A three-operator splitting perspective of a three-block ADMM for convex quadratic semidefinite programming and extensions

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.