- Research Article
- Open access
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Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems
Journal of Inequalities and Applications volume 2010, Article number: 842715 (2010)
Abstract
By using the concept of Fréchet differentiability of mapping, we present the Kuhn-Tucker optimality conditions for weakly efficient solution, Henig efficient solution, superefficient solution, and globally efficient solution to the vector equilibrium problems with constraints.
1. Introduction
Recently, some authors have studied the optimality conditions for vector variational inequalities. Giannessi et al. [1] turned the vector variational inequalities with constraints into vector variational inequalities without constraints and gave sufficient conditions for the efficient solutions and the weakly efficient solutions to the vector variational inequalities in . By using the concept of subdifferential of the function, Morgan and Romaniello [2] gave the scalarization and Kuhn-Tucker-like conditions for the weak vector generalized quasivariational inequalities in Hilbert space. Yang and Zheng [3] gave the optimality conditions for approximate solutions of vector variational inequalities in Banach space. On the other hand, some authors have derived the optimality conditions for weakly efficient solutions to vector optimization problems (see [4–19]).
Vector variational inequality problems and vector optimization problems, as well as several other problems, are special realizations of vector equilibrium problems (see [20, 21]); therefore, it is important to give the optimality conditions for the solution to the vector equilibrium problems for in this way we can turn the vector equilibrium problem with constraints to a corresponding scalar optimization problem without constraints, and we can then determine if the solution of the scalar optimization problem is a solution of the original vector equilibrium problem. Under the assumption of convexity, Gong [22] investigated optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to vector equilibrium problems with constraints and obtained that the weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems with constraints are equivalent to solution of corresponding scalar optimization problems without constraints, respectively. Qiu [23] presented the necessary and sufficient conditions for globally efficient solution under generalized cone-subconvexlikeness. Gong and Xiong [24] weakened the convexity assumptions in [22] and obtained necessary and sufficient conditions for weakly efficient solution, too.
In this paper, by using the concept of Fréchet differentiability of mapping, we study the optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to the vector equilibrium problems. We give Kuhn-Tucker necessary conditions to the vector equilibrium problems without convexity conditions and Kuhn-Tucker sufficient conditions with convexity conditions.
2. Preliminaries and Definitions
Throughout this paper, let be a real normed space, let
and
be real partially ordered normed spaces, and let
and
be closed convex pointed cones with
, where
denotes the interior of the set
.
Let and
be the topological dual spaces of
and
, respectively. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ1_HTML.gif)
be the dual cones of and
, respectively. Denote the quasi-interior of
by
, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ2_HTML.gif)
Let be a nonempty subset of
. The cone hull of
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ3_HTML.gif)
Denote the closure of by
and interior of
by
.
A nonempty convex subset of the convex cone
is called a base of
, if
and
. It is easy to see that
if and only if
has a base.
Let be a base of
. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ4_HTML.gif)
By the separation theorem of convex sets, we know .
Denote the closed unit ball of by
. Suppose that
has a base
. Let
. It is clear that
. The
will be used for the rest of the paper. For any
, denote
cone
, then cl
is a closed convex pointed cone, and
, for all
(see [25]).
Let be a nonempty open convex subset, and let
,
be mappings.
We define the constraint set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ5_HTML.gif)
and consider the vector equilibrium problems with constraints (for short, VEPC): find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ6_HTML.gif)
where is a convex cone in
.
Definition 2.1.
If , a vector
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ7_HTML.gif)
is called a weakly efficient solution to the VEPC.
For each , we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ8_HTML.gif)
Definition 2.2 (see [22]).
Let have a base
. A vector
is called a Henig efficient solution to the VEPC if there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ9_HTML.gif)
Definition 2.3 (see [22]).
A vector is called a globally efficient solution to the VEPC if there exists a point convex cone
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ10_HTML.gif)
Definition 2.4 (see [22]).
A vector is called a superefficient solution to the VEPC if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ11_HTML.gif)
where is closed unit ball of
.
