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Continuous-time Distributed Heavy-ball Algorithm for Distributed Convex Optimization over Undirected and Directed Graphs

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Abstract

This paper proposes second-order distributed algorithms over multi-agent networks to solve the convex optimization problem by utilizing the gradient tracking strategy, with convergence acceleration being achieved. Both the undirected and unbalanced directed graphs are considered, extending existing algorithms that primarily focus on undirected or balanced directed graphs. Our algorithms also have the advantage of abandoning the diminishing step-size strategy so that slow convergence can be avoided. Furthermore, the exact convergence to the optimal solution can be realized even under the constant step size adopted in this paper. Finally, two numerical examples are presented to show the convergence performance of our algorithms.

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References

  1. W. Wang, J. S. Huang, C. Y. Wen, H. J. Fan. Distributed adaptive control for consensus tracking with application to formation control of nonholonomic mobile robots. Automatica, vol.50, no.4, pp. 1254–1263, 2014. DOI: https://doi.org/10.1016/j.automatica.2014.02.028.

    Article  MathSciNet  MATH  Google Scholar 

  2. T. J. Zhang. Unmanned aerial vehicle formation inspired by bird flocking and foraging behavior. International Journal of Automation and Computing, vol.15, no. 4, pp.402–416, 2018. DOI: https://doi.org/10.1007/s11633-017-1111-x.

    Article  Google Scholar 

  3. Z. H. Deng, S. Liang, Y. G. Hong. Distributed continuous-time algorithms for resource allocation problems over weight-balanced digraphs. IEEE Transactions on Cybernetics, vol.48, no. 11, pp.3116–3125, 2018. DOI: https://doi.org/10.1109/TCYB.2017.2759141.

    Article  Google Scholar 

  4. M. Panda, B. Das, B. Subudhi, B. B. Pati. A comprehensive review of path planning algorithms for autonomous underwater vehicles. International Journal of Automation and Computing, vol.17, no. 3, pp. 321–352, 2020. DOI: https://doi.org/10.1007/s11633-019-1204-9.

    Article  Google Scholar 

  5. Z. Wang, D. Wang, D. B. Gu. Distributed optimal state consensus for multiple circuit systems with disturbance rejection. IEEE Transactions on Network Science and Engineering, vol.7, no.4, pp.2926–2939, 2020. DOI: https://doi.org/10.1109/TNSE.2020.3007472.

    Article  MathSciNet  Google Scholar 

  6. W. J. Feng, R. Jiang, G. L. Liu. Distributed power control in cooperative cognitive ad hoc networks. International Journal of Automation and Computing, vol.11, no. 4, pp.412–417, 2014. DOI: https://doi.org/10.1007/s11633-014-0807-4.

    Article  Google Scholar 

  7. W. Chen, Y. F. Fu. Cooperative distributed target tracking algorithm in mobile wireless sensor networks. Journal of Control Theory and Applications, vol. 9, no. 2, pp. 155, vol.9, no. 2, Article number 155, 2011. DOI: https://doi.org/10.1007/s11768-011-8124-8.

    Article  MathSciNet  Google Scholar 

  8. T. Yang, X. L. Yi, J. F. Wu, Y. Yuan, D. Wu, Z. Y. Meng, Y. G. Hong, H. Wang, Z. L. Lin, K. H. Johansson. A survey of distributed optimization. Annual Reviews in Control, vol.47, pp. 278–305, 2019. DOI: https://doi.org/10.1016/j.arcontrol.2019.05.006.

    Article  MathSciNet  Google Scholar 

  9. I. Lobel, A. Ozdaglar. Distributed subgradient methods for convex optimization over random networks. IEEE Transactions on Automatic Control, vol.56, no.6, pp.1291–1306, 2011. DOI: https://doi.org/10.1109/TAC.2010.2091295.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Nedic, A. Ozdaglar. Distributed subgradient methods for multi-agent optimization. IEEE Transactions on Automatic Control, vol.54, no. 1, pp.48–61, 2009. DOI: https://doi.org/10.1109/TAC.2008.2009515.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. C. Duchi, A. Agarwal, M. J. Wainwright. Dual averaging for distributed optimization: Convergence analysis and network scaling. IEEE Transactions on Automatic Control, vol.57, no.3, pp.592–606, 2012. DOI: https://doi.org/10.1109/TAC.2011.2161027.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Nedic, A. Ozdaglar, P. A. Parrilo. Constrained consensus and optimization in multi-agent networks. IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 922–938, 2010. DOI: https://doi.org/10.1109/TAC.2010.2041686.

