111

数理解析研究所講究録
第2065巻 2018年 111-119

GENERALIZED ALTERNATIVE THEOREMSBASED ON SET‐RELATIONSYUTO OGATA, YUTAKA SAITO, TAMAKI TANAKA (NIIGATA UNIVERSITY)
GUE MYUNG LEE, JAE HYOUNG LEE (PUKYONG NATIONAL UNIVERSITY)
ABSTRACT. This paper contains generalized alternative theorems via a set scalarization
method based on set‐relations. We introduce semidefinite system and its dual case as asupplementary result.

1. INTRODUCTION

Gordan’s theorem was appeared in 1873 [2]. This theorem focuses on the geometry offinitely many vectors and the origin. This paper is originally motivated by a generalized

Gordan’s theorem for vector‐valued functions [4] in 1986. Vector‐valued functions are able
to be replaced with matrices under a convex assumption. This was interestingly extended

to the case of set‐valued maps [7] in 1999 and [9] in 2000. They found that set‐valued
maps can be used in the form of alternative theorem with a similar convexity.
Their theorems based on some separation theorems require linear scalarizaions such as

bilinear forms. In 2005, Nishizawa, Onodsuka, and Tanaka gave generalization forms [8]by using subhnear scalarizations for vectors inspired by ones introduced in [3]. Sublinear
separation enables to eliminate assumptions of theorems related to hnearity and convexity.

Kuwano, Tanaka, and Yamada established sublinear scalarizations for sets [6] in 2009by use of set‐relations [5]. These functions are composed of six infimum types and six
supremum types. The number of the functions for set is much larger than that for vectors
which consist of just four types.

This comes from the complexity of ways where we

consider a set preceding another.

In this paper, we show 12 alternative theorems by using the scalarizing functions for
sets, considering the set‐relations between an image and a reference set. If the reference
set is the origin, our theorems will be the result of the previous research. This reference
set gives flexibility and let problems free themselves from the origin.
2010 Mathematics Subject Classification. Primary: 90\mathrm{C}29 ; Secondary: 90\mathrm{C}46, 49\mathrm{J}53, 54\mathrm{C}60.
Key words and phrases.

Alternative theorem, sublinear scalarization, set‐relations, set‐valued map.

This work was supported by JSPS KAKENHI Grant Number 26400194.

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2. PRELIMINARIES

Unless otherwise specified, we let X be a nonempty set, Y a real ordered topological

vector space with \leq c induced by a convex sohd (i.e., there exists nonempty interior) cone
C as follows: x\leq cy if y-x\in C for x, y\in Y , and F a set‐valued map from X to 2^{Y}.

Definition 2.1 (Kuroiwa, Tanaka, Ha (1997), [5]). Let A, B\in 2^{Y}\backslash \{\emptyset\}.
(i) A\displaystyle \leq_{c}^{(1)}B\Leftrightar ow^{\mathrm{d}\mathrm{e}\mathrm{f} A\subset\bigcap_{b\in B}(b-C)\Leftrightar ow B\subset\bigcap_{a\in A}(a+C) ;(ii) A\displaystyle \leq_{c}^{(2)}B\Leftrightar ow^{\mathrm{d}\mathrm{e}\mathrm{f} A\cap\bigcap_{b\in B}(b-C)\neq\emptyset ;(iii) A\leq_{C}^{(3)}B\Leftrightar ow^{\mathrm{d}\mathrm{e}\mathrm{f} B\subset(A+C) ;(iv) A\displaystyle \leq_{c}^{(4)}B\Leftrightar ow^{\mathrm{d}\mathrm{e}\mathrm{f} \bigcap_{a\in A}(a+C)\cap B\neq\emptyset ;(v) A\leq_{c}^{(5)}B\Leftrightar ow^{\mathrm{d}\mathrm{e}\mathrm{f} A\subset(B-C) ;(vi) A\leq_{c}^{(6)}B\Leftrightarrow^{\mathrm{d}\mathrm{e}\mathrm{f} A\cap(B-C)\neq\emptyset\Leftrightarrow(A+C)\cap B\neq\emptyset.
Definition 2.2 (Kuwano, Tanaka, Yamada, (2009), [6]). Let A, B\in 2^{Y}\backslash \{\emptyset\} and k\in intC.
For each j=1 ,6, scalarizing functions I_{k,B}^{(j)} and S_{k,B}^{(j)} from 2^{Y}\backslash \{\emptyset\} to \mathbb{R}\cup\{\pm\infty\} are de‐fined by I_{k,B}^{(j)}(A) :=\displaystyle \inf\{t\in \mathbb{R}|A\leq^{(}c^{j)}(tk+B

