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In this paper, we consider a class of possibly nonconvex, nonsmooth andnon-Lipschitz optimization problems arising in many contemporary applicationssuch as machine learning, variable selection and image processing. To solvethis class of problems, we propose a proximal gradient method withextrapolation and line search (PGels). This method is developed based on aspecial potential function and successfully incorporates both extrapolation andnon-monotone line search, which are two simple and efficient acceleratingtechniques for the proximal gradient method. Thanks to the line search, thismethod allows more flexibilities in choosing the extrapolation parameters andupdates them adaptively at each iteration if a certain line search criterion isnot satisfied. Moreover, with proper choices of parameters, our PGels reducesto many existing algorithms. We also show that, under some mild conditions, ourline search criterion is well defined and any cluster point of the sequencegenerated by PGels is a stationary point of our problem. In addition, byassuming the Kurdyka-${\L}$ojasiewicz exponent of the objective in our problem,we further analyze the local convergence rate of two special cases of PGels,including the widely used non-monotone proximal gradient method as one case.Finally, we conduct some numerical experiments for solving the $\ell_1$regularized logistic regression problem and the $\ell_{1\text{-}2}$ regularizedleast squares problem. Our numerical results illustrate the efficiency of PGelsand show the potential advantage of combining two accelerating techniques.

Proximal Gradient Method with Extrapolation and Line Search for a Class of Nonconvex and Nonsmooth Problems