Distributed continuous-time algorithm for nonsmooth aggregative optimization over weight-unbalanced digraphs

This paper studies the problem of distributed continuous-time aggregative optimization with set constraints under a weight-unbalanced digraph, where the nonsmooth objective function of each agent relies both on its own decision and on the aggregation of all agents’ decisions. To eliminate the impact of unbalanced digraphs, a consensus-based estimator that tracks the aggregation information is designed through a gradient rescaling technique. Considering that cost functions are nondifferentiable in many scenarios, such as electric power management that takes price caps into account, a novel distributed continuous-time optimization algorithm via generalized gradient is presented in a two-time scale. Moreover, the convergence of the algorithm is established through nonsmooth analysis and singular perturbation theory. Compared to the existing results, which depend on undirected graphs, the proposed strategy is applicable to general digraphs, which may be weight-unbalanced. Further, the assumption on the differentiability of objective functions is relaxed. Finally, two numerical examples are provided to verify the findings.