# Existence and Well-Posedness for Symmetric Vector Quasi-Equilibrium Problems

Authors: Kaihong Wang; Wenyan Zhang; Min Fang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

An existence result for the solution set of symmetric vector quasi-equilibrium problems that allows for discontinuities is obtained. Moreover, sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric
vector quasi-equilibrium problems are established.


---

## Body

## 1. Introduction

Well-posedness of optimization problems was first studied by Tykhonov [1] in 1966. Since then, the notion of well-posedness has been extended to different kinds of optimization problems (see [2–5]). In the book edited by Lucchetti and Revalski [6], Loridan gave a survey on some theoretical results of well-posedness, approximate solutions, and variational principles in vector optimization. Well-posedness for constrained optimization problems was first studied by Levitin and Polyak [7]. The study of Levitin-Polyak well-posedness for convex scalar optimization problems with functional constraints comes from [8]. Recently, this research was extended to general constrained vector optimization problems [9], generalized variational inequality problems with functional constraints [10], and vector equilibrium problems with functional constraints [11].In 2003, Fu [12] introduced the symmetric vector quasi-equilibrium problem (for short (
SVQEP
)) and studied existence conditions of (
SVQEP
). (
SVQEP
) is a generalization of the equilibrium problem, proposed by Blum and Oettli [13], and a unified model of several problems, for example, vector optimization problems, problems of vector Nash equilibria, vector variational inequalities, and vector complementarity problems. Farajzadeh [14] considered existence theorem of the solution of symmetric vector quasi-equilibrium problems in the Hausdorff topological vector space and answered the open question raised by Fu [12]. In [15], Li et al. obtained existence results for two classes of generalized vector quasi-equilibrium problems. Zhang [16] introduced generalized Levitin-Polyak well-posedness for (
SVQEP
) and obtained sufficient conditions for the generalized Levitin-Polyak well-posedness of (
SVQEP
).In this paper, we will introduce existence and well-posed theorem of(
SVQEP
) for discontinuous vector-valued mapping, which extend the corresponding result in [12] in metric space. Then, by using the conditions of the existence theorem of the solutions to (SVQEP) in [14], we obtain sufficient conditions for the generalized Levitin-Polyak well-posedness of (
SVQEP
), which improve the result of [16, Theorem 4.1].The paper is organized as follows. In Section2, we present some preliminary concepts. In Section 3, we prove the existence theorem of (
SVQEP
) that allows for discontinuities. In Section 4, we obtain generalized Levitin-Polyak well-posed results for (
SVQEP
).


## 2. Preliminaries

LetX and Y be real locally convex Hausdorff spaces, and let C and D be nonempty subsets of X and Y, respectively. Let Z be a real Hausdorff topological vector space and P
⊂
Z a closed convex and pointed cone with k
0
∈
int
⁡
P.Assume thatB
⊂
Z is a nonempty subset. A point b
∈
B is called a minimal point of B, if B
∩
(
b
-
P
)
=
{
b
}. A point b
∈
B is called a weak minimal point of B if B
∩
(
b
-
int
⁡
P
)
=
∅. A point b
∈
B is called an ε-weak minimal point of B if B
∩
(
b
-
ε
k
0
-
int
⁡
P
)
=
∅. By min
⁡
B, min
⁡
w
B, and ε
-
min
⁡
w
B we denote the sets of all minimal points, weak minimal points, and ε-weak minimal points of B, respectively. Obviously, min
⁡
B
⊆
min
⁡
w
B
⊆
ε
-
min
⁡
w
B.LetE be a nonempty subset of X and let h
:
E
→
Z be a vector-valued mapping. Consider the following vector-valued optimization problem:
(1)
(
E
,
h
)
:
minimize
h
(
x
)
,
x
∈
E
.
A point y
∈
E is called a (weak) minimizer of (
E
,
h
) if h
(
y
) is a minimal point (weak minimal point) of the set h
(
E
); namely, for every x
∈
E,
(2)
h
(
x
)
-
h
(
y
)
∉
-
P
∖
{
0
}
(
h
(
x
)
-
h
(
y
)
∉
-
int
⁡
P
)
.
The set of all (weak) minimizers of (
E
,
h
) is denoted by argmin
(
E
,
h
)(argwmin
(
E
,
h
)). Obviously, argmin
(
E
,
h
)
⊆
argwmin
(
E
,
h
).LetS
:
C
×
D
→
2
C and T
:
C
×
D
→
2
D be two set-valued mappings and let f
,
g
:
C
×
D
→
Z be two vector-valued mappings. Fu [12] defined a class of symmetric vector quasi-equilibrium problems (for short (
SVQEP
)), which consist in finding (
x
¯
,
y
¯
)
∈
C
×
D such that x
¯
∈
S
(
x
¯
,
y
¯
), y
¯
∈
T
(
x
¯
,
y
¯
), and
(3)
f
(
x
,
y
¯
)
-
f
(
x
¯
,
y
¯
)
∉
-
int
⁡
P
,
∀
x
∈
S
(
x
¯
,
y
¯
)
,
g
(
x
¯
,
y
)
-
g
(
x
¯
,
y
¯
)
∉
-
int
⁡
P
,
∀
y
∈
T
(
x
¯
,
y
¯
)
.
We call (
x
¯
,
y
¯
) a solution of (
SVQEP
) and denote by X
0
×
Y
0 the solution set of (
SVQEP
). The equilibrium problem contains optimization problem as special case (see [17, 18]). The problem is a generalization of quasi-optimization problem proposed by Crespi and Tan [19].Definition 1.
Let(
Z
,
P
) be an ordered topological vector space, let C be a nonempty convex subset of a vector space X, and let h
:
C
→
Z be a vector mapping. (i)
h is called P-convex if, for every x
1
,
x
2
∈
E and for every λ
∈
[
0,1
], one has
(4)
h
(
λ
x
1
+
(
1
-
λ
)
x
2
)
∈
λ
h
(
x
1
)
+
(
1
-
λ
)
h
(
x
2
)
-
P
.
(ii)
h is called proper P-quasiconvex if, for every x
1
,
x
2
∈
X and λ
∈
[
0,1
], one has either h
(
λ
x
1
+
(
1
-
λ
)
x
2
)
∈
h
(
x
1
)
-
P or h
(
λ
x
1
+
(
1
-
λ
)
x
2
)
∈
h
(
x
2
)
-
P.
(iii)
h is said to be natural P-quasiconvex on E if, for every x
1
,
x
2
∈
E, λ
∈
[
0,1
], there exists μ
∈
[
0,1
] such that
(5)
h
(
λ
x
1
+
(
1
-
λ
)
x
2
)
∈
μ
h
(
x
1
)
+
(
1
-
μ
)
h
(
x
2
)
-
P
.
(iv)
h is called strict P-convex if, for every x
1
,
x
2
∈
E with x
1
≠
x
2 and for every λ
∈
(
0,1
), one has
(6)
h
(
λ
x
1
+
(
1
-
λ
)
x
2
)
∈
λ
h
(
x
1
)
+
(
1
-
λ
)
h
(
x
2
)
-
int
⁡
P
.Remark 2.
It is clear that every strictP-convex mapping is convex, and every convex or proper P-quasiconvex mapping is natural P-quasiconvex. A vector mapping may be P-convex and not proper P-quasiconvex and conversely (see [20]). A vector mapping may be natural P-quasiconvex but neither P-convex nor proper P-quasiconvex.LetF
:
X
→
2
Y be a set-valued map. F is said to be upper semicontinuous (u
.
s
.
c. for short) at x
∈
X if, for any open set U
⊃
F
(
x
), there exists a neighborhood V of x such that
(7)
⋃
x
∈
V
‍
F
(
x
)
:
=
F
(
V
)
⊂
U
.
If F is u
.
s
.
c. at each point of X, then F is said to be u
.
s
.
c.   F is said to be lower semicontinuous (l
.
s
.
c. for short) at x
∈
X if, for any y
∈
F
(
x
) and any neighborhood U of y, there exists a neighborhood V of x such that ∀
x
′
∈
V; we have
(8)
F
(
x
′
)
∩
U
≠
∅
.
F is said to be l
.
s
.
c. if F is l
.
s
.
c. at every point of X. Moreover, F is said to be continuous if F is both l
.
s
.
c. and u
.
s
.
c. From [21], we can see that F is lower semicontinuous at x
0
∈
X if and only if, for any y
0
∈
F
(
x
0
) and any net {
x
n
} with x
n
→
x
0, there is a net {
y
n
} such that y
n
∈
F
(
x
n
) and y
n
→
y
0. F is said to be closed if the graph of F, that is, Graph
⁡
(
F
)
=
{
(
x
,
y
)
:
x
∈
X
,
y
∈
F
(
x
)
}, is a closed set in X
×
Y.LetF
:
X
→
2
X be a set-valued map. A point x
∈
X is called a fixed point of the set-valued map F if x
∈
F
(
x
).Lemma 3.
LetZ be a metric space, let x
∈
X, and let {
x
n
} be a sequence converging to x in X. Let a and z be two elements in Z such that z
-
a
∈
int
⁡
P and assume that there exist two sequences {
a
n
} and {
z
n
} valued in Z such that
(9)
lim
⁡
n
→
∞
d
(
z
,
z
n
-
P
)
=
0
,
lim
⁡
n
→
∞
d
(
a
,
a
n
+
P
)
=
0
,
where d
(
z
,
z
n
-
P
)
=
min
⁡
{
∥
z
-
q
n
∥
:
q
n
∈
z
n
-
P
} and d
(
a
,
a
n
+
P
)
=
min
⁡
{
∥
a
-
l
n
∥
:
l
n
∈
a
n
+
P
}. Then for all n
∈
N sufficiently large one has z
n
-
a
n
∈
int
⁡
P.The proof of Lemma3 is similar to that of Lemma  1 in [22]; for details, see [22].


