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To solve the separable convex optimization problem with linear constraints,Eckstein and Bertsekas introduced the generalized alternating direction methodof multipliers (in short, GADMM), which is an efficient and simple accelerationscheme of the aternating direction method of multipliers. Recently,\textbf{Fang et. al} proposed the linearized version of generalized alternatingdirection method of multipliers (in short, L-GADMM), where one of itssubproblems is approximated by a linearization strategy, and proved itsworst-case $\mathcal{O}(1/t)$ convergence rate measured by the iterationcomplexity in both ergodic and nonergodic senses. In this paper, we introducethe doubly linearized version of generalized alternating direction method ofmultipliers (in short, DL-GADMM), where both the $x$-subproblem and$y$-subproblem are approximated by linearization strategies. Based on the errorbound approach, we establish the linear convergence rate of both L-GADMM andDL-GADMM for separable convex optimization problem that the subdifferentials ofthe underlying functions are piecewise linear multifunctions. The results inthis paper extend, generalize and improve some known results in the literature.

The global linear convergence rate of the proximal version of the  generalized alternating direction method of multipliers for separable convex  programming