数理解析研究所講究録
1048 巻 1998 年 86-92

86

On Natural Criteria in Set-Valued
Optimization*
島根大学総合理工学部

大児 (Daishi Kuroiwa)

黒岩

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

1060 Nishikawatsu, Matsue, Shimane 690-8504, JAPAN

Abstract
We introduce some natural criteria of a minimization programming problem
whose objective function is a set-valued map. For such criteria, we define somesemicontinuities and prove certain theorems with respect to existence of solutions of
the problem. Also, we investigate certain duality problem for the set-valued minimization problem.

1.

Natural Criteria of Set-Valued Optimization

First, we define our set-valued minimization problem. Let $X$ be a topological space,a nonempty subset of $X,$ $(Y, \leq_{K})$ an ordered topological vector space with an ordering
convex cone $K$ , and $F$ a map from $X$ to
withfor each $x\in S$ . Our set-valued
minimization problem is the following:$(\mathrm{S}\mathrm{P})$$S$$2^{Y}$$(\mathrm{S}\mathrm{P})$Minimizesubject to$F(x)\neq\emptyset$$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 [9].

Definition 1.1. (l-Inequality&u-Inequalities)
For nonempty subsets $A,$ of ,
$B$

$Y$

$A\leq^{l}B\Leftrightarrow \mathrm{c}1(A+K)\supset \mathrm{c}1(B+K)$;*This research is partially supported by Grant-in-Aid for Scientific Research from the Ministry of

Education, Science and Culture of Japan, No. 09740146

87$A\leq^{u}B\Leftrightarrow \mathrm{c}1(A-K)\subset \mathrm{c}1(B-K)$

In these cases,$A$$A\leq^{l}B(\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}}.

.

is said to be smaller than $B$ with -inequality(resp.
. Also, a subset
of is said to be a 1-closed() is closed set of .
$l$

$Y$

$A$

A\leq^{u}B)$$A+K$ (resp. $\mathrm{A}-K$-inequality) if. -closed) if

$u$

$u$

$\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}}$$Y$

Note that $\mathrm{c}1(A+K)\supset \mathrm{c}1(B+K)$ is equivalent to $\mathrm{c}1(A+K)\supset B$ and also$\mathrm{c}1(B-K)$ is equivalent to $A\subset \mathrm{c}1(B-K)$ . Whenand are -closed set, $A\leq^{l}B$ if andimplies that ${\rm Min} A={\rm Min} B$ . If andonly if $A+K\supset B$ , and $A\leq^{l}B$ andimplies thatare -closed set, $A\leq^{u}B$ if and only if $A\subset B-K$ , and $A\leq^{u}B$ and${\rm Max} A={\rm Max} B$ .
By using the set relations above, we introduce two types criteria of minimal solutions.
is -closed map,
In this paper, when we consider -minimal solution, we assume thatthat is $F(x)$ is -closed for each $x\in X$ for simple consideration. Also we assume similarassumption when we consider -minimal solution.
$\mathrm{c}1(A-K)\subset$$A$$B\leq^{l}$

$B$

$l$

$A$

$A$

$A$

$B\leq^{u}$

$u$

$F$

$l$

$B$

$l$

$l$

$u$

Definition 1.2. (Minimal Solutions)ifis said to be -minimal solution of$F(x)\leq^{l}F(x_{0})$ and $x\in S$ implies $F(x_{0})\leq^{l}F(x)$ ;$\bullet$$x_{0}\in S$$\bullet$$x_{0}\in S$$l$$(\mathrm{S}\mathrm{P})$ifis said to be -minimal solution of$F(x)\leq^{u}F(X\mathrm{o})$ and $x\in S$ implies $F(x\mathrm{o})\leq^{u}F(x)$ .
$(\mathrm{S}\mathrm{P})$$u$

These concepts above are natural definitions for our set-valued optimization
criteria is based on comparisons between the objective set-values of $F$ .

$(\mathrm{S}\mathrm{P})$since theExample. (Vector-Valued Game)is
We consider a vector-valued two-person game; and are nonempty sets,
an ordered vector space, and is a map from $A\cross B$ to . Assume that player 1 chooses
first and player 2 chooses next. Player 1 chooses and player 2 chooses $f(a, b)$ is theloss for player 1. When player 2 is cooperative toward player 1, player 1 may choose a: .
-minimal solution of the following set-valued optimization problem$A$$(Y, \leq_{K})$

$B$

$Y$

$f$

$b,$

$a$

$(\mathrm{V}\mathrm{G})$$l$$(\mathrm{V}\mathrm{G})$Minimizesubject to$f(a, B)$$a\in A$.