## 3. Existence Results for (SVQEP)

Throughout this section, letX, Y, and Z be metric spaces, let the sets C
⊂
X and D
⊂
Y be nonempty, convex, and compact subsets, and let P
⊂
Z be a closed, convex pointed cone with int
⁡
P
≠
∅.Theorem 4.
Assume that(1)
S
:
C
×
D
→
2
C and T
:
C
×
D
→
2
D are continuous, and for each (
x
,
y
)
∈
C
×
D, S
(
x
,
y
), T
(
x
,
y
) are nonempty, closed convex subsets;
(2)
for any(
x
,
y
)
∈
S
(
x
,
y
)
×
T
(
x
,
y
), (
x
n
,
y
n
)
∈
S
(
x
n
,
y
n
)
×
T
(
x
n
,
y
n
) for all n
∈
N, and (
x
n
,
y
n
) converging to (
x
,
y
), the following conditions hold:
(10)
d
(
f
(
x
,
y
)
,
f
(
x
n
,
y
n
)
-
P
)
⟶
0
,
d
(
g
(
x
,
y
)
,
g
(
x
n
,
y
n
)
-
P
)
⟶
0
;
(3)
for any(
x
,
y
)
∈
S
(
x
,
y
)
×
T
(
x
,
y
) and {
y
n
}converging to y such that y
n
∈
T
(
x
n
,
y
n
), there exists x
¯
n
∈
S
(
x
n
,
y
n
) such that d
(
f
(
x
,
y
)
,
f
(
x
¯
n
,
y
n
)
+
P
)
→
0; for any (
x
,
y
)
∈
S
(
x
,
y
)
×
T
(
x
,
y
) and {
x
n
}converging to x such that x
n
∈
S
(
x
n
,
y
n
), there exists y
¯
n
∈
T
(
x
n
,
y
n
) such that d
(
g
(
x
,
y
)
,
g
(
x
n
,
y
¯
n
)
+
P
)
→
0;
(4)
f
(
S
(
x
,
y
)
,
y
) and g
(
x
,
T
(
x
,
y
)
) are compact sets;
(5)
for any fixedy
∈
D, f
(
x
,
y
) is proper P-quasiconvex in x; for any fixed x
∈
C, g
(
x
,
y
) is proper P-quasiconvex in y.
Then (SVQEP) has a solution.Proof.
Let us define set-valued mapsA
:
C
×
D
→
2
C and B
:
C
×
D
→
2
D by
(11)
A
(
x
,
y
)
=
{
u
∈
S
(
x
,
y
)
∣
f
(
u
,
y
)
∈
mi
n
w
f
(
S
(
x
,
y
)
,
y
)
}
,
hhhhhhhhhhhhhhhhhhhhhh
∀
(
x
,
y
)
∈
C
×
D
,
B
(
x
,
y
)
=
{
v
∈
T
(
x
,
y
)
∣
g
(
x
,
v
)
∈
mi
n
w
g
(
x
,
T
(
x
,
y
)
)
}
,
hhhhhhhhhhhhhhhhhhhhhhh
∀
(
x
,
y
)
∈
C
×
D
.
Similar to the proof of [12, Section 2 Theorem], the set A
(
x
,
y
) is nonempty and convex. We only need to show the following.
(I) For all(
x
,
y
)
∈
C
×
D, the set A
(
x
,
y
) is closed. In fact, let a sequence {
u
n
}
⊂
A
(
x
,
y
) and u
n
→
u; we need to show that u
∈
A
(
x
,
y
). It follows from u
n
∈
S
(
x
,
y
) and the closedness of S
(
x
,
y
) that u
∈
S
(
x
,
y
). Since
(12)
f
(
u
n
,
y
)
∈
mi
n
w
f
(
S
(
x
,
y
)
,
y
)
,
we have
(13)
f
(
u
n
,
y
)
-
f
(
z
,
y
)
∉
int
⁡
P
,
∀
z
∈
S
(
x
,
y
)
.