Definition 2.5.
Let be a real linear space, and let
be a real topological linear space. Let
be a nonempty subset of
, and let a mapping
and some
be given. If for some
the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ12_HTML.gif)
exists, then is called the Gâteaux derivative of
at
in the direction
. If this limit exists for each direction
, the mapping
is called Gâteaux differentiable at
.
Definition 2.6.
Let and
be real normed spaces, and let
be a nonempty open subset of
. Moreover, let a mapping
and some
be given. If there exists a continuous linear mapping
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ13_HTML.gif)
then is called the Fréchet derivative of
at
and
is called Fréchet differentiable at
.
Remark 2.7.
By [26, Lemma .18], we can see that if
is Fréchet differentiable at
, then
is Gâteaux differentiable at
and the Fréchet derivative of
at
is equal to the Gâteaux derivative of
at
in each direction
.
Definition 2.8.
Let and
be real linear spaces, let
be a pointed convex cone in
, and let
be a nonempty convex subset of
. A mapping
is called
-convex, if for all
and all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ14_HTML.gif)
Lemma 2.9 (see [18]).
Assume that pointed convex cone has a base
; then one has the following.
(i)For any ,
.
(ii)For any , there exists some
with
.
(iii), and when
is bounded and closed, then
, where
is the interior of
in
with respect to the norm of
.
3. Optimality Condition
In this section, we give the Kuhn-Tucker necessary conditions and Kuhn-Tucker sufficient conditions for weakly efficient solution, Henig efficient solution, globally efficient solution, and superefficient solution to the vector equilibrium problems with constraints.
Let be given. Denote the mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ15_HTML.gif)
By the proof of Theorem .20 in [26] or from the definition, we have the following lemma.
Lemma 3.1.
Let and
be real normed spaces, let
be a nonempty open convex subset of
, and let
be closed convex pointed cone in
. Assume that
is
-convex and
is Gâteaux differentiable at
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ16_HTML.gif)
Theorem 3.2.
Let ,
, and
be real normed spaces, and let
and
be closed convex pointed cones in
and
with
and
, respectively. Let
. Assume that
and
are Fréchet differentiable at
Furthermore, assume that there exists
such that
. If
is a weakly efficient solution to the VEPC, then there exist
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ17_HTML.gif)
Proof.
Assume that is a weakly efficient solution to the VEPC. Define the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ18_HTML.gif)
Since and
are linear operators, we can see that
is a nonempty open convex set. We claim that
. If not, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ19_HTML.gif)
From Remark 2.7, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ20_HTML.gif)
Since and
are open sets, there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ21_HTML.gif)
From ,
, and
, we can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ22_HTML.gif)
Since is a convex set,
. Thus, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ23_HTML.gif)
This contradicts that is a weakly efficient solution to the VEPC. Thus
. Noting that
is an open set, by the separation theorem of convex sets (see [26]), there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ24_HTML.gif)
Let . Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ25_HTML.gif)
It is clear that for every ,
,
,
we have
and
. By (3.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ26_HTML.gif)
We can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ27_HTML.gif)
Since is a closed convex cone,
. By the continuity of
, we can see that
for all
. That is,
. Similarly, we can show that
. We also have
. In fact, if
, from (3.10) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ28_HTML.gif)
By assumption, there exists such that
; thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ29_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ30_HTML.gif)
In particular, we have , and we get
. This is a contradiction. Thus,
. It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ31_HTML.gif)
for all ,
,
By (3.10), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ32_HTML.gif)
for all ,
,
Letting ,
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ33_HTML.gif)
It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ34_HTML.gif)
for all ,
,
. By (3.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ35_HTML.gif)
Letting , we obtain
. Noting that
and
, we have
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ36_HTML.gif)
From (3.19) and (3.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ37_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ38_HTML.gif)
This completes the proof.
Remark 3.3.
