Cooperative–Competitive Multiagent Systems for Distributed Minimax Optimization Subject to Bounded Constraints

This paper presents continuous-time multiagent systems for distributed minimax optimization subject to bounded constraints. All agents in the system are divided into two groups for minimization and maximization. The multiagent system features competitive intergroup interactions and cooperative intragroup interactions, both of which are based on the output information of agents. First, a proportional-integral (PI) intragroup interaction rule is utilized for consensus within each group in the system. With this interaction rule, the system is proved to be convergent to an optimal solution to the problem, under a certain requirement on the intergroup interactions. Second, another discontinuous intragroup interaction rule is introduced. It is proved that the system with such an interaction is still convergent to an optimal solution if the proportional gain exceeds a derived lower bound, without the previous requirement on the intergroup interactions. As a special case, the systems are further applied for distributed optimization. Finally, simulation results are presented to substantiate the theoretical results.

[1]  Sonia Martínez,et al.  Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication , 2014, Autom..

[2]  J. Cortés.  Discontinuous dynamical systems , 2008, IEEE Control Systems.

[3]  Deming Yuan,et al.  Distributed Primal-Dual Subgradient Method for Multiagent Optimization via Consensus Algorithms. , 2011, IEEE transactions on systems, man, and cybernetics. Part B, Cybernetics : a publication of the IEEE Systems, Man, and Cybernetics Society.

[4]  Jorge Cortés,et al.  Distributed event-triggered coordination for average consensus on weight-balanced digraphs , 2014, Autom..

[5]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[6]  Jing Wang,et al.  A control perspective for centralized and distributed convex optimization , 2011, IEEE Conference on Decision and Control and European Control Conference.

[7]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Multi-Agent Optimization , 2009, IEEE Transactions on Automatic Control.

[8]  Karl Henrik Johansson,et al.  Distributed Control of Networked Dynamical Systems: Static Feedback, Integral Action and Consensus , 2013, IEEE Transactions on Automatic Control.

[9]  Qingshan Liu,et al.  A One-Layer Projection Neural Network for Nonsmooth Optimization Subject to Linear Equalities and Bound Constraints , 2013, IEEE Transactions on Neural Networks and Learning Systems.

[10]  Magnus Egerstedt,et al.  Continuous-time proportional-integral distributed optimisation for networked systems , 2013, 1309.6613.

[11]  Yiguang Hong,et al.  Constrained Consensus Algorithms With Fixed Step Size for Distributed Convex Optimization Over Multiagent Networks , 2017, IEEE Transactions on Automatic Control.

[12]  Angelia Nedic,et al.  Subgradient Methods for Saddle-Point Problems , 2009, J. Optimization Theory and Applications.

[13]  Feng Liu,et al.  Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems , 2015, Autom..

[14]  Kok Lay Teo,et al.  Portfolio Optimization Under a Minimax Rule , 2000 .

[15]  ASHISH CHERUKURI,et al.  Saddle-Point Dynamics: Conditions for Asymptotic Stability of Saddle Points , 2015, SIAM J. Control. Optim..

[16]  Zheng Yan,et al.  A Neurodynamic Approach to Distributed Optimization With Globally Coupled Constraints , 2018, IEEE Transactions on Cybernetics.

[17]  Choon Yik Tang,et al.  Zero-gradient-sum algorithms for distributed convex optimization: The continuous-time case , 2011, Proceedings of the 2011 American Control Conference.

[18]  Sonia Martínez,et al.  On Distributed Convex Optimization Under Inequality and Equality Constraints , 2010, IEEE Transactions on Automatic Control.

[19]  Daniel W. C. Ho,et al.  A Layered Event-Triggered Consensus Scheme , 2017, IEEE Transactions on Cybernetics.

[20]  Ming Cao,et al.  Clustering in diffusively coupled networks , 2011, Autom..

[21]  Xinyi Le,et al.  A Two-Time-Scale Neurodynamic Approach to Constrained Minimax Optimization , 2017, IEEE Transactions on Neural Networks and Learning Systems.

[22]  Anna Scaglione,et al.  Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method , 2013, IEEE Transactions on Automatic Control.

[23]  Qingshan Liu,et al.  A Projection Neural Network for Constrained Quadratic Minimax Optimization , 2015, IEEE Transactions on Neural Networks and Learning Systems.

[24]  Karl Henrik Johansson,et al.  Nash Equilibrium Computation in Subnetwork Zero-Sum Games With Switching Communications , 2013, IEEE Transactions on Automatic Control.

[25]  Norman Biggs.  Algebraic Graph Theory: Index , 1974 .

[26]  E. Barron,et al.  Best response dynamics for continuous games , 2010 .

[27]  Qingshan Liu,et al.  Distributed Optimization Based on a Multiagent System in the Presence of Communication Delays , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[28]  Stephen P. Boyd,et al.  A minimax theorem with applications to machine learning, signal processing, and finance , 2007, CDC.

[29]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[30]  Magnus Egerstedt,et al.  Asynchronous Multiagent Primal-Dual Optimization , 2016, IEEE Transactions on Automatic Control.

[31]  Bahman Gharesifard,et al.  Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs , 2012, IEEE Transactions on Automatic Control.

[32]  T. Basar,et al.  H∞-0ptimal Control and Related Minimax Design Problems: A Dynamic Game Approach , 1996, IEEE Trans. Autom. Control..

[33]  Huijun Gao,et al.  Network-Induced Constraints in Networked Control Systems—A Survey , 2013, IEEE Transactions on Industrial Informatics.

[34]  Tamer Basar,et al.  Minimax estimation with intermittent observations , 2015, Autom..

[35]  Karl Henrik Johansson,et al.  Reaching an Optimal Consensus: Dynamical Systems That Compute Intersections of Convex Sets , 2011, IEEE Transactions on Automatic Control.

[36]  Qingshan Liu,et al.  A Multi-Agent System With a Proportional-Integral Protocol for Distributed Constrained Optimization , 2017, IEEE Transactions on Automatic Control.

[37]  Bahman Gharesifard,et al.  Distributed convergence to Nash equilibria in two-network zero-sum games , 2012, Autom..

[38]  Jorge Cortés,et al.  Distributed Saddle-Point Subgradient Algorithms With Laplacian Averaging , 2015, IEEE Transactions on Automatic Control.