  In this paper, two distributed multi-proximal primal-dual algorithms are
proposed to deal with a class of distributed nonsmooth resource allocation
problems. In these problems, the global cost function is the summation of local
convex and nonsmooth cost functions, each of which consists of one twice
differentiable function and multiple nonsmooth functions. Communication graphs
of underling multi-agent systems are directed and strongly connected but not
necessarily weighted-balanced. The multi-proximal splitting is designed to deal
with the difficulty caused by the unproximable property of the summation of
those nonsmooth functions. Moreover, it can also guarantee the smoothness of
proposed algorithms. Auxiliary variables in the multi-proximal splitting are
introduced to estimate subgradients of nonsmooth functions. Theoretically, the
convergence analysis is conducted by employing Lyapunov stability theory and
integral input-to-state stability (iISS) theory with respect to set. It shows
that proposed algorithms can make states converge to the optimal point that
satisfies resource allocation conditions.
