数理解析研究所講究録 1246 巻 2002 年 6-96

Efficiency of Set Optimization withWeighted Criteria
島根大学総合理工学部

黒岩

大史 (Daishi Kuroiwa)

Depar rment of Mathematics and Computer Science

Interdisciplinary Faculty of Science and Engineering, Shimane University

1060 Nishikawatsu, Matsue, Shimane 690-8504 , JAPAN1

Introduction

In this paper, we consider efficiency of set-valued optimization problems with weightedcriteria. Let $(E, \leq)$ be an ordered topological vector space, the ordering cone in $(E, \leq)$ ,is aclosed set. Also $C^{+}=\{x^{*}\in E^{*}|\langle x^{*}, x\rangle\geq 0,\forall x\in C\}$ and weand assume that
. Let $A$ be the family of all nonempty compact
choose aweight set $W$ , asubset ofconvex sets in , and anonempty subfamily of $A$ . Our purpose is to consider aboutminimal elements of with weighted criteria.
In this paPer, we introduce some concepts concerned with set-limit and cone-completeness,
to characterize existence of such minimal elements. Also we consider completeness of somemetric space including the whole space $A$ .
$C$

$C$

$C^{+}$

$E$

$B$

$B$

Definition 1.1 (1)$A\leq_{W}^{l}B$$A$$\neq A$

4夏f, B,$\overline{\langle z^{*},A+C\rangle}\supset\langle z^{*},B\rangle,\forall z^{*}\in W$$\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}$$\leq_{W}^{u}B$$\subset E$$\langle z^{*}$,$A\rangle$$\subset\overline{\langle z^{*},B-C\rangle}$,$\forall z^{*}\in W$

Definition 1.2 (Minimal for aFamily with Weight)is $(l, W)$ minimal in if $B_{0}\in B$ and condition
is $(u, W)$ minimal in if $B_{0}\in B$ and condition$B_{0}$$B_{0}$

$B$

$B$

$B\leq_{W}^{l}B_{0}$$B\leq_{W}^{u}B_{0}$implies
implies$B_{0}\leq_{W}^{l}B$.

$B_{0}\leq_{W}^{u}B$.

Similarly we can define (l,$W)$ -maximal and (u,$W)$ -maximal. In this paper we treat
only the (l, W) minimal notion

72Characterization of Efficiency
Definition 2.1 $((l, W)$ -Decreasing, $(l, W)$ -Complete, $(l, W)$ -Section)
Anet of sets
in $A$ is said to be $(l, W)$ -decreasing if$\{A_{\lambda}\}$$\lambda<\lambda’$

Asubfamily $V$
in $D$ such that$D$$\subset${$A_{\lambda’}\leq_{W}^{l}A_{\lambda}$$\Rightarrow$$\subset A$

is said to be $(l, W)$ -complete if there is no $(l, W)$ -decreasing net$A\in A|\exists\lambda$

Let $A\in A$ and $D$$\subset A$such that

$A\not\leq_{W}^{l}D_{\lambda}$$\{D_{\lambda}\}$}. Then the family$D(A)=\{D\in D |D\leq_{W}^{l}A\}$is called an $(l, W)$ section in 7)

Theorem 2.1 (Existence of (l, $W)$ -minimal sets)
B has an (l,$W)$ -minimal set if and only if B has anonempty (l,$W)$ -complete section
Definition 2.2 ( $W$-limit, $W$ -set limit)Let
be anet of , $x\in E$ , then$E$$\{a_{\lambda}\}_{\Lambda}$,$\lim_{\lambda^{\mathrm{W}}}a_{\lambda}\ni x\Leftrightarrow^{\mathrm{d}\mathrm{e}\mathrm{f}}\forall y^{*}\in W$

the set$\lim_{\lambda^{\mathrm{W}}}a_{\lambda}$$\langle y^{*}, a_{\lambda}\ranglearrow\langle y^{*}, x\rangle$

is called $W$ -limit of$\mathrm{L}\mathrm{i}\mathrm{m}\inf_{\lambda\in\Lambda}\mathrm{w}A_{\lambda}\ni x\Leftrightarrow^{\mathrm{d}\mathrm{e}\mathrm{f}}\exists\{a_{\lambda}\}$$\{a_{\lambda}\}$Also letsuch that$\mathrm{L}\mathrm{i}\mathrm{m}\sup_{\lambda\in\Lambda}\mathrm{w}^{A_{\lambda}}\ni x\Leftrightarrow^{\mathrm{d}\mathrm{e}\mathrm{f}}\exists\{a_{\lambda’}\}\subset\{a_{\lambda}\}$and$\{A_{\lambda}\}_{\lambda\in\Lambda}$$a_{\lambda}\in A_{\lambda}$

these are called $W$-lower and $W$-upper limits, resp.

