Fully Distributed Algorithm for Resource Allocation over Unbalanced Directed Networks without Global Lipschitz Condition

This note investigates the distributed optimal resource allocation problem of multi-agent systems over unbalanced directed networks under the relaxed condition that the gradients of local cost functions are locally Lipschitz. The objective is to cooperatively drive the decision variables of the agents to the optimal solution, which minimizes the sum of the local cost functions, while ensuring that the network resource constraints and local feasibility constraints are satisfied. A novel distributed algorithm is developed over unbalanced directed network topologies based on the topology balancing technique and adaptive control approach. The developed algorithm is fully distributed in the sense that it depends on neither the global Lipschitz continuity of the gradients nor prior global information about the network connectivity. By regarding the proposed algorithm as a perturbed system, its input-to-state stability with a vanishing perturbation is first established, and asymptotic convergence of the decision variables toward the optimal solution is then proved. The effectiveness of the proposed fully distributed algorithm is illustrated with two examples.