数理解析研究所講究録
1068 巻 1998 年 164-170

164

Lagrange Duality of Set-Valued
Optimization with Natural Criteria*
黒岩

大史 (Daishi Kuroiwa)

Department of Mathematics and Computer Science
Interdisciplinary Faculty of Science and Engineering, Shimane University

1060 Nishikawatsu, Matsue, Shimane 690-8504, JAPAN

Abstract
Set optimization problems with objective set-valued maps are considered, andsome criteria of solutions are defined. Also, cone lower semicontinuities of set-valued
maps are introduced, and existence theorems of solutions of such problems are established. Moreover, some duality results of these problems are investigated.

1

Introduction
We observe a set-valued optimization problem$(\mathrm{S}\mathrm{P})$Minimizesubject to$(\mathrm{S}\mathrm{P})$as follows:$F(x)$$x\in S$, andaan ordered vector space, $F$ a map from $X$ towhere $X$ is a set,
. This type of set-valued optimization
nonempty subset ofproblem has been developed as a generalization of vector-valued optimization problems forexamplearound twenty years. In many paper concerned with set-valued optimization([2, 5, 4, 6, 7, 11] , we can see that a minimal solution $x_{0}\in S$ is defined such as:$(\}’,$$\leq)$$2^{1}$$S$$\mathrm{D}\mathrm{o}\mathrm{m}(F)=\{x\in X|F(x)\neq\emptyset\}$$\mathrm{f}\mathrm{o}\mathrm{r}$$)$$F(x_{0})\cap{\rm Min}\cup F(_{X)}x\in S\neq\emptyset$

and this problem are often called (vector optimization with set-valued maps.’ However
because it
the criterion of solutions is sometimes not suitable for set-valued
is only based on simple comparisons between vectors though our problem is set-valued
optimization.
$\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}_{\mathrm{I}}\mathrm{n}\mathrm{i}_{\mathrm{Z}}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$supported by Grant-in-Aid for Scientific Research from the AIinistry of*This research is
Education, Science and Culture of.Japan, No. 09740146$\mathrm{P}^{\mathrm{a}\mathrm{l}\mathrm{t}}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$165

Our aim of this paper is to introduce some natural, suitable, and proper criteria based oncomparisons between values of the (set-valued) objective map for set-valued optimization,and investigate some properties concerned with the problem. In this paper, we call such
criteria based on the idea above natural criteria for set-valued optimization, see [9].
The organization of this paper is as follows: in Section 2, we formulate our set-valued optimization problem and define two types of notions of solutions. In Section 3, we introduce(natural) lower semicontinuities for set-valued maps, characterize such continuities, andderive some existence theorems of solutions by using the lower semicontinuities. Finally,
we show some duality theorems for our set-valued optimization in Section 4.

Natural Criteria of Set-Valued Optimization2. Let $X$ be a topologicalFirst, we redefine our set-valued minimization problem
space, $(Y, \leq_{K})$ an ordered topological space with an ordering convex cone $K,$ $F$ a map, and $S\subset \mathrm{D}\mathrm{o}\mathrm{m}(F)(=\{x\in X|F(x)\neq\emptyset\})$ . Our set-valued minimizationfrom $X$ toproblem is the following:$(\mathrm{S}\mathrm{P})$$2^{Y}$

Minimizesubject to$(\mathrm{S}\mathrm{P})$$F(x)$$x\in S$ .

To define notions of solutions for our problem, we introduce some relations between
two nonempty sets which like the order relation in topological vector spaces; though thenumber types of such relations is six, we treat two important relations of them, see [8].
In this paper, we define$A\leq^{l}B\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}A+K\supset B$$A\leq^{u}B\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}A\subset B-K$

,
,

with
is said to be smaller thanof . In these cases,
for nonempty subsets $A,$
-inequality(resp. -inequality) if $A\leq^{l}B$ (resp. $A\leq^{u}B$ ).
In the above notations, means lower and means upper. Note that $A\leq^{l}B$ is equivalent
and $A\leq^{u}B$ is equivalent to ${\rm Max} A={\rm Max} B$ .
to
By using the set relations above, we introduce two types criteria of minimal solutionsin the following definition. In this paper, when we consider -minimal solution, we assume
that $F$ is -closed map, that is $F(x)$ is -closed for each $x\in X$ for simple consideration. Also
we assume similar assumption, -closedness of $F$ , when we consider -minimal solution.
$B$

