# A Convex-Nonconvex Framework for Enhancing Minimization Induced Penalties

arXiv: https://arxiv.org/abs/2407.14819 · HTML: https://mlsanity.com/paper/2407.14819/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2024.

## Built on by (most cited first, 2 of 2)

- [Nonconvex Latent Optimally Partitioned Block-Sparse Recovery via Log-Sum and Minimax Concave Penalties](https://mlsanity.com/paper/2603.01304/) (2026)
- [Deep Unfolded Latent Optimally Partitioned-l2/l1 Networks for Data-driven Block-Sparse Recovery](https://mlsanity.com/paper/2606.12740/) (2026)

## Abstract (arXiv)

This paper presents a novel framework for nonconvex enhancement of minimization induced (MI) penalties while preserving the overall convexity of associated regularization models. MI penalties enable the adaptation to certain signal structures via minimization, but often underestimate significant components owing to convexity. To overcome this shortcoming, we design a generalized Moreau enhanced minimization induced (GME-MI) penalty by subtracting from the MI penalty its generalized Moreau envelope. While the proposed GME-MI penalty is nonconvex in general, we derive an overall convexity condition for the GME-MI regularized least-squares model. Moreover, we present a proximal splitting algorithm with guaranteed convergence to a globally optimal solution of the GME-MI model under the overall convexity condition. Numerical examples illustrate the effectiveness of the proposed framework.