基于有向图的分布式连续时间非光滑耦合约束凸优化分析

刘奕葶; 马铭莙; 付俊

doi:10.16383/j.aas.c210808

基于有向图的分布式连续时间非光滑耦合约束凸优化分析

doi: 10.16383/j.aas.c210808

1.
东北大学流程工业综合自动化国家重点实验室沈阳 110819

基金项目: 国家重点研发计划(2018AAA0101603)资助

详细信息

作者简介:
刘奕葶：东北大学流程工业综合自动化国家重点实验室硕士研究生. 主要研究方向为分布式优化, 耦合不等式路径约束. E-mail: 13840581163@163.com

马铭莙：东北大学流程工业综合自动化国家重点实验室博士研究生. 主要研究方向为分布式动态优化, 切换系统控制. E-mail: mingjun_mmj@163.com

付俊：东北大学流程工业综合自动化国家重点实验室教授. 主要研究方向为动态优化, 切换系统, 非线性控制. 本文通信作者. E-mail: junfu@mail.neu.edu.cn

计量
- 文章访问数: 621
- HTML全文浏览量: 201
- PDF下载量: 204
- 被引次数: 0
出版历程
- 收稿日期: 2021-10-14
- 网络出版日期: 2022-10-27
- 刊出日期: 2024-01-29

Distributed Continuous-time Non-smooth Convex Optimization Analysis With Coupled Constraints Over Directed Graphs

1.
State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819

Funds: Supported by National Key Research and Development Program of China (2018AAA0101603)

More Information

Author Bio:
LIU Yi-Ting　Master student at the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University. Her research interest covers distributed optimization and coupled inequality path constraint

MA Ming-Jun　Ph.D. candidate at the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University. Her research interest covers dynamic optimization of distributed systems and control of switched systems

FU Jun　Professor at the State Key Laboratory of Synthetical Automatic for Process Industries, Northeastern University. His research interest covers dynamic optimization, switching system, and nonlinear control. Corresponding author of this paper

摘要

摘要: 研究一类分布式优化问题, 其目标是在满足耦合不等式约束和局部可行集约束的情况下使非光滑全局代价函数值最小. 首先, 对原有的分布式连续时间投影算法进行拓展, 结合线性代数理论分析, 设计一个适用于强连通加权平衡有向通信网络拓扑图的算法. 其次, 在局部代价函数和耦合不等式约束函数是非光滑凸函数的假设条件下, 利用Moreau-Yosida函数正则化使目标函数和约束函数近似光滑可微. 然后, 根据强连通加权平衡有向图的分布式连续时间投影算法构造李雅普诺夫函数, 证明该算法下的平衡解是分布式优化问题最优解, 并对算法进行收敛性分析. 最后, 通过数值仿真验证算法的有效性.
- 多智能体网络 /
- 分布式优化 /
- 加权平衡有向图 /
- 耦合不等式约束
Abstract: In this paper, we study a class of distributed optimization problems whose objective is to minimize the value of a non-smooth global cost function while satisfying the coupling inequality constraint and the local feasible set constraint. First, we extend the original distributed continuous-time projection algorithm with linear algebraic theory analysis to design an algorithm for strongly connected weighted-balanced directed communication network topology graphs. Second, under the assumption that the local cost function and the coupled inequality constraint function are non-smooth convex functions, we use the Moreau-Yosida function regularization to make the objective function and the constraint function approximately smooth and differentiable. Then, the Lyapunov function is constructed according to the distributed continuous time projection algorithm of the strongly connected weighted equilibrium directed graph, the equilibrium solution under this algorithm is proved to be the optimal solution of the distributed optimization problem, as well as the convergence analysis of the algorithm is performed. Finally, the effectiveness of the algorithm is verified by numerical simulation.
- Multi-agent networks /
- distributed optimization /
- weight-balanced digraphs /
- coupled inequality constraints

HTML全文

图 1 加权平衡有向交互图

Fig. 1 Weight-balanced directed interaction graph

下载: 全尺寸图片幻灯片

图 2 状态$ x_{i} $的轨迹图

Fig. 2 Trajectories graph of state $ x_{i} $

下载: 全尺寸图片幻灯片

图 3 状态$ \tau_{i} $的轨迹图

Fig. 3 Trajectories graph of state $ \tau_{i} $

下载: 全尺寸图片幻灯片

图 4 状态$ \mu_{i} $的轨迹图

Fig. 4 Trajectories graph of state $ \mu_{i} $

下载: 全尺寸图片幻灯片

图 5 状态$ s_{i} $的轨迹图

Fig. 5 Trajectories graph of state $ s_{i} $

下载: 全尺寸图片幻灯片

参考文献(33)