$l$

$B$

$A$

$Y$

$u$

$l$

$u$

${\rm Min} A=\mathrm{b}\mathrm{I}\mathrm{i}\mathrm{n}B$

$l$

$l$

$l$

$u$

Definition 2.1 [9] An element $x_{0}\in S$ is said to beif(i) -minimal solution of$F(X_{0})\leq^{l}F(x)$ is satisfied;$F(x)\leq^{l}F(X_{0}),$for any $x\in S$ withif(ii) -minimal solution of
for any $x\in S$ with $F(x)\leq^{u}F(x_{0}),$ $F(x_{0})\leq^{u}F(x)$ is satisfied.
$l$

$u$

$(\mathrm{S}\mathrm{P})$$(\mathrm{S}\mathrm{P})$

$u$

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3Semicontinuities of Set-Values Maps and Existence
Theorems
To consider existence of solutions ofwere introduced in $[5, 9]$ .

$(\mathrm{S}\mathrm{P})$for our solutions, some cone semicontinuities

Definition 3.1 [9] A set-valued map is said to be -type lower semicontinuous on
for any -closed subset of $Y,$ $\mathcal{L}^{l}(A)=\{x\in S|F(x)\leq^{l}A\}$ is closed.
$F$

$l$

$S$

if

$A$

$l$

Definition 3.2 [9] A set-valued map $F$ is said to be -type quasi lower semicontinuous
at $x_{0}\in S$ if for each net
converges towith
is -decreasing, that is,for
Lim $\sup_{\lambda}(F(x\lambda)+K)$ is satisfied. A set-valuedmap
is said to be -type quasi lower semicontinuous on
if it is -type quasi lower
semicontinuous at each point of .
$l$$\{x_{\lambda}\}$$F(x_{\lambda}’)\leq^{l}F(x_{\lambda})$$\lambda<\lambda’,$

$F$

$l$

$\{F(x_{\lambda})\}$$x_{0}$$F(X_{0})\leq^{l}$

$S$

$l$

$l$

$S$

Definition 3.3 [5] A set-valued map $F$ is said to be upper -semicontinuous at $x_{0}\in S$ if
for any open set with $V\leq^{l}F(X_{0})$ , there exists a neighborhood ofsuch that $x\in U$$F$$V\leq^{l}F(x)$implies; A set-valued map
is said to be upper -semicontinuous on if it isupper -semicontinuous at each point of .
$IC$

$V$

$U$

$x_{0}$

$IC$

$IC$

$S$

$S$

Now we can see some characterization with respect to these lower semicontinuities.
Proposition 3.1 [9] Let $F$ be a -closed set-valued map. Then we have the following:$l$(i) upper $K$ -semicontinuity on implies -type lower semicontinuity on ;(ii) -type lower semicontinuity on implies -type quasi lower semicontinuity on
$S$

$S$

$l$

Also, if $X$ and

$Y$

$l$

are finite dimensional and(iv) -type lower semicontinuity on
$l$

$S$

$l$

$S$

$F$

$S$

.

is locally bounded, then we have

implies upper $K$ -semicontinuity on$S$.

