数理解析研究所講究録
第 1544 巻 2007 年 212-215

212

ON AN EQUIVALENCE RELATION BETWEENEFFICIENCY AND IDEAL EFFICIENCY

島根大学大学院総合理工学研究科 1 塗矢哲也 (Tetsuya Nuriya)
島根大学総合理工学部 2 黒岩大史 (Daishi Kuroiwa)
1 Interdisciplinary Graduate School of Science and Engineering,
2 Interdisciplinary Faculty of Science and Engineering,

Shimane University
Shimane University

Abstract

In previous researches, the set of all weakly efficient points and the setof all properly efficient points in vector optimization were represented byusing the sets of all efficient points with respect to convex cones. In thispaper, we consider a representation of the set of all ideal efficient points.
by using the sets of all efficient points, which is a representation similar tothe above.

1.Introduction
Let $E$ be a topological vector space and $K$ be a convex cone of $E$ which is not
the whole space. A binary relation
on $E$ is defined by$(\leq_{K}’)$.

$x\leq\kappa y\Leftrightarrow^{def}y\in x+K$$(x,y\in E)$

First, we state well-known notions of efficieney in vector optimization, see [6].

Definition 1. Let be a nonempty subset ofis not the whole space. We say that
$A$$E$

and $K$ a convex cone of $E$ which(1) $x\in A$ is an ideaZ efficient point of $A$ with respect to $K$ \‘if $y.\in x+K$ for all$y\in A$ ;(2) $x\in A.is$ an efficient point ofwith $x\in y+K$ ;$A$with respect to $K$ if $y\in x+K$ for all $y\in A$(3) $x\in A$ is a properly efficient point of with respect to $K$ if there erists aconvex cone of $E$ , which is not the whole space, such thatiscontained in intL and is an efficient point of with respect to , where$l(K)=K\cap(-K)$ ;
$A$

$L$

$K\backslash l(K)$

$x$

$A$

$L$

(4) Supposing that intK is nonempty, $x\in A$ is a weakly efficient point of $A$
with respect to $K$ if is an efficient point of $A$ with respect to intK;$x$$\cup\{\theta\}$

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The set of all effcient points (resp. properly efficient points, weakly effi cientpoints and ideal efficient points) of with respect to $K$ is denoted by ${\rm Min}(A|K)$(resp. $PrMin(A|K),$ $WMin(A|K)$ and IMin $(A|K)$ ).
$A$

WMin $(A|K)$ and $PrMin(A|K)$ can be represented by using the sets ofall eMcient points with respect to convex cones. Concretely, it is clear that
WMin $(A|K)={\rm Min}(A|intK\cup\{\theta\})$ , and a representation of the set of allproperly efficient points was given by Nieuwenhuis.
Theorem 1 ([1]). Let be a nonempty subset of $E$ and $K$ be a convex cone.
Assume that a sequenceof closed convex cones such that. Then, we haveandfor each $n\in N$ , and$A$$\{K_{n}\}_{n\in N}$$K\backslash \{\theta\}\subset intK_{\dot{n}}$$K_{n}\subset K_{n+1}$$K= \bigcap_{n\in N}K_{n}$$PrMin(A|K)=\bigcup_{n\in N}{\rm Min}(A|K_{n})$.

In this paper, we consider a representation of IMin $(A|K)$ , which is the sameas WMin $(A|K)$ and $PrMin(A|K)$ , by using the sets of all efficient points.
First, we state a representation of IMin $(A|K)$ when $K$ is closed, and we givescalarization for ideal efficiency by using the representation of IMin(A I $K$ ).

2Main results

Remember that a convex cone $K$ is said to be pointed if and only if $l(K)=\{\theta\}$ ,where is the origin of $E$ .
$\theta$

Theorem 2. Let be a nonempty subset of $E$ , and $K$ a closed pointed convex
cone of $E$ , which is not the whole space. Then the following equations hold:$A$