S_{k,B}^{(j)}(A) :=\displaystyle \sup\{t\in \mathbb{R}| (tk+B)\leq_{c}^{(j)}A\}.

3. GENERALIZED ALTERNATIVE THEOREMS

3.1. Infimum Types.

Theorem 3.1 (Infimum type (1)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in \mathrm{Y}, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty

subset in Y. If F is compact‐valued on X and V is compact, then exactly one of thefollowing two systems is consistent;(i) there exists x\in X such that F(x)\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(1)}V ;(ii) there exists k\in intC such that (I_{k,V}^{(1)}\circ F)(x)\geq 0 for all x\in X.
Proof. [(\mathrm{i})\Rightar ow \mathrm{n}\mathrm{o}\mathrm{t} (\mathrm{i}\mathrm{i})] Fix k \in intC. Assume that (i) is consistent. Then, there exists
x\in X such that y\displaystyle \in\bigcap_{v\in V}(v- intC) for all y\in F(x) .

Since \displaystyle \bigcap_{v\in V}(v- intC) is open, there

exists $\lambda$_{y}>0 such that y+$\lambda$_{y}k\displaystyle \in\bigcap_{v\in V}(v- intC) . Therefore, y\in S_{y} :=\displaystyle \bigcap_{v\in-$\lambda$_{y}k+V}(v-intC). This implies \{S_{y}\}_{y\in F(x)} is a cover of F(x) . Since F(x) is compact, there exists\{y_{1}, . . . , y_{n}\}\subset F(x) such that F(x)\displaystyle \subset\bigcup_{i\in I}S_{y_{l} where I :=\{1, . . . , n\}.
Taking \overline{\mathrm{A} :=\displaystyle \min\{$\lambda$_{y_{1} , . . . , $\lambda$_{y_{n} \}>0 , then F(x)\displaystyle \subset\bigcup_{i\in I}\bigcap_{v\in-$\lambda$_{y_{l} k+V}(v- intC) =\displaystyle \bigcap_{v\in-\overline{ $\lambda$}k+V}(v- intC) \displaystyle \subset\bigcap_{v\in-\overline{ $\lambda$}k+V}(v-C) . Thus, \displaystyle \inf\{t\in \mathbb{R}|F(x)\subset\bigcap_{v\in tk+V}(v-C)\}\leq-\overline{ $\lambda$}<0.

[Not (\mathrm{i})\Rightar ow(\mathrm{i}\mathrm{i}) ] Assume that (i) is not consistent. Then, for all x \inX,there exists\overline{y}\in F(x) such that \displaystyle \overline{y}\not\in\bigcap_{v\in V}(v- intC) . Since \displaystyle \bigcap_{v\in $\lambda$ k+V}(v-C)\subset\bigcap_{v\in V}(v- intC) for all113

k\in intC and $\lambda$<0 , then \displaystyle \overline{y}\not\in\bigcap_{v\in $\lambda$ k+V}(v-C) . Thus, \displaystyle \inf\{t\in \mathbb{R}|F(x)\subset\bigcap_{v\in tk+V}(v-C)\}

is non‐negative

口Remark 3.1. We let S := \displaystyle \bigcap_{v\in V}(v-C) . Then S+(-C) \subset S so that for all x \in X,

y\in F(x) , and k\in intC, there exists t_{y}>0 such that y\displaystyle \in\bigcap_{v\in t_{y}k+V}(v-C) . Thus, there
exists \overline{t}\geq 0 such that

(I_{k,V}^{(1)}\displaystyle \circ F)(x)=\inf\{t\in \mathbb{R}|F(x)\subset\bigcap_{v\in tk+V}(v-C)\}\leq\overline{t}<+\infty.