Moreover,is the real-field, and $F$ is a singleton map, then -type lower semicontinuity
and upper $K$ -semicontinuity are equivalent to to the ordinary lower semicontinuity of realvalued functions.
$Y$

$l$

Note that quasi lower semicontinuity is more weaker than another semicontinuities.
Now, we investigate -type lower semicontinuities of set-valued maps.
$u$Definition 3.4 [9] A set-valued map is said to be -type lower semicontinuous on
for any -closed subset of}’, $L^{u}(A)=\{x\in S|F(x)\leq^{u}A\}$ is closed.
$F$

$u$

$S$

if

$A$

$u$

Definition 3.5 [9] A set-valued map
is said to be -type quasi lower semicontinuous
at $x_{0}\in S$ if for each net
converges towith
is -decreasing, that is,for
Lim $\sup_{\lambda}(F(x\lambda)+IC)$ is satisfied. A set-valuedmap
is said to be -type quasi lower sernicontinuous on
if it is -type quasi lower
semicontinuous at each point of .
$F$$\{x_{\lambda}\}$$F(x_{\lambda}’)\leq^{u}F(x_{\lambda})$$F$$\lambda<\lambda’,$$u$$x_{0}$$\{F(x_{\lambda})\}$$u$$F(X_{0})\leq^{u}$

$S$

$u$

$S$

$u$

167

Definition 3.6 [5] A set-valued map $F$ is said to be lower $K$ -semicontinuous at $x_{0}\in S$ if
for any open set with, there exists a neighborhood of such that $x\in U$$V\cap(F(x)-K)\neq\emptyset$
implies; A set-valued map $F$ is said to be lower -semicontinuous on
if it is lower $K$ -semicontinuous at each point of .
$V$

$V\cap F(x_{0})\neq\emptyset$$U$$x_{0}$$I\zeta$$S$$S$

Now we can see some characterization with respect to these lower semicontinuities.
Proposition 3.2 [9] Let $F$ be a -closed set-valued map. Then we have the following:$u$(i) -type lower semicontinuity on
$u$

$S$

implies -type quasi lower semicontinuity on

$S$

$u$

.

Also, if $X$ and}’ are finite dimensional and $F$ is locally bounded, then we have(iii) -type lower semicontinuity on
$u$

$S$

is equivalent to lower $K$ -semicontinuity on$S$.

Moreover,is the real-field, and $F$ is a singleton map, then -type lower semicontinuity
and lower -semicontinuity are equivalent to to the ordinary lower semicontinuity of realvalued functions.
$Y$

$u$

$I\zeta$Now, we consider existence theorems for -type and -type minimal solutions.
$l$

$u$

Theorem 3.1 [9] Let $X$ be a topological space and}’ an ordered topological vector space.
If is a nonempty compact subset of $X$ and $F$ : $Sarrow 2^{1’}$ is a -type quasi lower semicontinuous and -closed set-valued map, then there exists a -type minimal solution of
.
$S$

$l$

$l$

$l$

$(\mathrm{S}\mathrm{P})$Theorem 3.2 [9] Let $X$ be a topological space and an ordered topological vector space.
If is a nonempty compact subset of $X$ and $F$ :is a -type quasi lower semi.
continuous and -closed set-valued map, then there exists a -type minimal solution of
$Y$

$S$

$Sarrow 2^{1^{r}}$

$u$

$u$

$(\mathrm{S}\mathrm{P})$$u$.

By using one of the above theorems, we can prove the following:
Corollary 3.1 Let $X$ be a topological space,a nonempty compact subset of $X$ , and: $Sarrow 2^{1’}$ is a lower semicontinuous, then there exists an element $x_{0}\in S$ such that
$S$

$f$

$f(x_{0})= \inf_{x\in S}f(x)$ .

Let
the topological dual space of 1
the null vector ofpolar cone of , that is, $K^{+}=\{y^{*}\in\}’*|\langle y^{*}, k\rangle\geq 0,$.
$\}’*\mathrm{b}\mathrm{e}$$r,$$\theta^{*}$$IC$$Y^{*}$. and$I\mathrm{c}’+$the positive$\forall k\in K\}$

Theorem 3.3 [9] Let
be a complete metric space, 1 an ordered locally convex
space with the cone . Also, $F$ be a map from $X$ tosatisfying the following conditions:$r$$(x_{\text{ノ}}.d)$$I\iota’$$\bullet$there exists

inf$2^{1^{r}}$$y^{*}\in I\mathrm{i}^{\prime+}\backslash \{\theta^{*}\}$$\langle y^{*}, F(.\mathcal{B})\rangle$

such thatis finite for each $x\in S$$F(x_{1})\leq^{l}F(x_{2}),$ $X1,$ $X_{2}\in S\Rightarrow \mathrm{i}\mathrm{I}\mathrm{l}\mathrm{f}\langle y^{*}.F(x_{2})\rangle$–inf$\langle y^{*}, F(x_{1})\rangle\geq d(x_{2,1}x)$

168

is -type lower semicontinuous and l-closed.

