Distributed Difference of Convex Optimization

TL;DR

This paper introduces DDC-Consensus, a distributed algorithm for nonconvex optimization over directed graphs, using smooth approximations of difference-of-convex functions, and proves its convergence to stationary points.

Contribution

The paper develops a novel distributed algorithm for nonconvex DC problems on directed graphs, with convergence guarantees and practical simulation validation.

Findings

Algorithm converges to stationary points.

Effective for directed graph topologies.

Simulation confirms practical performance.

Abstract

In this article, we focus on solving a class of distributed optimization problems involving $n$ agents with the local objective function at every agent $i$ given by the difference of two convex functions $f_i$ and $g_i$ (difference-of-convex (DC) form), where $f_i$ and $g_i$ are potentially nonsmooth. The agents communicate via a directed graph containing $n$ nodes. We create smooth approximations of the functions $f_i$ and $g_i$ and develop a distributed algorithm utilizing the gradients of the smooth surrogates and a finite-time approximate consensus protocol. We term this algorithm as DDC-Consensus. The developed DDC-Consensus algorithm allows for non-symmetric directed graph topologies and can be synthesized distributively. We establish that the DDC-Consensus algorithm converges to a stationary point of the nonconvex distributed optimization problem. The performance of the DDC-Consensus algorithm is evaluated via a simulation study to solve a nonconvex DC-regularized distributed least squares problem. The numerical results corroborate the efficacy of the proposed algorithm.