Theorem 3.2 (Infimum type (2)). Let X be a nonempty set, Y a topological vector
space, C a convex sohd cone in \mathrm{Y}, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty

subset in Y . If V is compact, then exactly one of the following two systems is consistent;(i) there exists x\in X such that F(x)\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(2)}V ;(ii) there exists k\in intC such that (I_{k,V}^{(2)}\circ F)(x)\geq 0 for all x\in X.
Proof. [(\mathrm{i})\Rightar ow \mathrm{n}\mathrm{o}\mathrm{t} (\mathrm{i}\mathrm{i})] Fix k\in intC. Assume that (i) is consistent. Then, there exist x\in X
and \overline{y}\in F(x) such that \displaystyle \overline{y}\in\bigcap_{v\in V}(v- intC) . Since \displaystyle \bigcap_{v\in V}(v- intC) is open, there exists\overline{ $\lambda$}>0 such that \displaystyle \overline{y}+\overline{ $\lambda$}k\in\bigcap_{v\in V}(v- intC) \displaystyle \subset\bigcap_{v\in V}(v-C) . Therefore, \displaystyle \overline{y}\in\bigcap_{v\in-\overline{ $\lambda$}k+V}(v-C) .
Then, F(x)\displaystyle \cap\bigcap_{v\in-\overline{ $\lambda$}k+V}(v-C) \neq \emptyset . Thus, \displaystyle \inf\{t\in \mathbb{R}|F(x)\cap\bigcap_{v\in tk+V}(v-C)\neq\emptyset\} \leq-\overline{ $\lambda$}<0.

[Not (\mathrm{i})\Rightar ow(\mathrm{i}\mathrm{i}) ] Assume that (i) is not consistent. Then, y \not\in \displaystyle \bigcap_{v\in V}(v- intC) for all
x\in X and y\in F(x) .

That means, y\displaystyle \not\in\bigcap_{v\in $\lambda$ k+V}(v-C) for all k\in intC and $\lambda$<0 . Thus,\displaystyle \inf\{t\in \mathbb{R}|F(x)\cap\bigcap_{v\in tk+V}(v-C)\neq\emptyset\} is non‐negative.

\squareRemark 3.2. We let S := \displaystyle \bigcap_{v\in V}(v-C) like Remark 3.1. For all x \in X, y \in F(x) ,and k \in intC, there exists \overline{t}> 0 such that y \in \displaystyle \bigcap_{v\in\overline{t}k+V}(v-C) . Thus, (I_{k,V}^{(2)}\circ F)(x)=\displaystyle \inf\{t\in \mathbb{R}|F(x)\cap\bigcap_{v\in tk+V}(v-C)\neq\emptyset\}\leq\overline{t}<+\infty.
Theorem 3.3 (Infimum type (3)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty
subset in Y . If V is compact, then exactly one of the following two systems is consistent;(i) there exists x\in X such that F(x)\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(3)}V ;(ii) there exists k\in intC such that (I_{k,V}^{(3)}\circ F)(x)\geq 0 for all x\in X.
Proof. [(\mathrm{i})\Rightar ow \mathrm{n}\mathrm{o}\mathrm{t} (\mathrm{i}\mathrm{i})] Fix k \in intC. Assume that (i) is consistent. Then, there exists
x\in X such that v\in F(x)+ intC for all v\in V .

Since F(x)+ intC is open, there exists$\lambda$_{v}>0 such that v-$\lambda$_{v}k\in F(x)+ intC. We let S_{v} :=$\lambda$_{v}k+F(x)+ intC then, \{S_{v}\}_{v\in V}

is a cover of V . Since V is compact, there exists \{v_{1}, . . . , v_{n}\}\subset V such that V\displaystyle \subset\bigcup_{i\in I}S_{v_{l}where I :=\{1, . . . , n\}.

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Taking \overline{$\lambda$} :=\displaystyle \min\{$\lambda$_{v_{1} , . . . , $\lambda$_{v_{n} \}>0 , then V\displaystyle \subset\bigcup_{i\in I}($\lambda$_{v_{l}}k+F(x)+ intC) =\overline{ $\lambda$}k+F(x)+intC \subset\overline{ $\lambda$}k+F(x)+C . Thus, \displaystyle \inf\{t\in \mathbb{R}|(tk+V)\subset(F(x)+C)\}\leq-\overline{ $\lambda$}<0.

