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

Recently, the ${\text{l}}_p$-norm regularization minimization problem $(P_p^{\lambda })$ has attracted great attention in compressed sensing. However, the ${\text{l}}_p$-norm $\mathrm{\parallel }x{\mathrm{\parallel }}_p^p$ 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 ${\text{l}}_p$ regularization minimization problems to approximate the NP-hard problem $(P_p^{\lambda })$. Inspired by the good performance of Half algorithm and $2\mathrm{/}3$ algorithm in some sparse signal recovery problems, two iterative thresholding algorithms are proposed to solve the problems $(P_{p,1\mathrm{/}2,ϵ}^{\lambda })$ and $(P_{p,2\mathrm{/}3,ϵ}^{\lambda })$ respectively. Numerical results show that our algorithms perform effectively in finding the sparse signal in some sparse signal recovery problems for some proper $p\in (0,1)$.