If there exists z
0
∈
S
(
x
,
y
) such that
(14)
f
(
u
,
y
)
-
f
(
z
0
,
y
)
∈
int
⁡
P
,
from condition (
2
), d
(
f
(
u
,
y
)
,
f
(
u
n
,
y
)
-
P
)
→
0; thus,
(15)
d
(
f
(
u
,
y
)
-
f
(
z
0
,
y
)
,
f
(
u
n
,
y
)
-
f
(
z
0
,
y
)
-
P
)
⟶
0
.
From Lemma 3, f
(
u
n
,
y
)
-
f
(
z
0
,
y
)
∈
int
⁡
P for n large enough. It is a contradiction.
(II)A is u.s.c., since C is a compact set and A
(
x
,
y
)
⊂
C. By [12, Lemma 2
(
ii
)], we need only to show that the set-valued map A is closed. Let a sequence {
(
x
n
,
y
n
)
}
⊂
C
×
D, (
x
n
,
y
n
)
→
(
x
,
y
) and u
n
∈
A
(
x
n
,
y
n
). Since u
n
∈
S
(
x
n
,
y
n
) and the set-valued map S is continuous, we have u
∈
S
(
x
,
y
). For any z
∈
S
(
x
,
y
), since S is l.s.c., there is a sequence {
z
n
}, z
n
∈
S
(
x
n
,
y
n
) such that z
n
→
z. Since f
(
u
n
,
y
n
)
∈
mi
n
w
f
(
S
(
x
n
,
y
n
)
,
y
n
), we get
(16)
f
(
u
n
,
y
n
)
-
f
(
z
n
,
y
n
)
∉
int
⁡
P
,
∀
z
n
∈
S
(
x
n
,
y
n
)
.
If there existsz
∈
A
(
x
,
y
) such that f
(
u
,
y
)
-
f
(
z
,
y
)
∈
int
⁡
P, from (ii),
(17)
d
(
f
(
u
,
y
)
,
f
(
u
n
,
y
n
)
-
P
)
⟶
0
.
From (iii), there exists z
¯
n
∈
S
(
x
n
,
y
n
) such that
(18)
d
(
f
(
z
,
y
)
,
f
(
z
¯
n
,
y
n
)
+
P
)
⟶
0
.
By Lemma 3, f
(
u
n
,
y
n
)
-
f
(
z
¯
n
,
y
n
)
∈
int
⁡
P, which contradicts (16).Remark 5.
It is clear that iff and g are continuous mappings and condition (
1
) holds, then conditions (
2
), (
3
), and (
4
) of Theorem 4 hold. The following example shows that Theorem 4 improves [12, Section 2. Theorem].Example 6.
Suppose thatX
=
Y
=
R, C
=
D
=
[
0,2
], and P
=
R
+
2 and let S
:
C
×
D
→
2
C and T
:
C
×
D
→
2
D be defined as S
(
x
,
y
)
=
C and T
(
x
,
y
)
=
D, respectively. For all (
x
,
y
)
∈
R
2, let
(19)
f
(
x
,
y
)
=
{
(
0,1
)
,
if
x
∈
[
0,1
)
,
(
y
,
1
)
,
if
x
=
[
1,2
]
,
g
(
x
,
y
)
=
{
(
1,0
)
,
if
y
∈
[
0,1
)
,
(
1
,
x
)
,
if
y
∈
[
1,2
]
.
It is clear that the mappings f and g are not continuous, but all the conditions of Theorem 4 hold.
Moreover, letx
*
=
(
1,1
); we can get from [23, Lemma 2.2] that the mappings f and g are natural P-quasiconvex but not demicontinuous (see [14, Definition 2.4]). Therefore, Theorem 4 is different from [14, Theorem 3.1].


