Linear Convergence of ADMM Under Metric Subregularity for Distributed Optimization

The alternating direction method of multipliers (ADMM) has seen much progress in the literature in recent years. Usually, linear convergence of distributed ADMM is proved under either second-order conditions or strong convexity. When both conditions fail, an alternative is expected to play the role. In this article, it is shown that distributed ADMM can achieve a linear convergence rate by imposing metric subregularity on a defined mapping. Furthermore, it is proved that both second-order conditions and strong convexity imply metric subregularity under reasonable conditions, e.g., the cost functions being twice continuously differentiable in a neighborhood. In addition, nonergodic convergence rates are presented as well for problems under consideration. Finally, simulation results are carried out to illustrate the efficiency of the proposed algorithm.