A Three-Operator Splitting Perspective of a Three-Block ADMM for Convex Quadratic Semidefinite Programming and Beyond

In recent years, several convergent variants of the multi-block alternating direction method of multipliers (ADMM) have been proposed for solving the convex quadratic semidefinite programming via its dual, which is inherently a 3-block separable convex optimization problem with coupled linear constraints. Among these multi-block ADMM-type algorithms, the modified 3-block ADMM in [Chang, XK, SY Liu and X Li (2016). Modified alternating direction method of multipliers for convex quadratic semidefinite programming. Neurocomputing, 214, 575–586] 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 in the objective function. In this paper, we lay the theoretical foundation of this phenomenon by interpreting this modified 3-block ADMM as a special implementation of the Davis–Yin 3-operator splitting [Davis, D and WT Yin (2017). A three-operator splitting scheme and its optimization applications. Set-Valued and Variational Analysis, 25, 829–858]. Based on this perspective, we are able to extend this modified 3-block ADMM to a generalized 3-block ADMM, in the sense of [Eckstein, J and DP Bertsekas (1992). On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming, 55, 293–318], which not only applies to the more general convex composite quadratic programming problems but also admits the flexibility of achieving even better numerical performance.