数理解析研究所講究録
1071 巻 1998 年 12-16

12On Duality of Set-Valued Optimization*
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, JAPANAbstract. An optimization problem which has set-valued objectives and inequality
constraints and its dual problems are defined and discussed.
Keywords. Set-valued analysis, vector optimization, set optimization.

1

Introduction and Preliminaries

Set-valued optimization is usually interpreted as vector optimization with set-valued
objectives, which called set-valued set optimization, and it has been investigated for about
twenty years. Against this type of set-valued optimization, set-valued set optimization has
been introduced and investigated in $[9, 10]$ , recently. These two are different thought theirsettings are same because criteria of solutions are different. Each optimization has various
applications for many fields of mathematics, economics, and so on.
In this paper, we discuss duality theory of set-valued set optimization. Now we mention
our setting. Let $X$ be a nonempty set,a topological vector space, a normed space,$K,$
pointed solid convex cones of $Y,$ , respectively, and $F:Xarrow 2^{Z},$ $G:Xarrow 2^{Y}$ with
$Y$

$L$

$Z$

$Z$

.

$\mathrm{D}\mathrm{o}\mathrm{m}(F)=\mathrm{D}\mathrm{o}\mathrm{m}(G)=X$

Our primal problem$(\mathrm{S}\mathrm{P})$$(\mathrm{S}\mathrm{P})$Minimizesubject tois the following:$F(x)$$G(x)\cap(-K)\neq\emptyset$

In set-valued vector optimization (see [2, 3, 4, 5, 6, 7, 11, 12, 13]), the aim is to find$x_{0}\in S=\{x\in X|G(x)\cap(-K)\neq\emptyset\}$ and $y_{0}\in F(X_{0})$ satisfying$y\mathrm{o}\in{\rm Min}\cup F(_{X)}x\in S^{\cdot}$*This research is partially supported by Grant-in-Aid for Scientific Research from the Ministry ofEducation, Science and Culture of Japan, No. 09740146

13In this paper, we investigate set-valued set optimization. To define this optimization,
we introduce two set relations as follows. For two nonempty sets and $B$ of $Z,-$$A$$A\leq_{L}^{l}B\Leftrightarrow^{\mathrm{d}\mathrm{e}\mathrm{f}}A+L\supset B$,$A\leq_{L}^{u}B\Leftrightarrow^{\mathrm{d}\mathrm{e}\mathrm{f}}A\subset B-L$.

, we can define two types of set-valued set optimization
Using these notations
and.
.
problems. In this paper, we use only.
$\leq_{L}^{l}$$\leq_{L}^{u}$$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}\underline{<}^{l}L$2Duality$\mathrm{o}\mathrm{f}\backslash$

For primal problem

Set-Valued Set. O.ptimization$(\mathrm{S}\mathrm{P})$, we define our notions of solutions based on set optimization.

Definition 2.1 (Solutions of Primal Problem) An element $x_{0}\in X$ is said to be(i) an -type feasible solution of(ii) an -type minimal solution of$l$$(\mathrm{S}\mathrm{P})$$l$$(\mathrm{S}\mathrm{P})$

if
if

$l$

$G(x)\leq_{L}^{l}\theta$

andimplies $F(X_{0})\leq_{L}^{l}F(x)$ .

$\Phi,$$\Phi(T)=${ $F(x)+T(y)|(x,$$\Phi_{w}(T)=$is -type feasible and there does not exist$l$$x_{0}$.

$\mathcal{L}(Y, Z)=$$\Phi_{w}$

{ :
$T$

$\mathcal{L}(Y, Z)arrow 2^{Z}$$y)\in \mathrm{G}\mathrm{r}(G)${ $F(x)+T(y)|(x,$if$(\mathrm{S}\mathrm{P})$$F(x)\leq_{\mathrm{i}\mathrm{n}\mathrm{t}L}^{l}F(X\mathrm{o})$Next we define dual problems. Let$\{T\in \mathcal{L}(Y, Z)|T(K)\subset L\}$ , and:$l$$x_{0}$$x\in X,$ $G(x)\leq_{L}^{l}\theta,$ $F(x)\leq_{L}^{l}F(x_{0})$(iii) an -type weak minimal solution of$x\in X$ such that;is -type feasible and$G(x)\leq_{L}^{l}\theta$$\mathrm{Y}arrow Z|T$

is a -type minimal solution of$y)\in \mathrm{G}\mathrm{r}(G)$Minimizesubject tofor$T\in \mathcal{L}_{+}(Y, Z)$

