数理解析研究所講究録 1415 巻 2005 年 105-109

105

PROPERLY EFFICIENT POINTS IN SET-VALUEDANALYSIS
集合値解析における真性有効点
黒岩大史 (DAISHI KUROIWA)

島根大学総合理工学部数理・情報システム学科
1.

$\mathrm{I}_{\mathrm{N}\mathrm{T}\mathrm{R}\mathrm{O}\mathrm{D}\mathrm{U}\mathrm{C}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}}$AND$\mathrm{p}_{\mathrm{R}\mathrm{E}\mathrm{L}\mathrm{I}\mathrm{M}\mathrm{I}\mathrm{N}\mathrm{A}\mathrm{R}\mathrm{I}\mathrm{E}\mathrm{S}}$Let $E$ be alocally convex topological vector space over the real number field, $K$ be aconvex cone of $E$ , and assume that $K$ is pointed and closed. By
on $E$ ;using $K$ , we can define avector ordering$\mathbb{R}$$\leq_{K}$$\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}$$x\leq_{K}y$$y-x\in K$ .

In this situation, we can consider notions of efficiency and proper efficiency invector optimization; for anonempty subset of $E$ ,$x\in A$ is said to be aminimal point of
if $a\leq_{K}x$with respect to$x\leq_{K}a$$a\inA$
withpointsofminimalofall;the set, thenfor some${\rmMin}(A|\leq_{K})$;is denoted byrespect to$x\in A$ is said to be aproperly minimal point ofifwith respect toand is asuch that
there exists aconvex cone; the set of all properly minimal
minimal point of with respect tois denoted by PrMin $(A |\leq_{K})$ ,points of with respect to$E$
where means the null vector of , and intL the set of all interior points of .
When we consider efficiency in set-valued optimization, there are two criteria;
one is for vector optimization, which is the typical one, see [2], and the other isfor set optimization, which is denned and researched recently, see $[3, 4]$ .
In this paper we introduce notions of proper efficiency for set optimization,and investigate them by an embedding idea. In section 2, we consider twobinary relations on certain families, and define notions proper efficiency basedon these relations. Also we characterize these relations by using positive polarcone. In section 3, to show an embedding theorem, we construct avector space,and introduce ametric on the space which consists an adequate topology.

.
.

$A$

$A$

$\leq_{K}$$A$$\leq_{K}$

$A$

$A$

$A$

$\leq_{K}$$K\subset \mathrm{i}\mathrm{n}\mathrm{t}L\cup\{\theta\}$$L\underline{\subset}E$$x$$\leq_{L}$$\leq_{K}$$L$$\theta$

2. ANOTION OF PROPER EFF1CIENCY$1\mathrm{N}$SET-VALUED OPTIMIZATION

We consider notions of efficiency for set-valued optimization in the sense of
be the family of all nonempty compact convex sets
set optimization. Let$\mathrm{C}(E)$.
;Secondary2000 Mathematics Subject Classification. Primary
Key words and phrases, set-valued optimization; set optimization; efficiency; proper effi$90\mathrm{C}29$$49\mathrm{J}53$

ciency ;embedding.
*This research is partially supported by Grant in-Aid for Young Scientists (B) from theMinistry of Education, Culture, Sports, Science and Technology, Japan, No. 15740061

106in $E$ ; we define two binary relations,$\leq_{K}^{l}$and$\leq_{K}^{u}$on$\mathrm{C}(E)$as follows: for$A$,$B\in \mathrm{C}(E)$$A\leq_{K}^{l}B\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}A+K\supset B$,$A\leq_{K}^{u}B\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}A\subset B-K$

see $[3, 4]$ . We have a result concerned with these relations by using the positivebe the positive polar cone, that is,polar cone of $K$ ; let$K^{+}$.

