# Constrained Optimization Involving Nonconvex ℓp Norms: Optimality Conditions, Algorithm and Convergence

arXiv: https://arxiv.org/abs/2110.14127 · HTML: https://mlsanity.com/paper/2110.14127/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 27 Oct 2021.

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

- [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)
- [Optimality condition and complexity analysis for linearly-constrained optimization without differentiability on the boundary](https://mlsanity.com/paper/1702.04300/) (2017)
- [Towards An Efficient Approach for the Nonconvex ℓp Ball Projection: Algorithm and Analysis](https://mlsanity.com/paper/2101.01350/) (2021)
- [Sparse Portfolio Selection via Quasi-Norm Regularization](https://mlsanity.com/paper/1312.6350/) (2013)
- [Efficient Dictionary Learning with Sparseness-Enforcing Projections](https://mlsanity.com/paper/1604.04767/) (2016)
- [Relating lp regularization and reweighted l1 regularization](https://mlsanity.com/paper/1912.00723/) (2019)

## Abstract (arXiv)

This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an ℓp norm (0<p<1) of the variables, which may appear in either the objective or the constraint. This kind of problems have strong applicability to a wide range of areas since usually the ℓp norm can promote sparse solutions. However, the nonsmooth and non-Lipschtiz nature of the ℓp norm often cause these problems difficult to analyze and solve. We provide the calculation of the subgradients of the ℓp norm and the normal cones of the ℓp ball. For both problems, we derive the first-order necessary conditions under various constraint qualifications. We also derive the sequential optimality conditions for both problems and study the conditions under which these conditions imply the first-order necessary conditions. We point out that the sequential optimality conditions can be easily satisfied for iteratively reweighted algorithms and show that the global convergence can be easily derived using sequential optimality conditions.