Calculus rules for derivatives of multimaps
type: Item
instance of: scholarly article
title: Calculus rules for derivatives of multimaps
zbMATH Open document ID: 1173.54008
author: Jean-Paul Penot
author: S. J. Li
author: Kaiwen Meng
published in: Set-Valued and Variational Analysis
publication date: 2009-08-11T00:00:00Z
review text: In the paper under review, the authors give calculus rules for derivatives of the compositions of two multifunctions. Let \(X\), \(Y\) and \(Z\) be normed vector spaces and \(F : X \rightrightarrows Y\) and \(G : Y \rightrightarrows Z\) be multimaps. The composition \((G \circ F) : X \rightrightarrows Z\) is defined by \((G \circ F) (x) = \bigcup_{y \in F(x)} G(y)\). For a point \(z= (x,y) \in \mathrm{cl} (\mathrm{gph}(F))\), the contingent derivative of \(F\) at \(z\) is the multimap \(DF(x,y) : X \rightrightarrows Y\) whose graph is the contingent cone to \(\mathrm{gph}(F)\) at \(z\) in \(X \times Y\), that is, \(\mathrm{gph} (DF(x,y)) = \lim \sup _{t \to +0} \frac{1}{t} (\mathrm{gph}(F) -z)\), where \(\mathrm{gph}(F)\) stands for the graph \(\{(x,y) \in X \times Y : y \in F(x) \}\) of \(F\), and \(\mathrm{cl}(A)\) for the closure of a subset \(A\) in a space. The multimap \(G ^{-1} : Z \rightrightarrows Y\) is defined by \(G^{-1} (z) = \{ y \in Y : z \in G(y)\}\). Define \(C : X \times Z \rightrightarrows Y\) by \(C(x,z) = F(x) \cap G^{-1}(y)\). The authors prove the following: Suppose \(Y\) is finite dimensional. Given \((x,z) \in \mathrm{gph}(G \circ F)\) such that \(C\) is directionally semi-compact at \((x,z)\) and \(DC((x,z) , y) (0,0) = \{0\}\) holds for every \(y \in \mathrm{cl}(C (x,z))\), then one has \[ D(G \circ F)(x,z) \subset \bigcup_{y \in \mathrm{cl}(C (x,z))} DG( y,z) \circ DF(x,y). \] If moreover, \(DF(x,y)(u) \cap DG(z,y)^{-1}(w) \subset DC((x,z),y)(u,w)\) holds for every \((u,w) \in X \times Z\) and \(y \in \mathrm{cl}(C (x,z))\), then the above inclusion is equality. Similar results for incident derivatives of multifunctions and calculus rules for sums of multimaps are also proved.
reviewed by: Takamitsu Yamauchi
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 5593754
zbMATH Keywords: calmness
zbMATH Keywords: contingent derivative: incident derivative
zbMATH Keywords: proto-differentiability
zbMATH Keywords: semi-differentiability
zbMATH Keywords: upper Lipschitz property
MaRDI profile type: Publication
full work available at URL: s11228-009-0105-4
OpenAlex ID: W4300292049
cites work: On proto-differentiability and strict proto-differentiability of multifunctions of feasible points in perturbed optimization problems
cites work: On the Calmness of a Class of Multifunctions
cites work: Q3122754
cites work: Sensitivity analysis for nonsmooth generalized equations
cites work: Lipschitzian inverse functions, directional derivatives, and applications in \(C^{1,1}\) optimization
cites work: Metric regularity: characterizations, nonsmooth variations and successive approximation∗
cites work: On proto-differentiability of generalized perturbation maps
cites work: Implicit multifunction theorems for the sensitivity analysis of variational conditions
cites work: Nonsingularity conditions for multifunctions
cites work: Lipschitzian multifunctions and a Lipschitzian inverse mapping theorem.
cites work: Compact nets, filters, and relations
cites work: Differentiability of Relations and Differential Stability of Perturbed Optimization Problems
cites work: Continuity of usual operations and variational convergences
cites work: An Implicit-Function Theorem for a Class of Nonsmooth Functions
cites work: Proto-differentiability of set-valued mappings and its applications in optimization
cites work: Variational Analysis
cites work: Contingent derivative of the perturbation map in multiobjective optimization
cites work: Tangent Cones and Quasi-Interiorly Tangent Cones to Multifunctions
cites work: Generalized Gradients and Applications
DOI: 10.1007/S11228-009-0105-4
description: scientific article; zbMATH DE number 5593754