[Not (\mathrm{i})\Rightar ow(\mathrm{i}\mathrm{i}) ] Assume that (i) is not consistent. Then, for all x\in X , there exists \overline{v}\in Vsuch that \overline{v}\not\in F(x)+ intC. Since - $\lambda$ k+F(x)+C\subset F(x)+ intC for all k\in intC and $\lambda$<0,

then \overline{v}\not\in- $\lambda$ k+F(x)+C . Thus, \mathrm{i}\mathrm{n}\mathrm{t}\{t\in \mathbb{R}| (tk+V) \subset (F(x)+C)\} is non‐negative.
Remark 3.3. We let S
k \in

:=

F(x)+C .

Then S+C

intC, there exists \overline{t} > 0 such that \overline{t}k+v \in\subsetS so that for all v \in

口

V and

F(x)+C . Thus, (I_{k,V}^{(3)}\circ F)(x)=\displaystyle \inf\{t\in \mathbb{R}|(tk+V)\subset(F(x)+C)\}\leq\overline{t}<+\infty.
Theorem 3.4 (Infimum type (4)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty
subset in Y . If F is compact‐valued on X , then exactly one of the following two systems
is consistent;(i) there exists x\in X such that F(x)\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(4)}V ;(ii) there exists k\in intC such that (I_{k,V}^{(4)}\circ F)(x)\geq 0 for all x\in X.

Proof. [(\mathrm{i})\Rightar ow \mathrm{n}\mathrm{o}\mathrm{t} (\mathrm{i}\mathrm{i})] Fix k\in intC. Assume that (i) is consistent. Then, there exist x\in X
and \overline{v}\in V such that \displaystyle \overline{v}\in\bigcap_{y\in F(x)}(y+ intC) . Since \displaystyle \bigcap_{y\in F(x)}(y+ intC) is open, there exists\overline{ $\lambda$}>0 such that -\displaystyle \overline{ $\lambda$}k+\overline{v}\in\bigcap_{y\in F(x)}(y+ intC) \displaystyle \subset\bigcap_{y\in F(x)}(y+C) .
Thus, \displaystyle \inf\{t\in \mathbb{R}|(tk+V)\cap\bigcap_{y\in F(x)}(y+C)\neq\emptyset\}\leq-\overline{ $\lambda$}<0.

[Not (\mathrm{i})\Rightar ow(\mathrm{i}\mathrm{i}) ] Assume that (i) is not consistent. Then, v\displaystyle \not\in\bigcap_{y\in F(x)}(y+ intC) for all
x\in X and v\in V .

Since \displaystyle \bigcap_{y\in F(x)}(- $\lambda$ k+y+C)\subset\bigcap_{y\in F(x)}(y+ intC) for all k\in intC andv\displaystyle \not\in\bigcap_{y\in F(x)}(- $\lambda$ k+y+C) .
Thus, \displaystyle \inf\{t\in \mathbb{R}| (tk+V)\cap\bigcap_{y\in F(x)}(y+C)\neq\emptyset\} is non‐negative.
$\lambda$<0 , thenRemark 3.4. We let S
V

and k

\in

:=

\displaystyle \bigcap_{y\in F(x)}(y+C) .

intC, there exists t_{v}there exists \overline{t} \geq>0Then S+C

such that t_{v}k+v\subset\in(\displaystyle \overline{t}k+V)\cap\bigcap_{y\in F(x)}(y+C) \neq \emptyset .
\displaystyle \inf\{t\in \mathbb{R}| (tk+V)\cap\bigcap_{y\in F(x)}(y+C)\neq\emptyset\}\leq\overline{t}<+\infty.
0 such that\squareS so that for all\displaystyle \bigcap_{y\in F(x)}(y+C) .
Thus,

v

\in

Then,

(I_{k,V}^{(4)}\mathrm{o}F)(x)=Theorem 3.5 (Infimum type (5)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty
subset in Y. If F is compact‐valued on X , then exactly one of the following two systems
is consistent;(i) there exists x\in X such that F(x)\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(5)}V ;(ii) there exists k\in intC such that (I_{k,V}^{(5)}\circ F)(x)\geq 0 for all x\in X.

