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 in its graph g p h M if the Clarke tangent cone to g p h M at is equal to the Bouligand contingent cone to g p h M at . In this paper we characterize, in several cases, this tangential regularity as the directional regularity of the scalar function _M defined by _M (x, y) : = d(y, M(x)). The results allow us to express, in a useful formula, the subdifferential of _M in terms of the normal cone to the graph of M.