Regularized dual gradient distributed method for constrained convex optimization over unbalanced directed graphs


This paper investigates a distributed optimization problem over a cooperative multi-agent time–varying network, where each agent has its own decision variables that should be set so as to minimize its individual objective subjected to global coupled constraints. Based on push-sum protocol and dual decomposition, we design a regularized dual gradient distributed algorithm to solve this problem, in which the algorithm is implemented in unbalanced time–varying directed graphs only requiring the column stochasticity of communication matrices. By augmenting the corresponding Lagrangian function with a quadratic regularization term, we first obtain the bound of the Lagrangian multipliers which does not require constructing a compact set containing the dual optimal set when compared with most of primal-dual based methods. Then, we obtain that the convergence rate of the proposed method can achieve the order of \(\mathcal {O}(\ln T/T)\) for strongly convex objective functions, where T is the number of iterations. Moreover, the explicit bound of constraint violations is also given. Finally, numerical results on the network utility maximum problem are used to demonstrate the efficiency of the proposed algorithm.

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  1. 1.

    Nedić, A., Ozdaglar, A.: Distributed subgradient methods for multi-agent optimization. IEEE Trans. Autom. Control 54(1), 48–61 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Nedić, A., Ozdaglar, A., Parrilo, P.: Constrainted consensus and optimization in multi-agent networks. IEEE Trans. Autom. Control 55(4), 922–938 (2010)

    Article  Google Scholar 

  3. 3.

    Jakovetic, D., Xavier, J., Moura, J.M.: Fast distributed gradient methods. IEEE Trans. Autom. Control 59(5), 1131–1146 (2014)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Nedić, A., Olshevsky, A: Distributed optimization over time-varing directed graphs. IEEE Trans. Autom. Control 3(60), 601–615 (2015)

    Article  Google Scholar 

  5. 5.

    Johansson, B., Keviczky, T., Johansson, M., Johansson, K.H.: Subgradient methods and consensus algorithms for solving convex optimization problems. In Proc. IEEE CDC, pp. 4185–4190. Cancun (2008)

  6. 6.

    Baingana, B., Mateos, G., Giannakis, G.: Proximal-gradient algorithms for tracking cascades over social networks. IEEE J. Selected Topics Signal Process. 8(4), 563–575 (2014)

    Article  Google Scholar 

  7. 7.

    Mateos, G., Giannakis, G.: Distributed recursiveleast-squares: Stability and performance analysis. IEEE Trans. Signal Process. 60(7), 3740–3754 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bolognani, S., Carli, R., Cavraro, G., Zampieri, S.: Distributed reactive power feedback control for voltage regulation and loss minimization. IEEE Trans. Autom. Control 60(4), 966–981 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Zhang, Y., Giannakis, G.: Distributed stochastic market clearing with high-penetration wind power and large-scale demand response. IEEE Trans. Power Syst. 31(2), 895–906 (2016)

    Article  Google Scholar 

  10. 10.

    Martinez, S., Bullo, F., Cortez, J., Frazzoli, E.: On synchronous robotic networks-Part I: Models, tasks, and complexity. IEEE Trans. Autom. Control 52(12), 2199–2213 (2007)

    Article  Google Scholar 

  11. 11.

    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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ram, S.S., Nedić, A., Veeravalli, V.V.: Distributed stochastic subgradient projection algorithms for convex optimization. J. Optim. Theory Appl. 147(3), 516–545 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Duchi, J.C., Agarwal, A., Wainwright, M.J.: Dual averaging for distributed optimization: Convergence analysis and network scaling. IEEE Trans. Autom. Control 57(3), 592–606 (2012)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Zhu, M., Martinez, S.: On distributed convex optimization under inequality and equality constraints. IEEE Trans. Autom. Control 57(1), 151–163 (2012)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Li, J., Wu, C., Wu, Z., Long, Q.: Gradient-free method for nonsmooth distributed optimization. J. Glob. Optim. 61(2), 325–340 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lorenzo, P., Scutari, G.: Netx: In-network nonconvex optimization. IEEE Trans. Signal Inf. Process. Netw. 2(2), 120–136 (2016)

    Article  Google Scholar 

  17. 17.

