# A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization

arXiv: https://arxiv.org/abs/2102.05433 · HTML: https://mlsanity.com/paper/2102.05433/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 68 references that are in the map.
- First posted on arXiv 10 Feb 2021.

## Builds on (most cited first, 17 of 17)

- [Global Convergence of ADMM in Nonconvex Nonsmooth Optimization](https://mlsanity.com/paper/1511.06324/) (2015)
- [Generalized Low Rank Models](https://mlsanity.com/paper/1410.0342/) (2014)
- [Inertial Proximal Alternating Linearized Minimization (iPALM) for Nonconvex and Nonsmooth Problems](https://mlsanity.com/paper/1702.02505/) (2017)
- [A globally convergent algorithm for nonconvex optimization based on block coordinate update](https://mlsanity.com/paper/1410.1386/) (2014)
- [Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm](https://mlsanity.com/paper/1510.06895/) (2015)
- [Global convergence of splitting methods for nonconvex composite optimization](https://mlsanity.com/paper/1407.0753/) (2014)
- [An Accelerated Linearized Alternating Direction Method of Multipliers](https://mlsanity.com/paper/1401.6607/) (2014)
- [The proximal alternating direction method of multipliers in the nonconvex setting: convergence analysis and rates](https://mlsanity.com/paper/1801.01994/) (2018)
- [Linearized ADMM for Non-convex Non-smooth Optimization with Convergence Analysis](https://mlsanity.com/paper/1705.02502/) (2017)
- [Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano](https://mlsanity.com/paper/1602.07283/) (2016)
- [Faster Stochastic Alternating Direction Method of Multipliers for Nonconvex Optimization](https://mlsanity.com/paper/2008.01296/) (2020)
- [Iteration-complexity of a Jacobi-type non-Euclidean ADMM for multi-block linearly constrained nonconvex programs](https://mlsanity.com/paper/1705.07229/) (2017)
- [Alternating Direction Method of Multipliers for A Class of Nonconvex and Nonsmooth Problems with Applications to Background/Foreground Extraction](https://mlsanity.com/paper/1506.07029/) (2015)
- [A Block Successive Upper Bound Minimization Method of Multipliers for Linearly Constrained Convex Optimization](https://mlsanity.com/paper/1401.7079/) (2014)
- [Linear Convergence of Adaptively Iterative Thresholding Algorithms for Compressed Sensing](https://mlsanity.com/paper/1408.6890/) (2014)
- [Stochastic Alternating Direction Method of Multipliers with Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1610.02758/) (2016)
- [Inertial nonconvex alternating minimizations for the image deblurring](https://mlsanity.com/paper/1907.12945/) (2019)

## Built on by (most cited first, 3 of 3)

- [Block Majorization Minimization with Extrapolation and Application to β-NMF](https://mlsanity.com/paper/2401.06646/) (2024)
- [Multiblock ADMM for nonsmooth nonconvex optimization with nonlinear coupling constraints](https://mlsanity.com/paper/2201.07657/) (2022)
- [A Linearized Alternating Direction Multiplier Method for Federated Matrix Completion Problems](https://mlsanity.com/paper/2503.12733/) (2025)

## Abstract (arXiv)

In this paper, we propose an algorithmic framework, dubbed inertial alternating direction methods of multipliers (iADMM), for solving a class of nonconvex nonsmooth multiblock composite optimization problems with linear constraints. Our framework employs the general minimization-majorization (MM) principle to update each block of variables so as to not only unify the convergence analysis of previous ADMM that use specific surrogate functions in the MM step, but also lead to new efficient ADMM schemes. To the best of our knowledge, in the nonconvex nonsmooth setting, ADMM used in combination with the MM principle to update each block of variables, and ADMM combined with inertial terms for the primal variables have not been studied in the literature. Under standard assumptions, we prove the subsequential convergence and global convergence for the generated sequence of iterates. We illustrate the effectiveness of iADMM on a class of nonconvex low-rank representation problems.