A variable-penalty alternating directions method for convex optimization
A variable-penalty alternating directions method for convex optimization
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DOI：
10.1007/bf02680549
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发表时间：
1998-09
影响因子：
2.7
通讯作者：
S. Kontogiorgis;R. Meyer
中科院分区：
文献类型：
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作者：
S. Kontogiorgis;R. Meyer
我们研究了一个广义版本的交替方向的方法，适用于最小化的两个凸函数的总和受到线性约束。该方法包括在每次迭代中连续求解两个优化问题，其中包含在目标函数的拉格朗日和邻近项。极小化器确定新的近似项，并简单更新拉格朗日项。我们证明了一个收敛定理，通过放松极小的唯一性假设，推广了已有的结果。另一个新颖之处是，我们允许惩罚矩阵，这些可能会改变每次迭代。这在应用中可能是有益的，因为它允许对问题的方法进行额外的调整，并且可以导致相对于固定惩罚的更快收敛。作为应用，我们推导了块角优化的分解格式，并给出了一类对偶块角问题的计算结果。
We study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints. The method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both Lagrangian and proximal terms. The minimizers determine the new proximal terms and a simple update of the Lagrangian terms follows. We prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers. Another novelty is that we allow penalty matrices, and these may vary per iteration. This can be beneficial in applications, since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties. As an application, we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems.