Exponentially Convergent Algorithm Design for Constrained Distributed Optimization via Non-smooth Approach

TL;DR

This paper introduces a novel distributed algorithm with exponential convergence for constrained optimization problems involving non-smooth functions, leveraging the exact penalty method and differential inclusions.

Contribution

It presents the first exponential convergence rate for a distributed algorithm addressing non-smooth constrained optimization problems.

Findings

Algorithm converges exponentially for strongly convex objectives.

Reformulation via exact penalty method simplifies the problem.

Applicable to non-smooth functions with set constraints.

Abstract

We consider minimizing a sum of non-smooth objective functions with set constraints in a distributed manner. As to this problem, we propose a distributed algorithm with an exponential convergence rate for the first time. By the exact penalty method, we reformulate the problem equivalently as a standard distributed one without consensus constraints. Then we design a distributed projected subgradient algorithm with the help of differential inclusions. Furthermore, we show that the algorithm converges to the optimal solution exponentially for strongly convex objective functions.