A Convergent 3-Block Semi-Proximal ADMM for Convex Minimization Problems with One Strongly Convex Block

In this paper, we present a semi-proximal alternating direction method of multipliers (sPADMM) for solving 3-block separable convex minimization problems with the second block in the objective being a strongly convex function and one coupled linear equation constraint. By choosing the semi-proximal terms properly, we establish the global convergence of the proposed sPADMM for the step-length tau is an element of (0, (1 + root 5)/2) and the penalty parameter sigma is an element of (0, +infinity). In particular, if sigma > 0 is smaller than a certain threshold and the first and third linear operators in the linear equation constraint are injective, then all the three added semi-proximal terms can be dropped and consequently, the convergent 3-block sPADMM reduces to the directly extended 3-block ADMM with tau is an element of (0, (1 + root 5)/2).