- Research Article
- Open access
- Published:
Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization
Journal of Inequalities and Applications volume 2010, Article number: 510838 (2010)
Abstract
By virtue of higher-order adjacent derivative of set-valued maps, relationships between higher-order adjacent derivative of a set-valued map and its profile map are discussed. Some results concerning stability analysis are obtained in parametrized vector optimization.
1. Introduction
Research on stability and sensitivity analysis is not only theoretically interesting but also practically important in optimization theory. A number of useful results have been obtained in scalar optimization (see [1, 2]). Usually, by stability, we mean the qualitative analysis, which is the study of various continuity properties of the perturbation or marginal
function
or map
of a family of parametrized optimization problems. On the other hand, by sensitivity, we mean the quantitative analysis, which is the study of derivatives of the perturbation function.
Some authors have investigated the sensitivity of vector optimization problems. In [3], Tanino studied some results concerning the behavior of the perturbation map by using the concept of contingent derivative of set-valued maps for general multiobjective optimization problems. In [4], Shi introduced a weaker notion of set-valued derivative TP-derivative
and investigated the behavior of contingent derivative for the set-valued perturbation maps in a nonconvex vector optimization problem. Later on, Shi also established sensitivity analysis for a convex vector optimization problem
see [5]
. In [6], Kuk et al. investigated the relationships between the contingent derivatives of the perturbation maps
i.e., perturbation map, proper perturbation map, and weak perturbation map
and those of feasible set map in the objective space by virtue of contingent derivative, TP-derivative and Dini derivative. Considering convex vector optimization problems, they also investigated the behavior of the above three kinds of perturbation maps under some convexity assumptions
see [7]).
On the other hand, some interesting results have been proved for stability analysis in vector optimization problems. In [8], Tanino studied some qualitative results concerning the behavior of the perturbation map in convex vector optimization. In [9], Li investigated the continuity and the closedness of contingent derivative of the marginal map in multiobjective optimization. In [10], Xiang and Yin investigated some continuity properties of the mapping which associates the set of efficient solutions to the objective function by virtue of the additive weight method of vector optimization problems and the method of essential solutions.
To the best of our knowledge, there is no paper to deal with the stability of higher-order adjacent derivative for weak perturbation maps in vector optimization problems. Motivated by the work reported in [3–9], in this paper, by higher-order adjacent derivative of set-valued maps, we first discuss some relationships between higher-order adjacent derivative of a set-valued map and its profile map. Then, by virtue of the relationships, we investigate the stability of higher-order adjacent derivative of the perturbation maps.
The rest of this paper is organized as follows. In Section 2, we recall some basic definitions. In Section 3, after recalling the concept of higher-order adjacent derivative of set-valued maps, we provide some relationships between the higher-order adjacent derivative of a set-valued map and its profile map. In Section 4, we discuss some stability results of higher-order adjacent derivative for perturbation maps in parametrized vector optimization.
2. Preliminaries
Throughout this paper, let and
be two finite dimensional spaces, and let
be a pointed closed convex cone with a nonempty interior
, where
is said to be pointed if
. Let
be a set-valued map. The domain and the graph of
are defined by
and
, respectively. The so-called profile map
is defined by
for all
At first, let us recall some important definitions.
Definition 2.1 (see [11]).
Let be a nonempty subset of
. An elements
is said to be a minimal point (resp. weakly minimal point) of
if
resp.,
. The set of all minimal points
resp., weakly minimal point
of
is denoted by
(resp.,
.
Definition 2.2 (see [12]).
A base for is a nonempty convex subset
of
with
such that every
,
has a unique representation
, where
and
.
Definition 2.3 (see [13]).
The weak domination property is said to hold for a subset of
if
Definition 2.4 (see [14]).
Let be a set-valued map from
to
.
(i) is said to be lower semicontinuous (l.s.c) at
if for any generalized sequence
with
and
, there exists a generalized sequence
with
such that
.
(ii) is said to be upper semicontinuous (u.s.c) at
if for any neighborhood
of
, there exists a neighborhood
of
such that
, for all
.
(iii) is said to be closed at
if for any generalized sequence
,
, it yields
.