    Article  MathSciNet  MATH  Google Scholar 

  13. K. I. Tsianos, S. Lawlor, M. G. Rabbat. Push-sum distributed dual averaging for convex optimization. In Proceedings of the 51st IEEE Conference on Decision and Control, IEEE, Maui, USA, pp. 5453–5458, 2012. DOI: https://doi.org/10.1109/CDC.2012.6426375.

    Google Scholar 

  14. C. G. Xi, U. A. Khan. Distributed subgradient projection algorithm over directed graphs. IEEE Transactions on Automatic Control, vol.62, no.8, pp.3986–3992, 2017. DOI: https://doi.org/10.1109/TAC.2016.2615066.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Jakovetić, J. Xavier, J. M. F. Moura. Fast distributed gradient methods. IEEE Transactions on Automatic Control, vol.59, no.5, pp. 1131–1146, 2014. DOI: https://doi.org/10.1109/TAC.2014.2298712.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Lee, A. Nedić. Asynchronous gossip-based random projection algorithms over networks. IEEE Transactions on Automatic Control, vol.61, no.4, pp.953–968, 2016. DOI: https://doi.org/10.1109/TAC.2015.2460051.

    Article  MathSciNet  MATH  Google Scholar 

  17. I. Lobel, A. Ozdaglar. Convergence analysis of distributed subgradient methods over random networks. In Proceedings of the 46th Annual Allerton Conference on Communication, Control, and Computing, IEEE, Monticello, USA, pp. 353–360, 2008. DOI: https://doi.org/10.1109/ALLERTON.2008.4797579.

    Google Scholar 

  18. K. Yuan, Q. Ling, W. T. Yin. On the convergence of decentralized gradient descent. SIAM Journal on Optimization, vol.26, no.3, pp. 1835–1854, 2016. DOI: https://doi.org/10.1137/130943170.

    Article  MathSciNet  MATH  Google Scholar 

  19. I. Matei, J. S. Baras. Performance evaluation of the consensus-based distributed subgradient method under random communication topologies. IEEE Journal of Selected Topics in Signal Processing, vol. 5, no. 4, pp. 754–771, 2011. DOI: https://doi.org/10.1109/JSTSP.2011.2120593.

    Article  Google Scholar 

  20. J. M. Xu, S. Y. Zhu, Y. C. Soh, L. H. Xie. Augmented distributed gradient methods for multi-agent optimization under uncoordinated constant stepsizes. In Proceedings of the 54th IEEE Conference on Decision and Control, IEEE, Osaka, Japan, pp. 2055–2060, 2015. DOI: https://doi.org/10.1109/CDC.2015.7402509.

    Google Scholar 

  21. S. Pu, W. Shi, J. M. Xu, A. Nedic. Push-pull gradient methods for distributed optimization in networks. IEEE Transactions on Automatic Control, vol.66, no. 1, pp. 1–16, 2021. DOI: https://doi.org/10.1109/TAC.2020.2972824.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Nedić, A. Olshevsky, W. Shi, C. A. Uribe. Geometrically convergent distributed optimization with uncoordinated step-sizes. In Proceedings of American Control Conference, IEEE, Seattle, USA, pp. 3950–3955, 2017. DOI: https://doi.org/10.23919/ACC.2017.7963560.

    Google Scholar 

  23. G. P. Chen, P. Yi, Y. G. Hong. Distributed optimization with projection-free dynamics. [Online], Available: https://arxiv.org/abs/2105.02450, 2021.

  24. M. J. Ye, G. Q. Hu, L. H. Xie, S. Y. Xu. Differentially private distributed Nash equilibrium seeking for aggregative games. IEEE Transactions on Automatic Control, to be published. DOI: https://doi.org/10.1109/TAC.2021.3075183.

  25. R. A. Freeman, P. Yang, K. M. Lynch. Stability and convergence properties of dynamic average consensus estimators. In Proceedings of the 45th IEEE Conference on Decision and Control, IEEE, San Diego, USA, pp. 338–343, 2006. DOI: https://doi.org/10.1109/CDC.2006.377078.

    Chapter  Google Scholar 

  26. M. J. Ye, G. Q. Hu. Game design and analysis for price-based demand response: An aggregate game approach. IEEE Transactions on Cybernetics, vol.47, no. 3, pp. 720–730, 2017. DOI: https://doi.org/10.1109/TCYB.2016.2524452.

