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The proximal generalized alternating direction method of multipliers(p-GADMM) is substantially efficient for solving convex composite programmingproblems of high-dimensional to moderate accuracy. The global convergence ofthis 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 theconvergence rate could be proved easily by mimicking the proximal ADMM, but wefind the relaxed points will certainly cause many difficulties for theoreticalanalysis. In this paper, we devote to exploring its convergence behavior andshow that the sequence generated by p-GADMM possesses Q-linear convergence rateunder some mild conditions. We would like to note that the proximal terms atthe subproblems are required to be positive definite, which is very common inmost practical implementations although it seems to be a bit strong.

Linear Convergence Rate Analysis of Proximal Generalized ADMM for Convex Composite Programming