Smoothing Accelerated Proximal Gradient Method with Fast Convergence Rate for Nonsmooth Convex Optimization Beyond Differentiability

We propose a smoothing accelerated proximal gradient (SAPG) method with fast convergence rate for finding a minimizer of a decomposable nonsmooth convex function over a closed convex set. The proposed algorithm combines the smoothing method with the proximal gradient algorithm with extrapolation k-1k+α-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{k-1}{k+\alpha -1}$$\end{document} and α>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > 3$$\end{document}. The updating rule of smoothing parameter μk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _k$$\end{document} is a smart scheme and guarantees the global convergence rate of o(lnσk/k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o(\ln ^{\sigma }k/k)$$\end{document} with σ∈(12,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (\frac{1}{2},1]$$\end{document} on the objective function values. Moreover, we prove that the iterates sequence is convergent to an optimal solution of the problem. We then introduce an error term in the SAPG algorithm to get the inexact smoothing accelerated proximal gradient algorithm. And we obtain the same convergence results as the SAPG algorithm under the summability condition on the errors. Finally, numerical experiments show the effectiveness and efficiency of the proposed algorithm.