    Article  Google Scholar 

  27. J. Koshal, A. Nedic, U. V. Shanbhag. Distributed algorithms for aggregative games on graphs. Operations Research, vol.64, no.3, pp.680–704, 2016. DOI: https://doi.org/10.1287/opre.2016.1501.

    Article  MathSciNet  MATH  Google Scholar 

  28. X. He, D. W. C. Ho, T. W. Huang, J. Z. Yu, H. Abu-Rub, C. J. Li. Second-order continuous-time algorithms for economic power dispatch in smart grids. IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol.48, no.9, pp. 1482–1492, 2018. DOI: https://doi.org/10.1109/TSMC.2017.2672205.

    Article  Google Scholar 

  29. A. J. Wang, T. Dong, X. F. Liao. Distributed optimal consensus algorithms in multi-agent systems. Neurocomputing, vol.339, pp. 26–35, 2019. DOI: https://doi.org/10.1016/j.neucom.2019.01.044.

    Article  Google Scholar 

  30. Y. S. Li, H. G. Zhang, B. N. Huang, J. Han. A distributed Newton-Raphson-based coordination algorithm for multi-agent optimization with discrete-time communication. Neural Computing and Applications, vol.32, no. 9, pp. 4649–4663, 2020. DOI: https://doi.org/10.1007/s00521-018-3798-l.

    Article  Google Scholar 

  31. N. T. Tran, Y. W. Wang, W. Yang. Distributed optimization problem for double-integrator systems with the presence of the exogenous disturbance. Neurocomputing, vol. 272, pp. 386–395, 2018. DOI: https://doi.org/10.1016/j.neucom.2017.07.005.

    Article  Google Scholar 

  32. Z. H. Deng, S. Liang, W. Y. Yu. Distributed optimal resource allocation of second-order multiagent systems. International Journal of Robust and Nonlinear Control, vol. 28, no. 14, pp. 4246–4260, 2018. DOI: https://doi.org/10.1002/rnc.4233.

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Varagnolo, F. Zanella, A. Cenedese, G. Pillonetto, L. Schenato. Newton-raphson consensus for distributed convex optimization. IEEE Transactions on Automatic Control, vol.61, no.4, pp.994–1009, 2016. DOI: https://doi.org/10.1109/TAC.2015.2449811.

    Article  MathSciNet  MATH  Google Scholar 

  34. H. Attouch, X. Goudou, P. Redont. The heavy ball with friction method, I. the continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system. Communications in Contemporary Mathematics, vol.2, no. 1, pp. 1–34, 2000. DOI: https://doi.org/10.1142/S0219199700000025.

    Article  MathSciNet  MATH  Google Scholar 

  35. S. L. Li, X. H. Nian, Z. H. Deng. Distributed optimization of second-order multi-agent systems with external disturbance over weight-balanced digraphs. In Proceedings of Chinese Control Conference, IEEE, Guangzhou, China, pp. 2006–2011, 2019. DOI: https://doi.org/10.23919/ChiCC.2019.8865509.

    Google Scholar 

  36. Z. H. Deng. Distributed algorithm design for resource allocation problems of second-order multiagent systems over weight-balanced digraphs. IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol.51, no.6, pp.3512–3521, 2021. DOI: https://doi.org/10.1109/TSMC.2019.2930672.

    Article  Google Scholar 

  37. D. Wang, J. J. Yin, W. Wang. Distributed randomized gradient-free optimization protocol of multiagent systems over weight-unbalanced digraphs. IEEE Transactions on Cybernetics, vol.51, no. 1, pp.473–482, 2021. DOI: https://doi.org/10.1109/TCYB.2018.2890140.

    Article  Google Scholar 

  38. D. Wang, J. L. Yin, W. Wang. Design of fixed step-size distributed optimization protocol of multiagent systems over weighted unbalanced digraphs. In Proceedings of the 8th International Conference on Information Science and Technology, IEEE, Cordoba, Spain, pp. 321–328, 2018. DOI: https://doi.org/10.1109/ICIST.2018.8426143.

    Google Scholar 

  39. Z. H. Li, Z. T. Ding, J. Y. Sun, Z. K. Li. Distributed adaptive convex optimization on directed graphs via continuous-time algorithms. IEEE Transactions on Automatic Control, vol.63, no.5, pp. 1434–1441, 2018. DOI: https://doi.org/10.1109/TAC.2017.2750103.

    Article  MathSciNet  MATH  Google Scholar 

  40. Y. N. Zhu, W. W. Yu, G. H. Wen, W. Ren. Continuous-time coordination algorithm for distributed convex optimization over weight-unbalanced directed networks. IEEE Transactions on Circuits and Systems II: Express Briefs, vol.66, no. 7, pp. 1202–1206, 2019. DOI: https://doi.org/10.1109/TCSII.2018.2878250.

