Distributed Inexact Consensus-Based ADMM Method for Multi-Agent Unconstrained Optimization Problem

Long Jian,Yiyi Zhao,Peng Li,Jiangping Hu

doi:10.1109/access.2019.2923269

Abstract

Recently, the alternating direction method of multipliers (ADMM) has been used effectively to solve the multi-agent unconstrained optimization problems, where the objective function is the sum of privately known local objective functions of agents. In this paper, first, with the help of the edge-node incidence matrix, an unconstrained optimization problem is transformed into an equivalent optimization problem with only equality constraint and, thus, can be dealt with the ADMM conveniently. Second, a novel distributed inexact consensus ADMM is proposed to enable the agents to reach consensus on the optimal solution of the optimization problem. At the same time, the analysis of the linear convergence of the proposed algorithm is also provided under some mild conditions. Finally, some simulation results are presented to demonstrate the better effectiveness of the proposed algorithm than the standard consensus-based ADMM algorithm.

Highlights

Distributed optimization, which cooperatively achieves optimal decisions for the local manipulation with private data and the diffusion of local information through a multi-agent network, has drawn more and more research attention in recent years
The inexact consensus (IC)-alternating direction method of multipliers (ADMM) is presented in Algorithm 1
NUMERICAL RESULTS This section examines the performance of the consensus ADMM (C-ADMM) and IC-ADMM algorithms, respectively, by using a numerical example

Summary

INTRODUCTION

Distributed optimization, which cooperatively achieves optimal decisions for the local manipulation with private data and the diffusion of local information through a multi-agent network, has drawn more and more research attention in recent years. L. Jian et al.: Distributed Inexact Consensus-Based ADMM Method for Multi-Agent Unconstrained Optimization Problem proved that the iterates of all agents converged to the same optimal point with probability 1. A distributed algorithm based on ADMM has been extended to solve the unconstrained or constrained optimization problems [20]–[22]. Introduced a new distributed ADMM optimization algorithm based an edge-node incidence matrix, which was a useful method to decompose the optimization problem with coupled constraints. Our work is closely related to some recent works [22], [25], which transformed the unconstrained optimization problem to a consensus-based constrained optimization problem with the help of an edge-node incidence matrix They proposed a distributed consensus ADMM (C-ADMM) to solve the optimization problem, and showed its convergence with a rate.

ALGEBRAIC GRAPH BASICS

DISTRIBUTED C-ADMM ALGORITHM

ALGORITHM CONVERGENCE ANALYSIS

NUMERICAL RESULTS

CONCLUSION