  This paper addresses a distributed convex optimization problem with a class
of coupled constraints, which arise in a multi-agent system composed of
multiple communities modeled by cliques. First, we propose a fully distributed
gradient-based algorithm with a novel operator inspired by the convex
projection, called the clique-based projection. Next, we scrutinize the
convergence properties for both diminishing and fixed step sizes. For
diminishing ones, we show the convergence to an optimal solution under the
assumptions of the smoothness of an objective function and the compactness of
the constraint set. Additionally, when the objective function is strongly
monotone, the strict convergence to the unique solution is proved without the
assumption of compactness. For fixed step sizes, we prove the non-ergodic
convergence rate of O(1/k) concerning the objective residual under the
assumption of the smoothness of the objective function. Furthermore, we apply
Nesterov's acceleration method to the proposed algorithm and establish the
convergence rate of O(1/k^2). Numerical experiments illustrate the
effectiveness of the proposed method.
