Approximation to a set-valued mapping. I: A proposal
type: Item
instance of: scholarly article
title: Approximation to a set-valued mapping. I: A proposal
zbMATH Open document ID: 0619.49005
DOI: 10.1007/BF01442236
published in: Applied Mathematics and Optimization
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 26E25
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 4005016
zbMATH Keywords: support function
zbMATH Keywords: directional derivative
zbMATH Keywords: set-valued mapping
zbMATH Keywords: first-order approximation
zbMATH Keywords: Hausdorff distance
zbMATH Keywords: tangent cones
author: V. F. Demyanov
author: Claude Lemaréchal
author: J. Zowe
reviewed by: Marcin Studniarski
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01442236
OpenAlex ID: W2033092352
description: scientific article; zbMATH DE number 4005016