Hierarchical Distributed Optimization Algorithm Design For Multi-cluster Networks

In distributed optimization,a group of agents collectively minimize the global objective function,and each agent can only communicate with its neighbors while the local objective function is its private information.In this thesis,the multi-cluster networks where all agents are sparsely distributed with different clusters are considered,and two hierarchical distributed optimization algorithms are developed as follows.(1)For the multi-cluster networks with undirected communication graph,a gossip-based distributed hierarchical algorithm is developed.Different from the existing hierarchical algorithms based on the circular method,a gossip scheme is utilized to accomplish the intercluster communication.Specifically,for each iteration cycle,only two neighbor clusters are randomly woken up to exchange their information while other clusters keep their latest states.Therefore,this mechanism relaxes the constraints on the inter-cluster graph.Technicalty,a new analysis method is adopted and some different techniques such as the non-expansiveness of projection operator,and together with the convergence results of the supermartingale are applied to study the intra-cluster consensus,based on which it is proved that estimates of all agents uniformly converge to the optimal solution with probability one.(2)For the multi-cluster networks with directed communication graph,a network floodingbased distributed hierarchical algorithm is developed.Different from the existing hierarchical algorithms over undirected graph,the proposed algorithm exploits the network flooding method to achieve the intra-cluster finite-time average consensus.Specifically,after a finite times of intra-cluster information diffusion,each agent obtains all initial states of the corresponding cluster and then computes the average value.Technically,some different techniques,such as the convergence results of convolution sequence are applied to study the optimal consensus among the leader agents,based on which it is proved that estimates of all agents uniformly converge to the optimal solution.Finally,some simulation examples are provided to verify the effectiveness of the proposed algorithms.