An extrapolated proximal iteratively reweighted method for nonconvex composite optimization problems

We consider a class of problems where the objective function is the sum of a smooth function and a composition of nonconvex and nonsmooth functions. Such optimization problems arise frequently in machine learning and data processing. The proximal iteratively reweighted method has been widely used and popularized in solving these problems. In this paper, we develop an extrapolated proximal iteratively reweighted method that incorporates two different flexible inertial steps at each iteration. We first prove the subsequential convergence of the proposed method under parameter constraints. Moreover, if the objective function satisfies the Kurdyka-Łojasiewicz property, the global convergence of the new method is established. In addition, we analyze the local convergence rate by making assumptions on the Kurdyka-Łojasiewicz exponent of the objective function. Finally, numerical results on lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_p$$\end{document} minimization and feature selection problems are reported to show the effectiveness and superiority of the proposed algorithm.