Distributed Nash Equilibrium Seeking for A Generalized Convex Game with Nonsmooth Objective Functions and Certain Nonsmooth Constraints

In this paper, distributed algorithms are proposed to find a Nash equilibrium for a generalized convex game with shared inequality and equality constraints and private inequality constraints that depend on the player itself. Both the objective functions and constraints could be nonsmooth and locally Lipschitz. Suppose that the players can communicate with their neighboring players and the communication graph can be represented by a connected undirected graph. By using the 1 penalty function, which is nondifferentiable, the constrained game is converted into an unconstrained game. The proposed differential inclusion exponentially converges to an Nash equilibrium of a strongly monotone game for centralized implementation, and exponentially converges to a η-neighborhood of an Nash equilibrium of a strongly monotone game for distributed implementation, with η being a positive constant that could be arbitrarily small. The distributed algorithm is based on a leader-following consensus scheme and the stability analysis of the algorithms uses nonsmooth analysis and singular perturbation for differential inclusion. A numerical example is given to show the effectiveness of the proposed algorithms.