Distributed Nash equilibrium seeking for constrained games

In this paper, the distributed Nash equilibrium seeking problem for the games with convex compact set constraint is studied. A distributed estimator based on the leader-follower consensus law is presented to estimate the states of the players. Each player communicates its estimations to its neighbors. The projections of the estimations for the states of the players onto the constraint sets are used in the Nash equilibrium seeking strategy. By synthesizing the projection term of the player's state and the estimator-projection-based gradient play term, the Nash equilibrium seeking algorithm is proposed. The pseudo-gradient of the cost function is only assumed to be strongly monotone in the compact constraint region not required to be globally strongly monotone. Also, the proposed Nash equilibrium seeking algorithm for the constrained games is feasible under any initial states of the players if the given conditions are satisfied. The convergence of the seeking algorithm is proved. Numerical simulations are provided to validate the effectiveness of the proposed seeking algorithm.