[1]	柴天佑. 自动化科学与技术发展方向. 自动化学报, 2018, 44(11): 1923-1930 doi: 10.16383/j.aas.2018.c180252 Chai Tian-You. Development directions of automation science and technology. Acta Automatica Sinica, 2018, 44(11): 1923-1930 doi: 10.16383/j.aas.2018.c180252
[2]	杨涛, 柴天佑. 分布式协同优化的研究现状与展望. 中国科学: 技术科学, 2020, 50(11): 1414-1425 doi: 10.1360/SST-2020-0040 Yang Tao, Chai Tian-You. Research status and prospects of distributed collaborative optimization. Scientia Sinica Technologica, 2020, 50(11): 1414-1425 doi: 10.1360/SST-2020-0040
[3]	Yang T, Yi X L, Wu J F, Yuan Y, Wu D, Meng Z Y, et al. A survey of distributed optimization. Annual Reviews in Control, 2019, 47: 278-305 doi: 10.1016/j.arcontrol.2019.05.006
[4]	Duan J D, Yang Y, Liu F. Distributed optimization of integrated electricity-natural gas distribution networks considering wind power uncertainties. International Journal of Electrical Power & Energy Systems, 2022, 135: Article No. 107460
[5]	Yi P, Hong Y G, Liu F. Distributed gradient algorithm for constrained optimization with application to load sharing in power systems. Systems & Control Letters, 2015, 83: 45-52
[6]	Palomar D P, Chiang M. Alternative distributed algorithms for network utility maximization: Framework and applications. IEEE Transactions on Automatic Control, 2007, 52(12): 2254-2269 doi: 10.1109/TAC.2007.910665
[7]	Yu J Q, Liu Q T, Zhao A J, Chen S Y, Gao Z K, Wang F, et al. A distributed optimization algorithm for the dynamic hydraulic balance of chilled water pipe network in air-conditioning system. Energy, 2021, 223: Article No. 120059 doi: 10.1016/j.energy.2021.120059
[8]	衣鹏, 洪奕光. 分布式合作优化及其应用. 中国科学: 数学, 2016, 46(10): 1547-1564 Yi Peng, Hong Yi-Guang. Distributed cooperative optimization and its applications. Scientia Sinica Mathematica, 2016, 46(10): 1547-1564
[9]	Marden J R, Roughgarden T. Generalized efficiency bounds in distributed resource allocation. IEEE Transactions on Automatic Control, 2014, 59(3): 571-584 doi: 10.1109/TAC.2014.2301613
[10]	Shi W, Ling Q, Wu G, Yin W T. EXTRA: An exact first-order algorithm for decentralized consensus optimization. SIAM Journal on Optimization, 2015, 25(2): 944-966 doi: 10.1137/14096668X
[11]	Qu G N, Li N. Harnessing smoothness to accelerate distributed optimization. IEEE Transactions on Control of Network Systems, 2018, 5(3): 1245-1260 doi: 10.1109/TCNS.2017.2698261
[12]	Nedić A, Olshevsky A, Shi W. Achieving geometric convergence for distributed optimization over time-varying graphs. SIAM Journal on Optimization, 2017, 27(4): 2597-2633 doi: 10.1137/16M1084316
[13]	刘秀华, 韩建, 魏新江. 基于中间观测器的多智能体系统分布式故障估计. 自动化学报, 2020, 46(1): 142-152 doi: 10.16383/j.aas.c180179 Liu Xiu-Hua, Han Jian, Wei Xin-Jiang. Intermediate observer based distributed fault estimation for multi-agent systems. Acta Automatica Sinica, 2020, 46(1): 142-152 doi: 10.16383/j.aas.c180179
[14]	Zhu Y N, Ren W, Yu W W, Wen G H. Distributed resource allocation over directed graphs via continuous-time algorithms. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2021, 51(2): 1097-1106 doi: 10.1109/TSMC.2019.2894862
[15]	Nedić A, Olshevsky A. Distributed optimization over time-varying directed graphs. IEEE Transactions on Automatic Control, 2015, 60(3): 601-615 doi: 10.1109/TAC.2014.2364096
[16]	Gharesifard B, Cortés J. Distributed continuous-time convex optimization on weight-balanced digraphs. IEEE Transactions on Automatic Control, 2014, 59(3): 781-786 doi: 10.1109/TAC.2013.2278132