We say that is l.s.c (resp., u.s.c, closed) on
if it is l.s.c (resp., u.s.c, closed) at each
.
is said to be continuous on
if it is both l.s.c and u.s.c on
.
Definition 2.5 (see [14]).
is said to be Lipschitz around
if there exist a real number
and a neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ1_HTML.gif)
where denotes the closed unit ball of the origin in
.
Definition 2.6 (see [14]).
is said to be uniformly compact near
if there exists a neighborhood
of
such that
is a compact set.
3. Higher-Order Adjacent Derivatives of Set-Valued Maps
In this section, we recall the concept of higher-order adjacent derivative of set-valued maps and provide some basic properties which are necessary in the following section. Throughout this paper, let be an integer number and
.
Definition 3.1 (see [15]).
Let and
be elements of
. The set
is called the
th-order adjacent set of
at
, if and only if, for any
, for any sequence
with
, there exists a sequence
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ2_HTML.gif)
Definition 3.2 (see [15]).
Let and
,
. The
th-order adjacent derivative
of
at
for vectors
is the set-valued map from
to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ3_HTML.gif)
Proposition 3.3.
Let and
,
. Then, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ4_HTML.gif)
Proof.
The proof follows on the lines of Proposition in [3] by replacing contingent derivative by
th-order adjacent derivative.
Note that the converse inclusion of (3.3) may not hold. The following example explains the case where we only take .
Example 3.4.
Let and
, let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ5_HTML.gif)
Let and
. For any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ6_HTML.gif)
Thus, for any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ7_HTML.gif)
Proposition 3.5.
Let and
,
. Assume that
has a compact base. Then, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ8_HTML.gif)
where is a closed convex cone contained in
.
Proof.
If the inclusion holds trivially. Thus, we suppose that
Let
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ9_HTML.gif)
Since ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ10_HTML.gif)
then it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ11_HTML.gif)
From (3.8) and the definition of th-order adjacent derivative, we have that for any sequence
with
, there exist sequences
with
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ12_HTML.gif)
Since is a closed convex cone contained in
,
has a compact base. It is clear that
is a compact base for
, where
is a compact base for
. In this proposition, we assume that
is a compact base of
. Since
, there exist
and
such that
. Since
is compact, we may assume without loss of generality that
.
Now, we show that Suppose that
Then, for some
, we may assume, without loss of generality, that
, for all
. Let
. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ13_HTML.gif)
By (3.11) and (3.12), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ14_HTML.gif)
From (3.13) and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ15_HTML.gif)
which contradicts (3.10). Therefore, and
Thus, it follows from (3.11) that
and the proof is complete.
Remark 3.6.
The inclusion of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ16_HTML.gif)
may not hold under the assumptions of Proposition 3.5. The following example explains the case where we only take
Example 3.7.
Let and
, let
and
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ17_HTML.gif)
Suppose that ,
. Then, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ18_HTML.gif)
Naturally, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ19_HTML.gif)
Thus, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ20_HTML.gif)
Proposition 3.8.
Let , and
,
, and let
has a compact base. Suppose that
fulfills the weak domination property for any
. Then, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ21_HTML.gif)
where is a closed convex cone contained in
.
Proof.
Let Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ22_HTML.gif)
By Proposition 3.5, we also have
Suppose that Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ23_HTML.gif)
From and Proposition 3.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ24_HTML.gif)
So, by (3.21), (3.22), and (3.23), which leads to a contradiction. Thus,
.
Conversely, let . Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ25_HTML.gif)
Suppose that . Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ26_HTML.gif)
Since fulfills the weak domination property for any
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ27_HTML.gif)
From (3.25) and (3.26), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ28_HTML.gif)
It follows from Proposition 3.5 and (3.27) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ29_HTML.gif)
which contradicts . Thus,
, and the proof is complete.
Obviously, Example 3.4 can also show that the weak domination property of is essential for Proposition 3.8.
Remark 3.9.
From Example 3.7, the equality of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ30_HTML.gif)
may still not hold under the assumptions of Proposition 3.8.
Proposition 3.10.
Let and
,
. Suppose that
is Lipschitz at
. Then,
is continuous on
Proof.
