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Linearized symmetric multi-block ADMM with indefinite proximal regularization and optimal proximal parameter
Calcolo ( IF 1.7 ) Pub Date : 2020-11-02 , DOI: 10.1007/s10092-020-00387-1
Xiaokai Chang , Jianchao Bai , Dunjiang Song , Sanyang Liu

The proximal term plays a significant role in the literature of proximal Alternating Direction Method of Multipliers (ADMM), since (positive-definite or indefinite) proximal terms can promote convergence of ADMM and further simplify the involved subproblems. However, an overlarge proximal parameter decelerates the convergence numerically, though the convergence can be established with it. In this paper, we thus focus on a Linearized Symmetric ADMM (LSADMM) with proper proximal terms for solving a family of multi-block separable convex minimization models, and we determine an optimal (smallest) value of the proximal parameter while convergence of this LSADMM can be still ensured. Our LSADMM partitions the data into two group variables and updates the Lagrange multiplier twice in different forms with suitable step sizes. The region of the proximal parameter, involved in the second group subproblems, is partitioned into the union of three different sets. We show the global convergence and sublinear ergodic convergence rate of LSADMM for the two cases, while a counter-example is given to illustrate that convergence of LSADMM can not be guaranteed for the remaining case. Theoretically, we obtain the optimal lower bound of the proximal parameter. Numerical experiments on the so-called latent variable Gaussian graphical model selection problems are presented to demonstrate performance of the proposed algorithm and the significant advantage of the optimal lower bound of the proximal parameter.



中文翻译:

具有不确定近端正则化和最佳近端参数的线性对称多块ADMM

近端项在近端交替方向乘数法(ADMM)的文献中起着重要作用,因为(正定或不确定)近端项可以促进ADMM的收敛,并进一步简化所涉及的子问题。但是,过大的近端参数虽然可以建立收敛,但在数值上会降低收敛速度。因此,在本文中,我们将重点放在具有适当近端项的线性对称ADMM(LSADMM)上,以解决一系列多块可分离凸最小化模型,并确定该LSADMM收敛时的近端参数的最佳(最小)值仍然可以保证。我们的LSADMM将数据划分为两个组变量,并以适当的步长以不同形式两次更新Lagrange乘数。第二组子问题中涉及的近端参数区域被划分为三个不同集合的并集。我们显示了两种情况下LSADMM的全局收敛性和亚线性遍历收敛速度,同时给出了一个反例以说明在其余情况下不能保证LSADMM的收敛性。从理论上讲,我们获得了近端参数的最佳下限。提出了对所谓的潜在变量高斯图形模型选择问题的数值实验,以证明所提出算法的性能以及近端参数的最佳下限的显着优势。我们显示了两种情况下LSADMM的全局收敛性和亚线性遍历收敛速度,同时给出了一个反例以说明在其余情况下不能保证LSADMM的收敛性。从理论上讲,我们获得了近端参数的最佳下限。提出了对所谓的潜在变量高斯图形模型选择问题的数值实验,以证明所提出算法的性能以及近端参数的最佳下限的显着优势。我们显示了两种情况下LSADMM的全局收敛性和亚线性遍历收敛速度,同时给出了一个反例以说明在其余情况下不能保证LSADMM的收敛性。从理论上讲,我们获得了近端参数的最佳下限。提出了对所谓的潜在变量高斯图形模型选择问题的数值实验,以证明所提出算法的性能以及近端参数最优下限的显着优势。

更新日期:2020-11-03
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