1. Introduction
The research of variational inequality problems is a part of development in the theory of optimization since optimization problems can often be specialized to the solution of variational inequality problems. It is very important to point out that these theories pertain to more than just optimization problems and there in lies much of their attractiveness. Several authors have presented numerous fascinating results on variational inequality problems; see cited references here [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12].
In 1984, Loridan [
13] studied the concept of
-efficient solutions for vector minimization problems where the function to be optimized has its values in the
space, which is a generalization of the classical problem for Pareto solution. Later in 1986, White [
14] extended
-optimality for scalar problems to vector maximization problems, or efficiency problems, with
m objective functions defined on a subset of
. In 1993, Burke et al. [
15] studied the concept of weak sharp minima for scalar optimization problem which was motivated by the application in convex and convex composite mathematical programming.
Recently, in 2016, Zhu [
16] suggested the necessary optimal conditions for the weak local sharp efficient solution of a constrained multi-objective optimization problem by using the generalized Fermat formula, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Fréchet subdifferentials, and the sum rule for Mordukhovich subdifferentials, and also got the some sufficient optimal conditions respectively for the local and global weak sharp efficient solutions of such a multi-objective optimization problem, by applying the approximate projection method, and some appropriate convexity and affineness conditions.
Motivated by the ideas of local sharp and weak local sharp efficient solutions, we define the local sharp vector variational type inequalities and Minty local sharp vector variational type inequalities, and establish the relations between local (or Minty local) sharp vector variational type inequality and vector optimization problems involving generated by locally Lipschitzian mappings.
2. Preliminaries
Throughout this paper,
denotes the
n-dimensional Euclidean space with a norm
Let
X be a nonempty convex subset of
The distance function
is defined by
A vector valued function
is said to be
-Lipschitz continuous if there exists a number
such that
Definition 1. Letbe a function. A lower semicontinuous functionis said to be approximate-convex atif for anythere existssuch that, for all, Definition 2. Letbe a function. A functionis said to be
- (i)
approximate-pseudoconvex type-I atif for anythere existssuch that, wheneverandthen - (ii)
approximate-pseudoconvex type-(strictly approximate-pseudoconvex type-) atif for anythere existssuch that, wheneverandthen - (iii)
approximate-quasiconvex type-I atif for anythere existssuch that, wheneverandthen - (iv)
approximate-quasiconvex type-(strictly approximate-quasiconvex type-) atif for anythere existssuch that, wheneverandthen
(VOP): A vector optimization problem (VOP) is formulated as follows:
where,
with
is a vector valued function.
Definition 3. - (i)
A vectoris said to be a local sharp efficient solution of (VOP), if for anythere exists a-neighborhood of, such that for all, - (ii)
A vectoris said to be a weak local sharp efficient solution of (VOP), if for any, there exists a-neighborhood of, such that for all,where
3. Local Sharp Vector Variational Type Inequalities
In this section, we consider local sharp and weak local sharp formulations of vector variational type inequality problems as follows:
(LSVVTI): For finding
there exists a
-neighborhood of
and for any
, such that
and
(WLSVVTI): For finding
there exists a
-neighborhood of
and for any
such that
and
where
We note that, if is a solution of (LSVVTI), then is also a solution of (WLSVVTI).
(
1) reduces to local sharp vector variational inequalities (
LSVVI): for finding
there exists a
-neighborhood of
and for any
, such that
and
In addition, (
2) reduces to weak local sharp vector variational inequalities (
WLSVVI) for finding
there exists a
-neighborhood of
and for any
such that
and
where
Again, we note that if
, then the solution of (LSVVI) is also a solution of (AVVI)
(defined by [
17]), but the converse need not be true:
Example: consider the function
where
and
If we take
then for any
there does not exist any
such that
that is,
is a solution of (AVVI)
. When
then for every
and
, we do not have
that is,
is not a solution of (LSVVI).
Unless otherwise stated, the following condition (C) is always assumed in this section.
- (C)
For the bi-function
and the mappings
for all
First of all, in this section, we give the relationship between the solutions of local sharp vector variational type inequalities (LSVVTI) and local sharp (or weak local sharp) efficient solutions of vector optimization problem (VOP).
Now we are at the stage of introducing and proving the main theorems:
Theorem 4. Let be a function and be locally Lipschitz and approximate η-convex at and satisfies the condition (C). If solves (LSVVTI), then it is a local sharp efficient solution of (VOP).
Proof. Contrary, assume that
is not a local sharp efficient solution of (VOP). Then, for any
and
, there exists
such that
it implies,
Since
is approximate
-convex at
there exists
such that for
we have
Hence, it follows from (
5) and (
6) that
Therefore, we have
it implies that
This is a contradiction to the fact that
solves (LSVVTI). □
In following theorem, we obtain the converse result of Theorem 4 by assuming the approximate -convexity of instead of .
Theorem 5. For each let η and be same as in Theorem 4, be approximate η-convex at , and satisfies the condition (C). Then the converse statement of Theorem 4 is true.
Proof. Suppose that
is not a solution of the (LSVVTI). Then, for any
and
there exists
and
, such that
it implies
Since
is approximate
-convex at
for any
there exists
such that for
we have
we can write it as
From (
7) and (
8), we have
Hence, we have
this implies that
which is a contradiction to the fact that
is a local sharp efficient solution of (VOP). □
In next theorem, the same result of Theorem 4 is obtained by substituting the strictly approximate -quasiconvex type-II condition instead of approximate -convexity condition on .
