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Randomized difference-based gradient-free algorithm for distributed resource allocation

Abstract

This paper considers a distributed resource allocation problem over time-varying networks. The objective of each agent in the network is to optimize the sum of separable convex functions subjected to resource constraints by observing its local objective function and the information exchanged with its adjacent neighbors. Thus, the problem lies in a distributed framework. In existing literature dealing with similar problems, the measurement of the gradients/subgradients of the objective functions has been applied in the algorithm design. In this paper, by adding stochastic dithers to the local objective functions and constructing randomized differences, we propose a distributed gradient-free algorithm for solving the problem, and show that the algorithm is strongly convergent; that is, the estimates generated from each agent almost certainly converge to the optimal resource allocation solution of the network. Finally, the effectiveness of the algorithm is validated by conducting numerical experiments.

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References

  1. Hetzer J, Yu D C, Bhattarai K. An economic dispatch model incorporating wind power. IEEE Trans Energy Convers, 2008, 23: 603–611

    Article  Google Scholar 

  2. Guo F, Wen C, Mao J, et al. Distributed economic dispatch for smart grids with random wind power. IEEE Trans Smart Grid, 2016, 7: 1572–1583

    Article  Google Scholar 

  3. Zhang Y, Giannakis G B. Efficient decentralized economic dispatch for microgrids with wind power integration. In: Proceedings of the 6th Annual IEEE Green Technologies Conference, 2014. 7–12

  4. Zhao C, Topcu U, Li N, et al. Design and stability of load-side primary frequency control in power systems. IEEE Trans Automat Contr, 2014, 59: 1177–1189

    MathSciNet  Article  Google Scholar 

  5. Mainland G, Parkes D C, Welsh M. Decentralized, adaptive resource allocation for sensor networks. In: Proceedings of the 2nd Conference on Symposium on Networked Systems Design & Implementation, 2005. 315–328

  6. Ozel O, Tutuncuoglu K, Yang J, et al. Transmission with energy harvesting nodes in fading wireless channels: optimal policies. IEEE J Sel Areas Commun, 2011, 29: 1732–1743

    Article  Google Scholar 

  7. Koopman B O. The optimum distribution of effort. J Oper Res Soc Am, 1953, 1: 52–63

    MathSciNet  MATH  Google Scholar 

  8. D’Amico A A, Sanguinetti L, Palomar D P. Convex separable problems with linear constraints in signal processing and communications. IEEE Trans Signal Process, 2014, 62: 6045–6058

    MathSciNet  Article  Google Scholar 

  9. Heal G M. Planning without prices. Rev Economic Studies, 1969, 36: 347–362

    Article  Google Scholar 

  10. Ibaraki T, Katoh N. Resource Allocation Problems: Algorithmic Approaches. Cambridge: MIT Press, 1988

    MATH  Google Scholar 

  11. Aybat N S, Hamedani E Y. A distributed ADMM-like method for resource sharing over time-varying networks. SIAM J Optim, 2019, 29: 3036–3068

    MathSciNet  Article  Google Scholar 

  12. Lakshmanan H, de Farias D P. Decentralized resource allocation in dynamic networks of agents. SIAM J Optim, 2008, 19: 911–940

    MathSciNet  Article  Google Scholar 

  13. Nedić A, Olshevsky A, Shi W. Improved convergence rates for distributed resource allocation. In: Proceedings of IEEE Conference on Decision and Control (CDC), 2018. 172–177

  14. Yi P, Lei J, Hong Y. Distributed resource allocation over random networks based on stochastic approximation. Syst Control Lett, 2018, 114: 44–51

    MathSciNet  Article  Google Scholar 

  15. Zhang J, You K, Cai K. Distributed dual gradient tracking for resource allocation in unbalanced networks. IEEE Trans Signal Process, 2020, 68: 2186–2198

    MathSciNet  Article  Google Scholar 

  16. Li X, Xie L, Hong Y. Distributed continuous-time algorithm for a general nonsmooth monotropic optimization problem. Int J Robust Nonlin Control, 2019, 29: 3252–3266

    MathSciNet  Article  Google Scholar 

  17. Conn A R, Scheinberg K, Vicente L N. Introduction to Derivative-Free Optimization. Philadelphia: SIAM, 2009

    Book  Google Scholar 

  18. Nesterov Y, Spokoiny V. Random gradient-free minimization of convex functions. Found Comput Math, 2017, 17: 527–566

    MathSciNet  Article  Google Scholar 

  19. Kiefer J, Wolfowitz J. Stochastic estimation of the maximum of a regression function. Ann Math Statist, 1952, 23: 462–466