    Li, J., Chen, G., Dong, Z., Wu, Z.: Distributed mirror descent method for multi-agent optimization with delay. Neurocomputing 177, 643–650 (2016)

    Article  Google Scholar 

  18. 18.

    Gharesifard, B., Cortes, J.: Distributed continuous-time convex optimization on weight-balanced digraphs. IEEE Trans. Autom. Control 59(3), 781–786 (2012)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Tsianos, K.I., Lawlor, S., Rabbat, M.G.: Consensus-based distributed optimization: Practical issues and applications in large-scale machine learning. In: 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1543–1550. IEEE (2012)

  20. 20.

    Nedić, A., Olshevsky, A: Stochastic gradient-push for strongly convex functions on time-varying directed graphs. IEEE Trans. Autom. Control 12(61), 3936–3947 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Bertsekas, D.P., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    Google Scholar 

  22. 22.

    Necoara, I., Suykens, J.A.: Application of smoothing technique to decomposition in convex optimization. IEEE Trans. Autom. Control 53(11), 2674–2679 (2008)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Li, J., Chen, G., Dong, Z., Wu, Z.: A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints. Comput. Optim. Appl. 64(3), 671–697 (2016)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Yuan, D., Xu, S., Zhao, H.: Distributed primal-dual subgradient method for multiagent optimization via consensus algorithms. IEEE Trans. Syst. Man Cybern. Part B (Cybernetics) 41(6), 1715–1724 (2011)

    Article  Google Scholar 

  25. 25.

    Aybat, N.S., Hamedani, E.Y.: Distributed primal-dual method for multi-agent sharing problem with conic constraints. In: 2016 50th Asilomar Conference on Signals, Systems and Computers, pp. 777–782. IEEE (2016)

  26. 26.

    Aybat, N.S., Hamedani, E.Y.: A distributed ADMM-like method for resource sharing under conic constraints over time-varying networks. arXiv:1611.07393 (2016)

  27. 27.

    Chang, T.H., Nedić, A., Scaglione, A.: Distributed constrained optimization by consensus-based primal-dual perturbation method. IEEE Trans. Autom. Control 59(6), 1524–1538 (2014)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Yuan, D., Ho, D.W.C., Xu, S.: Regularized primal-dual subgradient method for distributed constrained optimization. IEEE Trans. Cybern. 46(9), 2109–2118 (2016)

    Article  Google Scholar 

  29. 29.

    Khuzani, M.B., Li, N.: Distributed regularized primal-dual method: Convergence analysis and trade-offs. arXiv:1609.08262v3 (2017)

  30. 30.

    Falsone, A., Margellos, K., Garetti, S., Prandini, M.: Dual decomposition and proximal minimization for multi-agent distributed optimization with coupling constraints. Automatica 84, 149–158 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Low, S.H., Lapsley, D.E.: Optimization flow control. I. Basic algorithm and convergence. IEEE/ACM Trans. Network. 7, 861–874 (1999)

    Article  Google Scholar 

  32. 32.

    Beck, A., Nedić, A., Ozdaglar, A., Teboulle, M.: An O(1/k) Gradient method for network resource Aalocation problems. IEEE Trans. Cont. Net. Sys 1(1), 64–73 (2014)

    Article  Google Scholar 

  33. 33.

    Gu, C., Wu, Z., Li, J., Guo, Y.: Distributed convex optimization with coupling constraints over time-varying directed graphs. arXiv:1805.07916 (2018)

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This research was partially supported by the NSFC 11501070, 11671362 and 11871128, by the Natural Science Foundation Projection of Chongqing cstc2017jcyjA0788 and cstc2018jcyjAX0172, and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201800520).

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Correspondence to Jueyou Li.

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Gu, C., Wu, Z. & Li, J. Regularized dual gradient distributed method for constrained convex optimization over unbalanced directed graphs. Numer Algor 84, 91–115 (2020).

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  • Convex optimization
  • Distributed algorithm
  • Dual decomposition
  • Regularization
  • Multi-agent network