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Proof. [(\mathrm{i})\Rightar ow \mathrm{n}\mathrm{o}\mathrm{t} (\mathrm{i}\mathrm{i})] Fix k \in intC. Assume that (i) is consistent. Then, there exists
xX such that y \in\in

V- intC for all y \in

F(x) . Since V- intC is open, there exists$\lambda$_{y}>0 such that y+$\lambda$_{y}k\in V- intC. We let S_{y} :=-$\lambda$_{y}k+V- intC, then \{S_{y}\}_{y\in F(x)}is a cover of F(x) . Since F(x) is compact, there exists \{y_{1}, . . . , y_{n}\} \subset F(x) such that

F(x)\displaystyle \subset\bigcup_{i\in I}S_{y_{l} where I :=\{1, . . . , n\}.
Taking \overline{$\lambda$} :=\displaystyle \min\{$\lambda$_{y_{1} , . . . , $\lambda$_{y_{n} \}>0 , then F(x)\displaystyle \subset\bigcup_{i\in I}(-$\lambda$_{y_{t}}+V- intC) =-\overline{ $\lambda$}k+VintC \subset-\overline{ $\lambda$}k+V-C . Thus, \displaystyle \inf\{t\in \mathbb{R}|F(x)\subset tk+V-C\}\leq-\overline{ $\lambda$}<0.

[Not (\mathrm{i})\Rightar ow(\mathrm{i}\mathrm{i}) ] Assume that (i) is not consistent. Then, for all x \in\overline{y}\in F(x) such that \overline{y}\not\in V- intC. Since $\lambda$ k+V-C\subset$\lambda$<0 , then \overline{y}\not\in $\lambda$ k+V-C .

X,

there exists

V- intC for all k\in intC andThus, \displaystyle \inf\{t\in \mathbb{R}|F(x) \subset tk+V-C\} is non‐negative.

口Remark 3.5. We let S :=V-C . Then S+(-C)\subset S so that for all x\in X, y\in F(x) , andk\in intC, there exists t_{y}>0 such that y\in t_{y}k+V-C . Then, there exists \overline{t}\geq 0 such that

F(x)\subset\overline{t}k+V-C . Thus, (I_{k,V}^{(5)}\displaystyle \circ F)(x)=\inf\{t\in \mathbb{R}|F(x)\subset(tk+V-C)\}\leq\overline{t}<+\infty.
Theorem 3.6 (Infimum type (6)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty
subset in Y . Then, exactly one of the following two systems is consistent;(i) there exists x\in X such that F(x)\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(6)}V ;(ii) there exists k\in intC such that (I_{k,V}^{(6)}\mathrm{o}F)(x)\geq 0 for all x\in X.
Proof. [(\mathrm{i})\Rightar ow \mathrm{n}\mathrm{o}\mathrm{t} (\mathrm{i}\mathrm{i})] Fix k \in intC.
ist

x

\in

X

Assume that (i) is consistent.

and \overline{y} \in F(x) such that \overline{y} \inthat \overline{y}+\overline{ $\lambda$}k \in

V- intC \subset

V-C .

V- intC.

Then, there ex‐

Thus, there exists \overline{$\lambda$}

>

0

such

Then, F(x)\cap(-\overline{ $\lambda$}k+V-C) \neq \emptyset . Therefore,\displaystyle \inf\{t\in \mathbb{R}|F(x)\cap(tk+V-C)\neq\emptyset\}\leq-\overline{ $\lambda$}<0.
[Not (\mathrm{i})\Rightar ow(\mathrm{i}\mathrm{i}) ] Assume that (ii) is not consistent. Then, y\in V- intC for all x\in X andy\in F(x) . Since $\lambda$ k+V-C\subset V- intC for all k\in intC and $\lambda$<0 , then y\not\in $\lambda$ k+V-C.
Thus, \displaystyle \inf\{t\in \mathbb{R}|F(x)\cap(tk+V-C)\neq\emptyset\} is non‐negative.

