# Alternating Direction Method of Multipliers for A Class of Nonconvex and Nonsmooth Problems with Applications to Background/Foreground Extraction

arXiv: https://arxiv.org/abs/1506.07029 · HTML: https://mlsanity.com/paper/1506.07029/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2015.

## Built on by (most cited first, 4 of 4)

- [A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization](https://mlsanity.com/paper/2102.05433/) (2021)
- [A Riemannian ADMM](https://mlsanity.com/paper/2211.02163/) (2022)
- [ADMM Algorithms for Residual Network Training: Convergence Analysis and Parallel Implementation](https://mlsanity.com/paper/2310.15334/) (2023)
- [A Hybrid Registration and Fusion Method for Hyperspectral Super-resolution](https://mlsanity.com/paper/2407.05279/) (2024)

## Abstract (arXiv)

In this paper, we study a general optimization model, which covers a large class of existing models for many applications in imaging sciences. To solve the resulting possibly nonconvex, nonsmooth and non-Lipschitz optimization problem, we adapt the alternating direction method of multipliers (ADMM) with a general dual step-size to solve a reformulation that contains three blocks of variables, and analyze its convergence. We show that for any dual step-size less than the golden ratio, there exists a computable threshold such that if the penalty parameter is chosen above such a threshold and the sequence thus generated by our ADMM is bounded, then the cluster point of the sequence gives a stationary point of the nonconvex optimization problem. We achieve this via a potential function specifically constructed for our ADMM. Moreover, we establish the global convergence of the whole sequence if, in addition, this special potential function is a Kurdyka-Lojasiewicz function. Furthermore, we present a simple strategy for initializing the algorithm to guarantee boundedness of the sequence. Finally, we perform numerical experiments comparing our ADMM with the proximal alternating linearized minimization (PALM) proposed in [5] on the background/foreground extraction problem with real data. The numerical results show that our ADMM with a nontrivial dual step-size is efficient.