A Smooth Double Proximal Primal-Dual Algorithm for a Class of Distributed Nonsmooth Optimization Problem

TL;DR

This paper introduces a novel distributed primal-dual algorithm tailored for a class of nonsmooth convex optimization problems, effectively handling the sum of smooth and double nonsmooth functions with convergence guarantees.

Contribution

It proposes a new double proximal primal-dual method with convergence analysis for the SSDN problem, addressing the challenge of unproximable nonsmooth function sums.

Findings

Algorithm achieves consensus at the optimal point.

Convergence is proven via Lyapunov stability theory.

Simulations confirm the method's effectiveness.

Abstract

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.