  In this work, we consider solving a distributed optimization problem (DOP) in
a multi-agent network with multiple agent clusters. In each cluster, the agents
manage separable cost functions composed of possibly non-smooth components and
aim to achieve an agreement on a common decision of the cluster. The global
cost function is considered as the sum of the individual cost functions
associated with affine coupling constraints on the clusters' decisions. To
solve this problem, the dual problem is formulated by the concept of Fenchel
conjugate. Then an asynchronous distributed dual proximal gradient (Asyn-DDPG)
algorithm is proposed based on a cluster-based partial and mixed consensus
protocol, by which the agents are only required to communicate with their
neighbors with communication delays. An ergodic convergence result is provided,
and the feasibility of the proposed algorithm is verified by solving a social
welfare optimization problem in the simulation.
