On the distance function associated with a set-valued mapping
The author provides a nice approach for the characterization of a set-valued mapping \(M: E\rightrightarrows F\) (here \(E\) and \(F\) are assumed to be a Hausdorff topological vector space and a normed vector space, respectively) in terms of the distance function to the images, i.e., in terms of the (scalar) function NEWLINE\[NEWLINE\Delta_M(x, y):= d(y, M(x))= \inf\{\|y- z\|\mid z\in M(x)\}.NEWLINE\]NEWLINE Essential tools are special cone approximations of sets (the contingent cone \(K(S; x)\) and the Clarke tangent cone \(T(S; x)\)), the associated directional derivatives for scalar functions (the lower Hadamard directional derivative \(f^H(x; h)\) and the generalized Rockafellar directional derivative \(f^{\uparrow}(x; h)\)) and especially some associated directional derivatives for set-valued mappings. For this, the author introduces the so-called \(K\)-derivative \(D_KM((x,y); h)\) and the \(T\)-derivative \(D_TM((x, y);h)\) of the mapping \(M\) at the point \((x,y)\in \text{gph }M\) according to NEWLINE\[NEWLINE\begin{aligned} \text{gph }D_KM((x, y);\cdot) &= K(\text{gph }M; (x,y)),\\ \text{gph }D_T M((x, y(;\cdot) &= T(\text{gph }M; (x,y)).\end{aligned}NEWLINE\]NEWLINE Now, he is able to give some interesting relationships between the directional derivatives of the scalar functions \(\Delta_M\) and \(\Delta_{M_r}\) (using the \(r\)-enlargement \(M_r(x):= \{y\in F\mid\Delta_M(x, y)\leq r\}\) of the mapping \(M\)) and the distance functions to the images of the introduced set-valued derivatives of \(M\). In the same manner, he gives a description of the cone approximations to the sets \(\text{gph }M\) and \(\text{gph }M_r\) in terms (as level sets) of the directional derivatives of \(\Delta_M\) and \(\Delta_{M_r}\), respectively.NEWLINENEWLINENEWLINEAt the end of the paper, the given results are used for the study of the calmness property of a general perturbed optimization problem with a constraint by a set-valued mapping.
scientific article; zbMATH DE number 1690877