论文标题
对线性约束的复合凸优化的广义ADMM的收敛分析
Convergence Analysis of Generalized ADMM with Majorization for Linearly Constrained Composite Convex Optimization
论文作者
论文摘要
Xiao等人的乘数（ADMM）的广义交替方向方法。 [{\ tt数学。 prog。 Comput。，2018}]针对两个线性约束的复合凸编程问题，其中每个块的形式为“非滑动 +二次”。但是，在非二次（但平滑）的情况下，除非不再使用“非平滑 +平滑”的有利结构，否则该方法可能会失败。本文旨在通过使用主要的技术来近似增强的拉格朗日函数来解决此缺陷，以便将相应的子螺旋体分解为一些较小的问题，然后分别解决。此外，最近的对称高斯 - 塞德尔（SGS）分解定理可以确保较大的子问题与这些较小的子问题之间的等效性。本文的重点是收敛分析，也就是说，我们证明，所提出的方法生成的序列全球收敛到所考虑的问题的karush-kuhn-tucker点。最后，我们对一种模拟的凸复合优化问题进行了一些数值实验，这些实验表明该方法比其比较的方法更有效。
The generalized alternating direction method of multipliers (ADMM) of Xiao et al. [{\tt Math. Prog. Comput., 2018}] aims at the two-block linearly constrained composite convex programming problem, in which each block is in the form of "nonsmooth + quadratic". However, in the case of non-quadratic (but smooth), this method may fail unless the favorable structure of 'nonsmooth + smooth' is no longer used. This paper aims to remedy this defect by using a majorized technique to approximate the augmented Lagrangian function, so that the corresponding subprobllem can be decomposed into some smaller problems and then solved separately. Furthermore, the recent symmetric Gauss-Seidel (sGS) decomposition theorem guarantees the equivalence between the bigger subproblem and these smaller ones. This paper focuses on convergence analysis, that is, we prove that the sequence generated by the proposed method converges globally to a Karush-Kuhn-Tucker point of the considered problem. Finally, we do some numerical experiments on a kind of simulated convex composite optimization problems which illustrate that the proposed method is more efficient than its compared ones.
