DOI10.1109/acc.2016.7525353
  13. Shi, G., Anderson, B. D. O., Helmke, U., 10.1109/tac.2016.2612819, IEEE Trans. Automat. Control 62 (2017), 2659-2764. MR3660554DOI10.1109/tac.2016.2612819
  14. Shi, G., Johansson, K. H., 10.1016/j.automatica.2012.08.018, Automatica 48 (2012), 3018-3030. MR2995677DOI10.1016/j.automatica.2012.08.018
  15. Yi, P., Hong, Y., Liu, F., 10.1016/j.sysconle.2015.06.006, Systems Control Lett. 83 (2015), 45-52. Zbl1327.93033MR3373270DOI10.1016/j.sysconle.2015.06.006
  16. Zeng, X., Hui, Q., 10.1109/tac.2015.2409900, IEEE Trans. Automat. Control 60 (2015), 3083-3088. MR3419603DOI10.1109/tac.2015.2409900
  17. Zeng, X., Yi, P., Hong, Y., 10.1109/tac.2016.2628807, IEEE Trans. Automat. Control 62 (2017), 5227-5233. MR3708893DOI10.1109/tac.2016.2628807

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