## 4. Well-Posedness of (SVQEP)

In this section, we discuss the notion of generalized Levitin-Polyak well-posedness for(
SVQEP
).Definition 7.
A sequence{
(
x
n
,
y
n
)
}
⊂
C
×
D is called a Levitin-Polyak approximating solution sequence (in short LP sequence) for (
SVQEP
) if there exists {
ε
n
}
⊂
R
+ with ε
n
→
0 such that
(20)
d
(
x
n
,
S
(
x
n
,
y
n
)
)
≤
ε
n
,
d
(
y
n
,
T
(
x
n
,
y
n
)
)
≤
ε
n
,
f
(
x
,
y
n
)
-
f
(
x
n
,
y
n
)
+
ε
n
k
0
∉
-
int
⁡
P
,
∀
x
∈
S
(
x
n
,
y
n
)
,
g
(
x
n
,
y
)
-
g
(
x
n
,
y
n
)
+
ε
n
k
0
∉
-
int
⁡
P
,
∀
y
∈
T
(
x
n
,
y
n
)
.Definition 8.
The problem(
SVQEP
) is said to be generalized Levitin-Polyak well-posed (in short LP well-posed) if (i)
X
0
×
Y
0
≠
∅;
(ii)
for every LP sequence{
(
x
n
,
y
n
)
}, there exist a subsequence {
(
x
n
k
,
y
n
k
)
}
⊂
{
(
x
n
,
y
n
)
} and an element (
x
*
,
y
*
)
∈
X
0
×
Y
0 such that (
x
n
k
,
y
n
k
)
→
(
x
*
,
y
*
).Let us illustrate the notion of generalized LP well-posedness by some examples.Example 9.
LetX
=
Y
=
Z
=
R, C
=
D
=
[
0,1
], and P
=
R
+. For all (
x
,
y
)
∈
C
×
D, let f
(
x
,
y
)
=
x
+
y and g
(
x
,
y
)
=
y
-
x. Set-valued mappings S
:
C
×
D
→
2
C and T
:
C
×
D
→
2
D are defined by S
(
x
,
y
)
=
T
(
x
,
y
)
=
[
0,1
]; (
SVQEP
) is generalized LP well-posed.Example 10.
LetX
=
Y
=
Z
=
R, C
=
D
=
[
0,4
π
], and P
=
R
+. For all (
x
,
y
)
∈
C
×
D, let f
(
x
,
y
)
=
sin
x
y and g
(
x
,
y
)
=
cos
⁡
x
y. Set-valued mappings S
:
C
×
D
→
2
C and T
:
C
×
D
→
2
D are defined by S
(
x
,
y
)
=
[
2
π
,
4
π
] and T
(
x
,
y
)
=
[
1,2
]; (
SVQEP
) is not generalized LP well-posed.Remark 11.
(i) Generalized LP well-posedness of(
SVQEP
) implies that the set X
0
×
Y
0 is compact.
(ii) It is easy to see that the notion of well-posedness of(
SVQEP
) generalizes the notion of generalized LP well-posedness of vector equilibrium problem introduced in [20].Theorem 12.
Let{
ε
n
}
⊂
R
+ with ε
n
→
0. Under the assumptions of Theorem 4, (
SVQEP
) is generalized LP well-posed.Proof.
Let{
ε
n
}
⊂
R
+ with ε
n
→
0 and
(21)
h
h
h
h
h
h
h
h
h
h
h
i
d
(
x
n
,
S
(
x
n
,
y
n
)
)
≤
ε
n
,
f
(
x
,
y
n
)
-
f
(
x
n
,
y
n
)
+
ε
n
k
0
∉
-
int
⁡
P
,
hhhhhhhhhhhhhhhhh
∀
x
∈
S
(
x
n
,
y
n
)
;
h
h
h
h
h
h
h
h
h
h
h
i
d
(
y
n
,
T
(
x
n
,
y
n
)
)
≤
ε
n
,
g
(
x
n
,
y
)
-
g
(
x
n
,
y
n
)
+
ε
n
k
0
∉
-
int
⁡
P
,
hhhhhhhhhhhhhhhhh
∀
y
∈
T
(
x
n
,
y
n
)
.
Since S
,
T are continuous and compact-valued, there exist a subsequence {
(
x
n
k
,