Theorem 6. Let η and be the same as in Theorem 4, be a strictly approximate η-quasiconvex type-II at , for each , and satisfies the condition (C). If solves (LSVVTI), then it is a local sharp efficient solution of (VOP).
Proof. Assume that
is not a local sharp efficient solution of (VOP). Then, for any
and
, there exists
such that
it implies,
Since
is a strictly approximate
-quasiconvex type-II at
for any
there exists
such that by setting
we have
implies that
This means that
is not a solution of (LSVVTI). □
In the following theorem, we can get the the generalization of Theorem 5 by assuming the strictly approximate -pseudoconvex type-II condition on .
Theorem 7. For each let η and be same as in Theorem 4, be a strictly approximate η-pseudoconvex type-II at , and satisfies the condition (C). If is a weak local sharp efficient solution of (VOP), then it is also a solution of (LSVVTI).
Proof. Suppose that
is not a solution of (LSVVTI). Then, for any
and
there exists
and
, such that
Hence, we have,
and we can rewrite as
Since
is a strictly approximate
-pseudoconvex type-II at
for any
there exists
such that, for
we have
Therefore, we have
this implies that
Therefore, we show that
is a local weak sharp efficient solution of (VOP). This completes the proof. □
4. Minty Local Sharp Vector Variational Type Inequalities
In this section, we present relationship between the solutions of Minty local sharp vector variational type inequalities (MLSVVTI) and local sharp (or weak local sharp) efficient solutions of vector optimization problem (VOP).
Now, we consider Minty local sharp and Minty weak local sharp formulations of vector variational type inequality problems as follows:
(MLSVVTI): Finding
there exists a
-neighborhood of
and any
such that
and
(MWLSVVTI): For finding
there exists a
-neighborhood of
and any
such that
and
where
Theorem 8. For each let η and be same as in Theorem 4, be approximate η-convex at , and satisfies the condition (C). If solves (MLSVVTI), then is a local sharp efficient solution of (VOP).
Proof. Suppose that
is not a local sharp efficient solution of (VOP). Then, for any
and
there exists
such that
it implies,
Since
is approximate
-convex at
for any
there exists
such that, for
we have
It follows from (
14) and (
15), we have
that is,
implies that
which is a contradiction to the fact that
solves (MLSVVTI). This completes the proof. □
In following theorem, we can get the converse result of Theorem 8 by assuming the approximate -convexity of instead of .
Theorem 9. For each let η and be same as in Theorem 8, be approximate η-convex at , and satisfies the condition (C). If is a local sharp efficient solution of (VOP), then solves (MLSVVTI).
Proof. Suppose that
is not a solution of the (MLSVVTI). Then, for any
and
there exists
and
such that
it implies,
Since
is approximate
-convex at
for any
there exists
such that, for
we have
we can rewrite as
Combining (
16) and (
17), we have
Hence, we have
implies that
This is a contradiction to the fact that
is a local sharp efficient solution of (VOP). □
In following theorem, we can get same result of Theorem 8 by assuming the strictly approximate -quasiconvex type-II condition insted of approximate -convexity on .
Theorem 10. For each let η and be same as in Theorem 8, be a strictly approximate η-quasiconvex type-II at and satisfies the condition (C). If solves (MLSVVTI), then is a local sharp efficient solution of (VOP).
Proof. Assume that
is not a local sharp efficient solution of (VOP). Then, for any
and
there exists
such that
it implies,
Hence, we can rewrite as
Since
is a strictly approximate
-quasiconvex type-II at
for any
there exists
such that, for
we have
That is,
implies that
which is a contradiction to the fact that
solves (MLSVVTI). □
The following theorem is an improvement of the Theorem 9 for the weak local sharp efficient solution of (VOP).
Theorem 11. For each let η and be same as in Theorem 8, be a strictly approximate η-pseudoconvex type-II at and satisfies the condition (C). If is a weak local sharp efficient solution of (VOP), then solves (MLSVVTI).
Proof. On the contrary, assume that
is not a solution of (MLSVVTI). Then, for any
and
there exists
and
such that
Hence, we obtain
it implies,
Since
is a strictly approximate
-pseudoconvex type-II at
for any
there exists
such that, for
we have
This implies that
hence, we have
This is a contradiction to the fact that
is a weak local sharp efficient solution of (VOP). □
5. Conclusions
In this paper, we formulate local (Minty local) sharp vector variational type inequality problems and establish the relationship between local (Minty local) sharp vector variational type inequality and vector optimization problems involving locally Lipschitzian functions; that is, in Theorems 4–7, we give the necessary or sufficient conditions between the local sharp vector variational type inequality (LSVVTI) and vector optimization problems (VOP), and in Theorems 8–11, we give the necessary or sufficient conditions between the Minty local sharp vector variational type inequality (MLSVVTI) and vector optimization problems (VOP), by using the approximate -convexity, strictly approximate -quasiconvex type-II condition, and strictly approximate -pseudoconvex type-II condition at
The results of our research in this paper are generalized, extended, and improved studies of concepts of
-efficient solutions for vector minimization problems [
13],
-optimality for scalar problems to vector maximization problems, or efficiency problems [
14], weak sharp minima for scalar optimization problem [
15], weak local sharp efficient solution of a constrained multi-objective optimization, and the local and global weak sharp efficient solutions of such a multi-objective optimization problem [
16].