Generalized Nash Equilibrium Seeking Research Articles

Optimizing Fixed-Time Generalized Nash Equilibrium Seeking in Multi-Unmanned Aerial Vehicle Games

Optimizing Fixed-Time Generalized Nash Equilibrium Seeking in Multi-Unmanned Aerial Vehicle Games

Distributed generalized Nash equilibrium seeking strategy design for multi‐cluster games of high‐order players

AbstractIn this paper, a multi‐cluster game (MCG) of autonomous players is explored. Different from the existing MCGs, the high‐order dynamics of players is taken into account in our problem. Owing to the high‐order dynamics of players, existing strategies for MCGs are unable to tackle our problem. For purpose of seeking the variational generalized Nash equilibrium (vGNE) of the game, a distributed strategy is designed on the basis of gradient descent and state feedback, in which a distributed estimator is embedded for the players to estimate the decisions of other players. Furthermore, the asymptotical convergence of the strategy is analyzed via Lyapunov stability theory and variational analysis. Finally, the effectiveness of our method is verified through a numerical simulation.

Hierarchical Robust Generalized Nash Equilibrium Seeking of High-Order Uncertain Nonlinear Systems.

This article investigates the distributed generalized Nash equilibrium (GNE) seeking problem of noncooperative games (NGs) for high-order strict-feedback nonlinear multiagent systems (MASs). In particular, the feasible action set of each agent is not only subject to local set and inequality constraints but also coupled through an equality constraint with other agents. This constraint structure is more general and covers most of the constraints in the GNE seeking literature. To accomplish the concerned GNE seeking objective, we propose a novel hierarchical GNE seeking approach in this article to decouple the distributed GNE seeking algorithm design into two layers. First, we construct a distributed primary-dual GNE estimator to generate virtual reference signals that converge to the GNE. Then, with the output of the estimator as the reference signal, we develop an adaptive tracking controller to solve the resultant tracking problems under output constraints. To overcome the negative effects of the disturbances, novel compensating terms associated with smooth functions and positive integrable time-varying functions are incorporated in the controller design, which thereby realizes the exact GNE seeking in the presence of nonvanishing mismatched disturbances. At last, an example is given to support the theoretical analysis of the proposed algorithms.

Distributed NE and GNE seeking for heterogeneous multi-player games over directed communication networks.

The non-cooperation game with heterogeneous dynamics is of both theoretical significance and practical relevance because of its extensive penetration into various fields, such as the game confrontation composed of unmanned aerial vehicles and unmanned vehicles or the power generation systems with varied turbine assemblies. To solve such a game problem, this paper investigates distributed Nash equilibrium (NE) and generalized Nash equilibrium (GNE) seeking problems for heterogeneous multi-player systems in non-cooperation games. First, by incorporating the output regulation technique, a distributed NE seeking strategy is designed for heterogeneous multi-player games over undirected communication networks. Then, the GNE seeking problem subject to coupling constraints among multiple players is further studied and a distributed GNE seeking strategy is developed by utilizing the Lagrange multiplier and the output regulation technique. Second, in the case of directed communication networks, both distributed NE and GNE seeking strategies are proposed by employing new type of auxiliary variables to compensate for the unbalanced networks. Compared with some existing results on distributed NE and GNE seeking strategies, both the dynamics of players and communication networks are extended further. Finally, numerical simulations are presented to show the effectiveness of the NE and GNE seeking strategies.

Generalized Nash Equilibrium Seeking for Directed Nonsmooth Multicluster Games via a Distributed Lipschitz Algorithm

Generalized Nash Equilibrium Seeking for Directed Nonsmooth Multicluster Games via a Distributed Lipschitz Algorithm

Generalized Nash Equilibrium Seeking for Noncooperative Game With Different Monotonicities by Adaptive Neurodynamic Algorithm.

This article proposes a novel adaptive neurodynamic algorithm (ANA) to seek generalized Nash equilibrium (GNE) of the noncooperative constrained game with different monotone conditions. In the ANA, the adaptive penalty term, which acts as trajectory-dependent penalty parameters, evolves based on the degree of constraints violation until the trajectory enters the action set of noncooperative game. It is shown that the trajectory of the ANA enters the action set in finite time benefited from the adaptive penalty term. Moreover, it is proven that the trajectory exponentially (or polynomially) converges to the unique GNE when the pseudo-gradient of cost function in noncooperative game satisfies strong (or "generalized" strong) monotonicity. To the best of our knowledge, this is the first time to study the polynomial convergence of GNE seeking algorithm. Furthermore, when the pseudo-gradient mentioned above satisfies monotonicity in general, based on Tikhonov regularization method, a new ANA for finding its ε -generalized Nash equilibrium ( ε -GNE) is proposed, and the related exponential convergence of the algorithm is established. Finally, the river basin pollution game and 5G base station location game are given as examples to showcase the algorithm's effectiveness.