    Google Scholar 

  41. W. Y. Yu, P. Yi, Y. G. Hong. A gradient-based dissipative continuous-time algorithm for distributed optimization. In Proceedings of the 35th Chinese Control Conference, IEEE, Chengdu, China, pp. 7908–7912, 2016. DOI: https://doi.org/10.1109/ChiCC.2016.7554612.

    Google Scholar 

  42. C. Godsil, G. Royle. Algebraic Graph Theory. New York, USA: Springer, 2001. DOI: https://doi.org/10.1007/978-1-4613-0163-9.

    Book  MATH  Google Scholar 

  43. S. S. Kia, J. Cortés, S. Martínez. Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication. Automatica, vol.55, pp. 254–264, 2015. DOI: https://doi.org/10.1016/j.automatica.2015.03.001.

    Article  MathSciNet  MATH  Google Scholar 

  44. R. T. Rockafellar. Convex Analysis. Princeton, USA: Princeton University Press, 1970.

    Book  MATH  Google Scholar 

  45. R. Xin, U. A. Khan. Distributed heavy-ball: A generalization and acceleration of first-order methods with gradient tracking. IEEE Transactions on Automatic Control, vol. 65, no. 6, pp. 2627–2633, 2020. DOI: https://doi.org/10.1109/TAC.2019.2942513.

    Article  MathSciNet  MATH  Google Scholar 

  46. F. Z. Zhang. The Schur Complement and its Applications Boston, USA: Springer, 2005. DOI: https://doi.org/10.1007/b105056.

    Book  MATH  Google Scholar 

  47. B. Gharesifard, J. Cortés. Distributed continuous-time convex optimization on weight-balanced digraphs. IEEE Transactions on Automatic Control, vol. 59, no. 3, pp. 781–786, 2014. DOI: https://doi.org/10.1109/TAC.2013.2278132.

    Article  MathSciNet  MATH  Google Scholar 

  48. Y. N. Zhu, W. Ren, W. W. Yu, G. H. Wen. Distributed resource allocation over directed graphs via continuous-time algorithms. IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol.51, no. 2, pp. 1097–1106, 2021. DOI: https://doi.org/10.1109/TSMC.2019.2894862.

    Article  Google Scholar 

  49. R. Olfati-Saber, R. M. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, vol.49, no. 9, pp. 1520–1533, 2004. DOI: https://doi.org/10.1109/TAC.2004.834113.

    Article  MathSciNet  MATH  Google Scholar 

  50. S. N. Chow, J. A. Yorke. Lyapunov theory and perturbation of stable and asymptotically stable systems. Journal of Differential Equations, vol.15, no. 2, pp. 308–321, 1974. DOI: https://doi.org/10.1016/0022-0396(74)90082-5.

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Wang, N. Elia. Control approach to distributed optimization. In Proceedings of the 48th Annual Allerton Conference on Communication, Control, and Computing, IEEE, Monticello, USA, pp. 557–561, 2010. DOI: https://doi.org/10.1109/ALLERTON.2010.5706956.

    Google Scholar 

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Acknowledgements

This work was supported by National Nature Science Foundation of China (Nos. 61663026, 62066026, 61963028 and 61866023) and Jiangxi NSF (No. 20192BAB 207025).

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  1. School of Science, Nanchang University, Nanchang, 330031, China

    Hao-Ran Yang & Wei Ni

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  1. Hao-Ran Yang
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Correspondence to Wei Ni.

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Colored figures are available in the online version at https://link.springer.com/journal/11633

Hao-Ran Yang received the B. Sc. degree in statistics from Lingnan Normal University, China in 2015. He is currently a master student in Nanchang University, China.

His research interests include distributed optimization and multi-agent systems. E-mail: 13410901483@163.com ORCID iD: 0000-0001-5238-2723

Wei Ni received the Ph. D. degree in systems science from Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China in 2010. He is currently an associate professor with School of Science, Nanchang University, China.

His research interests include control of switched and impulsive systems, and complex systems. E-mail: niwei@ncu.edu.cn (Corresponding author) ORCID iD: 0000-0002-1410-8317

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Yang, HR., Ni, W. Continuous-time Distributed Heavy-ball Algorithm for Distributed Convex Optimization over Undirected and Directed Graphs. Mach. Intell. Res. 19, 75–88 (2022). https://doi.org/10.1007/s11633-022-1319-2

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