Since is Lipschitz at
, there exist a real number
and a neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ31_HTML.gif)
First, we prove that is l.s.c. at
. Indeed, for any
. From the definition of
th-order adjacent derivative, we have that for any sequence
with
, there exists a sequence
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ32_HTML.gif)
Take any and
. Obviously,
,
for any
sufficiently large. Therefore, by (3.30), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ33_HTML.gif)
So, with (3.31), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ34_HTML.gif)
We may assume, without loss of generality, that . Thus, by (3.33),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ35_HTML.gif)
It follows from (3.34) that for any sequence with
,
, there exists a sequence
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ36_HTML.gif)
Obviously, . Hence,
is l.s.c. on
We will prove that is u.s.c. on
In fact, for any
, we consider the neighborhood
of
. Let
and
. From the definition of
, we have that for any sequence
with
, there exists a sequence
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ37_HTML.gif)
Take any . Obviously,
,
for any
sufficiently large. Therefore, by (3.30), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ38_HTML.gif)
Similar to the proof of l.s.c., there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ39_HTML.gif)
Thus, Hence,
is u.s.c. on
and the proof is complete.
4. Continuity of Higher-Order Adjacent Derivative for Weak Perturbation Map
In this section, we consider a family of parametrized vector optimization problems. Let be a set-valued map from
to
, where
is the Banach space of perturbation parameter vectors,
is the objective space, and
is considered as the feasible set map in the objective space. In the optimization problem corresponding to each parameter valued
, our aim is to find the set of weakly minimal points of the feasible objective valued set
. Hence, we define another set-valued map
from
to
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ40_HTML.gif)
The set-valued map is called the weak perturbation map. Throughout this section, we suppose that
is a closed convex cone contained in
.
Definition 4.1 (see [11]).
is said to be
-minicomplete by
near
if
, for any
where
is a neighborhood of
.
Remark 4.2.
Since , the
-minicompleteness of
by
near
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ41_HTML.gif)
Hence, if is
-minicomplete by
near
, then, for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ42_HTML.gif)
The following lemma palys a crucial role in this paper.
Lemma 4.3.
Let and
,
, and let
have a compact base. Suppose that the following conditions are satisfied:
(i) fulfills the weak domination property for any
;
(ii) is Lipschitz at
;
(iii) is
-minicomplete by
near
.
Then, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ43_HTML.gif)
Proof.
We first prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ44_HTML.gif)
In fact, from Proposition 3.5, Proposition 3.8, and the -minicompleteness of
by
near
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ45_HTML.gif)
Thus, result (4.5) holds.
Now, we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ46_HTML.gif)
In fact, assume that . Then, for any sequence
with
, there exists a sequence
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ47_HTML.gif)
Suppose that Then, there exists
such that
Thus, for the preceding sequence
there exists a sequence
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ48_HTML.gif)
Obviously, for any
sufficiently large. Therefore, by (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ49_HTML.gif)
So, with (4.9), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ50_HTML.gif)
Since and
,
, for
sufficiently large. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ51_HTML.gif)
which contradicts (4.8). Then, This completes the proof.
The following example shows that the -minicompleteness of
is essential in Lemma 4.3, where we only take
Example 4.4 ( is not
-minicomplete by
near
).
Let ,
and
, and let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ52_HTML.gif)
Then, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ53_HTML.gif)
Suppose that ,
. Then,
is Lipschitz at
, and for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ54_HTML.gif)
fulfills the weak domination property. We also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ55_HTML.gif)
On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ56_HTML.gif)
Thus, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ57_HTML.gif)
Theorem 4.5.
Let and
,
. Then,
is closed on
Proof.
From the definition of , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ58_HTML.gif)
Since is closed set,
is closed on
, and the proof is complete.
Theorem 4.6.
Let and
. If
is a compact space, then
is u.s.c. on
Proof.
Since is a compact space, it follows from Corollary
of Chapter
in [14] and Theorem 4.5 that
is u.s.c. on
Thus, the proof is complete.
Theorem 4.7.
Let . Suppose that
is a compact set and the assumptions of Lemma 4.3 are satisfied. Then,
is u.s.c. at
.
Proof.
It follows from Lemma 4.3 and Theorem 4.5 that is closed. By Proposition 3.10, we have that
is u.s.c. at
. Since
is a compact set, it follows from Theorem
of Chapter
in [14] that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ59_HTML.gif)
is u.s.c. at , and the proof is complete.
