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

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.