Distributed online generalized Nash Equilibrium learning in multi-cluster games: A delay-tolerant algorithm

This paper addresses the problem of distributed online generalized Nash equilibrium (GNE) learning for multi-cluster games with delayed feedback information. Specifically, each agent in the game is assumed to be informed a sequence of local cost functions and constraint functions, which are known to the agent with time-varying delays subsequent to decision-making at each round. The objective of each agent within a cluster is to collaboratively optimize the cluster's cost function, subject to time-varying coupled inequality constraints and local feasible set constraints over time. Additionally, it is assumed that each agent is required to estimate the decisions of all other agents through interactions with its neighbors, rather than directly accessing the decisions of all agents, i.e., each agent needs to make decision under partial-decision information. To solve such a challenging problem, a novel distributed online delay-tolerant GNE learning algorithm is developed based upon the primal-dual algorithm with an aggregation gradient mechanism. The system-wise regret and the constraint violation are formulated to measure the performance of the algorithm, demonstrating sublinear growth with respect to time under certain conditions. Finally, numerical results are presented to verify the effectiveness of the proposed algorithm.