Iterative reweighted methods for ℓ1−ℓp minimization

In this paper, we focus on the $\ell_1-\ell_p$ minimization problem with $0stationary points of $\ell_1-\ell_p$ minimization, and hence of its local minimizers. In algorithms, based on three locally Lipschitz continuous $\epsilon$-approximation to $\ell_p$ norm, we design several iterative reweighted $\ell_1$ and $\ell_2$ methods to solve those approximation problems. Furthermore, we show that any accumulation point of the sequence generated by these methods is a generalized first-order stationary point of $\ell_1-\ell_p$ minimization. This result, in particular, applies to the iterative reweighted $\ell_1$ methods based on the new Lipschitz continuous $\epsilon$-approximation introduced by Lu \cite{Lu14}, provided that the approximation parameter $\epsilon$ is below a threshold value. Numerical results are also reported to demonstrate the efficiency of the proposed methods.