Convergence rate analysis of proximal iteratively reweighted $$\ell _1$$ methods for $$\ell _p$$ regularization problems

Abstract

In this paper, we focus on the local convergence rate analysis of the proximal iteratively reweighted \(\ell _1\) algorithms for solving \(\ell _p\) regularization problems, which are widely applied for inducing sparse solutions. We show that if the Kurdyka–Łojasiewicz property is satisfied, the algorithm converges to a unique first-order stationary point; furthermore, the algorithm has local linear convergence or local sublinear convergence. The theoretical results we derived are much stronger than the existing results for iteratively reweighted \(\ell _1\) algorithms.