  This technical note studies a class of distributed nonsmooth convex consensus
optimization problem. The cost function is a summation of local cost functions
which are convex but nonsmooth. Each of the local cost functions consists of a
twice differentiable convex function and two lower semi-continuous convex
functions. We call this problem as single-smooth plus double-nonsmooth (SSDN)
problem. Under mild conditions, we propose a distributed double proximal
primal-dual optimization algorithm. The double proximal operator is designed to
deal with the difficulty caused by the unproximable property of the summation
of those two nonsmooth functions. Besides, it can also guarantee that the
proposed algorithm is locally Lipschitz continuous. An auxiliary variable in
the double proximal operator is introduced to estimate the subgradient of the
second nonsmooth function. Theoretically, we conduct the convergence analysis
by employing Lyapunov stability theory. It shows that the proposed algorithm
can make the states achieve consensus at the optimal point. In the end,
nontrivial simulations are presented and the results demonstrate the
effectiveness of the proposed algorithm.