$K^{+}=\{x^{*}\in E^{*}|\langle x^{*}, x\rangle\geq 0,\forall x\in K\}$

Proposition 1. For each , $B\in \mathrm{C}(E)$ , the following three assertions areequivalent:(1) $A\leq_{K}^{l}B$ ,,(2) $A+C\leq_{K}^{l}B+C$ for some(3) inf
for all
and also the following three assertions are equivalent:(4) $A\leq_{K}^{u}B$ ,,(5) $A+C\leq_{K}^{u}B+C$ for some$y^{*}\in K^{+}$(6)
allfor
Note that Proposition 1 holds when $K$ is a nontrivial closed convex cone of$E$ .
, see $[3, 4]$ . Let $A$ be a
Now we introduce notions of efficiency on. $X\in A$ is said to be, that isnonempty subfamily of$A\leq_{K}^{l}X$$A$
iffor some $A\in A$ , thenrespect
withtopointofa minimal$X\leq_{K}^{l}A$ ; the set of all minimal points of $A$ with respect toi denoted byandand the set. Ako efficiency with respect to$A$$C\in \mathrm{C}(E)$$\langle y^{*}, A\rangle\leq\inf\langle y^{*}, B\rangle$$y^{*}\in K_{j}^{+}$$C\in \mathrm{C}(E)$$\sup\langle y^{*}, A\rangle\leq\sup\langle y^{*}, B\rangle$$\mathrm{C}(E)$$\mathrm{C}(E)$$\emptyset\neq A\subset \mathrm{C}(E)$$\leq_{K}^{l}$$\leq_{K}^{l}$${\rm Min}(A|\leq_{K}^{l})$$\mathrm{s}$${\rm Min}(A|\leq_{K}^{u})$$\leq_{K}^{u}$defined.
.
Next we define proper efficiency on$A$. $X\in A$ is said to beDefinition 1. Let be a nonempty subfamily of$A$a properly minimal point of with respect
if there exists a convex cone
and $X$ is a minimal point of $A$ with respect tosuch that
denoted; the set of all properly minimal point of $A$ with respect.
by PrMin
areand the set PrMin
Also the proper efficiency with respect todefined.
These notions of efficiency and proper efficiency are generalizations of onesin vector-valued optimization.
Proposition 2. Let $A$ be a nonempty subfamily of $C(E)$ , and assume thatis singleton for any $A\in A$ , then the following three assertions are equivalent:(1) $x\in{\rm Min}(\cup A|\leq_{K})$ ,(2) $\{x\}\in{\rm Min}(A|\leq_{K}^{l})$ ,(3) $\{x\}\in{\rm Min}(A|\leq_{K}^{u})$ ,and also the following three assertions are equivalent:
,(4)
,(5)
,(6)
$where\cup A=$ { $a\in E|\exists A\in A$ such that $a\in A$ }.
$\mathrm{C}(E)$$\mathrm{C}(E)$$to\leq_{K}^{l}$$L\subseteq E$$K\subset \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{L}$$\leq_{L}^{l}$$\{\theta\}$$to\leq_{K}^{l}$

$s$

$is$

$(A|\leq_{K}^{l})$$\leq_{K}^{u}$$(A |\leq_{K}^{u})$$A$$x\in \mathrm{P}\mathrm{r}\mathrm{M}\mathrm{i}\mathrm{n}(\cup A|\leq_{K})$$\{x\}\in \mathrm{P}\mathrm{r}\mathrm{M}\mathrm{i}\mathrm{n}(A|\leq_{K}^{l})$$\{x\}\in \mathrm{P}\mathrm{r}\mathrm{M}\mathrm{i}\mathrm{n}(A|\leq_{K}^{u})$107, $K=\mathbb{R}_{+r}^{2}A=\{At |t\in[-\sqrt{2}, \sqrt{2}]\}$ where $A_{t}=$$\{(x, y)\in E|x^{2}+y^{2}\leq 1, x+y=t\}$ . Then we have

Example 1. Let$E=\mathbb{R}^{2}$${\rm Min}(A|\leq_{K}^{l})=\{A_{t}|t\in[-\sqrt{2}, -1]\}$PrMin$(A |\leq_{K}^{l})$,$=\{At |t\in[-\sqrt{2}, -1)\}$,${\rm Min}(A|\leq_{K}^{u})=\mathrm{P}\mathrm{r}\mathrm{M}\mathrm{i}\mathrm{n}(A|\leq_{K}^{u})=\{At |t=-\sqrt{2}\}$.

