A distributed proximal-point algorithm for Nash equilibrium seeking in generalized potential games with linearly coupled cost functions

In this paper, we consider distributed Nash equilibrium (NE) seeking in potential games over a multi-agent network, where each agent can not observe the actions of all its rivals. Based on the best response dynamics, we design a distributed NE seeking algorithm by incorporating the non-smooth finite-time average tracking dynamics, where each agent only needs to know its own action and exchange information with its neighbours through a communication graph. We give a sufficient condition for the Lipschitz continuity of the best response mapping for potential games, and then prove the convergence of the proposed algorithm based on the Lyapunov theory. Numerical simulations are given to verify the result and illustrate the effectiveness of the algorithm.

In this article, we introduce first-order and zeroth-order Nash equilibrium seeking dynamics with fixed-time and practical fixed-time convergence certificates for noncooperative games having finitely many players. The first-order algorithms achieve exact convergence to the Nash equilibrium of the game in a finite time that can be additionally upper bounded by a constant that is independent of the initial conditions of the actions of the players. Moreover, these fixed-time bounds can be prescribed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> by the system designer under an appropriate tuning of the parameters of the algorithms. When players have access only to measurements of their cost functions, we consider a class of distributed multitime scale zeroth-order model-free adaptive dynamics that achieve semiglobal practical fixed-time stability, qualitatively preserving the fixed-time bounds of the first-order dynamics as the time scale separation increases. Moreover, by leveraging the property of fixed-time input-to-state stability, further results are obtained for mixed games where some of the players implement different seeking dynamics. Fast and slow switching communication graphs are also incorporated using tools from hybrid systems. We consider potential games as well as general nonpotential strongly monotone games. Numerical examples illustrate our results.

In this paper, we consider a composite optimization problem with linear coupling constraints in a multi-agent network. In this problem, the agents cooperatively optimize a strongly convex cost function which is the linear sum of individual cost functions composed of smooth and possibly non-smooth components. To solve this problem, we propose an asynchronous penalized proximal gradient (Asyn-PPG) algorithm, a variant of classical proximal gradient method, with the presence of the asynchronous updates of the agents and uniform communication delays in the network. Specifically, we consider a slot-based asynchronous network (SAN), where the whole time domain is split into sequential time slots and each agent is permitted to execute multiple updates during a slot by accessing the historical state information of the agents. By the Asyn-PPG algorithm, an explicit convergence rate can be guaranteed based on deterministic analysis. The feasibility of the proposed algorithm is verified by solving a consensus-based distributed regression problem and a social welfare optimization problem in the electricity market.

Distributed Nash Equilibrium (NE) seeking for networked games has been widely investigated in recent years. In particular, convergence speed and communication cost are two of the critical concerns for the design of the seeking algorithms. To achieve fast convergence while saving communication resources, this paper investigates the appointed-time distributed NE seeking problem for networked games under a discrete-time communication scenario. By utilizing the idea of motion planning, a new continuous-time distributed NE seeking algorithm, which ensures convergence to NE at the prescribed time, is proposed based on sampled-data information exchange. Compared with the existing related works, the established algorithm has advantages on both the communication cost and convergence speed (i.e., appointed-time convergence). Finally, the effectiveness of the proposed method is verified via numerical simulations.

This paper presents a study on Nash equilibrium seeking in noncooperative games within multi-robot systems, a topic of increasing importance in diverse sectors including civil, security, and military. Unlike conventional approaches where players can directly observe the actions of others, our method assumes limited visibility, where players can only communicate through an undirected and connected communication graph. We introduce a novel distributed control approach, integrating gradient play with a consensus protocol. This method facilitates effective Nash equilibrium seeking by leveraging information shared among neighboring robots in heterogeneous linear dynamic systems. The proposed solution employs a high-level distributed Nash equilibrium-seeking algorithm, serving as an optimal reference generator for each robot to track the Nash equilibrium, and an advanced output regulation technique, aiming to regulate the output (e.g., position) of the robots with respect to the obtained references. Theoretical analysis confirms the convergence of our algorithm through Lyapunov stability analysis. The effectiveness and practical applicability of our approach are validated through numerical simulations and empirical testing with physical robots, highlighting its efficacy and utility in real-world scenarios.

We consider the generalized Nash equilibrium (GNE) problem via distributed computation. Specifically, we consider a partial-decision information setting where each agent has no direct access to the decisions of all others while its cost function depends on them. Instead, each agent is assumed to exchange information with its neighbors via a communication network. To enhance communication efficiency, we propose a distributed event-triggered scheme such that each agent can independently determine when to transmit information to its neighbors. Thus, a fully-distributed, event-triggered GNE seeking algorithm is designed by combining event-triggered scheme, consensus, and projected-pseudo-gradient dynamics. Through primal-dual analysis, we prove convergence to a variational GNE by recasting the overall scheme as an inexact forward-backward iteration.