$l$

$l$

$(\mathrm{S}\mathrm{D})$$(w\mathrm{S}\mathrm{D})$$(x, y)\in \mathrm{G}\mathrm{r}(G)$Maximizesubject toMaximize
subject to$(\mathrm{S}\mathrm{D})$and$(w\mathrm{S}\mathrm{D})$$\Phi(T)$$T\in \mathcal{L}_{+}(Y, Z)$$\Phi_{w}(T)$$T\in \mathcal{L}_{+}(Y, Z)$$(\mathrm{S}\mathrm{P}_{\tau})$},is a -type weak minimal solution of$F(x)+T(y)$. Now, we set$\mathcal{L}_{+}(Y, Z)=$

defined bywhere$(\mathrm{S}\mathrm{P}_{T})$is linear},as follows:$(\mathrm{S}\mathrm{P}_{\tau})$},

14
Definition 2.2 (Solutions of Dual Problem) An element(i) an -type feasible solution of
$l$

and

$T_{0}\in \mathcal{L}_{+}(Y, Z)$$(\mathrm{S}\mathrm{D})$$\Phi(T)\neq\emptyset$(ii) an -type maximal solution ofthat$l$$T_{1}\in \mathcal{L}_{+}(\mathrm{Y}, Z),$

if;

if

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

$A_{1}\in\Phi(T_{1}),$ $A_{0}\leq_{L1}^{l}A$(iii) an -type weak maximal solution ofsuch that$l$there do not exist

is said to be$T_{0}\in \mathcal{L}(\mathrm{Y}, Z)$$(w\mathrm{S}\mathrm{D})$$T_{1}\in \mathcal{L}_{+}(Y, Z),$

if

imply
$T_{0}$$A_{1}\leq_{L}^{l}A_{0}$is feasible and there exists$A_{1}\in\Phi_{w}(\tau_{1})$such that$A_{0}\in\Phi_{w}(\tau_{0})$$A_{0}\leq_{\mathrm{i}\mathrm{n}\mathrm{t}L}^{l}A_{1}$Proposition 2.1 (Weak Duality)Let
be an -type feasible solution ofan -type feasible solution of$F(x_{1})+T_{1}(y_{1})\in\Phi(T_{1})$an element ofsatisfying. Then,$l$$x_{0}$$(\mathrm{S}\mathrm{P}),$$(x_{1}, y_{1})$$T_{1}$.

$l$$(\mathrm{S}\mathrm{D})$, and$\mathrm{G}\mathrm{r}(G)$$F(X_{0})\leq_{L}^{l}F(x_{1})+T_{1}(y_{1})$imply $F(x_{1})+T_{1}(y_{1})\leq_{L}^{l}F(x_{0})$ .

Now we have one of main theorems of this paper.
Theorem 2.1 (Strong Duality)
Let the following assumptions are satisfied:(H1): $F$ is nonempty compact convex values;(H2): for each$x_{1},$$x_{2}\in X,$$(x, y)\in \mathrm{G}\mathrm{r}(G)$$y_{1}\in G(x_{1}),$$y_{2}\in G(x_{2}),$$\lambda\in(0,1)$, there exists

such that$F(x)\leq_{L}^{l}(1-\lambda)F(x_{1})+\lambda F(x_{2})$$\{$$y\leq_{K}(1-\lambda)y1+\lambda y_{2}$(H3): Slater condition: there is $x’\in X$ such that $G(x’)\cap$ (-int $K$ )Then for each minimal solution ofsuch that the following is satisfied:$x_{0}$(i)$1/\mu$is affine on int(ii) for each$a\in \mathrm{i}\mathrm{n}\mathrm{t}L$$(\mathrm{S}\mathrm{P})$$y_{0}^{*}\in K^{+}\backslash \{\theta\}$

and

$L$

, there does not exist$F(x)+T_{a}(y)\leq_{\mathrm{i}\mathrm{n}\mathrm{t}L}^{l}F(x_{0})$where, there exist$T_{a}(y)=\langle y_{0}^{*}, y\rangle\mu(a)a,$$y\in Y$.

$(x, y)\in \mathrm{G}\mathrm{r}(G)$$\neq\emptyset$

such that$\mu$.