In vector optimization, we have ${\rm Min}(\cup A|\leq_{K})=\{(x, y)\in E|x^{2}+y^{2}=1$ , $x\leq$
and $\mathrm{P}\mathrm{r}\mathrm{M}\mathrm{i}\mathrm{n}(\cup A|\leq_{K})=\{(x, y)\in E|x^{2}+y^{2}=1, x<0, y<0\}$ .
,$0$$y\leq 0\}_{\mathrm{J}}$3. AN INVESTIGATION OF PROPER EFFICIENCY 1NSET-VALUED
OPTIMIZATION

will be used. The similar argument
In this section, only binary relation.
will be available for relation
To study proper efficiency in set optimization, we consider an embedding; weis embedded, . [1].
will construct a vector space in which
be defined byTheorem 1. Let a binary relation$\leq_{K}^{l}$$\leq_{K}^{u}$$\mathrm{c}.\mathrm{f}$$\mathrm{C}(E)$$\mathcal{V}$$\simeq on\mathrm{C}(E)^{2}$$(A_{1}, B_{1})\simeq(A_{2}, B_{2})\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}A_{1}+B_{2}+K=A_{2}+B_{1}+K$

for$(A_{1}, B_{1})$,$(A_{2}, B_{2})\in \mathrm{C}(E)^{2}$

We denote the quotient space.

an equivalence relation on$Then\simeq is$$\mathrm{C}(E)^{2}$.

that is$\mathrm{C}(E)^{2}/\simeq as\mathcal{V}_{f}$$\mathcal{V}=\{[(A, B)]|(A, B)\in \mathrm{C}(E)^{2}\}$,,where $[(A, B)]=\{(A’, B’)\in \mathrm{C}(E)^{2}|(A, B)\simeq(A’, B’)\}$ . Let addition andscalar multiplication in the quotient space as follows:$[(A_{1}, B_{1})]+[(A_{2}, B_{2})]=[(A_{1}+A_{2}, B_{1}+B_{2})]$ ,$\mathcal{V}$$\lambda\cdot[(A, B)]=\{$$if\lambda\geq 0if\lambda<0$$[(\lambda A,\lambda B)][((-\lambda)B,(-\lambda)A)]$

is a vector space over .
Then, and denote
The null vector in isnotation: Let be a convex cone in $E$ , and let.

$\mathbb{R}$$(\mathcal{V}, +, )$$[(\{\theta\}, \{\theta\})]$$\mathcal{V}$$\Theta$. We define the following$L$$\mu(L):=\{[(A, B)]\in \mathcal{V}|B+L\supset A\}$,

then we can check $\mu(L)$ is a convex cone in , and especially, $\mu(K)$ is a pointed
convex cone in . Generally, we can induce order relations in for an arbitrary
convex cone in since is regarded as a general ordered vector space.
,in is defined as follows: for $[(A_{1}, B_{1})]$ ,binary relation$\mathcal{V}$$\mathcal{V}$$\mathcal{V}$$\mathrm{A}$$\mathcal{K}$$\mathcal{V}$$\mathcal{V}$$\leq\kappa$$[(A_{2}, B_{2})]\in \mathcal{V}$$\mathcal{V}$,$[(A_{1}, B_{1})]\leq\kappa[(A_{2}, B_{2})]\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}[(A_{2}, B_{2})]-[(A_{1}, B_{1})]\in \mathcal{K}$and also an efficiency in$\mathcal{V}$is defined as follows: for$\mathcal{U}\subset \mathcal{V}$${\rm Min}(\mathcal{U}|\leq\kappa)=\{U\in \mathcal{U}|U’\in \mathcal{U}, U’\leq\kappa U\Rightarrow U\leq\kappa U’\}$.