Now, we give an example to illustrate Theorem 4.7, where we also take
Example 4.8.
Let ,
, and
, and let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ60_HTML.gif)
Then, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ61_HTML.gif)
Suppose that ,
,
and
. Obviously,
has a compact base,
is Lipschitz at
and
is
-minicomplete by
near
. By direct calculating, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ62_HTML.gif)
fulfill the weak domination property, which is a strong property for a set-valued map. We also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ63_HTML.gif)
Thus, the conditions of Theorem 4.7 are satisfied. Obviously, both and
are u.s.c at
.
Theorem 4.9.
Let . Suppose that
is uniformly compact near
and the assumptions of Lemma 4.3 are satisfied. Then,
is l.s.c. at
.
Proof.
By Lemma 4.3, it suffices to prove that is l.s.c. at
. Let
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ64_HTML.gif)
By Proposition 3.10, we have that is l.s.c. at
. Then, there exists a sequence
with
such that
. Since
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ65_HTML.gif)
Then, for any sequence with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ66_HTML.gif)
then it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ67_HTML.gif)
Since is uniformly compact near
, we may assume, without loss of generality, that
. It follows from the closedness of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510838/MediaObjects/13660_2010_Article_2180_Equ68_HTML.gif)
From (4.28) and is closed, we have
. Then, it follows from (4.25) and (4.29) that
. Thus,
is l.s.c. at
, and the proof is complete.
It is easy to see that Example 4.8 can also illustrate Theorem 4.9.
References
Bonnans JF, Shapiro A: Perturbation Analysis of Optimization Problems, Springer Series in Operations Research. Springer, New York, NY, USA; 2000:xviii+601.
Fiacco AV: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Mathematics in Science and Engineering. Volume 165. Academic Press, Orlando, Fla, USA; 1983:xii+367.
Tanino T: Sensitivity analysis in multiobjective optimization. Journal of Optimization Theory and Applications 1988, 56(3):479–499. 10.1007/BF00939554
Shi DS: Contingent derivative of the perturbation map in multiobjective optimization. Journal of Optimization Theory and Applications 1991, 70(2):385–396. 10.1007/BF00940634
Shi DS: Sensitivity analysis in convex vector optimization. Journal of Optimization Theory and Applications 1993, 77(1):145–159. 10.1007/BF00940783
Kuk H, Tanino T, Tanaka M: Sensitivity analysis in vector optimization. Journal of Optimization Theory and Applications 1996, 89(3):713–730. 10.1007/BF02275356
Kuk H, Tanino T, Tanaka M: Sensitivity analysis in parametrized convex vector optimization. Journal of Mathematical Analysis and Applications 1996, 202(2):511–522. 10.1006/jmaa.1996.0331
Tanino T: Stability and sensitivity analysis in convex vector optimization. SIAM Journal on Control and Optimization 1988, 26(3):521–536. 10.1137/0326031
Li Shengjie: Sensitivity and stability for contingent derivative in multiobjective optimization. Mathematica Applicata 1998, 11(2):49–53.
Xiang SW, Yin WS: Stability results for efficient solutions of vector optimization problems. Journal of Optimization Theory and Applications 2007, 134(3):385–398. 10.1007/s10957-007-9214-0
Sawaragi Y, Nakayama H, Tanino T: Theory of Multiobjective Optimization, Mathematics in Science and Engineering. Volume 176. Academic Press, Orlando, Fla, USA; 1985:xiii+296.
Holmes RB: Geometric Functional Analysis and Its Applications. Springer, New York, NY, USA; 1975:x+246.
Luc DT: Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems. Volume 319. Springer, Berlin, Germany; 1989:viii+173.
Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xi+518.
Aubin J-P, Frankowska H: Set-Valued Analysis, Systems & Control: Foundations & Applications. Volume 2. Birkhäuser, Boston, Mass, USA; 1990:xx+461.
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (Grant no. 10871216) and Chongqing University Postgraduates Science and Innovation Fund (Grant no. 201005B1A0010338).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Sun, X., Li, S. Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization. J Inequal Appl 2010, 510838 (2010). https://doi.org/10.1155/2010/510838
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/510838