[17]	Zeng X L, Yi P, Hong Y G, Xie L H. Distributed continuous-time algorithms for nonsmooth extended monotropic optimization problems. SIAM Journal on Control and Optimization, 2018, 56(6): 3973-3993 doi: 10.1137/17M1118609
[18]	时侠圣, 杨涛, 林志赟, 王雪松. 基于连续时间的二阶多智能体分布式资源分配算法. 自动化学报, 2021, 47(8): 2050-2060 doi: 10.16383/j.aas.c200968 Shi Xia-Sheng, Yang Tao, Lin Zhi-Yun, Wang Xue-Song. Distributed resource allocation algorithm for second-order multi-agent systems in continuous-time. Acta Automatica Sinica, 2021, 47(8): 2050-2060 doi: 10.16383/j.aas.c200968
[19]	杨涛, 徐磊, 易新蕾, 张圣军, 陈蕊娟, 李渝哲. 基于事件触发的分布式优化算法. 自动化学报, 2022, 48(1): 133-143 doi: 10.16383/j.aas.c200838 Yang Tao, Xu Lei, Yi Xin-Lei, Zhang Sheng-Jun, Chen Rui-Juan, Li Yu-Zhe. Event-triggered distributed optimization algorithms. Acta Automatica Sinica, 2022, 48(1): 133-143 doi: 10.16383/j.aas.c200838
[20]	Yi P, Hong Y G, 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 doi: 10.1016/j.automatica.2016.08.007
[21]	Deng Z H, Liang S, Hong Y G. Distributed continuous-time algorithms for resource allocation problems over weight-balanced digraphs. IEEE Transactions on Cybernetics, 2018, 48(11): 3116-3125 doi: 10.1109/TCYB.2017.2759141
[22]	Lu J, Tang C Y. Zero-gradient-sum algorithms for distributed convex optimization: The continuous-time case. IEEE Transactions on Automatic Control, 2012, 57(9): 2348-2354 doi: 10.1109/TAC.2012.2184199
[23]	Kia S S, Cortés J, Martínez S. Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication. Automatica, 2015, 55: 254-264 doi: 10.1016/j.automatica.2015.03.001
[24]	Zeng X L, Yi P, Hong Y G. Distributed continuous-time algorithm for constrained convex optimizations via nonsmooth analysis approach. IEEE Transactions on Automatic Control, 2017, 62(10): 5227-5233 doi: 10.1109/TAC.2016.2628807
[25]	Yang S F, Liu Q S, Wang J. A multi-agent system with a proportional-integral protocol for distributed constrained optimization. IEEE Transactions on Automatic Control, 2017, 62(7): 3461-3467 doi: 10.1109/TAC.2016.2610945
[26]	Liang S, Zeng X L, Hong Y G. Distributed nonsmooth optimization with coupled inequality constraints via modified Lagrangian function. IEEE Transactions on Automatic Control, 2018, 63(6): 1753-1759 doi: 10.1109/TAC.2017.2752001
[27]	Cherukuri A, Mallada E, J. Cortés. Asymptotic convergence of constrained primal-dual dynamics. Systems & Control Letters, 2016, 87: 10-15
[28]	Li X X, Xie L H, Hong Y G. Distributed continuous-time nonsmooth convex optimization with coupled inequality constraints. IEEE Transactions on Control of Network Systems, 2020, 7(1): 74-84 doi: 10.1109/TCNS.2019.2915626
[29]	Jia W W, Qin S T. Distributed optimization over directed graphs with continuous-time algorithm. In: Proceedings of the Chinese Control Conference (CCC). Guangzhou, China: IEEE, 2019. 1911−1916
[30]	Liu Q S, Wang J. A second-order multi-agent network for bound-constrained distributed optimization. IEEE Transactions on Automatic Control, 2015, 60(12): 3310-3315 doi: 10.1109/TAC.2015.2416927
[31]	Loxton R C, Teo K L, Rehbock V, Yiu K F C. Optimal control problems with a continuous inequality constraint on the state and the control. Automatica, 2009, 45(10): 2250-2257 doi: 10.1016/j.automatica.2009.05.029
[32]	Bolte J. Continuous gradient projection method in Hilbert spaces. Journal of Optimization Theory and Applications, 2003, 119(2): 235-259 doi: 10.1023/B:JOTA.0000005445.21095.02
[33]	Ruszczyński A. Nonlinear Optimization. Princeton: Princeton University Press, 2006.