Neural-Network-Based Distributed Generalized Nash Equilibrium Seeking for Uncertain Nonlinear Multiagent Systems

Neural-Network-Based Distributed Generalized Nash Equilibrium Seeking for Uncertain Nonlinear Multiagent Systems

Distributed generalized Nash equilibrium seeking: event-triggered coding-decoding-based secure communication

Distributed generalized Nash equilibrium seeking: event-triggered coding-decoding-based secure communication

Generalized Nash Equilibrium Seeking for Noncooperative Games With Heterogeneous Individual Dynamics

Generalized Nash Equilibrium Seeking for Noncooperative Games With Heterogeneous Individual Dynamics

Distributed GNE Seeking Strategy for Second-Integrator Multiplayer Systems Over Directed Topologies.

This article studies the generalized Nash equilibrium (GNE) seeking problem of second-integrator multiplayer systems. In particular, each player is endowed with an individual payoff function with respect to collective decision variables, and simultaneously, a coupling inequality constraint and a set constraint are imposed to each player. The players communicate with their local neighbors over a directed topology. To begin with, a distributed-observer-based seeking strategy is synthesized by leveraging a proper composite variable. It is first demonstrated using nonsmooth analysis that the established distributed observer enables each player to accurately estimate the decision variables of others in terms of a strongly connected topology condition. Upon this basis, all the decision variables are then shown to converge to the expected GNE asymptotically borrowing from convex theory. In addition, three extension results are also given under the built GNE seeking framework. First, under the postulation that the velocity information is unavailable, a velocity-free distributed GNE seeking strategy is synthesized for second-integrator systems by implementing a proper auxiliary dynamics. Second, we consider nonlinear Euler-Lagrange systems with unknown inertia parameters and synthesize an improved distributed GNE seeking strategy resorting to an adaptation technique. Third, we focus on integrator chain systems and synthesize a modified distributed GNE seeking strategy using a new composite variable based on a proper coordinate transformation. For three extension cases, we all show in detail the achievement of the GNE seeking objective. Finally, a practical example is simulated to confirm the built GNE seeking results.

Distributed generalized Nash equilibrium seeking: A backward-reflected-forward-backward-based algorithm

This paper studies non-cooperative games, in which each player’s local objective function depends on its own decisions as well as those of other players. In these games, each player’s local objective function is differentiable, its decision is constrained by a local feasibility constraint, and the decisions of all players are coupled with an equality constraint. By analyzing the variational problem of the games, a distributed generalized Nash equilibrium seeking algorithm with fixed step sizes is developed based on the backward-reflected-forward–backward splitting. Each player performs a backward step followed by a forward–backward step, and the pseudo-gradient is evaluated at a reflection term. Moreover, the convergence of the proposed distributed algorithm is analyzed under standard assumptions using the operator theory and the convex analysis theory. Finally, the simulation results verify the effectiveness of the algorithm and the correctness of the theoretical analysis.

Optimal Selection and Tracking Of Generalized Nash Equilibria in Monotone Games

A fundamental open problem in monotone game theory is the computation of a specific generalized Nash equilibrium (GNE) among all the available ones, e.g. the optimal equilibrium with respect to a system-level objective. The existing GNE seeking algorithms have in fact convergence guarantees toward an arbitrary, possibly inefficient, equilibrium. In this paper, we solve this open problem by leveraging results from fixed-point selection theory and in turn derive distributed algorithms for the computation of an optimal GNE in monotone games. We then extend the technical results to the time-varying setting and propose an algorithm that tracks the sequence of optimal equilibria up to an asymptotic error, whose bound depends on the local computational capabilities of the agents.

Distributed event-triggered generalized Nash equilibrium seeking for aggregative games on unbalanced digraphs

Distributed event-triggered generalized Nash equilibrium seeking for aggregative games on unbalanced digraphs

Distributed Computation of Stochastic GNE With Partial Information: An Augmented Best-Response Approach

In this paper, we focus on the stochastic generalized Nash equilibrium problem (SGNEP) which is an important and widely-used model in many different fields. In this model, subject to certain global resource constraints, a set of self-interested players aim to optimize their local objectives that depend on their own decisions and the decisions of others and are influenced by some random factors. We propose a distributed stochastic generalized Nash equilibrium seeking algorithm in a partial-decision information setting based on the Douglas-Rachford operator splitting scheme, which relaxes assumptions in the existing literature. The proposed algorithm updates players' local decisions through augmented best-response schemes and subsequent projections onto the local feasible sets, which comprise most of the computational workload. The projected stochastic subgradient method is applied to provide approximate solutions to the augmented best-response subproblems for each player. The Robbins-Siegmund theorem is leveraged to establish the main convergence results to a true Nash equilibrium using the proposed inexact solver. Finally, we illustrate the validity of the proposed algorithm via two numerical examples, i.e., a stochastic Nash-Cournot distribution game and a multi-product assembly problem with the two-stage model.

