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.