An Implementable Splitting Algorithm for the ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document}

The split feasibility problem (SFP) captures a wide range of inverse problems, such as signal processing, image reconstruction, and so on. Recently, applications of $$\ell _1$$ℓ1-norm regularization to linear inverse problems, a special case of SFP, have been received a considerable amount of attention in the signal/image processing and statistical learning communities. However, the study of the $$\ell _1$$ℓ1-norm regularized SFP still deserves attention, especially in terms of algorithmic issues. In this paper, we shall propose an algorithm for solving the $$\ell _1$$ℓ1-norm regularized SFP. More specifically, we first formulate the $$\ell _1$$ℓ1-norm regularized SFP as a separable convex minimization problem with linear constraints, and then introduce our splitting method, which takes advantage of the separable structure and gives rise to subproblems with closed-form solutions. We prove global convergence of the proposed algorithm under certain mild conditions. Moreover, numerical experiments on an image deblurring problem verify the efficiency of our algorithm.