# Nonconvex and Nonsmooth Sparse Optimization via Adaptively Iterative Reweighted Methods

arXiv: https://arxiv.org/abs/1810.10167 · HTML: https://mlsanity.com/paper/1810.10167/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2018.

## Built on by (most cited first, 10 of 10)

- [Compressed Sensing: A Discrete Optimization Approach](https://mlsanity.com/paper/2306.04647/) (2023)
- [Deep Unfolding of Iteratively Reweighted ADMM for Wireless RF Sensing](https://mlsanity.com/paper/2106.03686/) (2021)
- [Avoiding strict saddle points of nonconvex regularized problems](https://mlsanity.com/paper/2401.09274/) (2024)
- [An Iteratively Reweighted Method for Sparse Optimization on Nonconvex ℓp Ball](https://mlsanity.com/paper/2104.02912/) (2021)
- [Constrained Optimization Involving Nonconvex ℓp Norms: Optimality Conditions, Algorithm and Convergence](https://mlsanity.com/paper/2110.14127/) (2021)
- [A Screening Strategy for Structured Optimization Involving Nonconvex ℓq,p Regularization](https://mlsanity.com/paper/2208.02161/) (2022)
- [Anderson acceleration for iteratively reweighted ℓ1 algorithm](https://mlsanity.com/paper/2403.07271/) (2024)
- [Efficient Low-rank Identification via Accelerated Iteratively Reweighted Nuclear Norm Minimization](https://mlsanity.com/paper/2406.15713/) (2024)
- [Alternating Iteratively Reweighted ℓ1 and Subspace Newton Algorithms for Nonconvex Sparse Optimization](https://mlsanity.com/paper/2407.17216/) (2024)
- [The Effective Number of Nonzeros: Theory and Regularization for Sparse Recovery](https://mlsanity.com/paper/2603.13826/) (2026)

## Abstract (arXiv)

We propose a general formulation of nonconvex and nonsmooth sparse optimization problems with convex set constraint, which can take into account most existing types of nonconvex sparsity-inducing terms, bringing strong applicability to a wide range of applications. We design a general algorithmic framework of iteratively reweighted algorithms for solving the proposed nonconvex and nonsmooth sparse optimization problems, which solves a sequence of weighted convex regularization problems with adaptively updated weights. First-order optimality condition is derived and global convergence results are provided under loose assumptions, making our theoretical results a practical tool for analyzing a family of various reweighted algorithms. The effectiveness and efficiency of our proposed formulation and the algorithms are demonstrated in numerical experiments on various sparse optimization problems.