\squareRemark 3.6. We let S := V-C like Remark 3.5. For all x \in X, y \in F(x) , andk\in intC, there exists t_{y} >0 such that y\in t_{y}k+V-C .

Then, there exists \overline{t}\geq 0 suchthat F(x)\cap(\overline{t}k+V-C)\neq\emptyset . Thus, (I_{k,\mathcal{V} ^{(6)}\displaystyle \mathrm{o}F)(x)=\inf\{t\in \mathbb{R}|F(x)\cap(tk+V-C)\neq\emptyset\}\leq\overline{t}<+\infty.

3.2. Supremum Types. We obtain supremum types conversely by replacing an ordering
cone C in Infimum types with -C . In addition, just like six remarks described above,(S_{k,V}^{(\mathrm{j})}\circ F)(x)>-\infty for all x\in X and j=1 , . . . , 6.

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Theorem 3.7 (Supremum type (1)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty

subset in Y . If F is compact‐valued on X and V is compact, then exactly one of thefollowing two systems is consistent;(i) there exists x\in X such that V\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(1)}F(x) ;(ii) there exists k\in intC such that (S_{k,V}^{(1)}\circ F)(x)\leq 0 for all x\in X.
Theorem 3.8 (Supremum type (2)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty
subset in Y . If V is compact, then exactly one of the following two systems is consistent;(i) there exists x\in X such that V\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(2)}F(x) ;(ii) there exists k\in intC such that (S_{k,V}^{(2)}\circ F)(x)\leq 0 for all x\in X.

Theorem 3.9 (Supremum type (3)). Let X be a nonempty set, Y a topological vector
space, C a convex sohd cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty

subset in Y . If V is compact, then exactly one of the following two systems is consistent;(i) there exists x\in X such that V\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(3)}F(x) ;(ii) there exists k\in intC such that (S_{k,V}^{(3)}\circ F)(x)\leq 0 for all x\in X.
Theorem 3.10 (Supremum type (4)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in \mathrm{Y}, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty

subset in Y . If F is compact‐valued on X , then exactly one of the following two systems
is consistent;(i) there exists x\in X such that V\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(4)}F(x) ;(ii) there exists k\in intC such that (S_{k,V}^{(4)}\circ F)(x)\leq 0 for all x\in X.

Theorem 3.11 (Supremum type (5)). Let X be a nonempty set, Y a topological vector
space, C a convex solid cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty

subset in Y . If F is compact‐valued on X , then exactly one of the following two systems
is consistent;(i) there exists x\in X such that V\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(5)}F(x) ;(ii) there exists k\in intC such that (S_{k,V}^{(5)}\mathrm{o}F)(x) \leq 0 for all x\in X.
Theorem 3.12 (Supremum type (6)). Let X be a nonempty set, Y a topological vector
space, C a convex sohd cone in Y, F : X \rightarrow 2^{Y} a set‐valued map, and V a nonempty

subset in Y . Then, exactly one of the following two systems is consistent;(i) there exists x\in X such that V\leq_{\mathrm{i}\mathrm{n}\mathrm{t}C}^{(6)}F(x) ;117(ii) there exists k\in intC such that (S_{k,V}^{(6)}\circ F)(x)\leq 0 for all x\in X.
4. POSITIVE SEMI‐DEFINITE SYSTEMLet S^{n} be the set of n\times n symmetric matrices, S_{+}^{n} the set of n\times n positive semidefinite
symmetric matrices, S_{++}^{n} the set of n\times n positive definite symmetric matrices, F:X\rightarrow
2^{S^{n}} a set‐valued map, and V a nonempty subset in S^{n} . We denote the trace of A\in S^{n}by \mathrm{t}\mathrm{r}(A) :=\displaystyle \sum_{i=1}^{n}a_{i } . Some notions are from pp.9‐10 of [1].
Theorem 4.1. Exactly one of the following two systems is consistent;(i) There exists x\in X such that F(x)\cap(V-S_{++}^{n})\neq\emptyset ;(ii) There exists K\in S_{++}^{n} such that\displaystyle \inf\{t\in \mathbb{R}|F(x)\cap(tK+V-S_{+}^{n})\neq\emptyset\}\geq 0 for all x\in X.
This theorem is simply verified by S_{++}^{n}=\mathrm{i}\mathrm{n}\mathrm{t}S_{+}^{n}.