Definition 2.3 $((l, W)$ and $(u, W)$ -Set limits)$\mathrm{A}\mathrm{C}\mathrm{A}$Lim$\inf_{\mathrm{W}}^{u}A_{\lambda}=\mathrm{L}\mathrm{i}\mathrm{m}$

AHA

Lim

$\sup_{\mathrm{W}}^{l}A_{\lambda}=\mathrm{L}\mathrm{i}\mathrm{m}$$\lambda\in\Lambda$

Lim$\inf_{\mathrm{W}}(A_{\lambda}-C)$

AEA$\sup_{\mathrm{W}}^{u}A_{\lambda}=\mathrm{L}\mathrm{i}\mathrm{m}$AHA$\sup \mathrm{w}(A_{\lambda}+C)$$\lambda\in\Lambda$$\sup_{\mathrm{W}}(A_{\lambda}-C)$

AHA,VAbe anet of $A$ , $x\in E$ , then$\in\Lambda$

and$\lim_{\lambda^{\mathrm{W}}}a_{\lambda}\ni x$:asubnet such that a2$\lim_{\lambda’}\mathrm{w}a_{\lambda}’\ni x$$\mathrm{L}\mathrm{i}\mathrm{m}\inf_{\lambda\in\Lambda}\iota A_{\lambda}=\mathrm{L}\mathrm{w}\mathrm{i}\mathrm{m}\inf_{\mathrm{W}}(A_{\lambda}+C)$.

$\in A_{\lambda}$,VA 6$\Lambda$

8Proposition 2.1 If$A\leq_{W}^{l}A_{\lambda}$$A_{\lambda}$

is $(l, W)$ -decreasing then$\Leftrightarrow$$A \leq_{W\mathrm{W}}^{\iota\iota}\mathrm{L}\mathrm{i}\mathrm{m}\inf_{\lambda\in\Lambda}A_{\lambda}$

Theorem 2.2 The following are equivalent:$\bullet$$\bullet$$\bullet$$\bullet$$B$

has an $(l, W)$ -minimal set$B$

has anonempty $(l, W)$ -complete section

There exists $A_{0}\in A$ such that $B(A_{0})=\{B\in B|B\leq_{W}^{l}A_{0}\}$ is $(l, W)$ -complete

For any $(l, W)$ -decreasing net
Lim$\{B_{\lambda}\}$in$B$, there exists $A_{0}\in A$ such that

$A_{0}\leq_{W}^{l}$$\inf_{\mathrm{W}\lambda\in\Lambda}^{l}B_{\lambda}$

Corollary 2.1 Let $F$ be aset-valued map from asubset $X$ of atopological space into $E$ .
If $X$ is compact and$x_{\lambda}arrow x_{0}$,$\{F(x_{\lambda})\}$$\Rightarrow F(x_{0})\leq_{W}^{l}$: $(l, W)$ -decreasingLim$\inf_{\mathrm{W}\lambda\in\Lambda}^{l}F(x_{\lambda})$then there is an $(l, W)$ -minimal set in $\{F(x)|x\in X\}$ .

3CompletenessIn this section, we consider about completeness of metric space (.A
as follows:
define aquotient space$/\equiv_{W}^{l}$, ). At first we$d$$A/\equiv_{W}^{l}$$A/\equiv_{W}^{l}=\{[A]|A\in A\}$,

where $[A]=\{B\in A|A\equiv_{W}^{l}B\}$ for each $A\in A$ . In this space, we define an order relation.
,For $[A]$ ,$[B]\in A/\equiv_{W}^{l}$$[A]\leq_{W}^{l}[B]\Leftrightarrow^{\mathrm{d}\mathrm{e}\mathrm{f}}A\leq_{W}^{l}B$Then$\leq_{W}^{l}$is an order relation on$[B]\in A/\equiv_{W}^{l}$$A/\equiv_{W}^{l}$. Next, we define ametric on the space. For $[A]$ ,,$d([A], [B])= \sup_{y^{*}\in W}|\min\langle y^{*}, A\rangle-\min\langle y^{*}, B\rangle|$.
Then is ametric on
Now we have aquestion. Is$A/\equiv_{W}^{l}$

$d$

$d$

,
Counterexample 3.1. Then, $x_{2}\leq n$ ,converges to any elements of, $d(A_{n}, A_{0})arrow 0$ as $narrow\infty$ )$E=\mathrm{R}^{2}$$x_{1}$$1\leq x_{1}x_{2}\}$complete?

