Solving Nonsmooth Resource Allocation Problems with Feasibility Constraints through Novel Distributed Algorithms

The distributed non-smooth resource allocation problem over multi-agent networks is studied in this paper, where each agent is subject to globally coupled network resource constraints and local feasibility constraints described in terms of general convex sets. To solve such a problem, two classes of novel distributed continuous-time algorithms via differential inclusions and projection operators are proposed. Moreover, the convergence of the algorithms is analyzed by the Lyapunov functional theory and nonsmooth analysis. We illustrate that the first algorithm can globally converge to the exact optimum of the problem when the interaction digraph is weight-balanced and the local cost functions being strongly convex. Furthermore, the fully distributed implementation of the algorithm is studied over connected undirected graphs with strictly convex local cost functions. In addition, to improve the drawback of the first algorithm that requires initialization, we design the second algorithm which can be implemented without initialization to achieve global convergence to the optimal solution over connected undirected graphs with strongly convex cost functions. Finally, several numerical simulations verify the results.