y
n
k
)
}
⊂
{
(
x
n
,
y
n
)
} and an element (
x
0
,
y
0
)
∈
S
(
x
0
,
y
0
)
×
T
(
x
0
,
y
0
) such that {
(
x
n
k
,
y
n
k
)
}
→
(
x
0
,
y
0
). If there exists x
-
∈
S
(
x
0
,
y
0
), such that f
(
x
-
,
y
)
∈
f
(
x
0
,
y
0
)
-
int
⁡
P, from Theorem 4(ii),
(22)
d
(
f
(
x
0
,
y
0
)
,
f
(
x
n
k
,
y
n
k
)
-
ε
n
k
k
0
-
P
)
⟶
0
.
From Theorem 4(iii), there exists x
-
n
k
∈
S
(
x
n
k
,
y
n
k
) such that
(23)
d
(
f
(
x
-
,
y
)
,
f
(
x
-
n
k
,
y
n
k
)
+
P
)
⟶
0
.
By Lemma 3,
(24)
f
(
x
-
n
k
,
y
n
k
)
-
f
(
x
n
k
,
y
n
k
)
+
ε
n
k
k
0
∈
-
int
⁡
P
when n
k is large enough, which is a contradiction. Therefore,
(25)
f
(
x
,
y
0
)
-
f
(
x
0
,
y
0
)
∉
-
int
⁡
P
,
∀
x
∈
S
(
x
0
,
y
0
)
.
Similarly,
(26)
g
(
x
0
,
y
)
-
g
(
x
0
,
y
0
)
∉
-
int
⁡
P
,
∀
y
∈
T
(
x
0
,
y
0
)
.
From Definition 7, (
SVQEP
) is generalized LP well-posed.Definition 13 (see [23]).
LetX be a topological space, and let Z be a topological vector space. A function f
:
X
→
Z is said to be demicontinuous if
(27)
f
-
1
(
M
)
=
{
x
∈
X
:
f
(
x
)
∈
M
}
is closed in X for each closed half space M
⊂
Z.Lemma 14 (see [23]).
LetX be a topological space, Z a topological vector space, and f
:
X
→
Z a demicontinuous function. Then for any x
*
∈
Z
*, the composite function x
*
∘
f is continuous, where Z
* is the topological dual space of Z.LetP
⊂
Z be a closed convex and pointed cone with int
⁡
P
≠
∅ and
(28)
T
=
{
x
*
∈
Z
*
:
x
*
separates
the
sets
P
and
-
int
⁡
P
}
.
By [24, p. 165, Theorem 2], there exists x
*
∈
Z
* such that
(29)
∀
x
∈
P
,
x
*
(
x
)
≥
0
,
∀
x
∈
-
int
⁡
P
,
x
*
(
x
)
<
0
.
We get that T
≠
∅.Lemma 15.
If(
x
*
,
x
)
<
0 for all x
*
∈
T, then x
∈
-
int
⁡
P.Proof.
If we assume that there existsx
∉
-
int
⁡
P such that (
x
*
,
x
)
<
0 for all x
*
∈
T, then we have
(30)
-
int
⁡
P
⋂
‍
{
λ
x
+
(
1
-
λ
)
p
:
λ
∈
[
0,1
]
,
p
∈
P
}
=
∅
.
If not, there exist y
∈
int
⁡
P, λ
∈
[
0,1
], and p
∈
P such that -
y
∈
λ
x
+
(
1
-
λ
)
p. Thus,
(31)
x
=
-
1
λ
(
λ
·
y
λ
+
(
1
-
λ
)
p
)
∈
-
int
⁡
P
.
It is a contradiction. Thus, (30) holds. By [24, p. 165, Theorem 2], there exists x
*
′
∈
Z
* such that, for all y
∈
-
int
⁡
P,
(32)
x
*
′
(
y
)
<
0
,
and for all y
∈
{
λ
x
+
(
1
-
λ
)
p
:
λ
∈
[
0,1
]
,
p
∈
P
},
(33)
x
*
′
(
y
)
≥
0
.
Then, x
*
′
(
x
)
≥
0 and x
*
′
∈
T. This, however, contradicts the fact that (
x
*
,
x
)
<
0 for all x
*
∈
T.Theorem 16.
Assume that(1)
S
:
C
×
D
→
2
C and T
:
C
×
D
→
2
D are continuous, and for each (
