Convergence Rate Analysis for the Alternating Direction Method of Multipliers with a Substitution Procedure for Separable Convex Programming
本文在前期工作基础上，将交替方向乘子法与高斯回代或前代过程结合，提出通用算法框架，并从收缩角度证明其收敛性，在误差界条件下证明局部线性收敛速度，否则给出最坏情况迭代复杂度估计。
Recently, in He et al. [He BS, Tao M, Yuan XM (2012) Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22(2):313–340], we have showed the first possibility of combining the Douglas-Rachford alternating direction method of multipliers (ADMM) with a Gaussian back substitution procedure for solving a convex minimization model with a general separable structure. This paper is a further study on this theme. We first derive a general algorithmic framework to combine ADMM with either a forward or backward substitution procedure. Then, we show that convergence of this framework can be easily proved from the contraction perspective, and its local linear convergence rate is provable if certain error bound condition is assumed. Without such an error bound assumption, we can estimate its worst-case convergence rate measured by the iteration complexity.