  In this paper, the distributed resource allocation problem on strongly
connected and weight-balanced digraphs is investigated, where the decisions of
each agent are restricted to satisfy the coupled network resource constraints
and heterogeneous general convex sets. Moreover, the local cost function can be
non-smooth. In order to achieve the exact optimum of the nonsmooth resource
allocation problem, a novel continuous-time distributed algorithm based on the
gradient descent scheme and differentiated projection operators is proposed.
With the help of the set-valued LaSalle invariance principle and nonsmooth
analysis, it is demonstrated that the algorithm converges asymptotically to the
global optimal allocation. Moreover, for the situation where local constraints
are not involved and the cost functions are differentiable with Lipschitz
gradients, the convergence of the algorithm to the exact optimal solution is
exponentially fast. Finally, the effectiveness of the proposed algorithms is
illustrated by simulation examples.