When player 2 is non-cooperative, to be exact, player 2 wills player l’s loss, then player 1.
should choose a -minimal solution of$u$2.

$(\mathrm{V}\mathrm{G})$Natural Semicontinuity of Set-Valued Mapsfor our solutions, remember classical results
To consider existence of solutions ofwith respect to existence of solution of some minimization problems:$(\mathrm{S}\mathrm{P})$88(i) Letbe a topological space, $D$ a compact set in $Z$ , and a lower semicontinuous
real-valued function on $D$ . Then, attains its minimum on $D$ .
(ii) Let $Z$ be a complete metric space,:a lower semicontinuous andproper function which is bounded from below. Then there exists $z_{0}\in Z$ such that$f(z)\geq f(z_{0})-\epsilon d(z, z_{0})$ for all $z\in Z$ . (Ekeland’s variational theorem, [1])(iii) Let $Z$ be a Banach space, $C$ a closed convex cone in $Z,$
, which is the topological dual space of $Z,$
for some, and $D$ a nonempty
closed subset of $Z$ such thatis bounded from below on $D$ . Then,.
(Phelps’ extreme theorem, [1])
$Z$

$f$

$f$

$Zarrow \mathrm{R}\cup\{\infty\}$$f$$C\subset\{z\in Z|\langle z, z^{*}\rangle+\epsilon||z||\geq 0\}$$z^{*}\in Z^{*}$$\epsilon>0$$z^{*}$${\rm Min} D\neq\emptyset$

We can find that some of theorems are cohcerned with concept of the lower semicontinuityof real-valued functions. For set-valued maps, we know the usual lower semicontinuity; aset-valued map from $X$ to is said to be lower semicontinuous at
if for any $y\in F(x_{0})$and for any net, there exists a net of elements
withconverging
to . However, this notion is a generalization of the continuity of real-valued functions,then it is not a generalization of the lower semicontinuity and not suitable for our purpose
to use this definition. Therefore, in this section, we define some lower semicontinuities
of set-valued maps which are generalizations of the lower semicontinuities of real-valued
functions and based on our natural criteria. Remember the lower semicontinuities of realvalued functions; a real-valued function on a topological space $X$ is said to be lower
semicontinuous on a subset of $X$ if one of the following is satisfied:
$F$

$Y$

$\{x_{\lambda}\}$$x_{0}$$y_{\lambda}\in F(x_{\lambda})$$x_{\lambda}arrow x_{0}$

$y$

$f$

$S$

(A) for each $x_{0}\in S$ and for any, there exists a neighborhood$X$insuch that $x\in x_{0}+U$ implies that $f(x_{0})-\epsilon<f(x)$ ;$\epsilon>0$(B) for each $x_{0}\in S$ , if a net(C) for$\alpha\in \mathrm{R},$$\{x_{\lambda}\}$satisfies$\mathcal{L}(\alpha)=\{x\in S|f(x)\leq\alpha\}$$x_{\lambda}arrow x_{0}$

then

$U$

of the null vector$f(X_{0})\leq\varliminf_{\lambda}f(X_{\lambda})$;is closed.

We introduce our lower semicontinuities as generalizations the above. To this end, we
define the upper limit and the lower limit of, see [2].
$\{A_{\lambda}\}$Definition 2.1. (LimFor$\{A_{\lambda}\}\subset 2^{Y},$$\mathrm{L}\mathrm{i}\mathrm{m}_{\lambda}$

inf$\inf_{\lambda}A_{\lambda}$$(\Lambda, <)$$A_{\lambda}=\mathrm{t}\mathrm{h}\mathrm{e}$$\mathrm{L}\mathrm{i}\mathrm{m}_{\lambda}\sup A_{\lambda}=\mathrm{t}\mathrm{h}\mathrm{e}$&Lim$\sup_{\lambda}A_{\lambda}$): a directed set,

set of limit points of$\{a_{\lambda}\},$

set of cluster points of$a_{\lambda}\in A_{\lambda;}$$\{a_{\lambda}\},$$a_{\lambda}\in A_{\lambda}$.