$F:Sarrow 2^{Y}$$\bullet$$l$Then, there exists a -type minimal solution of$l$$(\mathrm{S}\mathrm{P})$.

an ordered locally convex
Theorem 3.4 [9] Let (X, ) be a complete metric space,satisfying the following conditions:space with the cone $K$ . Also, $F$ be a map from $X$ to
$Y$

$d$

$2^{Y}$

there exists$\bullet$infsuch that$y^{*}\in K^{+}\backslash \{\theta^{*}\}$$\langle y^{*}, F(X)\rangle$

is finite for each $x\in S$$F(x_{1})\leq^{u}F(x_{2})jx_{1},$ $X2 \in S\Rightarrow\sup\langle y^{*}, F(x_{2})\rangle-\sup\langle y^{*}, F(x_{1})\rangle\geq d(x_{21}, x)$$F:Sarrow 2^{1^{\gamma}}$$\bullet$is -type lower semicontinuous and u-closed.
$u$Then, there exists a -type minimal solution of$u$$(\mathrm{S}\mathrm{P})$.

Using one of the above theorems, we can show Phelps’ theorem, see [1]:be a Banach space, $D$ a closed nonempty subset of $Y$ , andCorollary 3.2 Let
a convex cone of $Y$ . If there exist $y^{*}\in K^{+}$ and $\alpha>0$ such that

$I\zeta$$(\mathrm{I}^{\nearrow}, ||\cdot||)$

(i)
(ii)

$\langle y^{*}, \cdot\rangle$

is bounded from below on $D$ and$I\zeta\subset\{y\in \mathrm{I}’|\langle y^{*}, y\rangle+\alpha||y||\leq 0\}$

.

Then

${\rm Min} D=\mathrm{E}\mathrm{x}\mathrm{t}_{K}D\neq\emptyset$

4

Duality of Set-Valued Optimization.

In this section, we consider a -type set-valued minimization problem with an inequality.
and its dual problem
constraint$l$$(\mathrm{S}\mathrm{D})$$(\mathrm{S}\mathrm{P})$$F(x)$$l-\mathrm{I}_{\mathrm{V}}\mathrm{I}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}$$(\mathrm{S}\mathrm{P})$subject to$(\mathrm{S}\mathrm{D})$$G(x)\leq^{l}\theta$$\Phi(T)$$l-\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}_{\mathrm{I}}\mathrm{n}\mathrm{i}_{\mathrm{Z}\mathrm{e}}$subject to$T\in \mathcal{L}_{+}(1’, Z)$

where, $X$ is a nonempty set, $(Y, \leq_{K}),$ $(Z, \leq_{L})$ ordered vector spaces with ordering cones$K,$, respectively, $F$ : $Xarrow 2^{Z},$ $G$ : $Xarrow 2^{1^{r}},$ $\mathcal{L}(Y, Z)=$ { : $Yarrow Z|T$ is iinear},
$T$

$L$

$\mathcal{L}_{+}(1^{r}, Z)=\{T\in \mathcal{L}(]^{\nearrow}, Z)|T(IC)\subset L\},$$\Phi$:$\mathcal{L}(1’, Z)arrow 2^{Z}$

defined by $\Phi(T)=l-{\rm Min}\{F(X)+T(y)|(x, y)\in \mathrm{G}\mathrm{r}(G)\}$ .

Definition 4.1 (Solutions)(i) an -feasible solution of$l$$\mathrm{G}\mathrm{r}(G)=\{(x, y)\in X\cross Y|y\in G(x)\}$$x_{0}$$(\mathrm{S}\mathrm{P})$is said to be

if$G(x)\leq^{l}\theta$

:

and

169

(ii) an -solution of$l$if$(\mathrm{S}\mathrm{P})$$x_{0}$

is -feasible and$l$implies $F(X_{0})\leq^{l}F(x)$ .

