Convergence analysis of generalized ADMM with majorization for linearly constrained composite convex optimization

The generalized alternating direction method of multipliers (ADMM) of Xiao et al. (Math Prog Comput 10:533–555, 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 subproblems 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 evidently efficient.