Abstract A classical approach to solving two-block separable convex optimization could be the symmetric alternating direction method of multipliers (S-ADMM). However, its convergence may not be guaranteed for a general multi-block case without additional assumptions. Bai et al. proposed a variant of S-ADMM entitled the generalized symmetric ADMM (GS-ADMM), in which the variables are regrouped into two groups firstly. The two groups of variables are updated in a Gauss-Seidel scheme, while the variables within each group are updated in a Jacobi scheme and the Lagrangian multipliers are updated two times. In order to derive its convergence property, the authors add a special proximal term to each subproblem. In this paper, inspired by the partial PPA block-wise ADMM (PPBADMM) [32] proposed by Shen et al., we propose a partially proximal S-ADMM (PPSADMM). In PPSADMM, the special proximal term is only added to the subproblems in the first group as PPBADMM. We perform an extension step on all variables with a fixed step size at the end of each iteration. Without stringent assumptions, we establish the global convergence result and the O ( 1 / t ) convergence rate in the ergodic sense for PPSADMM. Its numerical performance is justified on two types of problems.

中文翻译：

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用于具有线性约束的可分离凸优化的部分近端 S-ADMM

摘要 解决两块可分离凸优化的经典方法可能是乘法器的对称交替方向法（S-ADMM）。但是，对于没有额外假设的一般多块情况，可能无法保证其收敛。白等人。提出了 S-ADMM 的一种变体，称为广义对称 ADMM（GS-ADMM），其中变量首先重新分组为两组。两组变量以高斯-赛德尔方案更新，而每组内的变量以雅可比方案更新，拉格朗日乘子更新两次。为了推导出其收敛性，作者为每个子问题添加了一个特殊的近端项。在本文中，受到Shen 等人提出的部分PPA 分块ADMM (PPBADMM) [32] 的启发，我们提出了一种部分近端 S-ADMM (PPSADMM)。在 PPSADMM 中，特殊近端项仅作为 PPBADMM 添加到第一组中的子问题中。我们在每次迭代结束时以固定步长对所有变量执行扩展步骤。在没有严格假设的情况下，我们建立了 PPSADMM 遍历意义上的全局收敛结果和 O ( 1 / t ) 收敛率。它的数值性能在两类问题上是合理的。