Continuity of Solution Mappings for Parametric Set Optimization Problems under Partial Order Relations ()
1. Introduction
Set-valued optimization which is a generalization of vector optimization has been studied and applied in many fields, such as engineering, mathematical finance, medicine, robust and fuzzy optimization; see [1] [2] [3] [4] and the references therein.
As we know, for set-valued optimization problems, there are two types of criteria which are vector optimization criterion and set optimization criterion. Based on the vector optimization criterion, discussing the continuity of solution set mapping for set-valued vector optimization problems is very similar to discuss vector variational inequalities or vector equilibrium problems [5]. However, this criterion is not always suitable for all types of set-valued optimization problems. To surmount the shortcoming, Kuroiwa [6] [7] introduced and discussed the set optimization criterion which is based on a comparison among the values of objective set-valued mapping. The method seems to be more natural than classical methods, whenever one needs to consider preferences over sets. Many research results have been studied for parametric set optimization problems under different kinds of set order relations, such as the optimality conditions, convexity, well-posedness, existence, duality theory and algorithms; see [8] - [15] and the references therein.
It is well-known that stability analysis plays a key role in optimization theories and related applications [16] [17] [18] [19]. Especially, the continuity of solution mappings to perturbed optimization problem is one of the important contents. The semicontinuity of solution mappings for parametric set optimization problems was firstly investigated by Xu and Li in [20]. They obtained the continuity of the weak minimal solution set mapping and the minimal solution set mapping to a parametric u-set optimization problem under strong assumption conditions. Subsequently, Xu and Li [21] studied the continuity of the minimal solution set mappings to parametric set optimization problems by the semicontinuity of the u-lower level mappings. In terms of the continuity of the level mappings, Khoshkhabar-amiranloo [16] discussed the stability of the minimal solutions of set optimization problems by using the C-Hausdorff continuity instead of the continuity in the sense of Berge. Recently, Chen et al. [22] used another proof method to get the semicontinuity of parametric set optimization problems with lower set less order relation under some strong assumption conditions. Through the above papers, we can find that they all need the continuity assumptions of set-valued objective mappings when they proved the semicontinuity of solution mappings for parametric set optimization problems.
Recently, Karaman [23] introduced two new order relations on family of sets called m1 and m2 set order relations which are partial order relations and are smaller than the vector order relations introduced in [24]. Preechasilp and Wangkeeree [25] gave some sufficient conditions for the semicontinuity of the solution mapping to a parametric set optimization problem with (converse) m1-property assumptions under m1 set order relation. Since it is difficult to check the (converse) m1-property assumptions, this limits the applications of stability of set optimization problems (See Example 5.1 and 5.2). To the best of our knowledge, there are few stability results on the solution mapping to parametric set optimization problems in terms of nonlinear scalarization methods.
From what has been mentioned above, the first aim of our paper is to get the semicontinuity of solution mappings to parametric set optimization problems when the set-valued objective mapping is not continuous. The second aim of the paper is to obtain the semicontinuity of solution mappings to parametric set optimization problems via nonlinear scalarization methods. Hence, we define two types of new monotonicity of the set-valued mapping by using two nonlinear scalarization functions under partial order relation. The third aim of this paper is to investigate the semicontinuity of solution mapping of parametric set optimization problems under weaker and simpler assumptions. We give some sufficient conditions for the semicontinuity and closedness of solution mappings for parametric set optimization problems. In our results, we have no (converse) m1-property assumptions for objective mappings and the assumption (iv) of Theorem 4.2 in [21] and [26]. Moreover, the continuity of the objective mapping is replaced by some monotonicity assumptions. So, our results do not need any additional assumptions to get the lower semicontinuity of solution mapping of parametric set optimization problems. Our results extend and improve the corresponding ones of [21] [25] [26].
The rest of this paper is organized as follows. In Section 2, some basic concepts and preliminary results about the semicontinuity of set-valued mappings are introduced. In Section 3, by using the nonlinear scalarization functions, two types of new monotonicity of the set-valued mapping are defined. In Section 4, the semicontinuity and closedness of the solution mappings for the parametric set optimization problems are discussed without any continuities of the set-valued objective mapping. In Section 5, we give the comparisons between our results in Section 4 and ones in [16] [21] [22] [25] [26]. In Section 6, we give some concluding remarks.