In general, LimLim
and if equality holds, it is said to beconverges to the set. From the above notation, condition
is presented by$\{f(x_{0})\}\leq^{u}$
Lim
or
Lim
. From this, todefine several kinds of lower semicontinuity, we use notion Lim sup.
$\inf_{\lambda}A_{\lambda}\subset$$\sup_{\lambda}A_{\lambda}$$\{A_{\lambda}\}$$f(X_{0})\leq\varliminf_{\lambda}f(X\lambda)$$\{f(x_{0})\}\leq^{l}$$\sup_{\lambda}(f(x_{\lambda})+\mathrm{R}_{+})$$\sup_{\lambda}(f(x\lambda)-\mathrm{R}_{+})$

89
Definition 2.2. ( -type Lower Semicontinuity)
A set-valued map $F$ is said to be$l$-type (A) lower semicontinuous at $x_{0}\in S$ if
for any net
withand for any open set
impliessuch that;$l$$\bullet$$\{x_{\lambda}\}$$x_{\lambda}arrow x_{0}$$\hat{\lambda}<\lambda$$\hat{\lambda}$

$U$

with

$U\leq^{l}F(X\mathrm{o})$, there exists$U\leq^{l}F(X_{\lambda})$-type (B) lower semicontinuous at $x_{0}\in S$ if
for any net
with
Lim
$l$

$\bullet$$\{x_{\lambda}\}$$x_{\lambda}arrow x_{0},$$F(x_{0})\leq^{l}$$\sup_{\lambda}(F(X_{\lambda})+K)$;-type (C) lower semicontinuous on if
for any -closed subset of $Y,$ $\mathcal{L}^{l}(A)=\{x\in S|F(x)\leq^{l}A\}$ is closed.
$S$

$l$

$\bullet$

$A$

$l$

A set-valued map-type (A) (resp.
$l$

$F$

$(\mathrm{B})$) lower semicontinuous onis said to be -type (A) (resp.
) lower semicontinuous at each point of .
$l$$(\mathrm{B})$$S$

if it is$S$

These concepts are generalizations of lower semicontinuity of real-valued functions, however, the following concept is more weaker than the lower semicontinuity.
Definition 2.3. ( -type Demi-Lower Semicontinuity)
A set-valued map $F$ is said to be -type demi-lower semicontinuous at $x_{0}\in S$ if for each net
implies
with
and
Lim
.
A set-valued map $F$ is said to be -type demi-lower semicontinuous on
if it is l-typedemi-lower semicontinuous at each point of .
$l$

$l$

$\{x_{\lambda}\}$$\lambda<\lambda’$$x_{\lambda}arrow x_{0}$$F(x_{\lambda^{;}})\leq^{l}F(x_{\lambda}),$$F(x_{0})\leq^{l}$$\sup_{\lambda}(F(X_{\lambda})+K)$
$S$

$l$

$-$

$S$

Now we can see some characterization with respect to these lower semicontinuities.
Proposition 2.1. We have the following:(i) -type (A) l.s.c. on(ii) -type (B) l.s.c. on(iii) -type (C) l.s.c. on$l$$S\Rightarrow l$$l$$S\Rightarrow l$$l$$S\Rightarrow l$-type (B) l.s.c. on ;-type (C) l.s.c. on ;-type demi-l.s. . on .
$S$

$S$

$\mathrm{c}$$S$

Also, if is finite dimensional and $F$ is locally bounded then, -type (A), (B), and (C)lower semicontinuities are equivalent.
$Y$

$l$

Now, we investigate -type lower semicontinuity of set-valued maps.
$u$Definition 2.4. ( -type Lower Semicontinuity) A set-valued map $F$ is said to be$u$$\bullet$-type (A) lower semicontinuous atif
withand for any open set withfor any net, there exists
such that $(F(x_{\lambda})-K)\cap U\neq\emptyset$ ;$u$$x_{0}$$\{x_{\lambda}\}$$\lambda$$\bullet$$U$$x_{\lambda}arrow x_{0}$$F(x_{0})\cap U\neq\emptyset$$\lambda’>\lambda$-type (B) lower semicontinuous atwithfor any net$u$$\{x_{\lambda}\}$$x_{\lambda}arrow x_{0},$$x_{0}$if$F(X_{0})\leq^{u}$

Lim $\sup_{\lambda}(F(X\lambda)-K)$ ;, for any

90
-type (C) lower semicontinuous on if
for any subset of $Y,$ $\mathcal{L}^{u}(A)=\{x|F(x)\leq^{u}A\}$ is closed.
$S$

$u$

$\bullet$$A$

A set-valued map$u$-type (A) (resp.

is said to be -type (A) (resp.
) lower semicontinuous on) lower semicontinuous at each point of .

