An algorithm for the minimization of nonsmooth nonconvex functions using inexact evaluations and its worst-case complexity

An adaptive regularization algorithm using inexact function and derivatives evaluations is proposed for the solution of composite nonsmooth nonconvex optimization. It is shown that this algorithm needs at most O(|log(ϵ)|ϵ-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|\log (\epsilon )|\,\epsilon ^{-2})$$\end{document} evaluations of the problem’s functions and their derivatives for finding an ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}-approximate first-order stationary point. This complexity bound therefore generalizes that provided by Bellavia et al. (Theoretical study of an adaptive cubic regularization method with dynamic inexact Hessian information. arXiv:1808.06239, 2018) for inexact methods for smooth nonconvex problems, and is within a factor |log(ϵ)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\log (\epsilon )|$$\end{document} of the optimal bound known for smooth and nonsmooth nonconvex minimization with exact evaluations. A practically more restrictive variant of the algorithm with worst-case complexity O(|log(ϵ)|+ϵ-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|\log (\epsilon )|+\epsilon ^{-2})$$\end{document} is also presented.