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# Communications in Mathematical Sciences

## Volume 18 (2020)

### Number 1

### An efficient and globally convergent algorithm for $\ell_{p,q} - \ell_r$ model in group sparse optimization

Pages: 227 – 258

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n1.a10

#### Authors

#### Abstract

Group sparsity has lots of applications in various data science related problems. It combines the underlying sparsity and group structure of the variables. A general and important model for group sparsity is the $\ell_{p,q} - \ell_r$ optimization model with $p \geq 1, 0 \lt q \lt 1, 1 \leq r \leq \infty$, which is applicable to different types of measurement noises. It includes not only the non-smooth composition of $\ell_q (0 \lt q \lt 1)$ and $\ell_p (p \geq 1)$, but also the non-smooth $\ell_1 / \ell_{\infty}$ fidelity term. In this paper, we present a nontrivial extension of our recent work to solve this general group sparse minimization model. By a motivating proposition, our algorithm is naturally designed to shrink the group support and eliminate the variables gradually. It is thus very fast, especially for large-scale problems. Combined with a proximal linearization, it allows an inexact inner loop implemented by scaled alternating direction method of multipliers (ADMM), and still has global convergence. The algorithm gives a unified framework for the full parameters. Many numerical experiments are presented for various combinations of the parameters $p,q,r$. The comparisons show the advantages of our algorithm over others in the existing works.

#### Keywords

group sparse, $\ell_{p,q}-\ell_r$ model, non-Lipschitz optimization, Laplace noise, Gaussian noise, uniform distribution noise, lower bound theory, Kurdyka–Łojasiewicz (KL) property

#### 2010 Mathematics Subject Classification

49M05, 65K10, 90C26, 90C30

Received 13 July 2019

Accepted 17 September 2019

Published 1 April 2020