$F$

$u$

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

if it is$S$$(\mathrm{B})$Definition 2.5. ( -type Demi-Lower Semicontinuity) A set-valued map is said tobe -type demi-lower semicontinuous at
if for any net
withandimpliesLim. A set-valued map
is said
to be -type demi-lower semicontinuous on if it is -type demi-lower semicontinuous ateach point of .
$F$

$u$

$u$

$\{x_{\lambda}\}$$x_{0}$$F(x_{\lambda};)\leq^{u}F(x_{\lambda}),$$F(x_{0})\leq^{u}$$\sup_{\lambda}(F(X_{\lambda})-K)$
$S$

$u$

$\lambda<\lambda’$$x_{\lambda}arrow x_{0}$

$F$

$u$

$S$

Proposition 2.2. ( -type Lower$u$(i) -type (B) l.s.c. on(ii) -type (C) l.s.c. on$\mathrm{S}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}$)$\mathrm{W}\mathrm{e}$have the following:-type (C) l.s.c. on ;-type demi-l.s. . on .

$u$

$S\Rightarrow u$$u$$S\Rightarrow u$

$S$

$S$

$\mathrm{c}$Also, if is finite dimensional and $F$ is locally bounded then, -type (A), (B), and (C)lower semicontinuities are equivalent.
$Y$

$u$

Existence Theorems for Two Types Semicontinuities of Set-Valued Maps

3.

Theorem 3.1. (Existence of -type Solutions 1)
Let $X$ be a topological space and an ordered topological vector space. If is a nonempty
compact subset of $X$ and $F$ : $Sarrow 2^{Y}$ is a -type demi-lower semicontinuous. set-valued
map, then there exists a -minimal solution of
.
$l$

$Y$

$S$

$l$

$l$

$(\mathrm{S}\mathrm{P})$In the rest of the paper, let
be the topological dual space of $Y,$, andthe null vector of .
$Y^{*}$$0,$ $\forall k\in K\}$

$K^{+}=\{y^{*}\in Y^{*}|\langle y^{*}, k\rangle\geq$$Y^{*}$$\theta^{*}$

Theorem 3.2. (Existence of -type Solutions 2)
Let (X, ) be a complete metric space,
an ordered locally convex space with the cone
$K$ . Also, $F$ be a map from $X$ tosatisfying the following conditions:
$l$

$Y$

$d$

$2^{Y}$$\bullet$there exists
inf$y^{*}\in K^{+}\backslash \{\theta^{*}\}$$\langle y^{*}, F(\cdot)\rangle$:

such that$Sarrow \mathrm{R}$$F(x_{1})\leq^{l}F(X_{2}),$ $X1,$ $x_{2} \in S\Rightarrow\inf\langle y^{*}, F(x_{2})\rangle$$\bullet$$F:Sarrow 2^{Y}$

is -type (C) lower semicontinuous.
$l$Then, there exists a -minimal solution of$l$$(\mathrm{S}\mathrm{P})$

.

–inf

$\langle y^{*}, F(x_{1})\rangle\geq d(x_{21}, x)$

91
Theorem 3.3. (Existence of -type Solutions 1)
Let $X$ be a topological space and an ordered topological vector space. If is a nonempty
compact subset of $X$ and $F$ : $Sarrow 2^{Y}$ is a -type demi-lower semicontinuous. set-valued.
map, then there exists a -minimal solution of
$u$

$S$

$Y$

$u$

$(\mathrm{S}\mathrm{P})$$u$Moreover, we can show the following theorem in similar way of Theorem 3.2..

Theorem 3.4. (Existence of -type Solutions 2)
an ordered locally convex space with the cone
Let (X, ) be a complete metric space,$K$ . Also,satisfying the following conditions:be a map from $X$ to
$u$

$Y$

$d$

$2^{Y}$$F$$\bullet$there exists$y^{*}\in K^{+}\backslash \{\theta^{*}\}$.

