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

TL;DR

This paper introduces a semi-proximal ADMM for 3-block convex minimization problems with a strongly convex second block, establishing convergence under specific conditions and simplifying to a direct 3-block ADMM in certain cases.

Contribution

The paper develops a convergent semi-proximal ADMM for 3-block problems with strong convexity and linear constraints, extending existing methods and providing conditions for simplification.

Findings

Proves global convergence for step-length τ in (0, (1+√5)/2)

Identifies conditions where semi-proximal terms can be omitted

Shows the simplified ADMM converges under injectivity assumptions

Abstract

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 $\tau \in (0, (1+\sqrt{5})/2)$ and the penalty parameter $\sigma\in (0, +\infty)$. 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 $\tau \in (0, (1+\sqrt{5})/2)$.