$x\in X,$ $G(x)\leq^{l}\theta,$ $F(x)\leq^{l}F(X\mathrm{o})$$T_{0}$

is said to be(i) a feasible solution of$T_{0}\in \mathcal{L}_{+}(Y, Z)$(ii) an -solution of$l$if$(\mathrm{S}\mathrm{D})$and$(\mathrm{S}\mathrm{D})$if$\Phi(T)\neq\emptyset$;

is feasible and there exists $A_{0}\in\Phi(T_{0})$ such that

$T_{0}$$T_{1}\in \mathcal{L}_{+}(Y, Z),$ $A_{1}\in\Phi(T),$ $A_{0}\leq^{l}A_{1}$implies$A_{1}\leq^{l}A_{0}$Proposition 4.1 (Weak Duality)
an -feasible solution ofLet
be an -feasible solution ofsatisfying $F(x_{1})+T_{1}(y_{1})\in\Phi(T_{1})$ . Then,element of$l$$x_{0}$$(\mathrm{S}\mathrm{P}),$$l$$T_{1}$$(\mathrm{S}\mathrm{D})$, and$(x_{1}, y_{1})$an$\mathrm{G}\mathrm{r}(G)$implies $F(x_{1})+T_{1}(y_{1})\leq^{l}F(X\mathrm{o})$$F(X_{0})\leq^{l}F(x_{1})+T_{1}(y_{1})$Definition 4.2 (Lagrangian Function) For $x\in X,$ $y\in Y,$$T\in \mathcal{L}(Y, Z)$,$L(x, y, T)=F(x)+T(y)$ .

In usual,$y$is an element of $G(x)$ .

.Definition 4.3 (Saddle Point. )is said to be an -saddle point of$l$$(x_{0}, y_{0}, T_{0})\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$(i) $L(x,$$y,$$\tau_{0}\mathrm{I}\leq^{l}L(x_{0}, y_{0}, \tau_{0}),$

$L$

if

$(x, y)\in \mathrm{G}\mathrm{r}(G)\Rightarrow L(x0, y_{0}, \tau_{0})\leq^{l}L(x,$ $y,$ $\tau_{0)}$(ii) $L(x_{0}, y_{0}, T_{0})\leq^{l}L(X0, y0, T),$ $T\in \mathcal{L}_{+}(Y, Z)\Rightarrow L(x_{0,y_{0},)}T\leq^{l}L(X_{0,y_{0},T_{0})}$
Theorem 4.1 Assume thatcondition: for each$x\in \mathrm{D}\mathrm{o}\mathrm{m}(F)$$\bullet$$\langle y^{*}, y\rangle>0$for each

(ii)
(iii)

$y0\leq\theta$

and$T_{0}(y_{0})=\theta$$T_{0}$$(\mathrm{s}\mathrm{P})$is an -solution of$(\mathrm{S}\mathrm{D})$$l$

Theorem 4.2$y_{0}\leq\theta$

and

$l$

, then we have.

$(x_{0}, y_{0}, T_{0})\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$$T_{0}(y_{0})=\theta$$L$;(i) $L(x, y, \tau_{0})\leq^{l}L(x_{0}, y_{0}, T_{0}),$(ii)

is an -saddle point of;is an -solution of$l$$x_{0}$;.

$(.c_{0,}.y_{0,o} T)\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$(i)$y^{*}\in IC^{+}$$y\in I\zeta\backslash \{\theta\}$$\bullet\inf_{y\in F(x)}\langle y^{*}, y\rangle>-\infty$Ifis closed, is solid, and $F$ satisfies the following boundedsuch that
there exists$L$$I\zeta$.

is an -saddle point of
$l$

$L$

if and only if$(x, y)\in \mathrm{G}\mathrm{r}(G)\Rightarrow L(x_{0}, y0, \tau 0)\leq^{l}L(x,$ $y,$ $T_{0)}$170

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