Proximal Dynamics in Multiagent Network Games

Abstract

Abstract-We consider dynamics and protocols for agents seeking an equilibrium in a network game with proximal quadratic cost coupling. We adopt an operator theoretic perspective to show global convergence to a network equilibrium, under the assumption of convex cost functions with proximal quadratic couplings, time-invariant and time-varying communication graph along with convex local constraints, and time-invariant communication graph along with convex local constraints and separable convex coupling constraints. We show that proximal dynamics generalize opinion dynamics in social networks and are applicable to distributed tertiary control in power networks. Advantageously, distributed computation and communication setups allow each agent to keep its own data private and exchange information with selected agents only. Typically, in networked multi-agent systems, the state (or decision) variables of each agent evolve as a result of local decision making, e.g. local optimization subject to private constraints, and distributed communication with some other agents, according to a communication graph. It then follows naturally that the aim of the agents is to reach a collective equilibrium state, where no agent can benefit from updating its state variables. I. INTRODUCTIONLiterature overview: In this paper, we study network games with proximal, hence quadratic, cost coupling between neighboring agents, that are related to the literature of distributed multi-agent equilibrium seeking in network games and distributed multi-agent optimization.Network games among agents with convex compact local constraints have been considered in [14] under the assumption of strongly convex quadratic cost functions and time-invariant communication graph; in [15] [16], under the assumption of differentiable cost functions with Lipschitz continuous gradient, strictly monotone pseudo-gradient game mapping (hence strictly convex cost functions), and undirected, possibly time-varying, communication graph. Multi-agent games with convex compact local and also coupling constraints have been considered in [17] under the assumption of strongly convex twice differentiable cost functions with bounded gradients, with strictly increasing congestion cost term.

Multipopulation Aggregative Games:Equilibrium Seeking via Mean-Field Control and Consensus

“…Distributed optimization and computational game theory for large-scale multi-agent systems have been strong recent research areas in systems and control [1] with a variety of applications, e.g. in wireless communication [2], transmission networks [3], social networks [4] and smart grids [5].…”

Leader–Follower Network Aggregative Game With Stochastic Agents’ Communication and Activeness

“…summation or weighted sum), the network game is known as network aggregative game (NAG) [3]. Many applications can be studied via this framework including, power system [4], opinion dynamics [5], communication system [6], provision of public goods [7] and criminal networks [8].…”

“…Therein, some algorithms are proposed which M. Shokri and H. Kebriaei converge to the Nash equilibrium point using the best response functions while at each iteration, agents communicate with their neighbors and update their strategies. Furthermore, in [13], a Nash seeking dynamics is utilized for agents with proximal quadratic cost functions whose best responses are in the form of proximal operator. In [14], the cost function of the agents is considered in general form.…”

“…From other aspects, compared to aggregative games which consider the average strategy of whole population as a common coupling term among the agents [9]- [11], in this paper, the local aggregative term is studied in which only neighbors of each follower, as well as the leader, affect the follower's cost function. Compared to the literature of NAGs, those consider the local aggregative term, we have studied a general strongly convex cost function instead of quadratic one [12], [13]. In contrast with the papers which have utilized the best response dynamics as the agents' decision update rule [12]- [14], we have used the projected sub-gradient method for optimization to cope with the limited computational capabilities of the agents.…”

Network games with dynamic players: Stabilization and output convergence to Nash equilibrium

“…In Parise et al (2015), distributed iterative strategies are proposed for agents to converge to an NE in network games with quadratic cost and convex constraints. Grammatico (2018) investigates the convergence of different equilibrium seeking proximal dynamics for multiagent network games with convex cost functions, time-varying communication graph, and coupling constraints. Our work focuses on NE seeking in a class of network games, since in cooperative control of large networks, it can be computational difficult or unrealistic for each agent to consider the actions of all others.…”