Continuous-Time Distributed Proximal Gradient Algorithms for Nonsmooth Resource Allocation Over General Digraphs

This paper studies a nonsmooth resource allocation problem with network resource constraints and local set constraints, where the interaction graphs among agents are generally strongly connected digraphs. First, we design a centralized continuous-time proximal gradient algorithm, where each agent uses the global Lagrangian multipliers and the global values of constraint functions. For the case that the agents' private information could not be leaked and the global Lagrangian multipliers are not available, the agents are endowed with some additional variables to estimate those global information via consensus protocols. Then, we construct a class of continuous-time distributed proximal gradient algorithms by using a two-time scale mechanism to integrate the proposed proximal gradient algorithm and consensus protocols. By adopting Lyapunov stability theory and convex optimization theory, we prove that the decision variables asymptotically converge to the optimal solution of the nonsmooth resource allocation problem. Finally, numerical simulations are applied to illustrate the effectiveness of the proposed algorithms.