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 et al. (Math Program Comput 10(4):533–555, 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 technical conditions. We would like to note that the proximal parameters at the subproblems are required to be positive definite, which is common in most practical implementations although it seems to be a bit strong.

乘法器的近端广义交替方向法 (p-GADMM) 对于解决高维到中等精度的凸复合规划问题非常有效。该方法的全局收敛性是由Xiao等人建立的。(Math Program Comput 10(4):533–555, 2018)，但没有给出其收敛速度。人们可能想当然地认为通过模仿近端 ADMM 可以很容易地证明收敛速度，但我们发现松弛点肯定会给理论分析带来许多困难。在本文中，我们致力于探索其收敛行为，并证明p-GADMM生成的序列在某些技术条件下具有Q线性收敛速度。我们要注意的是，子问题的近端参数需要是正定的，