Distributed Optimization over Weakly Connected Digraphs with Matrix Weights

This article introduces a novel approach to solve the distributed optimization problems (DOPs) within matrix weighted weakly connected directed topologies. Including matrix weights alongside directed networks complicates the analysis of the associated Laplacian matrix properties. To address these complexities, we find a sufficient condition to ensure that the optimization holds true. For the properties of weakly connected digraphs, we define an extended almost NSi (ANSi) partition method by extending the NSi graph partition. The partition identifies those nodes that cannot effectively exchange gradient information due to network topology constraints by detecting unidirectional connectivity components in the gradient information transfer path. These nodes are unable to receive complete gradient updates ’in time’, resulting in global optimisation not being achieved. To tackle this problem, we design a corresponding inverse feedback control to ensure strong connectivity throughout the topology. Finally, we develop a distributed optimization control based on an improved PI optimization algorithm that allows the system to achieve optimization without initial constraints exponentially.