Convergence of distributed approximate subgradient method for minimizing convex function with convex functional constraints

In this paper, we investigate the distributed approximate subgradient-type method for minimizing a sum of differentiable and non-differentiable convex functions subject to nondifferentiable convex functional constraints in a Euclidean space. We establish the convergence of the sequence generated by our method to an optimal solution of the problem under consideration. Moreover, we derive a convergence rate of order $ \mathcal{O}(N^{1-a}) $ for the objective function values, where $ a\in (0.5, 1) $. Finally, we provide a numerical example illustrating the effectiveness of the proposed method.