Approximation to a set-valued mapping. I: A proposal
Let \(F: {\mathbb{R}}^ m\to 2^{{\mathbb{R}}^ n}\) be a set-valued mapping with nonempty convex compact values and let x, \(d\in {\mathbb{R}}^ m\) be fixed. The problem considered in the paper is to find another convex compact multivalued map \(t\mapsto S(x+td)\) for \(t\geq 0\) which provides a first- order approximation to F at x in direction d, in the following sense: \[ (1)\quad \lim_{t\downarrow 0,h\to d}t^{-1}d_ H(S(x+td),F(x+th))=0, \] where \(d_ H\) denotes the Hausdorff distance. The authors propose to define S as follows: \[ (2)\quad S(x+td):=\{z\in {\mathbb{R}}^ n| <z,q>\leq \sigma (x,q)+t\sigma_ d'(x,q)\text{ for all } q\in {\mathbb{R}}^ n\}, \] where \[ \sigma (x,q):=\sup \{<z,q>| \quad z\in F(x)\}\quad and\quad \sigma_ d'(x,q):=\lim_{t\downarrow 0,h\to d}t^{-1}[\sigma (x+th,q)-\sigma (x,q)]. \] Some conditions are presented under which the mapping S defined by (2) satisfies condition (1). A comparison with some other approximations obtained via tangent cones is also given.
scientific article; zbMATH DE number 4005016