In this paper, we propose a distributed algorithm for optimization problems that involve a separable, possibly nonconvex objective function subject to convex local constraints and linear coupling constraints. The method is based on the accelerated distributed augmented Lagrangians (ADAL) algorithm that was recently developed by the authors to address convex problems. Here, we extend this line of work in two ways. First, we establish convergence of the method to a local minimum of the problem, using assumptions that are common in the analysis of nonconvex optimization methods. To the best of our knowledge, this is the first work that shows convergence to local minima specifically for a distributed augmented Lagrangian (AL) method applied to nonconvex optimization problems; distributed AL methods are known to perform very well when used to solve convex problems. Second, we propose a more general and decentralized rule to select the stepsizes of the method. This improves on the authors’ original ADAL method, where the stepsize selection used global information at initialization. Numerical results are included to verify the correctness and efficiency of the proposed distributed method.

This paper extends the recently introduced ALADIN algorithm to non-convex continuous-time optimal control problems with nonlinear dynamics and linear coupling constraints. The algorithm alternates between solving a convexified local problem in a distributed manner and a linearized quadratic problem on a centralized entity while using the solution of both for an update step. This paper presents an analysis of the local convergence of the algorithm and shows a quadratic convergence rate. Furthermore, a globalization strategy is presented that ensures global convergence. The paper closes with a numerical evaluation of the algorithm.

This article investigates the distributed Nash equilibrium seeking problem of quadratic time‐varying games with Euler–Lagrange (EL) players, where external disturbances and parametric uncertainties are involved. A gradient‐based hierarchical algorithm consisting of a game layer and a control layer is proposed. Specifically, in the game layer, EL players communicate with neighbors through a graph to reach the consensus on potential aggregate values, which will be employed to calculate the gradient of each player's objective function, and then, a gradient‐based sliding mode controller is developed to track time‐varying gradient in the control layer. Thus, the convergence results are hierarchically obtained through the Lyapunov stability method. In addition, the hierarchical control strategy is extended to address the constrained problems through the utilization of a smooth penalty function. By appropriately choosing control parameters, the Nash equilibrium seeking errors can be arbitrarily small. The relation between the optimal solutions of the original problem and the dual one is further discussed. Finally, the proposed methods are numerically verified.

This brief studies the formation placement problems for multi-cluster system with convergent time requirements based on a time-varying Nash equilibrium (NE) seeking approach. In order to determine the placement of each agent that subject to formation constraints in each cluster within a prescribed setting time, a basic centralized time-varying NE seeking algorithm based on time transformation approach is designed. In addition, by applying a prescribed-time distributed observer, an extended distributed time-varying NE seeking algorithm is proposed. Compared with existing results, the convergence time can be appointed in advance according to specific mission requirements. Finally, the effectiveness of proposed algorithms is verified by numerical simulations.

Transverse beam sizes and quasi emittances for linearly coupled optics

This paper investigates the Nash equilibrium (NE) seeking problem of the game in which the dynamics of all players are affected by external disturbances and the communication network among players may be influenced by persistent attacks. To solve this problem, a distributed robust switched algorithm is proposed. This distributed switched strategy consists of two modules: One is the internal-model based dynamics that can be used to compensate the effects caused by external disturbances; the other is gradient-based optimization module which can utilize the local information to estimate the players' decisions and dynamically seek the optimal solution. By utilizing average dwell-time and time-ratio constraints to characterize the network attacks, the proposed switched strategy is modeled into a hybrid dynamical system. Moreover, employing the hybrid systems approach, we prove that the decisions of all players can exponentially converge to the NE of the considered game. Finally, an example is provided to illustrate the theoretical results.

This article investigates the Nash equilibrium (NE) seeking problems for multicoalition games with heterogeneous general linear dynamics over unbalanced digraphs. The coalition can be regarded as a virtual player but the true decision-makers are the actual players themselves which can only access to their own cost functions. To deal with the unbalanced digraphs, the left eigenvector is estimated and its prescribed-time convergence is illuminated with the time transfer approach. Then the NE seeking part and the output regulation part are designed to adapt the dynamics of the players. The initial values of the auxiliary vectors are selected to avoid utilizing the out-degree information. The steady state of the closed-loop system is analyzed and the prescribed-time convergence is proved based on the Lyapunov method. Finally, the simulation results of both the mobile sensor connectivity game and the electricity market game are presented to show the effectiveness of the proposed algorithm.

In this paper, we study the generalized Nash equilibrium (GNE) seeking problem for an N-coalition game with inequality constraints. First, using full decision information, a finite-time average consensus-based approach is proposed. It is demonstrated that the algorithm converges to a GNE (specifically, a variational equilibrium) of the game. Then, using the finite-time consensus tracking technique, we propose a distributed algorithm that only requires neighboring action information for each agent. The solution by using partial decision information covers the distributed GNE seeking solutions of generalized non-cooperative games as a special case.

A payoff-based learning approach for Nash equilibrium seeking in continuous potential games