  The decision making and management of many engineering networks involves
multiple parties with conflicting interests, while each party is constituted
with multiple agents. Such problems can be casted 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 together to optimize the payoff function
of the cluster. In a large-scale network, the information of agents in a
cluster can not be available immediately for agents beyond this cluster, which
raise challenges to the existing Nash equilibrium seeking algorithms. Hence, we
consider a partial-decision information scenario in generalized Nash
equilibrium seeking for multi-cluster games in a distributed manner. We
reformulate the problem as 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 fully 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 resorting to the fixed point theory by providing a sufficient
condition. We discuss its potential application in Energy Internet with
numerical studies.