: int$Larrow(\mathrm{O}, \infty)$

15

3Lagrangian Duality of Set-Valued Set Optimization
In this paper, we define Lagrangian set-valued map$L:X\cross Y\cross \mathcal{L}(\mathrm{Y}, Z)arrow 2^{Z}$

as$L(x, y, T)=F(x)+T(y)$for $x\in X,$ $y\in Y,$ $T\in \mathcal{L}(Y, Z)$ , and we define concepts of saddle point the following
definition. Such definitions are different from ordinary definitions in set-valued vector
optimization.
Definition 3.1 (Saddle Point)
A tripleis said to be an -type saddle point offollowing two conditions (i) and (ii) are satisfied:$(x_{0}, y_{0}, T_{0})\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$(i) $L(x, y, T_{0})\leq_{L}^{l}L(x_{0,yT}0,0),$$l$$(X, y)\in \mathrm{G}\mathrm{r}(G)\Rightarrow L(x_{0}, y_{0}, \tau_{0})\leq_{L}^{l}L(_{X},$(ii) $L(x_{0}, y_{0,0}\tau)\leq_{L}^{l}L(x_{0}, y_{0}, \tau),$$L$

if the.

$y,$ $T_{0))}$$T\in \mathcal{L}_{+}(Y, z)\Rightarrow L(x_{0}, y_{0}, T)\leq_{L}^{l}L(x_{0}, y_{0}, T_{0})$.

Definition 3.2 (Weak Saddle Point)
A tripleis said to be an -type weak saddle point of
the following two conditions (i) and (ii) are satisfied:$(x_{0}, y_{0}, T_{0})\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$(i) there does not(ii) there does not$(x, y)\in \mathrm{G}\mathrm{r}(G)$$T\in \mathcal{L}_{+}(Y, Z)$

Note that a triple-type minimal solution oftion 3.2 if and only if$l$such thatsuch that$L(x, y, T_{0})\leq_{\mathrm{i}\mathrm{n}\mathrm{t}L}^{l}L(X_{0,y_{0}}, T_{0})$;$L(x_{0}, y_{0}, T_{0})\leq_{\mathrm{i}\mathrm{n}\mathrm{t}L}^{l}L(x0, y0, T)$.
$(x_{0,y_{0}})$$(\mathrm{S}\mathrm{P}_{T})$$(x_{0,y_{0}})$$L(x_{0}, y_{00}, \tau)\in\Phi_{w}(T_{0})$ifsatisfies (i) of Definition 3.1 if and only if
is an, or equivalently, $L(x_{0,y_{0}}, T_{0})\in\Phi(T_{0})$ , and (i) of Definiis an -type weak minimal solution of, or equivalently,$(x_{0}, y_{0}, \tau_{0})$

$l$

$L$

$l$

$(\mathrm{S}\mathrm{P}_{T})$.

Theorem 3.1 Assume that $K$ is closed, is solid, and $F$ satisfies the following bounded
condition: for each
there exists $y^{*}\in K^{+}$ such that$L$

$x\in \mathrm{D}\mathrm{o}\mathrm{m}(F)$$\bullet$$\langle y^{*}, y\rangle>0$for each$y\in K\backslash \{\theta\}$$\bullet\inf_{y\in F(x})\langle yy*,\rangle>-\infty$If.

$(x_{0}, y_{0}, T_{0})\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$

(i)

(ii)

(iii)

$y\mathrm{o}\leq\theta$

and$T_{0}(y_{0})=\theta$;

is an -type saddle point of , then we have;is an -type minimal solution of$l$$x_{0}$$T_{0}$$L$$l$is an -type maximal solution of$l$$(\mathrm{s}\mathrm{P})$;$(\mathrm{S}\mathrm{D})$.

Corollary 3.1 Let the same assumption of Theorem 3.1 is fulfilled. Then, $(x_{0,y_{0}}, T_{0})\in$
is an -type saddle point of if and only if$\mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$$l$(i) $L(x, y, \tau_{0})\leq_{L}^{l}L(x_{0}, y_{0}, T_{0}),$$L$$(x, y)\in \mathrm{G}\mathrm{r}(G)\Rightarrow L(x_{0,y\mathrm{o}}, T_{0})\leq_{L}^{l}L(x,$$y,$ $\tau_{0)}$16(ii)$y0\leq\theta$

and$T_{0}(y\mathrm{o})=\theta$.

Theorem 3.2 Let the assumptions of Theorem 2.1 is satisfied. Then for each minimal, there existofsolutionand :such that the following
is satisfied:$y_{0}^{*}\in K^{+}\backslash \{\theta\}$$(\mathrm{S}\mathrm{P})$$x_{0}$(i) $1/\mu$ is affine on int(ii) for eachwhere$a\in \mathrm{i}\mathrm{n}\mathrm{t}L,$$\mu$$\mathrm{i}\mathrm{n}\mathrm{t}Larrow(\mathrm{O}, \infty)$$L$

is a weak saddle point of$y\in Y$ .

$(x_{0}, y0, \tau)a$$T_{a}(y)=\langle y_{0}^{*}, y\rangle\mu(a)a,$$L$

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