2. Preliminaries
Throughout this paper, let X, Y and Z be real normed linear spaces, and let
be a nonempty subset of Z,
and
are closed, convex and pointed cones. Let
be the family of all nonempty subsets of Y, and
be the family of all nonempty bounded subsets of Y. Let
be the topological dual space of X and
be the dual cone of K, defined by
Denote the quasi-interior of
by
, i.e.,
Proposition 2.1. [24]
(i) If K is a closed convex cone in a real locally convex linear space X, then
.
(ii) Let
. If
, then
.
We now recall some order relations on
. The first one is lower set less (
) and upper set less (
) order relations on
.
Definition 2.1. [24] Let
be arbitrarily closen sets.
(i) The upper set less order relations (
) is defined by
,
(ii) The lower set less order relations (
) is defined by
.
(
) and (
) order relations have been extensively studied in many literature. Recently, by using properties of Minkowski difference, Karaman et al. established the following new order relations on a family of sets.
Definition 2.2. [23] Let
.
The set
is called Minkowski (Pontryagin) difference of B and A.
Moreover, when
, the strictly version of
and
are defined by
Remark 2.1. It is easily seen that, for
,
implies
and
implies
. However, the converse is not true. The detailed counter-example sees in [23].
Let M be a nonempty set of X and
be a set-valued mapping. The set optimization problem is defined as follows:
Definition 2.3. [23] An element
is said to be
(i) a m1-minimal solution of (SOP) if there does not exist any
with
such that
, that is, either
or
for any
;
(ii) a m2-minimal solution of (SOP) if there does not exist any
with
such that
, that is, either
or
for any
;
If the set M and the mapping F are perturbed by a parameter
which varies over a set
, we definite the following parametric set optimization problem (PSOP):
where
,
, F and M are set-valued mappings,
. For each
, we denote
and
by the m1-minimal solution mapping and m2-minimal solution mapping to (PSOP), respectively. That is,
is defined as follows:
In this paper, we assume that
for each
,
. Next, we will discuss the upper and lower semicontinuities of
at a point
. Now, let us recall some basic definitions and their properties.
Definition 2.4. [27] Suppose that
is a set-valued mapping, and
is given.
(i)
is called lower semicontinuous (l.s.c) at
iff for any open set
with
, there exists a neighborhood
of
such that
, for all
.
(ii)
is called upper semicontinuous (u.s.c) at
iff for any open set
with
, there exists a neighborhood
of
such that
, for all
.
(iii)
is called closed at
iff for each sequence
,
, it follows that
.
is said to be l.s.c (resp. u.s.c) on
, if and only if it is l.s.c (resp. u.s.c) at each
.
is said to be continuous on
if and only if it is both l.s.c and u.s.c on
.
Proposition 2.2. [27] [28]
(i)
is l.s.c at
if and only if for any sequence
with
and any
, there exists
such that
.
(ii) If
has compact values at
(i.e.,
is a compact set), then
is u.s.c at
if and only if for any sequence
with
and any
, there exist
and a subsequence
of
, such that
.
3. Monotonicity via Scalarization
In this section, new monotonicity definitions will be introduced by the scalarization method. When we want to obtain the semicontinuity results of solution mappings of (SOP), we always use some assumptions that include the continuity of the objective mapping F (see [16] [21] [22] ). In this paper, we will get the semicontinuity results of solution mappings of (PSOP) without the continuity of the objective mapping F by using a new monotonicity condition. Firstly, we recall two nonlinear scalarization functions and some properties of these functions.
Definition 3.1. [23] Let the functions
and
as
for all
, where
.
Proposition 3.1. [23] Let
and
be finite. Then, the following conditions are satisfied:
(i) when
is compact and C is closed,
(ii)
Next, two types of new monotonicity of the set-valued mapping will be defined by the above nonlinear scalarization functions.
Definition 3.2. Let
. A set-valued mapping
is said to be
(i) m1-monotonically increasing on
iff
, one has that
(ii) m2-monotonically increasing on
iff
, one has that
Remark 3.1. If G is a single-valued mapping and
,
. It is easy to see that
. Therefore, k and e are all positive real numbers. Obviously,
and
, thus
Then,
Noting that
and
, we have that
We can see that G is reduced to the well-known monotonically increasing function.
Definition 3.3. Let
. A set-valued mapping
is said to be
(i) m1-monotonically decreasing on
iff
, one has that
(ii) m2-monotonically decreasing on
iff
, one has that
Similarly, if all the conditions in Remark 2 hold, we also can get that
This shows that G is reduced to the well-known monotonically decreasing function. Next, we give an example to illustrate Definitions 3.2 and 3.3.
