Further Study on the Convergence Rate of Alternating Direction Method of Multipliers with Logarithmic-quadratic Proximal Regularization

In the literature, the combination of the alternating direction method of multipliers (ADMM) with the logarithmic-quadratic proximal (LQP) regularization has been proved to be convergent and its worst-case convergence rate in the ergodic sense has been established. In this paper, we focus on a convex minimization model and consider an inexact version of the combination of the ADMM with the LQP regularization. Our primary purpose is to further study its convergence rate; and establish its worst-case convergence rates measured by the iteration complexity in both the ergodic and non-ergodic senses. In particular, existing convergence rate results for this combination are subsumed by the new results.