# An Iteratively Reweighted Method for Sparse Optimization on Nonconvex ℓp Ball

arXiv: https://arxiv.org/abs/2104.02912 · HTML: https://mlsanity.com/paper/2104.02912/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 34 references that are in the map.
- First posted on arXiv 7 Apr 2021.

## Builds on (most cited first, 7 of 7)

- [Non-convex Optimization for Machine Learning](https://mlsanity.com/paper/1712.07897/) (2017)
- [Nonconvex and Nonsmooth Sparse Optimization via Adaptively Iterative Reweighted Methods](https://mlsanity.com/paper/1810.10167/) (2018)
- [Sharp Time--Data Tradeoffs for Linear Inverse Problems](https://mlsanity.com/paper/1507.04793/) (2015)
- [Learning Compact Neural Networks with Regularization](https://mlsanity.com/paper/1802.01223/) (2018)
- [Towards An Efficient Approach for the Nonconvex ℓp Ball Projection: Algorithm and Analysis](https://mlsanity.com/paper/2101.01350/) (2021)
- [Structured Sparsity: Discrete and Convex approaches](https://mlsanity.com/paper/1507.05367/) (2015)
- [Inexact Primal-Dual Gradient Projection Methods for Nonlinear Optimization on Convex Set](https://mlsanity.com/paper/1911.07758/) (2019)

## Abstract (arXiv)

This paper is intended to solve the nonconvex ℓp-ball constrained nonlinear optimization problems. An iteratively reweighted method is proposed, which solves a sequence of weighted ℓ1-ball projection subproblems. At each iteration, the next iterate is obtained by moving along the negative gradient with a stepsize and then projecting the resulted point onto the weighted ℓ1 ball to approximate the ℓp ball. Specifically, if the current iterate is in the interior of the feasible set, then the weighted ℓ1 ball is formed by linearizing the ℓp norm at the current iterate. If the current iterate is on the boundary of the feasible set, then the weighted ℓ1 ball is formed differently by keeping those zero components in the current iterate still zero. In our analysis, we prove that the generated iterates converge to a first-order stationary point. Numerical experiments demonstrate the effectiveness of the proposed method.