Kybernetika 60 no. 4, 427-445, 2024

Distributed dual averaging algorithm for multi-agent optimization with coupled constraints

Zhipeng Tu and Shu LiangDOI: 10.14736/kyb-2024-4-0427

Abstract:

This paper investigates a distributed algorithm for the multi-agent constrained optimization problem, which is to minimize a global objective function formed by a sum of local convex (possibly nonsmooth) functions under both coupled inequality and affine equality constraints. By introducing auxiliary variables, we decouple the constraints and transform the multi-agent optimization problem into a {variational inequality problem} with a set-valued monotone mapping. We propose a distributed dual averaging algorithm to find the weak solutions of the variational inequality problem with an $O(1/\sqrt{k})$ convergence rate, {where $k$ is the number of iterations.} Moreover, we show that weak solutions are also strong solutions that match the optimal primal-dual solutions to the considered optimization problem. {A numerical example is} given for illustration.

Keywords:

distributed optimization, variational inequality, coupled constraints, dual averaging, multi-agent networks

Classification:

90C33, 68W15

References:

  1. A. Auslender and R. Correa: Primal and dual stability results for variational inequalities. Comput. Optim. Appl. 17 (2000), 117-130.   DOI:10.1023/A:1026594114013
  2. A. Auslender and M. Teboulle: Projected subgradient methods with non-Euclidean distances for non-differentiable convex minimization and variational inequalities. Math. Program. 120 (2009), 27-48.   DOI:10.1007/s10107-007-0147-z
  3. D. P. Bertsekas and J. N. Tsitsiklis: Parallel and Distributed Computation: Numerical Methods. Prentice hall Englewood Cliffs, NJ 1989.   CrossRef
  4. J. M. Borwein and Q. J. Zhu: Techniques of Variational Analysis. Springer Science and Business Media, New York 2004.   CrossRef
  5. T. H. Chang, A. Nedić and A. Scaglione: Distributed constrained optimization by consensus-based primal-dual perturbation method. IEEE Trans. Automat. Control 59 (2014), 1524-1538.   DOI:10.1109/TAC.2014.2308612
  6. G. Chen, G. Xu, W. Li and Y. Hong: Distributed mirror descent algorithm with Bregman damping for nonsmooth constrained optimization. IEEE Trans. Automat. Control (2023), 1-8.   DOI:10.1109/tac.2023.3244995
  7. A. Cherukuri and J. Cortés: Distributed generator coordination for initialization and anytime optimization in economic dispatch. IEEE Trans. Control Network Syst. 2 (2015), 226-237.   DOI:10.1109/TCNS.2015.2399191
  8. J. C. Duchi, A. Agarwal and M. J. Wainwright: Dual averaging for distributed optimization: Convergence analysis and network scaling. IEEE Trans. Automat. Control 57 (2011), 592-606.   DOI:10.1109/TAC.2011.2161027
  9. F. Facchinei and J. S. Pang: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer Science and Business Media, 2007.   CrossRef
  10. I. Gutman and W. Xiao: Generalized inverse of the Laplacian matrix and some applications. Bulletin (Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathématiques) (2004), 15-23.   CrossRef
  11. J. B. Hiriart-Urruty and C. Lemaréchal: Convex Analysis and Minimization Algorithms I: Fundamentals. Springer Science and Business Media, 2013.   CrossRef
  12. B. Johansson, T. Keviczky, M. Johansson and K. H. Johansson: Subgradient methods and consensus algorithms for solving convex optimization problems. In: 47th IEEE Conference on Decision and Control, IEEE 2008, pp. 4185-4190.   DOI:10.1109/cdc.2008.4739339
  13. J. Koshal, A. Nedić and U. V. Shanbhag: Multiuser optimization: Distributed algorithms and error analysis. SIAM J. Optim. 21 (2011), 1046-1081.   DOI:10.1137/090770102
  14. S. Liang, X. Zeng and Y. Hong: Distributed nonsmooth optimization with coupled inequality constraints via modified Lagrangian function. IEEE Trans. Automat. Control 63 (2017), 1753-1759.   DOI:10.1109/TAC.2017.2752001
  15. S. Liang, L. Wang and G. Yin: Distributed smooth convex optimization with coupled constraints. IEEE Trans. Automat. Control 65 (2019), 347-353.   DOI:10.1109/TAC.2019.2912494
