Henig proper efficient points and generalized Henig proper efficient points
type: Item
instance of: scholarly article
title: Henig proper efficient points and generalized Henig proper efficient points
zbMATH Open document ID: 1187.46060
publication date: 2009-11-11T00:00:00Z
review text: Let \(X\) be a real locally convex space (l.c.s) and \(S\) a closed convex pointed cone in \(X\). A~convex subset \(\Theta\) of \(X\) is called a base for \(S\) if \(0\notin\) cl\((\Theta)\) and \(S=\)cone\((\Theta)\). A point \(\bar x\) in a subset \(A\) of \(X\) is called an efficient point of \(A\) with respect to the order given by \(S\) if \((A-\bar x)\cap(-S)=\{0\}\). If \(X\) is a normed space with closed unit ball \(B\) then \(\delta:=\)dist\((0,\Theta) >0, \) so that \(\Theta+\varepsilon B\) is a base of the cone \(S_\varepsilon(\Theta)=\)cone\((\Theta+\varepsilon B)\), for every \(\varepsilon\) with \(0<\varepsilon <\delta\). A~point \(\bar x\in A\) is called a Henig proper efficient point for \(A\) provided there exists \(0<\varepsilon<\delta\) such that cl\([\)cone\((A-\bar x)\cap(-\)cl\(S_\varepsilon(\Theta))]=\{0\}\). These can be extended to l.c.s.~by taking a continuous linear functional \(f_\Theta\) on \(X\) such that \(\delta:=\inf f_\Theta(\Theta)>0\). Then \(V_\Theta= \{ x\in X : |f_\Theta(x)| <\delta/2\}\) is a convex neighborhood of 0. It follows that for any convex 0-neighborhood \(V\subset V_\Theta,\, \Theta + V\) is a base of the cone cone\((\Theta +V)\). One defines a proper efficient point \(\bar x \in A\) with respect to \(\Theta\) (written as \(\bar x\in HE(A,\Theta)\)) as above, by replacing \(S_\varepsilon\) by \(S_V(\Theta)\). A point \(\bar x\in A\) is called Henig proper efficient (generalized Henig proper efficient) provided that \(\bar x\in \bigcap_{\Theta\in B(S)} E(A,\Theta)\) (respectively, \(\bar x\in \bigcup_{\Theta\in B(S)} HE(A,\Theta))\), where \(B(S)\) denotes the family of all bases of the cone \(S\). The paper under review is concerned with with the relations between the Henig proper efficient points and generalized Henig proper efficient points. Among other results, the author obtains a necessary and sufficient condition in order that generalized Henig proper efficient points be Henig proper efficient.
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 90C48
zbMATH DE Number: 5629512
zbMATH Keywords: locally convex space
zbMATH Keywords: ordered vector spaces
zbMATH Keywords: base of a cone
zbMATH Keywords: efficient point
zbMATH Keywords: Henig efficient point
zbMATH Keywords: generalized Henig efficient point
author: Jinghui Qiu
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: s10114-008-6418-7
OpenAlex ID: W2398109746
cites work: Proper efficiency with respect to cones
cites work: Super Efficiency in Vector Optimization
cites work: Proper efficiency in locally convex topological vector spaces
cites work: Q5613420
cites work: Q5569793
cites work: Q5736344
cites work: Ordered linear spaces
cites work: Weakly compact sets. Lectures held at S.U.N.Y., Buffalo, in spring 1978
cites work: Weak countable compactness implies quasi-weak drop property
cites work: On weak drop property and quasi-weak drop property
DOI: 10.1007/S10114-008-6418-7
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5629512