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

arXiv: https://arxiv.org/abs/1410.7933 · HTML: https://mlsanity.com/paper/1410.7933/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2014.

## Built on by (most cited first, 2 of 2)

- [Convex Augmentation for Total Variation Based Phase Retrieval](https://mlsanity.com/paper/2205.00834/) (2022)
- [AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as its Third Constraint Block](https://mlsanity.com/paper/2608.14396/) (2026)

## Abstract (arXiv)

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 τ ∈ (0, (1+√5)/2) and the penalty parameter σ∈ (0, +∞). In particular, if σ>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 τ ∈ (0, (1+√5)/2).