Example 3.1. Let
,
,
,
and
. Let
. A set-valued mapping
is defined by
It follows from a direct computation that
and
. For every
, we have that
. Then
and
Hence, G is m1-monotonic decreasing on
. When
, we can find that G is m1-monotonic increasing on
.
4. Semicontinuity of Solution Set Mapping
In this section, we shall discuss the upper and lower semicontinuities of the m1-minimal solution set mappings
(
) of (PSOP) at
.
Lemma 4.1. Assume that
is m1-monotonically decreasing and bounded with compact values on
for each
. Then,
Proof. “Necessity”. It follows from the linearity of k, Proposition 2.1 (i) and
that
for all
. Since
is m1-monotonically decreasing on
for each
, we have that
. This together with the compactness of
and Proposition 3.1(i) leads to
.
“Sufficiency”. By
and Proposition 3.1 (i), we get that
. Since
is m1-monotonically decreasing on
for each
, we have that
for all
. It follows from the linearity of
and Proposition 2.1 (i) that
.
Lemma 4.2. Assume that
is m1-monotonically increasing with compact values on
for each
. Then, for any
, one has
Proof. Since
and
is compact valued on
for each
, we get that
by Proposition 3.1 (i). Besides
is m1-monotonicaly increasing on
, so we have that
for all
. It follows from Proposition 2.1(ii) that
.
Lemma 4.3. Assume that
is m1-monotonically increasing with compact values on
for each
. Then, for any
, one has
Proof. It follows from the linearity of k, Proposition 2.1 (i) and
that
for some
. Since
is m1-monotonically increasing on
for each
, we have that
, which implies that
by the compact values of
and Proposition 3.1 (i).
Theorem 4.1. Let
. Suppose that the following conditions are satisfied:
(i) M is continuous at
and
is a compact convex set;
(ii)
is m1-monotonically increasing with compact values on
for each
.
Then,
is u.s.c and closed at
.
Proof. Suppose that
is not u.s.c at
. Then there exists a neighborhood
of
, and for any neighborhood V of
, there exists
such that
. Thus, there exists a sequence
with
such that
,
. This implies that there exists
such that
(1)
It follows from
that
. Since M is u.s.c with compact values at
and by Proposition 2.2 (ii), there exist
and a subsequence
of
such that
. Without any loss of generality, we assume that
.
Now, we prove that
. Suppose that
, then there exists
with
such that
(2)
As M is l.s.c at
and by Proposition 2.2 (i), there exists
such that
. It follows from (2) and Lemma 4.2 that
that is,
. Noting that
and
, we get that
. Since
, we have that
. It follows from the closedness of K that
for n large enough. By Lemma 4.3, we can get that
(3)
On the other hand, from
, we have that there exists
such that
by Proposition 2.1 (i). Since
is strictly m1-monotonicaly increasing on
for each
, we have
and so
by Proposition 3.1 (i). This implies that
. Combining with (3), we have
, which contradicts
. Therefore,
. It follows form
that
. Noting that
, we have
for n large enough, which contradicts with (1). Therefore,
is u.s.c at
.
Next, we show that
is closed at
. Take
with
and
. Since
and M is u.s.c with compact values at
, we obtain that
. Then by virtue of the same proof as above, we get that
. Furthermore,
is compact by assumption (i) and
.
Now, we give an example to illustrate Theorem 4.1.
Example 4.1 Let
,
,
,
for each
,
and
. Let
,
and set-valued mapping
as follows:
It is easy to see that the assumptions in Theorem 4.1 are satisfied. It follows from a direct computation that
and
when
. Hence,
is u.s.c and closed at
.
Theorem 4.2. Let
. Suppose that the following conditions are satisfied:
(i) M is continuous at
and
is a compact convex set;
(ii)
is m1-monotonic decreasing and bounded with compact values on
for each
.
Then,
is l.s.c at
.
Proof. Suppose that
is not l.s.c at
. Then there exist
and a neighborhood
of
, for any neighborhood U of
, there exists
such that
. Hence, there exists a sequence
with
such that
(4)
Since
and M is l.s.c at
, there exists
such that
. Therefore,
for n large enough.
