  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.
