  The multi-cluster games are addressed in this paper, where all players team
up with the players in the cluster that they belong to, and compete against the
players in other clusters to minimize the cost function of their own cluster.
The decision of every player is constrained by coupling inequality constraints,
local inequality constraints and local convex set constraints. Our problem
extends well-known noncooperative game problems and resource allocation
problems by considering the competition between clusters and the cooperation
within clusters at the same time. Besides, without involving the resource
allocation within clusters, the noncooperative game between clusters, and the
aforementioned constraints, existing game algorithms as well as resource
allocation algorithms cannot solve the problem. In order to seek the
variational generalized Nash equilibrium (GNE) of the multi-cluster games, we
design a distributed algorithm via gradient descent and projections. Moreover,
we analyze the convergence of the algorithm with the help of variational
analysis and Lyapunov stability theory. Under the algorithm, all players
asymptotically converge to the variational GNE of the multi-cluster game.
Simulation examples are presented to verify the effectiveness of the algorithm.