The condition that there exists such that
is a generalized Slater constraint condition. In fact, from the proof of Theorem 3.2, we obtain that there exists
such that
and
.
On the other hand, if is Gâteaux differentiable at
, and
is
-convex on
, and
then
. In fact, by Lemma 3.1, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ39_HTML.gif)
Theorem 3.4.
Let ,
, and
be real normed spaces, let
be a closed convex pointed cone with
, and let
be a closed convex pointed cone with
, and let
. Assume that
and
are Gâteaux differentiable at
,
is
-convex on
, and
is
-convex on
. If there exist
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ41_HTML.gif)
then is a weakly efficient solution to the VEPC.
Proof.
Since the mappings and
are Gâteaux differentiable at
,
is
-convex on
, and
is
-convex on
, from Lemma 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ42_HTML.gif)
From , and (3.26), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ43_HTML.gif)
By (3.27), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ44_HTML.gif)
We will show that is a weakly efficient solution to the VEPC. If not, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ45_HTML.gif)
From and the above statement, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ46_HTML.gif)
Noticing , we have
; so
because of
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ47_HTML.gif)
This contradicts (3.30). Hence, is a weakly efficient solution to the VEPC.
From Theorems 3.2 and 3.4, and Remark 2.7, we get the following corollary.
Corollary 3.5.
Let ,
, and
be real normed spaces, let
be a closed convex pointed cone with
, and let
be a closed convex pointed cone with
, and let
. Assume that
and
are Fréchet differentiable at
,
is
-convex on
, and
is
-convex on
. Furthermore, assume that there exists
such that
. Then
is a weakly efficient solution to the VEPC if and only if there exist
,
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ48_HTML.gif)
The concept of weakly efficient solution to the vector equilibrium problem requires the condition that the ordering cone has an nonempty interior. If the ordering cone has an empty interior, we cannot discuss the property of weakly efficient solutions to the vector equilibrium problem. However, if the ordering cone has a base, we can give necessary conditions and sufficient conditions for Henig efficient solutions and globally efficient solutions to the vector equilibrium problems with constraints.
Theorem 3.6.
Let ,
, and
be real normed spaces, let
be a closed convex pointed cone in
with a base, let
be a closed convex pointed cone in
with
, and let
. Assume that
and
are Fréchet differentiable at
. Furthermore, Assume that there exists
such that
. If
is a Henig efficient solution to the VEPC, then there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ49_HTML.gif)
Proof.
Assume that is a Henig efficient solution to the VEPC. By the definition, there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ50_HTML.gif)
Define the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ51_HTML.gif)
It is clear that is a nonempty open convex set and
. By the separation theorem of convex sets, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ52_HTML.gif)
Let . Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ53_HTML.gif)
Hence, for every ,
,
,
, we have
and
; this implies that
,
. In a way similar to the proof of Theorem 3.2, we have
. By Lemma 2.9, we can see that
. It is clear that
for all
,
,
By (3.38), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ54_HTML.gif)
In a way similar to the proof of Theorem 3.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ55_HTML.gif)
From (3.40) and (3.41), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ56_HTML.gif)
This completes the proof.
Theorem 3.7.
Let ,
, and
be real normed spaces, let
be a closed convex pointed cone in
with a base, let
be a closed convex pointed cone in
with
, and let
. Assume that
and
are Gâteaux differentiable at
,
is
-convex on
, and
is
-convex on
. If there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ57_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ58_HTML.gif)
then is a Henig efficient solution to the VEPC.
Proof.
From ,
, (3.43), and (3.44), in a way similar to the proof of Theorem 3.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ59_HTML.gif)
We will show that is a Henig efficient solution to the VEPC; that is, there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ60_HTML.gif)
Suppose to the contrary that for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ61_HTML.gif)
Then by , and by Lemma 2.9, there exists some
such that
. For this
we have
Thus, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ62_HTML.gif)
By , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ63_HTML.gif)
Notice , we have
, and thus we obtain
because of
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ64_HTML.gif)
This contradicts (3.45). Hence, is a Henig efficient solution to the VEPC.