$\sup\langle y^{*}, F(\cdot)\rangle\wedge$.
$\bullet$such that$Sarrow \mathrm{R}$$F(x_{1})\leq^{u}F(x_{2}),$ $X_{1},$ $x_{2} \in S\Rightarrow\sup\langle y^{*}, F(x_{2})\rangle-\sup\langle y^{*}, F(x_{1})\rangle\geq d(x_{2,1}x)$$F:Sarrow 2^{Y}$

is -type (C) lower semicontinuous.
$u$Then, there exists a -minimal solution of$u$$(\mathrm{S}\mathrm{P})$.

Duality Problem for Set-Valued Optimization

4.

In this section, we introduce a duality problem for our -type set-valued minimization
with $S=\{x\in X|G(x)\leq\theta\}$ , and we show some properties between these
problem:and its dual problem
problems. First, we redefine our set-valued problem$l$$(\mathrm{S}\mathrm{P})$$(\mathrm{D}\mathrm{P})$$(\mathrm{S}\mathrm{P})$-Minimizesubject to$l$$(\mathrm{S}\mathrm{P})$-Maximizesubject to$F(x)$$G(x)\leq\theta$$\Phi(T)$$l$$(\mathrm{S}\mathrm{D})$$T\in \mathcal{L}^{+}(Y, Z)$

where$\bullet$$\bullet$$X$$(Y, \leq_{K}),$$\bullet F$$\bullet$: a nonempty set,$(Z, \leq_{L})$: $Xarrow 2^{Z},$$\mathcal{L}(Y, Z)\equiv$$G$: ordered vector spaces with an ordering cones $K,$

: $Xarrow 2^{Y}$ ;{ $T:Yarrow Z|T$ is linear},$\mathcal{L}^{+}(Y, Z)\equiv\{T\in \mathcal{L}(Y, z)|T(K)\subset L\}$$\bullet$$\Phi$:$\mathcal{L}(Y, Z)arrow 2^{Z}$;

defined by.

$\Phi(T)\equiv l-{\rm Min}\{F(X)+T(y)|(x, y)\in \mathrm{G}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}(c)\}$$L$, respectively;

92
Proposition 4.1. (Weak Duality) Let
be a feasible solution ofa feasible, andsolution ofan element of Graph $(c)$ satisfying $F(x_{1})+T(y_{1})\in\Phi(T)$ .
Then,
$x$

$(\mathrm{S}\mathrm{P}),$$T$$(x_{1}, y_{1})$$(\mathrm{S}\mathrm{D})$$F(x)\leq^{l}F(x_{1})+T(y_{1})$implies$F(x_{1})\leq^{l}F(x)$$\{$$T(y_{1})=\theta$

Corollary 4.1. Let be a feasible solution ofThen $F(x)=F(x)+T(\theta)\in\Phi(T)$ .
$x$.

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

and

$T$

a feasible solution of$(\mathrm{S}\mathrm{D})$.

References
1. H. Attouch and H. Riahi, Stability Results for Ekeland’s -Variational Principle and Cone
Extremal Solutions, Math. Oper. Res. 18 (1993), 173-201.
$\epsilon$

2. J. P. Aubin and H. Frankowska, “Set-Valued Analysis,” Birkh\"auser, Boston, 1990.
3. J. M. Borwein, Proper efficient points for maximizations with respect to cones, SIAM J.
Control Optim. 15 (1977), 57-63.
4. H. W. Corley, Existence and Lagrangian Duality for Maximizations of Set-Valued Functions,
J. Optim. Theo. Appl. 54 (1987), 489-501.

5. H. W. Corley, Optimality Conditions for Maximizations of Set-Valued Functions, J. Optim.
Theo. Appl. 58 (1988), 1-10.

6. H. Kawasaki, Conjugate Relations and Weak Subdifferentials of Relations, Math. Oper. Res.
6 (1981), 593-607.
7. H. Kawasaki, A Duality Theorem in Multiobjective Nonlinear Programming, Math. Oper.
Res. 7 (1982) 95-110.

8. D. Kuroiwa, Convexity for Set-valued Maps, Appl. Math. Letters 9 (1996), 97-101.
9. D. Kuroiwa, T. Tanaka, and T. X. Duc Ha, On Cone Convexity of Set-Valued Maps, Nonlinear Analysis, Theory, Methods & Applications 30 (1997), 1487-1496.
10. T. Tanino and Y. Sawaragi, Conjugate Maps and Duality in Multiobjective Optimization,
J. Optim. Theo. Appl. 31 (1980), 473-499.
11. P. L. Yu, Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision
Problems with Multiobjectives, J. Optim. Theo. Appl. 14 (1974), 319-377.