# Convergence Study on the Proximal Alternating Direction Method with Larger Step Size

The alternating direction method of multipliers (ADMM) is a popular method for the separable convex programming with linear constraints, and the proximal ADMM is its important variant. Previous studies show that the relaxation factor $\gamma\in (0, \frac{1+\sqrt{5}}{2})$ by Fortin and Glowinski for the ADMM is also valid for the proximal ADMM. In this paper, we further demonstrate that the feasible region of $\gamma$ depends on the proximal term added on the second subproblem, and can be enlarged when the proximal factor is positive. We derive the exact relationship between the relaxation factor $\gamma$ and the proximal factor. Finally, we prove the global convergence and derive a worst-case $O(1/t)$ convergence rate in the ergodic sense for this generalized scheme.