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 (ADMM) 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 semi-proximal ADMM for the step-length $\tau \in (0,(1+\sqrt{5})\mathrm{/}2)$ and the penalty parameter $\sigma \in (0,+\mathrm{\infty })$. 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 semi-proximal ADMM reduces to the directly extended $3$-block ADMM with $\tau \in (0, (1+\sqrt{5})/2)$.