Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method

TL;DR

This paper introduces a consensus-based distributed primal-dual perturbation algorithm for solving complex constrained optimization problems in networked systems, effectively handling both smooth and non-smooth constraints.

Contribution

It presents a novel distributed primal-dual perturbation method that estimates global functions via consensus and converges to an optimal solution, even with non-smooth constraints.

Findings

Algorithm converges to an optimal primal-dual solution.

Handles both smooth and non-smooth inequality constraints.

Demonstrated effectiveness in smart grid demand response.

Abstract

Various distributed optimization methods have been developed for solving problems which have simple local constraint sets and whose objective function is the sum of local cost functions of distributed agents in a network. Motivated by emerging applications in smart grid and distributed sparse regression, this paper studies distributed optimization methods for solving general problems which have a coupled global cost function and have inequality constraints. We consider a network scenario where each agent has no global knowledge and can access only its local mapping and constraint functions. To solve this problem in a distributed manner, we propose a consensus-based distributed primal-dual perturbation (PDP) algorithm. In the algorithm, agents employ the average consensus technique to estimate the global cost and constraint functions via exchanging messages with neighbors, and meanwhile use a local primal-dual perturbed subgradient method to approach a global optimum. The proposed PDP method not only can handle smooth inequality constraints but also non-smooth constraints such as some sparsity promoting constraints arising in sparse optimization. We prove that the proposed PDP algorithm converges to an optimal primal-dual solution of the original problem, under standard problem and network assumptions. Numerical results illustrating the performance of the proposed algorithm for a distributed demand response control problem in smart grid are also presented.