The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention in many areas because of its efficiency and easy implementation. To theoretically guarantee convergence of the linearized ADMM, the step size for the linearized subproblems, or the reciprocal of the linearization parameter, should be sufficiently small. On the other hand, small step sizes decelerate the convergence numerically. Hence, it is interesting to probe the optimal (largest) value of the step size that guarantees convergence of the linearized ADMM. This analysis is lacked in the literature. In this paper, we provide a rigorous mathematical analysis for finding this optimal step size of the linearized ADMM and accordingly set up the optimal version of the linearized ADMM in the convex programming context. The global convergence and worst-case convergence rate measured by the iteration complexity of the optimal version of linearized ADMM are proved as well.

中文翻译：

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最优线性化乘法器交替方向的凸规划

乘法器的交变方向法（ADMM）被广泛应用于各个领域；针对不同应用场景量身定制的不同变体也已经在文献中进行了深入研究。其中，线性化ADMM的效率高且易于实施，因此在许多领域受到了特别广泛的关注。为了从理论上保证线性化ADMM的收敛性，线性化子问题的步长或线性化参数的倒数应足够小。另一方面，较小的步长会在数值上降低收敛速度。因此，探寻能够确保线性化ADMM收敛的最佳步长（最大）值很有趣。文献中缺乏这种分析。在本文中，我们提供了严格的数学分析，以找到线性化ADMM的最佳步长，从而在凸编程环境中设置线性化ADMM的最佳版本。并证明了线性化ADMM最优版本的迭代复杂度所度量的全局收敛性和最坏情况的收敛速度。