$\{[A_{n}]\}$$A/\equiv_{W}^{l}$$x_{1}x_{2}\}$, $W=[(1,0), (0,1)]$ , $A_{n}=\{(x_{1}, x_{2})\in E|0\leq$, butdoes notis aCauchy sequence in. (For example, $A_{0}=\{(x_{1}, x_{2})\in E|0\leq x_{1}$ , $x_{2},1\leq$$C=\mathrm{R}_{+}^{2}$$A/\equiv_{W}^{l}$$\{[A_{n}]\}$

9
How conditions assure the completeness? Concerning the question, we have the following two theorems.
Theorem 3.1$K$ of $E$ such that$\{[A_{n}]\}$Proof. Let$\mu_{A_{n}}$is aCauchy sequence infor each .

$A_{n}\subset K$:$Warrow \mathrm{R}$$A/\equiv_{W}^{l}$, and there exists acompact subset$n$defined by$\mu_{A_{n}}(y^{*}):=\inf_{a\in A_{n}}\langle y^{*}, a\rangle$, $y^{*}\in W$such that:then there exists acontinuous function$W$. For $y^{*}\in W$ , there exists $a_{y}*\in K$ such thatuniformly on$A_{0}:=\{a_{y^{*}}|y^{*}\in W\}$ , thenAlso we have c\^o10\in A, and then we conclude the proof.
$\mu_{0}$$Warrow \mathrm{R}$converges to. Let.
$\mu 0$$\mu_{A_{n}}$$\mu_{0}(y^{*})=\langle y^{*}, a_{y}*\rangle$$\mu_{0}(y^{*})=\inf_{a\in A_{0}}\langle y^{*}, a\rangle=\inf_{a\in \mathrm{c}\mathrm{o}A_{0}}\langle y^{*}, a\rangle=\inf_{a\in\overline{\mathrm{c}\mathrm{o}}A_{0}}\langle y^{*}, a\rangle\square$, and there exists acompact subset
is aCauchy sequence inTheorem 3.2$K$ of $E$ and asequence $\{x_{n}\}\subset E$ such that $x_{n}+A_{n}\subset K$ for each. Assume that$A$$E$$C^{+}-C^{+}=E^{*}$ andconverges some element of .
is reflexive, then$A/\equiv_{W}^{l}$$\{[A_{n}]\}$$n$$\{[A_{n}]\}$

Proof. Let$\mu_{A_{n}}$:$Warrow \mathrm{R}$defined by$\mu_{A_{n}}(y^{*}):=\inf_{a_{n}}\langle y^{*}, a\rangle$, $y^{*}\in W$

converges tosuch that:then there exists acontinuous function
uniformly on $W$ . From condition $x_{n}+A_{n}\subset K$ , there exists $M$ such that
forfor each $y^{*}\in W$ and , and by assumption $C^{+}-C^{+}=E^{*}$ , we haveand . Using uniform boundedness theorem, we have $||x_{n}||\leq M$ for eacheachconverges toand $x_{0}\in E$ such that
. Then we can choose asubsequence
weakly.
. Let $A_{0}:=$For $y^{*}\in W$ , there exists $a_{y}*\in K$ such that$\{a_{y^{*}}-x_{0}|y^{*}\in W\}$ , thenfor, then we complete the proof.
each $y^{*}\in W$ . Also we have$\mu_{0}$$Warrow \mathrm{R}$$\mu_{0}$$\mu_{A_{n}}$$|\langle y^{*}, x_{n}\rangle|\leq M$$|\langle y^{*}, x_{n}\rangle|\leq M$$n$$y^{*}\in E^{*}$$n$$\{x_{n’}\}$$n$$\{x_{n’}\}$$x_{0}$$\langle y^{*}, x_{0}\rangle+\mu_{0}(y^{*})=\langle y^{*}, a_{y}*\rangle$$\mu_{0}(y^{*})=\inf_{a\in A_{0}}\langle y^{*}, a\rangle=\inf_{a\in \mathrm{c}\mathrm{o}A_{0}}\langle y^{*}, a\rangle=\inf_{a\in\overline{\mathrm{c}\mathrm{o}}A_{0}}\langle y^{*}, a\rangle$$\square$$\overline{\mathrm{c}\mathrm{o}}A_{0}\in A$

References[1] D. Kuroiwa, “On Weighted Criteria of Set Optimization,” RIMS Kokyuroku, 2001.
[2] D. Kuroiwa, “Existence Theorems of Set Optimization with Set-Valued Maps,” Journal ofInformations&Optimization Sciences, to appear.

.

[3] D. T. Luc, “Theory of Vector Optimization,” Lecture Note in Econom. and Math. Syste ms319, Springer, Berlin, 1989