As a consequence of the embedding, we have an important result for researchof efficiency in set optimization

108

be defined by $\varphi(A)=[(A, \{\theta\})]$
Proposition 3. Let a function :$A\in{\rmMin}(A|\leq_{K}^{l})$. Thenif and only if
for any$\mathrm{C}(E)arrow \mathcal{V}$$\varphi$$\varphi(A)\in{\rm Min}(\varphi(A)|\leq_{\mu(K)}$$A\in \mathrm{C}(\mathrm{E})$$)$.

To consider a notion of proper efficiency in , we introduce a topology in .
To our purpose, we would like to find a topology in satisping the following
condition:then $\mu(K)\subset$
be a convex cone withIfpoints
allofinterior
thesetholds, whereis$\mu(L)$in the topology on .
ofWe usually consider the following norm; assume that $W$ is a weak* compact, a functional . on is defined by, for each $[(A, B)]\in \mathcal{V}$ ,base of$\mathcal{V}$$\mathcal{V}$$\mathcal{V}$$L\subseteq E$$K\subset \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{L}\cup\{\theta\}_{7}$$\mathrm{i}\mathrm{n}\mathrm{t}\mu(L)$$\mathrm{i}\mathrm{n}\mathrm{t}\mu(L)\mathrm{U}\{\ominus\}$$\mathcal{V}$$K^{+}$$\mathcal{V}$$||$$||$$\langle y^{*}, B\rangle|$$||[(A, B)]||= \sup_{y^{*}\in W}|\inf\langle y^{*}, A\rangle-\inf$.

Then this is a norm on , . [4]. However, this norm is inadequate.
, $A=\{(0,0)\}$ , and $B=[(1, -1), (-1,1)]$ ,,Example 2. Let$(1,$$-1)$
to $(1, 1)$ . In this situation, for any convex conesegment ffom
the.
though,with, then we can choose a positive number 5 such
Indeed, if
that $[(A, B)]+\delta N$ is included in $\mu(L)$ , where $N$ $=\{[(C, D)]\in \mathcal{V}|||[(C, D)]||\leq$. Also we can show $[(N, \{\theta\})]\in N$ where $N=\{(x, y)\in \mathbb{R}^{2}|x^{2}+y^{2}\leq 1\}$
when $W=\{(x, y)\in \mathbb{R}_{+}^{2}|x^{2}+y^{2}=1\}$ . However $[(A, B)]+\delta[(N, \{\theta\})]\not\in$$\mu(L)\cup\{\Theta\}$ because $B+L\not\supset A+\delta N$ .
Now we consider the following metric on ; this consists an adequate
topology.
Theorem 2. Let $P$ be a compact convex base of $K$ , an element of $P$ , anddoes not meet $\{\lambda p|\lambda\in[0,1)\}$ . For $\lambda\in[0,1)$ , letassume that, $[(C, D)]\in \mathcal{V}$ ,). For
cone$\mathcal{V}$$\mathrm{c}.\mathrm{f}$$E=\mathbb{R}^{2}$$K=\mathbb{R}_{+}^{2}$$l_{\dot{7}}ne$$K\subset \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{L}1l$$L\subsetarrow E$$\{\#\}$$[(A, B)]\in\mu(_{\backslash }K)$$[(A, B)]\not\in \mathrm{i}\mathrm{n}\mathrm{t}\mu(L)\cup\{\Theta\}$$[(A, B)]\in \mathrm{i}\mathrm{n}\mathrm{t}\mu(L)\mathrm{U}\{\Theta\}$

