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.

在本文中，我们考虑了具有多块变量和可分离结构的原型凸优化问题。通过将具有合适的近端参数的对数二次近似（LQP）正则化器添加到每个第一个分组子问题中，我们开发了一种基于局部LQP的乘数交替方向方法（ADMM-LQP）。对偶变量更新两次，步长比经典区域大$（0，\frac{\mathrm{1个}+\sqrt{5}}{\mathrm{2个}}）$。使用预测校正方法分析ADMM-LQP生成的迭代的属性，我们建立了它的全局收敛性和亚线性收敛率。$Ø（\mathrm{1个}/Ť）$在新的遍历和非遍历意义上，其中T表示迭代索引。我们还将算法扩展到非光滑复合凸优化，并建立与我们的ADMM-LQP相似的收敛结果。