The Equivalence of Three Types of Error Bounds for Weakly and Approximately Convex Functions

We start by establishing the equivalence of three types of error bounds: weak sharp minima, level-set subdifferential error bounds and Łojasiewicz (for short Ł) inequalities for weakly convex functions with exponent α∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0,1]$$\end{document} and approximately convex functions. Then we apply these equivalence results to a class of nonconvex optimization problems, whose objective functions are the sum of a convex function and a composite function with a locally Lipschitz function and a smooth vector-valued function. Finally, applying a characterization for lower-order regularization problems, we show that the level-set subdifferential error bound with exponent 1 and the Ł inequality with exponent 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}$$\end{document} hold at a local minimum point.