Continuous-Time Penalty Methods for Nash Equilibrium Seeking of a Nonsmooth Generalized Noncooperative Game.

In this article, we propose centralized and distributed continuous-time penalty methods to find a Nash equilibrium for a generalized noncooperative game with shared inequality and equality constraints and private inequality constraints that depend on the player itself. By using the $\ell _{1}$ penalty function, we prove that the equilibrium of a differential inclusion is a normalized Nash equilibrium of the original generalized noncooperative game, and the centralized differential inclusion exponentially converges to the unique normalized Nash equilibrium of a strongly monotone game. Suppose that the players can communicate with their neighboring players only and the communication topology can be represented by a connected undirected graph. Based on a leader-following consensus scheme and singular perturbation techniques, we propose distributed algorithms by using the exact $\ell _{1}$ penalty function and the continuously differentiable squared $\ell _{2}$ penalty function, respectively. The squared $\ell _{2}$ penalty function method works for games with smooth constraints and the exact $\ell _{1}$ penalty function works for certain scenarios. The proposed two distributed algorithms converge to an $\eta$ -neighborhood of the unique normalized Nash equilibrium and an $\eta$ -neighborhood of an approximated Nash equilibrium, respectively, with $\eta$ being a positive constant. For each $\eta >0$ and each initial condition, there exists an $\varepsilon ^*$ such that for each $0< \varepsilon < \varepsilon ^*$ , the convergence can be guaranteed where $\varepsilon$ is a parameter in the algorithm. [ABSTRACT FROM AUTHOR]