IMin $(A|K)={\rm Min}(A+(-K)^{c}\cup\{\theta\}|K)$ .
$A_{1}=^{\iota}\{(x, y)\cdot|x\geq 0, y\leq 0, ||(x, y)||\leq 1\}$
Example 1. Let
and $A_{2}=\{(x, y)|\Vert(x, y)\Vert\leq 1\}$ , whereis the set of all nonnegative real
numbers. Then,$E=\mathbb{R}^{2},$$K=\mathbb{R}_{+}^{2},$$\mathbb{R}_{+}$$A_{1}+(-K)^{c}\cup\{\theta\}$$=$$(0, -1)+\mathbb{R}^{2}\backslash (-\mathbb{R}_{+}^{2}\backslash \{(0,0)\})$$A_{2}+(-K)^{c}\cup\{\theta\}$$=$$(-1, -1)+\mathbb{R}^{2}\backslash (-\mathbb{R}_{+}^{2})$,,and we can check

IMin $(A_{1}|K)$
IMin $(A_{2}|K)$$={\rm Min}(A_{1}+(-K)^{c}\cup\{\dot{\theta}\}|K)=\{(0, -1)\}$$={\rm Min}(A_{2}+(-K)^{c}\cup\{\theta\}|K)=\emptyset$

.

,

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Figure of Example 1.

Next, we give a scalarization for ideal efficiency by using the representationof the set of all ideal efficient points. In the previous researches, we obtained thefollowing scalarization for weak efficiehcy.
Definition 2 ([5]). A functionto $K$ if$f$:$Earrow \mathbb{R}$is said to be monotonic with respect$x\leq\kappa y,x\neq y\Rightarrow f(x)<f(y),$ .

and $fi_{8}$ said to be weakly monotonic with respect to $K$ if$x\leq Ky\Rightarrow f(x)\leq f(y)$Theorem 3 ([5]). Let
convex cone.. Then,(1)$A$,$x\leq int_{K\cup\{\theta\}}y,$$x\neq y\Rightarrow f(x)<f(y)$be a nonempty subset of$\mathbb{R}^{n}$.

and $K$ be a closed pointed

is a weakly efficient point of with respect to $K$ if and only if there exzsts
a continuous weakly monotonic function $f$ from $E$ to 8uch that $f(x)$ isthe minimum of $f$ on .
$A$

$x$

$\mathbb{R}$

$A$

(2)

is a pmperly efficient point of $A$ with respect to $K$ if and only if there exist
intL and a continuous. a closed pointed convex coneof $E$ with$E$ to$f$suchthat is monotonic with respect to andfunction from$f(x)$ is the minimum ofon .
$x$

$L$

$f$

$K\backslash \{\theta\}\subset$$L$$\mathbb{R}$$f$$A$

By using Theorem 2 and 3, we have the following theorem.

215Theorem 4. Let
be a nonempty subset of $E$ and $K$ a closed pointed convex
cone. The following assertions are true:$A$

is an ideal efficient point of with respect tocontinuous weakly monotonic function $f$ from $E$ tominimum of on $A+(-K)^{c}\cup\{\theta\}$ ;(1) if$A$$x$then there exists asuch that $f(x)$ is the$K_{f}$$\mathbb{R}$$f$(2) if there vist a closed pointed convex cone of $E$ withanda continuous function $f$ from $E$ to such that $f$ is monotonic utth respect
to and $f(x)$ is the minimum of on $A+(-c1L)^{c}U\{\theta\}$ , then is an ideal
efficient point of with respect to $K$ .
$L$

$e$

$K\backslash \{\theta\}\subset intL$$\mathbb{R}$

$L$

$f$

$x$

$A$

References[1] J. W. Nieuwenhuis, Properly efficient and efficient solutions for vector maximization problems in Euclidean space, Joumal of Mathematical Analysis and Applications 84 (1981), 311-317.
[2] M. I. Henig, A cone separation theorem, Joumal of Optimization Theory andApplications 36 (1982) 3, 451-455.
.

[3] M. I. Henig, Proper efficiency with respect to cones, Joumal of Optimization
Theory and Applications 36 (1982) 3, 387-407.
$-$[4] S. Helbig, Approximation of the efficient point set by perturbation of the orderingcone, Methods and Models of Operations Research 36 (1982) 3, 387-407.

[5] D. T. Luc, On scalarizing method in vector optimization, Lecture Notes in Economic and Math.Systems 273 (1986), 46-51.
. [6]D. T. Luc, Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems 319. Springer- Verlag, Berlin, 1989.