Scalarization of Tangential Regularity of Set-Valued Mappings

A set-valued mapping M from a topological vector space E into a normed vector space F is tangentially regular at a point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {\bar x,\bar y} \right) $$ \end{document} in its graph g p h M if the Clarke tangent cone to g p h M at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {\bar x,\bar y} \right) $$ \end{document} is equal to the Bouligand contingent cone to g p h M at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {\bar x,\bar y} \right) $$ \end{document}. In this paper we characterize, in several cases, this tangential regularity as the directional regularity of the scalar function ΔM defined by ΔM(x, y) : = d(y, M(x)). The results allow us to express, in a useful formula, the subdifferential of ΔM in terms of the normal cone to the graph of M.