  A distributed nonsmooth robust resource allocation problem with cardinality
constrained uncertainty is investigated in this paper. The global objective is
consisted of local objectives, which are convex but nonsmooth. Each agent is
constrained in its private convex set and has only the information of its
corresponding local objective. The resource allocation condition is subject to
the cardinality constrained uncertainty sets. By employing the duality theory
of convex optimization, a dual problem of the robust resource allocation
problem is presented. For solving this dual problem, a distributed primal-dual
projected algorithm is proposed. Theoretically, the convergence analysis by
using stability theory of differential inclusions is conducted. It shows that
the algorithm can steer the multi-agent system to satisfy resource allocation
condition at the optimal solution. In the end, a nontrivial simulation is shown
and the results demonstrate the efficiency of the proposed algorithm.