Corollary 3.8.
Let ,
, and
be real normed spaces, and
be a closed convex pointed cone with a base, let
be a closed convex pointed cone with
, and let
. Assume that
and
are Fréchet differentiable at
,
is
-convex on
, and
is
-convex on
. Furthermore, assume that there exists
such that
. Then
is a Henig efficient solution to the VEPC if and only if there exist
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ65_HTML.gif)
Remark 3.9.
If has a bounded closed base
, in view of Lemma 2.9, we have
; besides,
is a superefficient solution to the VEPC if and only if
is a Henig efficient solution to the VEPC (see [22]). Hence, by Corollary 3.8, we have the following corollary.
Corollary 3.10.
Let ,
, and
be real normed spaces, let
be a closed convex pointed cone with a bounded closed base, let
be a closed convex pointed cone with
, and let
. Assume that
and
are Fréchet differentiable at
,
is
-convex on
, and
is
-convex on
. Furthermore, assume that there exists
such that
. Then
is a superefficient efficient solution to the VEPC if and only if there exists
(with respect to the norm topology of
),
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ66_HTML.gif)
Theorem 3.11.
Let ,
, and
be real normed spaces, let
be a closed convex pointed cone in
with a base, let
be a closed convex pointed cone in
with
, and let
. Assume that
and
are Fréchet differentiable at
, and there exists
such that
. If
is a globally efficient solution to the VEPC, then there exist
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ67_HTML.gif)
Proof.
Assume that is a globally efficient solution to the VEPC. By the definition, there exists a pointed convex cone
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ68_HTML.gif)
Define the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ69_HTML.gif)
Since is a convex cone, we know that
is a nonempty open convex set and
. By the separation theorem of convex sets, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ70_HTML.gif)
Let . Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ71_HTML.gif)
Hence, for every ,
,
,
, we have
and
. This implies that
,
, and therefore
because of
. It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ72_HTML.gif)
for all ,
,
By (3.56), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ73_HTML.gif)
for all ,
,
Letting , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ74_HTML.gif)
In a way similar to the proof of Theorem 3.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ75_HTML.gif)
From (3.60) and (3.61), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ76_HTML.gif)
This completes the proof.
Theorem 3.12.
Let ,
, and
be real normed spaces, let
be a closed convex pointed cone in
with a base, let
be a closed convex pointed cone in
with
, and let
. Assume that
and
are Gâteaux differentiable at
,
is
-convex on
, and
is
-convex on
. If there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ77_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ78_HTML.gif)
then is a globally efficient solution to the VEPC.
Proof.
From , (3.63), and (3.64), in a way similar to the proof of Theorem 3.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ79_HTML.gif)
We will show that is a globally efficient solution to the VEPC; that is, there exists a pointed convex cone
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ80_HTML.gif)
Suppose to the contrary that for any pointed convex cone with
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ81_HTML.gif)
By , we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ82_HTML.gif)
We have , and
is a pointed convex cone. By (3.67), there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ83_HTML.gif)
By the definition of , we have that
. Noticing
, we have
, and so
because of
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ84_HTML.gif)
This contradicts (3.65). Hence, is a globally efficient solution to the VEPC.
Corollary 3.13.
Let ,
, and
be real normed spaces, let
be a closed convex pointed cone with a base, let
be a closed convex pointed cone with
, and let
. Assume that
and
are Fréchet differentiable at
,
is
-convex on
, and
is
-convex on
. Furthermore, assume that there exists
such that
. Then
is a globally efficient solution to the VEPC if and only if there exist
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F842715/MediaObjects/13660_2010_Article_2272_Equ85_HTML.gif)
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Acknowledgments
This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province (2008GZS0072), China.
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Wei, ZF., Gong, XH. Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems. J Inequal Appl 2010, 842715 (2010). https://doi.org/10.1155/2010/842715
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DOI: https://doi.org/10.1155/2010/842715