New directions in distributed Nash equilibrium seeking based on monotone operator theory
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Control Theory and Technology http://link.springer.com/journal/11768 DOI https://doi.org/10.1007/s11768-020-0109-z
New directions in distributed Nash equilibrium seeking based on monotone operator theory Peng YI† , Tongyu WANG Department of Control Science & Engineering, Tongji University, Shanghai 201804, China; Shanghai Institute of Intelligent Science and Technology, Tongji University, Shanghai 201804, China
There has been an increasing research interest in modeling, optimization and control of various multi-agent networks that have wide applications in industry, defense, security, and social areas, such as computing clusters, interconnected micro-grid systems, unmanned vessel swarms [1], power systems [2], multiple UAV systems [3] and sensor networks [4]. For non-cooperative agents that only concern selfish profit-maximizing, the decision making problem can be modelled and solved with the help of game theory, while Nash equilibrium (NE) seeking is at the core to solve the non-cooperative multi-agent games [5–7]. Distributed NE seeking methods are appealing compared with the center-based methods in large-scale networks due to its scalability, privacy protection, and adaptability. Recently, monotone operator theory is explored for distributed NE seeking, which is shown to provide an uniform framework for various algorithms in different scenarios. It has been gradually developing into a cutting-edge research field, with the prospect and necessity of future in-depth research. In non-cooperative multi-agent games, each agent has different characteristics and pursues maximizing its own benefit. Hence, there is no centralized manager that can force all agents to adopt specified strategies to optimize the overall benefits. Under the NE, no player can decrease its cost by unilaterally changing its local decision to another feasible one. To seek an NE, the agent is required to optimize its own objective function given the opponent’s countermeasures. Therefore, various optimization-based methods have been investigated for distributed NE seeking, such as the gradient flow method and the best response method. Monotone operator theory is an important branch of nonlinear analysis and mathematical programming, and is also closely connected with the variational analysis, convex optimization and fixed point iterations [8]. It has been gaining momentum in the fields of large-scale optimization, signal processing, and machine learning [9]. Particularly, the operator splitting method for finding zeros of the sum of multiple monotone operators embodies the “Divide and Conquer” idea, customizes the iteration rules based on the properties of operators, and thus can efficiently solve complex composite optimization problems. The monotone operator theory also provides unified theoretical tools for the analysis of various optimization algorithms. For example, the projected gradient method, the proximal gradient method, the primal-dual method and the ADMM method all can be treated as specific operator splitting methods, whil
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