  The generalized alternating direction method of multipliers (ADMM) of Xiao et
al. [{\tt Math. Prog. Comput., 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 subprobllem 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 more efficient than its
compared ones.
