Iterative thresholding algorithm based on non-convex method for modified lp-norm regularization minimization

• Main Authors: Cui, Angang, Peng, Jigen, Li, Haiyang, Wen, Meng, Xiong, Jiajun
• Format: Journal Article
• Language: English
• Published: 25.04.2018
• Subjects: Mathematics - Optimization and Control
• DOI: 10.48550/arxiv.1804.09385

Recently, the$\l_{p}$ -norm regularization minimization problem$(P_{p}^{\lambda})$has attracted great attention in compressed sensing. However, the$\l_{p}$ -norm$\|x\|_{p}^{p}$in problem$(P_{p}^{\lambda})$is nonconvex and non-Lipschitz for all$p\in(0,1)$ , and there are not many optimization theories and methods are proposed to solve this problem. In fact, it is NP-hard for all$p\in(0,1)$and$\lambda>0$ . In this paper, we study two modified$\l_{p}$regularization minimization problems to approximate the NP-hard problem$(P_{p}^{\lambda})$ . Inspired by the good performance of Half algorithm and$2/3$algorithm in some sparse signal recovery problems, two iterative thresholding algorithms are proposed to solve the problems$(P_{p,1/2,\epsilon}^{\lambda})$and$(P_{p,2/3,\epsilon}^{\lambda})$respectively. Numerical results show that our algorithms perform effectively in finding the sparse signal in some sparse signal recovery problems for some proper$p\in(0,1)$ .