  16. S. Liang, L. Wang and G. Yin: Distributed dual subgradient algorithms with iterate-averaging feedback for convex optimization with coupled constraint. IEEE Trans. Cybernetics 51 (2019), 2529-2539.   DOI:10.1109/TCYB.2019.2933003
  17. Q. Liu and J. Wang: A second-order multi-agent network for bound-constrained distributed optimization. IEEE Trans. Automat. Control 60 (2015), 3310-3315.   DOI:10.1109/TAC.2015.2416927
  18. Y. Lou, Y. Hong and S. Wang: Distributed continuous-time approximate projection protocols for shortest distance optimization problems. Automatica 69 (2016), 289-297.   DOI:10.1016/j.automatica.2016.02.019
  19. A. Nedić and A. Ozdaglar: Approximate primal solutions and rate analysis for dual subgradient methods. SIAM J. Optim. 19 (2009), 1757-1780.   DOI:10.1137/070708111
  20. Y. Nesterov: Introductory Lectures on Convex Optimization: A Basic Course. Springer Science and Business Media, 2003.   CrossRef
  21. Y. Nesterov: Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Program. 109.2-3 (2007), 319-344.   DOI:10.1007/s10107-006-0034-z
  22. Y. Nesterov: Primal-dual subgradient methods for convex problems. Math. Program. 120 (2009), 221-259.   DOI:10.1007/s10107-007-0149-x
  23. Y. Nesterov and V. Shikhman: Dual subgradient method with averaging for optimal resource allocation. Europ. J. Oper. Res. 270 (2018), 907-916.   DOI:10.1016/j.ejor.2017.09.043
  24. M. Rabbat and R. Nowak: Distributed optimization in sensor networks. In: Proc. 3rd International Symposium on Information Processing in Sensor Networks, 2004, pp. 20-27.   CrossRef
  25. S. B. Regina and A. Iusem: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, New York 2003.   CrossRef
  26. R. T. Rockafellar: Characterization of the subdifferentials of convex functions. Pacific J. Math. 17 (1966), 497-510.   DOI:10.2140/pjm.1966.17.497
  27. R. T. Rockafellar: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149 (1970), 75-88.   DOI:10.1090/S0002-9947-1970-0282272-5
  28. A. Ruszczynski: Nonlinear Optimization. Princeton University Press, 2011.   CrossRef
  29. I. R. Shafarevich and A. O. Remizov: Linear Algebra and Geometry. Springer Science and Business Media, 2012.   CrossRef
  30. Z. Tu and W. Li: Multi-agent solver for non-negative matrix factorization based on optimization. Kybernetika 57 (2021), 60-77.   DOI:10.14736/kyb-2021-1-0060
  31. L. Xiao and S. Boyd: Optimal scaling of a gradient method for distributed resource allocation. J. Optim. Theory Appl. 129 (2006), 469-488.   DOI:10.1007/s10957-006-9080-1
  32. L. Xiao, S. Boyd and S. J. Kim: Distributed average consensus with least-mean-square deviation. J. Parallel Distributed Comput. 67 (2007), 33-46.   DOI:10.1016/j.jpdc.2006.08.010
  33. J. C. Yao: Variational inequalities with generalized monotone operators. Math. Oper. Res. 19 (1994), 691-705.   DOI:10.1287/moor.19.3.691
  34. P. Yi, Y. Hong and F. Liu: Distributed gradient algorithm for constrained optimization with application to load sharing in power systems. Systems Control Lett. 83 (2015), 45-52.   DOI:10.1016/j.sysconle.2015.06.006
  35. P. Yi, Y. Hong and F. Liu: Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems. Automatica 74 (2016), 259-269.   DOI:10.1016/j.automatica.2016.08.007
  36. P. Yi and L. Pavel: A distributed primal-dual algorithm for computation of generalized Nash equilibria via operator splitting methods. In: 2017 IEEE 56th Annual Conference on Decision and Control, IEEE 2017, pp. 3841-3846.   DOI:10.1109/cdc.2017.8264224
  37. X. Zeng, S. Liang, Y. Hong and J. Chen: Distributed computation of linear matrix equations: An optimization perspective. IEEE Trans. Automat. Control 64 (2018), 1858-1873.   DOI:10.1109/TAC.2018.2847603
  38. X. Zeng, P. Yi and Y. Hong: Distributed continuous-time algorithm for constrained convex optimizations via nonsmooth analysis approach. IEEE Trans. Automat. Control 62 (2016), 5227-5233.   DOI:10.1109/TAC.2016.2628807
  39. Y. Zhang and M. Zavlanos: A consensus-based distributed augmented Lagrangian method. In: 2018 Conference on Decision and Control, IEEE 2018, pp. 1763-1768.   DOI:10.1109/cdc.2018.8619512
  40. M. Zhu and S. Martínez: On distributed convex optimization under inequality and equality constraints. IEEE Trans. Automat. Control 57 (2011), 151-164.   DOI:10.1109/TAC.2011.2167817