Henig proper efficient points and generalized Henig proper efficient points
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.
scientific article; zbMATH DE number 5629512