  In this paper, we consider a prototypical convex optimization problem with
multi-block variables and separable structures. By adding the Logarithmic
Quadratic Proximal (LQP) regularizer with suitable proximal parameter to each
of the first grouped subproblems, we develop a partial LQP-based Alternating
Direction Method of Multipliers (ADMM-LQP). The dual variable is updated twice
with relatively larger stepsizes than the classical region
$(0,\frac{1+\sqrt{5}}{2})$. Using a prediction-correction approach to analyze
properties of the iterates generated by ADMM-LQP, we establish its global
convergence and sublinear convergence rate of $O(1/T)$ in the new ergodic and
nonergodic senses, where $T$ denotes the iteration index. We also extend the
algorithm to a nonsmooth composite convex optimization and establish {similar
convergence results} as our ADMM-LQP.
