Distributed Difference of Convex Optimization
Vivek Khatana, Murti V. Salapaka
TL;DR
The paper tackles distributed DC optimization over directed graphs, where each agent holds a local objective of the form $f_i(x)-g_i(x)$. It introduces a smooth surrogate $F_\mu$ built from Moreau envelopes and develops the DDC-Consensus algorithm, which combines local gradient steps with a finite-time $\eta$-consensus to enforce agreement across agents. Theoretical results show convergence to a stationary point of the original nonconvex problem and quantify the impact of consensus error on the updates. A finite-time consensus protocol enables distributed synthesis over non-symmetric topologies, and numerical experiments on a DC-regularized distributed least squares problem validate the approach and compare it against inexact proximal and mixing baselines. The work advances scalable, directed-network optimization for nonconvex DC problems with provable convergence and practical performance gains.
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.