$1\}$

$d$

$\mathcal{V}$

$p$

$P$

$-\lambda p$ $+$$K_{\lambda}=$$\lfloor(\lceil A, B)]$$\mathrm{P}$$d([(A, B)], [(C, D)])= \min\{1, e(A+D, B+C)\}$

where$e(A, B)= \inf\{\lambda\in[0,1)|A+K_{\lambda}=B+K_{\lambda}\}$,then As a metric on .
Lemma 1. Under the same assumptions in Theorem 2,(1)is a closed convex cone of $E$ for each $\lambda\in[0,1)$ ,(2)if $0\leq\lambda<\mu<1$ ,(3)for any $\lambda\in[0,1)$ ,with $e(A, B)<1$ .
(4) $A-\downarrow- K_{e(A,B)}=B+K_{e(A,B)}$ , for any ,Definition 2. Let be a pointed convex cone in and a nonempty subset
if there existsof V. is a properly minimal point of with respect
point ofisaminimalandsuch thatconvex cone$to\leqc$points
is the set of all interior, wherewith respectof on thetopology de ined by the metric in V. The set of all properly minimal points of.
with respect
is denoted by PrMin
Then we have the following results$d$$\mathcal{V}$$K_{\lambda}$$K_{\lambda}\subset K_{\mu}$$\bigcap_{\mu\in(\lambda,1)}K_{\mu}=K_{\lambda}$$A$$B\in \mathrm{C}(E)$$U$$to\leq\kappa$$\mathcal{U}$$\mathcal{L}\subsetarrow \mathcal{V}$$\mathcal{K}\subset \mathrm{i}\mathrm{n}\mathrm{t}\mathcal{L}\mathrm{U}\{8\}$$d$$to\leq\kappa$

$a$

$U$

$L$

$\mathrm{i}\mathrm{n}\mathrm{t}\mathcal{L}$$\mathcal{U}$$\mathcal{U}$$\mathcal{U}$$\mathcal{V}$$\mathcal{K}$$(\mathcal{U}|\leq\kappa)$

103

satisfying
Lemma 2. Assume that there exists a convex coneB)]\in\mathcal{V}|B\leq_{L}^{l}A\}$$\mu(L)=\{[(A,.
, where. Then) is \^a so
is nonempty, then PrMin(\mbox{\boldmath $\varphi$}(A)Theorem 3. If PrMin
nonempty.
, that is foris sequentially compact inTheorem 4. Assume that
$A0\inA$ suchandsubsequenceaexists
thereofeach)$e(A_{n’},A)arrow0$
PrMin(\mbox{\boldmath$\varphi$}(A). Thenthat$K\subset$$L\subsetneq E$$\mathrm{i}\mathrm{n}\mathrm{t}L\cup\{\theta\}$$\mu(K)\subset \mathrm{i}\mathrm{n}\mathrm{t}\mu(L)\cup\{\ominus\}$$(A |\leq_{K}^{l})$$|\leq_{\mu(K)}$$(\mathcal{V}, d)$$\varphi(A)$$\{A_{n}\}$$\{A_{n’}\}$$\{A_{n}\}_{n\in \mathrm{N}}\subset A$$|\leq_{\mu(K)}$REFERENCES[1] H. R\emptyset a&tr\"om, An embedding theorem for spaces of convex sets. Proc. Amer Math Soc.
3 (1952), 165-169.
[2] D. T. Luc, Theory of vector optimization. Lecture Notes in Economics and MathematicalSystems, 319. Sprrnger- Veflag, Berlin, 1989.
[3] D. Kuroiwa, Existence Theorems of Set Optimization with Set-Valued Maps. Joumd ofInfomatiom Optimization Sciences 24 (2003), 73-84.
[4] D. Kuroiwa, Existence of efficient points of set optimization with weighted criteria. Joufnal of Nonlinear and Convex Analysis 4 (2003), 117-124.
[5] D. Kuroiwa and T. Nuriya, Proper efficiency and embedding theorem in set optimization,
preprint.
$6^{(}$

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