x
,
y
)
∈
C
×
D, S
(
x
,
y
), T
(
x
,
y
) are nonempty, closed convex subsets;
(2)
f
,
g
:
C
×
D
→
Z are demicontinuous;
(3)
for any fixedy
∈
D, f
(
x
,
y
) is natural P-quasiconvex in x; for any fixed x
∈
C, g
(
x
,
y
) is natural P-quasiconvex in y.
Then(
SVQEP
) is generalized LP well-posed.Proof.
From [14, Theorem 3.1], X
0
×
Y
0
≠
∅. Let {
ε
n
}
⊂
R
+ with ε
n
→
0 and
(34)
h
h
h
h
h
h
h
h
h
h
h
h
d
(
x
n
,
S
(
x
n
,
y
n
)
)
≤
ε
n
,
f
(
x
,
y
n
)
+
ε
n
k
0
-
f
(
x
n
,
y
n
)
∉
-
int
⁡
P
,
hhhhhhhhhhhhhhhhh
∀
x
∈
S
(
x
n
,
y
n
)
;
(35)
h
h
h
h
h
h
h
h
h
h
h
h
d
(
y
n
,
T
(
x
n
,
y
n
)
)
≤
ε
n
,
g
(
x
n
,
y
)
+
ε
n
k
0
-
g
(
x
n
,
y
n
)
∉
-
int
⁡
P
,
hhhhhhhhhh
hhhhhhh
∀
y
∈
T
(
x
n
,
y
n
)
.
Since S
,
T are continuous and compact-valued, there exist a subsequence {
(
x
n
k
,
y
n
k
)
}
⊂
{
(
x
n
,
y
n
)
} and an element (
x
0
,
y
0
)
∈
S
(
x
0
,
y
0
)
×
T
(
x
0
,
y
0
) such that (
x
n
k
,
y
n
k
)
→
(
x
0
,
y
0
). If there exists x
-
∈
S
(
x
0
,
y
0
), such that
(36)
f
(
x
-
,
y
0
)
∈
f
(
x
0
,
y
0
)
-
int
⁡
P
,
for allx
*
∈
T, we have
(37)
x
*
(
f
(
x
-
,
y
0
)
-
f
(
x
0
,
y
0
)
)
<
0
.
Then,
(38)
x
*
(
f
(
x
-
,
y
0
)
)
<
x
*
(
f
(
x
0
,
y
0
)
)
.
Since S is l.s.c. at x
-, there exists x
-
n
∈
S
(
x
n
,
y
n
) such that x
-
n
→
x
-. By Lemma 14, x
*
(
f
(
x
-
n
k
,
y
n
k
)
)
+
x
*
(
ε
n
k
k
0
)
<
x
*
(
f
(
x
n
k
,
y
n
k
)
) when n
k is large enough. By Lemma 15,
(39)
f
(
x
-
n
k
,
y
n
k
)
-
f
(
x
n
k
,
y
n
k
)
+
ε
n
k
k
0
∈
-
int
⁡
P
,
which contradicts (34). Therefore, (
SVQEP
) is generalized LP well-posed.LetK
⊂
R
n be a closed set, let f
:
K
⊆
R
n
→
R
l, and consider a cone C
⊆
R
l to be closed, convex, pointed, and with nonempty interior. We consider the following vector optimization problem.(
VP
)
Findx
0
∈
K such that
(40)
f
(
x
)
-
f
(
x
0
)
∉
-
int
⁡
C
,
∀
x
∈
K
.As a consequence of Theorem16, we have the following well-posed result for the solution of (
VP
).Corollary 17.
LetK
⊂
R
n be a compact set, and let f
:
R
n
→
R
l be demicontinuous and natural C-quasiconvex on K. Then, (
VP
) is generalized LP well-posed.Remark 18.
(i) From Remark2, every strict P-convex mapping is natural P-quasiconvex. Then Theorem 16 improves [16, Theorem 4.1].
(ii) From [23, Lemma 2.1
(
iii
)], every natural C-quasiconvex function is C-quasiconvex. In the assumption of f being continuous, Corollary 17 is a special case of [18, Theorem 4.2].



---

*Source: 750709-2014-07-01.xml*