Now, we prove that
. Suppose that
, then there exists
with
such that
(5)
Since M is u.s.c with compact values at
, there exists
and a subsequence
of
such that
. Without any loss of generality, we assume that
. It follows from (5) and Lemma 4.1 that
Noting that
, we get that
. By the closedness of K, we have that
. Again from Lemma 4.1, then we get that
(6)
Noting that
, then
which contradicts with (6). Hence,
for n large enough. This contradicts with (4). Therefore,
is l.s.c at
.
Now, we give an example to illustrate Theorem 4.2.
Example 4.2. Let
,
,
,
for each
,
and
. Let
,
and set-valued mapping
as follows:
It is easy to see that
is m1-monotonic decreasing on
. It follows from a direct computation that
for each
. Hence,
is l.s.c at
.
Based on the similar technique as perviously shown, we also can get the following Corollaries.
Corollary 4.1. Let
. Suppose that the following conditions are satisfied:
(i) M is continuous at
and
is a compact convex set;
(ii)
is m2-monotonic increasing and bounded with compact values on
for each
.
Then,
is u.s.c and closed at
.
Corollary 4.2 Let
. Suppose that the following conditions are satisfied:
(i) M is continuous at
and
is a compact convex set;
(ii)
is m2-monotonic decreasing and bounded with compact values on
for each
.
Then,
is l.s.c at
.
In this section, we shall give the comparisons between the results (Theorem 4.1 and Theorem 4.2) in Section 4 and ones in [16] [21] [22] [25] [26].
Remark 5.1. The proof of Theorem 4.1 is different from one of Theorem 3.4 of Preechasilp and Wangkeeree [25]. Moreover, in Theorem 4.1, we do not use converse m1-property for set-valued objective mapping F, which is applied in Theorem 3.4 of Preechasilp and Wangkeeree [25]. And the order relations we used are smaller than the set less order relations in [16] [21] [22] [26].
The following example is given to show that the converse m1-property in [25] is unnecessary in Theorem 4.1 for this paper.
Example 5.1. Let
,
,
,
for each
,
and
. Let
,
and set-valued mapping
as follows:
It is easy to see that the assumptions in Theorem 4.1 are satisfied. It follows from a direct computation that
for each
. Hence,
is u.s.c at
. Besides, F does not satisfy the converse m1-property at
with respect to
. Indeed, we obtain that
. However, for any sequences
with
,
with
and
with
, one has
Remark 5.2. It is worth noting that the proof of Theorem 4.2 is very different from one in Theorem 3.4 of Preechasilp and Wangkeeree [25]. In Theorem 4.2, we do not use m1-property for set-valued objective mapping F, which is applied in Theorem 3.8 of Preechasilp and Wangkeeree [25]. Moreover, we do not use the assumption (iv) in Theorem 4.2 of Xu and Li [21] and Theorem 4.2 of Liu and Wei [26] either. And the order relations we used are smaller than the set less order relations in [16] [21] [22] [26].
The following example is given to show that the m1-property in [25] and the assumption (iv) in Theorem 4.2 of [21] and Theorem 4.2 of [26] are dispensable in some cases.
Example 5.2. Let
,
,
,
for each
,
and
. Let
,
and set-valued mapping
as follows:
It is easy to see that the assumptions in Theorem 4.2 are satisfied. It follows from Example 4.5 of [26] that the assumption (iv) in Theorem 4.2 of [21] and Theorem 4.2 of [26] does not hold. Besides, F does not satisfy the m1-property at
with respect to
. Indeed, we obtain that
. However, for any sequences
with
,
with
and
with
, one has
By a direct computation that
for each
. Hence,
is l.s.c at
.
6. Conclusions
By using two scalarizing functions which are proposed by partial order relations, some new monotonicity concepts of set-valued mapping are introduced in this paper. The continuity of the solution mapping for a parametric optimization problem is obtained under these kinds of monotonicity concepts and some suitable assumptions which do not need the continuity of the set-valued objective mapping. Especially, when we give the lower semicontinuity of the solution mapping, we do not use assumptions presented in [21] [25] [26]. And the order relations we used are smaller than the set less order relations in [16] [21] [22] [26].
The findings of the paper are summarized as follows.
(i) We put forward some new monotonicity concepts of set-valued mapping.
(ii) Under new assumptions which are different from ones presented in [21] [25] [26], we give the lower semicontinuity and the upper semicontinuity of the solution mapping of the problem (PSOP) with partial order relations.
Acknowledgements
The author would like to express their deep gratitude to the anonymous referees for their valuable comments and suggestions, which helped to improve the paper.