Distributed Primal–Dual Algorithms for Stochastic Generalized Nash Equilibrium Seeking Under Full and Partial-Decision Information

Distributed Primal–Dual Algorithms for Stochastic Generalized Nash Equilibrium Seeking Under Full and Partial-Decision Information

Distributed event‐triggered generalized Nash equilibrium seeking in multi‐coalition noncooperative games with coupling constraints

AbstractThis paper is concerned with a multi‐coalition noncooperative game with coupling equality constraints. Each coalition is a player consisted of multiple agents in noncooperative games and desire to minimize its own objective function based on local information. Each agent as actual decision maker in the same coalition is to optimize the objective function of the coalition cooperately. To seek a generalized Nash equilibrium (GNE) of the multi‐coalition game, a distributed continuous‐time algorithm is developed. Moreover, to further reduce the communication among agents and coalitions, an event‐triggered mechanism (ETM) is introduced for the multi‐coalition game. By using ETM, a novel distributed GNE seeking algorithm is proposed, where agents and coalitions are allowed to exchange estimation information with neighbors only when the triggering condition is satisfied. Remarkably, the proposed event‐triggered scheme introduces internal variables to regulate its threshold dynamically, which excludes Zeno behavior. By Lyapunov analysis, it is proved that the coalitions' decision variables converge to a GNE in both algorithms. Finally, the effectiveness of the proposed methods is validated by numerical simulations.

Generalized multi-cluster game under partial-decision information with applications to management of energy internet

The decision making and management of many engineering networks involves multiple parties with conflicting interests, while each party is constituted by multiple agents. Such problems can be cast as a multi-cluster game. Each cluster is treated as a self-interested player in a non-cooperative game where agents in the same cluster cooperate to optimize the cost function of the cluster. In a large-scale network, the information of agents in a cluster can not be available immediately to agents beyond this cluster, which raises challenges to the existing Nash equilibrium seeking algorithms. Hence, we consider a partial-decision information scenario in multi-cluster games. We reformulate the problem by finding zeros of the sum of preconditioned monotone operators by the primal-dual analysis and graph Laplacian matrix. Then a distributed generalized Nash equilibrium seeking algorithm is proposed without requiring full awareness of its opponent clusters’ decisions based on a forward-backward-forward method. With the algorithm, each agent estimates the strategies of all the other clusters by communicating with neighbors via an undirected network. We show that the derived operators can be monotone when the communication strength parameter is sufficiently large. We prove the algorithm convergence by providing a sufficient condition of the fixed point theory. We discuss its potential application in Energy Internet with numerical studies.

An adaptive generalized Nash equilibrium seeking algorithm under high-dimensional input dead-zone

In this paper, a novel adaptive generalized Nash equilibrium (GNE) seeking algorithm is designed, in order to address the non-cooperative game with private inequality constraints under high-dimensional input dead-zone. That is to say, the dead-zone dynamics may be thought of as a generic high-dimensional convex set, and the introduction of two methods distinguishes our works in seeking the GNE of non-cooperative games. On the one hand, a two-time-scale structure based on singular perturbation method is led into the design of GNE seeking algorithm, where the fast dynamics part rapidly eliminates the influence of input dead-zone, and the slow dynamics part drives the players’ action to the GNE. On the other hand, adaptive penalty method is utilized to ensure the player’s action enters the inequality constraints set without a prior estimation of centralized information for penalty parameters. The algorithm in this paper realizes complete distribution and parameter independence, making it easy to apply in practical programming. At last, several numerical examples regarding the electricity markets are employed to verify the effectiveness of the theoretical results.

Generalized Nash equilibrium seeking for networked noncooperative games with a dynamic event-triggered mechanism

The generalized Nash equilibrium (GNE) seeking problem of networked noncooperative games is addressed in this paper, where the decision set of each agent is coupled with others via shared constraints. To improve the communication efficiency among agents, a semi-decentralized dynamic event-triggered GNE seeking algorithm is proposed under the full-decision information scenario. Further, considering the partial-decision scenario, where each agent only has the direct interaction with its neighbors via a network, a distributed dynamic event-triggered GNE seeking algorithm is proposed on the basis of a consensus protocol and an operator splitting method. Some sufficient conditions are derived to guarantee the convergence to a variational GNE. Finally, examples on Nash-Cournot games and task allocation games are demonstrated to verify the potential and validity of theoretical results.

Semi-Decentralized Generalized Nash Equilibrium Seeking in Monotone Aggregative Games

We address the generalized Nash equilibrium seeking problem for a population of agents playing aggregative games with affine coupling constraints. We focus on semi-decentralized communication architectures, where there is a central coordinator able to gather and broadcast signals of aggregative nature to the agents. By exploiting the framework of monotone operator theory and operator splitting, we first critically review the most relevant available algorithms and then design two novel schemes: (i) a single-layer, fixed-step algorithm with convergence guarantee for general (non cocoercive, non-strictly) monotone aggregative games and (ii) a single-layer proximal-type algorithm for a class of monotone aggregative games with linearly coupled cost functions. We also design novel accelerated variants of the algorithms via (alternating) inertial and over-relaxation steps. Finally, we show via numerical simulations that the proposed algorithms outperform those in the literature in terms of convergence speed.