Corollary 4.1. Let f : \mathbb{R}^{m}\rightarrow S^{n} be a function. Then, exactly one of the following two
systems is consistent;(i) There exists x\in \mathbb{R}^{m} such that f(x)\in-S_{++}^{n} ;(ii) There exists B\in S_{++}^{n} such that\displaystyle \inf\{t\in \mathbb{R}|f(x)\in tB-S_{+}^{n}\}\geq 0 for all x\in \mathbb{R}^{m}.
Let A\in S^{n} and B\in S_{++}^{n} be given. Consider the following semidefinite optimization
problem and its dual problem:(SDP)\displaystyle \inf\{\mathrm{t}\mathrm{r}(-AX) |\mathrm{t}\mathrm{r}((-B)X)=-1, X\in S_{+}^{n}\}.
\displaystyle \sup\{-t|-tB+S=-A, S\in S_{+}^{n}\}.

(SDD)

Notice that \displaystyle \inf\{t\in \mathbb{R}|A\in tB-S_{+}^{n}\}=-\sup\{-t\in \mathbb{R}|-tB+S=-A, S\in S_{+}^{n}\} forall A\in S^{n} and B\in S_{++}^{n} , and we can calculate \displaystyle \inf\{t\in \mathbb{R}|A\in tB-S_{+}^{n}\} with Matlab asa semidefinite optimization problem.

Lemma 4.1. Problems (SDP) and (SDD) are strictly feasible, that is, there exists \tilde{X}\in
S_{++}^{n} such that \mathrm{t}\mathrm{r}((-B)\tilde{X})=-1 , and there exist \tilde{t}\in \mathbb{R} and \tilde{S}\in S_{++}^{n} such that -\tilde{t}B+\tilde{S}=-A.

Proof. Let X :=\displaystyle \sum_{i=1}^{n}u_{i}q_{i}q_{i}^{\mathrm{T} \in S_{++}^{n} where u_{i} are positive and \{q_{i}\}_{i=1,\ldots,n} is a base of \mathbb{R}^{n}.
Then, \mathrm{t}\mathrm{r}(BX)\displaystyle \mathrm{t}\mathrm{r}(B(\sum_{i=1}^{n}u_{i}q_{i}q_{i}^{\mathrm{T} )\displaystyle \sum_{i=1}^{n}u_{i}\mathrm{t}\mathrm{r}(Bq_{i}q_{i}^{\mathrm{T} )\tilde{X} :=(1/\mathrm{t}\mathrm{r}(BX))X\in S_{++}^{n} satisfies \mathrm{t}\mathrm{r}(B\tilde{X})=1.
===\displaystyle \sum_{i=1}^{n}u_{i}q_{i}^{\mathrm{T} Bq_{i} > 0 . Thus,

Let B\in S_{++}^{n} . If A=$\theta$_{S^{n}} , we have -tB+\tilde{S}=$\theta$_{S^{n}} by letting \tilde{S} :=tB for all t>0.
Now we assume that A \neq $\theta$_{S^{n} . Since S_{++}^{n} is open, there exists an open neighborhood

118

N($\theta$_{S^{n} ) such that B+U \subset S_{++}^{n} .

U \in

There exists \tilde{t} > 0 such that(-1/\tilde{t})A \in

U.

Therefore, we can find \tilde{S} :=\tilde{t}B-A=\tilde{t}(B-(1/\tilde{t})A)\in S_{++}^{n}.

\squareLemma 4.2. The optimal values of (SDP) and (SDD) coincide and their optimal solutions
are nonempty.

Proof. Since (SDP) and (SDD) are strictly feasible, it follows from Corollary 2.1 in p.28

of [10] that the optimal values of (SDP) and (SDD) coincide and their optimal solutions\squareare nonempty.

