Distributed Proximal Algorithms for Nonsmooth Optimization: Unified Convergence Analysis

This paper studies two classes of nonsmooth distributed optimization problems with coupled constraints, in which the local cost function of each agent consists of a Lipschitz differentable function and a nonsmooth function. By applying the primal-dual method and proximal operation, we propose two discrete-time distributed algorithms to solve the nonsmooth resource allocation problem and optimal consensus problem, respectively. Different from some previous results with decreasing step-sizes, the proposed algorithms are developed by using the constant step-sizes, which achieves a faster convergence rate. Moreover, we find that these two distributed proximal algorithms have the same structure and can be formulated in a unified framework. A unified convergence analysis is shown that these two algorithms achieve exact convergence to an optimal solution with an ergodic convergence rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(1/k)$</tex-math></inline-formula>. Finally, a simulation example is presented to demonstrate the effectiveness of the proposed algorithms.