论文标题
凸复合编程的近端广义ADMM的线性收敛速率分析
Linear Convergence Rate Analysis of Proximal Generalized ADMM for Convex Composite Programming
论文作者
论文摘要
乘数的近端广义交替方向方法（P-GADMM）基本上有效地解决了高维至中度精度的凸复合编程问题。该方法的全球融合是由Xiao，Chen＆Li [Math [Math。程序。计算，2018年]，但没有给出其收敛速度。可以理所当然的是，通过模仿近端ADMM可以轻松证明融合率，但是我们发现轻松的观点肯定会在理论分析中造成许多困难。在本文中，我们致力于探索其收敛行为，并表明P-GADMM产生的序列在某些轻度条件下具有Q线性收敛速率。我们想指出的是，子问题的近端术语必须是积极的，这在大多数实际实现中非常普遍，尽管它似乎有些强。
The proximal generalized alternating direction method of multipliers (p-GADMM) is substantially efficient for solving convex composite programming problems of high-dimensional to moderate accuracy. The global convergence of this method was established by Xiao, Chen & Li [Math. Program. Comput., 2018], but its convergence rate was not given. One may take it for granted that the convergence rate could be proved easily by mimicking the proximal ADMM, but we find the relaxed points will certainly cause many difficulties for theoretical analysis. In this paper, we devote to exploring its convergence behavior and show that the sequence generated by p-GADMM possesses Q-linear convergence rate under some mild conditions. We would like to note that the proximal terms at the subproblems are required to be positive definite, which is very common in most practical implementations although it seems to be a bit strong.