Remark 4.1. Let A\in S^{n} and B\in S_{++}^{n} be given. Then, h_{S_{+}^{n}}(A;B) :=\displaystyle \min\{t\in \mathbb{R} |A\intB-S_{+}^{n}\displaystyle \}=\max\{\mathrm{t}\mathrm{r}(AX)|\mathrm{t}\mathrm{r}(BX)=1, X\in S_{+}^{n}\}.
From Corollary 4.1, Lemma 4.2, and Remark 4.1, we obtain the following corollary.

Corollary 4.2. Let f : \mathbb{R}^{m} \rightarrow S^{n} be a map. Then, exactly one of the following two
systems is consistent;(i) There exists x\in \mathbb{R}^{m} such that f(x)\in-S_{++}^{n} ;(ii) There exists B\in S_{++}^{n} such that\displaystyle \max\{\mathrm{t}\mathrm{r}(f(x)X)|\mathrm{t}\mathrm{r}(BX)=1, X\in S_{+}^{n}\}\geq 0 for all x\in \mathbb{R}^{m}.
This implies a test giving a criterion for satisfaction or dissatisfaction of Slater’s con‐dition.

5. CONCLUSION

We have extended some concepts of alternative theorems to set‐valued case. Our theo‐rems give a natural generalization to ones having been researched since before. In addition,these result can be used when we are to make sure a set precedes another. Note that there
are several ways of reduction of scalarizing functions to ease calculation of their values.
REFERENCES[1] J. M. Borwein and A. S. Lewis, Convex analysis and nonlinear optimization, Springer, New York,2000.

[2] P. Gordan, Über die Auflösung hnearer Gleichungen mit reellen coefficienten, Mathematische An‐
nalen, 6 (1873) 23‐28.

[3] A. Göpfert, H. Riahi, C. Tammer, and C. Zàlinescu, Variational methods in partially ordered spaces,
Springer, New York, 2003.

[4] V. Jeyakumar, A generalization of a minimax theorem of Fan via a theorem of the alternative, J.
Optim. Theory Appl. 48 (1986) 525‐533.

119[5] D. Kuroiwa, T. Tanaka, and T. X. D. Ha, On convexity of set‐valued maps, Nonlinear Anal. 30(1997), 1487‐1496.
[6] I. Kuwano, T. Tanaka, and S. Yamada, Charactenzaton of nonhnear scalarizing function for set‐valued maps, in Proceedings of the Asian Conference on Nonlinear Analysis and optimization,

S. Akashi, W. Takahashi, and T. Tanaka (eds.), Yokohama Publishers, Yokohama, 2009, pp.193‐204.

[7] Z. Li, A theorem of the alternative and its application to the optimization of set‐valued maps, J.
Optim. Theory Appl. 100 (1999) 365‐375.

[8] S. Nishizawa, M. Onodsuka, and T. Tanaka, Alternative theorems for set‐valued maps based on anonhnear scalarization, Pacific J. Optim. 1 (2005) 147‐159.

[9] X. M. Yang, X. Q. Yang, and G. Y. Chen, Theorems of the alternative and optimization with set‐valued maps, J. Optim. Theory Appl. 107 (2000) 627‐640.

[10] E. de Klerk: Aspects of semidefinite programming: intemor point algorithms and selected applications,
Kluwer Academic Publishers, Dordrecht (2002).
GRADUATE SCHOOL OF SCIENCE AND TECHNOLOGY, NIIGATA UNIVERSITY, NiiCATA, JAPAN\mathrm{E} ‐mail address (Yuto Ogata):y‐ogata@m.sc.niigata‐u.ac.jp\mathrm{E} ‐mail address (Yutaka Saito):

ysaito@m.sc.niigata‐u.ac.jp\mathrm{‐mailE}address (Tamaki Tanaka):

tamakidmath. sc. niigta-\mathrm{u} . ac. jp
新潟大学大学院自然科学研究科

小形優人,齋藤裕,田中環

DEPARTMENT OF APPLIED MATHEMATICS, PUKYONG NATIONAL UNIVERSITY, BUSAN, KOREA

E‐‐mail address (Gue Myung Lee): gmlee@pknu.ac.kr\mathrm{E} ‐mail address (Jae Hyoung Lee): \mathrm{m}\mathrm{c}7558 $\Phi$naver. com

釜慶大学校応用数学科

GUE MYUNG LEE, JAE HYOUNG LEE