    MathSciNet  Article  Google Scholar 

  20. Chen H F, Duncan T E, Pasik-Duncan B. A Kiefer-Wolfowitz algorithm with randomized differences. IEEE Trans Automat Contr, 1999, 44: 442–453

    MathSciNet  Article  Google Scholar 

  21. Spall J C. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans Automat Contr, 1992, 37: 332–341

    MathSciNet  Article  Google Scholar 

  22. Hajinezhad D, Hong M, Garcia A. Zeroth order nonconvex multi-agent optimization over networks. 2017. ArXiv:1710.09997

  23. Pang Y, Hu G. A distributed optimization method with unknown cost function in a multi-agent system via randomized gradient-free method. In: Proceedings of the 11th Asian Control Conference (ASCC), 2017. 144–149

  24. Sahu A K, Jakovetic D, Bajovic D, et al. Distributed zeroth order optimization over random networks: a Kiefer-Wolfowitz stochastic approximation approach. In: Proceedings of IEEE Conference on Decision and Control (CDC), 2018. 4951–4958

  25. Wang Y, Zhao W, Hong Y, et al. Distributed subgradient-free stochastic optimization algorithm for nonsmooth convex functions over time-varying networks. SIAM J Control Optim, 2019, 57: 2821–2842

    MathSciNet  Article  Google Scholar 

  26. Yuan D M, Ho D W C. Randomized gradient-free method for multiagent optimization over time-varying networks. IEEE Trans Neural Netw Learn Syst, 2015, 26: 1342–1347

    MathSciNet  Article  Google Scholar 

  27. Poveda J, Quijano N. Distributed extremum seeking for real-time resource allocation. In: Proceedings of 2013 American Control Conference, 2013. 2772–2777

  28. Ramírez-Llanos E, Martínez S. Gradient-free distributed resource allocation via simultaneous perturbation. In: Proceedings of the 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2016. 590–595

  29. Ramírez-Llanos E, Martínez S. Distributed and robust resource allocation algorithms for multi-agent systems via discrete-time iterations. In: Proceedings of the 54th IEEE Conference on Decision and Control (CDC), 2015. 1390–1395

  30. Kushner H, Yin G G. Stochastic Approximation and Recursive Algorithms and Applications. New York: Springer, 2003

    MATH  Google Scholar 

  31. Boyd S, Boyd S P, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004

    Book  Google Scholar 

  32. Ruszczynski A. Nonlinear Optimization. Princeton: Princeton University Press, 2006

    Book  Google Scholar 

  33. Yi P, Hong Y, Liu F. Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems. Automatica, 2016, 74: 259–269

    MathSciNet  Article  Google Scholar 

  34. Feijer D, Paganini F. Stability of primal-dual gradient dynamics and applications to network optimization. Automatica, 2010, 46: 1974–1981

    MathSciNet  Article  Google Scholar 

  35. Yi P, Hong Y, Liu F. Distributed gradient algorithm for constrained optimization with application to load sharing in power systems. Syst Control Lett, 2015, 83: 45–52

    MathSciNet  Article  Google Scholar 

  36. Lei J, Chen H F, Fang H T. Asymptotic properties of primal-dual algorithm for distributed stochastic optimization over random networks with imperfect communications. SIAM J Control Optim, 2018, 56: 2159–2188

    MathSciNet  Article  Google Scholar 

  37. Aysal T C, Yildiz M E, Sarwate A D, et al. Broadcast gossip algorithms for consensus. IEEE Trans Signal Process, 2009, 57: 2748–2761

    MathSciNet  Article  Google Scholar 

  38. Srivastava K, Nedić A, Stipanović D M. Distributed constrained optimization over noisy networks. In: Proceedings of IEEE Conference on Decision and Control (CDC), 2010. 1945–1950

  39. Zhang Q, Zhang J F. Distributed parameter estimation over unreliable networks with markovian switching topologies. IEEE Trans Automat Contr, 2012, 57: 2545–2560

    MathSciNet  Article  Google Scholar 

  40. Chen H F. Stochastic Approximation and Its Applications. Dordrecht: Kluwer Academic Publishers, 2002

    MATH  Google Scholar 

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Acknowledgements

This work was supported in part by National Key Research and Development Program of China (Grant No. 2018YFA0703800) and National Natural Science Foundation of China (Grant No. 61822312).

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Correspondence to Wenxiao Zhao.

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Geng, X., Zhao, W. Randomized difference-based gradient-free algorithm for distributed resource allocation. Sci. China Inf. Sci. 65, 142205 (2022). https://doi.org/10.1007/s11432-020-3147-2

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  • DOI: https://doi.org/10.1007/s11432-020-3147-2

Keywords

  • resource allocation
  • distributed algorithm
  • randomized difference