  In recent years, several convergent multi-block variants of the alternating
direction method of multipliers (ADMM) have been proposed for solving the
convex quadratic semidefinite programming via its dual, which is naturally a
3-block separable convex optimization problem with one coupled linear equality
constraint. Among of these ADMM-type algorithms, the modified 3-block ADMM in
[Chang et al., Neurocomput. 214: 575--586 (2016)] bears a peculiar feature that
the augmented Lagrangian function is not necessarily to be minimized with
respect to the block-variable corresponding to the quadratic term of the
objective function. In this paper, we lay the theoretical foundation of this
phenomena by interpreting this modified 3-block ADMM as a realization of a
3-operator splitting framework. Based on this perspective, we are able to
extend this modified 3-block ADMM to a generalized 3-block ADMM, which not only
applies to the more general convex composite quadratic programming setting but
also admits the potential of achieving even a better numerical performance.
