Multi-cluster distributed optimization in open multi-agent systems over directed graphs with acknowledgement messages

In this paper, we tackle the problem of distributed optimization over directed networks in open multi-agent systems (OMAS), where agents may dynamically join or leave, causing persistent changes in network topology and problem dimension. These disruptions not only pose significant challenges to maintaining convergence and stability in distributed optimization algorithms, but could also break the network topology into multiple clusters, each one associated with its own set of objective functions. To address this, we propose a novel Open Distributed Optimization Algorithm with Gradient Tracking (OPEN-GT), which employs: (a) a dynamic mechanism for detecting active out-neighbors through acknowledgement messages, and (b) a fully distributed max-consensus procedure to spread information regarding agent departures, in possibly unbalanced directed networks. We show that when all active agents execute OPEN-GT, the optimization process in each formed cluster remains consistent, while the agents converge to their cluster-wide optimal solution if there exists a time after which the network remains unchanged. Finally, we validate our approach in a simulated environment with dynamically changing agent populations, demonstrating its resilience to network variations and its ability to support distributed optimization under OMAS dynamics.