Scalarization of tangential regularity of set-valued mappings
Having a set-valued mapping \(M:E\rightarrow F\) from the Hausdorff topological vector space \(E\) into the normed space \(F\), one associates the scalar mapping \(\Delta _{M}:E\times F\rightarrow \overline{{R}},\) \(\Delta _{M}(x,y)=d(y,M(x)).\) The authors are mainly interested by the relationships between the tangential regularity of \(M\) at \((\overline{x},\overline{y})\in\text{gph} M\) (i.e., the coincidence of Clarke and Bouligand tangent cones of \(\text{gph} M\) at \((\overline{x},\overline{y})\)) and the directional regularity of \(\Delta _{M}\) at \((\overline{x},\overline{y})\) (i.e., the coincidence of Rockafellar and lower Hadamard directional derivatives of \(\Delta _{M}\) at \((\overline{x},\overline{y})\)). It is shown that \(\Delta _{M} \) is directionally regular at \((\overline{x},\overline{y})\) when \(M\) is tangentially regular at \((\overline{x},\overline{y}),\) the converse being true if \(M\) is in the class \({\mathcal{K}}(\overline{x},\overline{y}),\) in particular if \(F\) is finite dimensional. Many other interesting results are also stated.
scientific article; zbMATH DE number 1331786