Distributed Difference of Convex Optimization

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 fi and gi 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.