Tangent cone and contingent cone to the intersection of two closed sets
type: Item
instance of: scholarly article
title: Tangent cone and contingent cone to the intersection of two closed sets
zbMATH Open document ID: 1201.49016
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-07-13T00:00:00Z
review text: Let \(X\) be a normed vector space and \(C\subset X\) a nonempty closed set; let \(T_C(x)\) denote the tangent cone to \(C\) at \(x\in C\). For \(X=\mathbb R^n\), \textit{R. T. Rockafellar} [Nonlinear Anal., Theory Methods Appl.\ 3, 145--154 (1979; Zbl 0443.26010)] has shown that, if \(x\in C_1\cap C_2\), then \[ T_{C_1}(x)\cap T_{C_2}(x)\subset T_{C_1\cap C_2}(x), \] provided that \(T_{C_1}(x)\cap \mathrm{int}\,T_{C_2}(x)\neq \emptyset\). This last hypothesis is essential as illustrated by example 1.1 of this paper. The author studies sufficient conditions for the equation above to hold. He also studies the analogous inclusion for the contingent cones of the sets \(C_i\), and gives interesting characterizations of the cones, involving Clarke and Dini derivatives of the constraint functions. Finally, he presents new constraint qualifications for the Karush-Kuhn-Tucker conditions.
reviewed by: Gabriel Larotonda
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 5736444
zbMATH Keywords: tangent cone
zbMATH Keywords: contingent cone
zbMATH Keywords: hypertangent cone
zbMATH Keywords: hypercontingent vector
zbMATH Keywords: Mangasarian-Fromovitz constraint qualification
zbMATH Keywords: Dini derivative
zbMATH Keywords: Clarke derivative
zbMATH Keywords: Karush-Kuhn-Tucker conditions
author: Zili Wu
MaRDI profile type: Publication
full work available at URL: j.na.2010.04.042
OpenAlex ID: W2017043023
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DOI: 10.1016/J.NA.2010.04.042
description: scientific article; zbMATH DE number 5736444