Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 510838, 15 pages
doi:10.1155/2010/510838
Research Article
Stability Analysis for Higher-Order Adjacent
Derivative in Parametrized Vector Optimization
X. K. Sun and S. J. Li
College of Mathematics and Science, Chongqing University, Chongqing 400030, China
Correspondence should be addressed to X. K. Sun, [email protected]
Received 29 March 2010; Accepted 3 August 2010
Academic Editor: Jong Kim
Copyright q 2010 X. K. Sun and S. J. Li. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
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
F are defined by DomF{x ∈ X : Fx
/
∅} and GraphF{x, y ∈ X × Y : y ∈
Fx,x ∈ DomF}, respectively. The so-called profile map F K : X ⇒ Y is defined by
F Kx : FxK, for all x ∈ Dom
F.
At first, let us recall some important definitions.
Definition 2.1 see 11.LetQ be a nonempty subset of Y. An elements y ∈ Q is said to be
a minimal point resp. weakly minimal point of Q if Q − y ∩ −K{0}resp., Q − y ∩
− int K∅. The set of all minimal points resp., weakly minimal point of Q is denoted by
Min
K
Q resp., WMin
K
Q.
Definition 2.2 see 12.AbaseforK is a nonempty convex subset B of Kwith 0
/
∈ B such that
every k ∈ K, k
/
0 has a unique representation k αb, where b ∈ B and α>0.
Definition 2.3 see 13. The weak domination property is said to hold for a subset H of Y if
H ⊆ WMin
K
H int K ∪{0}.
Definition 2.4 see 14.LetF be a set-valued map from X to Y .
i F is said to be lower semicontinuous l.s.c at
x ∈ X if for any generalized sequence
{x
n
We say that F is l.s.c resp., u.s.c, closed on X if it is l.s.c resp., u.s.c, closed at each x ∈ X.
F is said to be continuous on X if it is both l.s.c and u.s.c on X.
Definition 2.5 see 14. F is said to be Lipschitz around
x ∈ X if there exist a real number
M>0 and a neighborhood N
x of x such that
F
x
1
⊆ F
x
2
M
x
1
− x
2
B
Y
, ∀x
1
,x
2
∈ N
m−1
is called the mth-order adjacent set of C at x, u
1
, ,u
m−1
,ifandonly
if, for any x ∈ T
bm
C
x, u
1
, ,u
m−1
, for any sequence {h
n
}⊆R
\{0} with h
n
→ 0, there
exists a sequence {x
n
}⊆X with x
n
→ x such that
x h
n
u
1
h
,v
m−1
of F at x, y for vectors
u
1
,v
1
, ,u
m−1
,v
m−1
is the set-valued map from X to Y defined by
Graph
D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
T
bm
m−1
,
D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
K ⊆ D
bm
F K
x, y,u
1
,v
1
, ,u
m−1
3
if x>0.
3.4
Let
x, y0, 0 ∈ GraphF and u
1
,v
1
u
2
,v
2
1, 0. For any x>0, we have
D
b2
F
x, y,u
1
,v
1
x
{
0
}
{
1
}
,D
b3
F K
x, y,u
1
,v
1
,u
2
,v
2
x
R.
3.5
Thus, for any x>0, we have
D
b2
F K
x, y,u
1
,v
1
,u
2
,v
2
x
/
⊆ D
b3
F
x, y,u
1
,v
1
,u
2
,v
2
x
K.
,v
1
, ,u
m−1
,v
m−1
x
⊆ D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
.
3.7
where
x
/
∅. Let y
0
∈
WMin
K
D
bm
F
Kx, y, u
1
,v
1
, ,u
m−1
,v
m−1
x. Then,
y
0
∈ D
bm
F
K
, ,u
m−1
,v
m−1
x
⊆ Min
K
D
bm
F
K
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
Journal of Inequalities and Applications 5
From 3.8 and the definition of mth-order adjacent derivative, we have that for any sequence
{h
n
}⊆R
\{0} with h
n
→ 0, there exist sequences {x
n
,y
n
} with x
n
,y
n
→ x, y
0
and
{
k
n
}⊆
K such that
y h
n
v
1
. 3.11
Since
K is a closed convex cone contained in int K ∪{0},
K has a compact base. It is clear
that B ∩
K is a compact base for
K, where B is a compact base for K. In this proposition, we
assume that
B is a compact base of
K. Since
k
n
∈
K, there exist α
n
> 0andb
n
∈
B such that
n
εh
m
n
/α
n
k
n
∈
K.
Then, we have
k
n
− k
n
∈
K.
3.12
By 3.11 and 3.12,weobtainthat
y h
n
v
1
··· h
m−1
From 3.13 and
k
n
/h
m
n
ε/α
n
k
n
εb
n
→ εb
/
0, we have
y
0
− εb ∈ D
bm
F
K
x, y,u
1
,v
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
x, and the proof is complete.
Remark 3.6. The inclusion of
WMin
K
D
bm
F K
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
y ∈ R
2
: y
x
3
,x
3
. 3.16
6 Journal of Inequalities and Applications
Suppose that
x, y0, 0, 0 ∈ GraphF, u
1
,v
1
u
2
,v
2
1, 0, 0. Then, for any
x ∈ X,
D
b2
F
x, y,u
1
{
1, 1
}
,
D
b2
F K
x, y,u
1
,v
1
x
y
1
,y
2
∈ R
2
,y
2
∈ R
2
: y
1
≥ 1,y
2
≥ 1
.
3.17
Naturally, we have
WMin
K
D
b2
F K
x, y,u
1
,v
1
x
1
,v
1
,u
2
,v
2
x
y
1
,y
2
∈ R
2
: y
1
≥ 1,y
2
1
∪
x
/
⊆ D
b2
F
x, y,u
1
,v
1
x
,
WMin
K
D
b3
F K
x, y,u
1
,v
1
,u
2
∈ X × Y, i 1, 2, ,m − 1, and let K
has a compact base. Suppose that Px : D
bm
F
Kx, y, u
1
,v
1
, ,u
m−1
,v
m−1
x fulfills the
weak domination property for any x ∈ DomD
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
. Then, for any
x ∈ DomD
bm
Fx, y,u
1
,v
D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
,
3.20
where
K is a closed convex cone contained in int K ∪{0}.
Proof. Let y
0
∈ WMin
K
D
bm
F
Kx, y, u
. 3.21
By Proposition 3.5, we also have y
0
∈ D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
x.
Journal of Inequalities and Applications 7
Suppose that y
0
/
∈ WMin
K
D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
m−1
x and Proposition 3.3, we have
y
∈ D
bm
F
K
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
. 3.23
So, by 3.21, 3.22,and3.23, y
0
/
∈ WMin
K
∈ WMin
K
D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
x. Then,
y
0
∈ D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
F
Kx, y, u
1
,v
1
, ,u
m−1
,v
m−1
x. Then, there exists y
∈
D
bm
F
Kx, y, u
1
,v
1
, ,u
m−1
,v
m−1
x such that
y
0
− y
1
,v
1
, ,u
m−1
,v
m−1
x
. 3.26
From 3.25 and 3.26, we have
y
0
− k − k
∈ WMin
K
D
bm
F
K
x, y,u
1
,v
x
,
3.28
which contradicts y
0
∈ WMin
K
D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
x.Thus,y
0
∈
WMin
K
D
bm
F
Kx, y, u
x
WMin
K
D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
3.29
may still not hold under the assumptions of Proposition 3.8.
Proposition 3.10. Let
x, y ∈ GraphF and u
i
,v
i
∈ X × Y, i 1, 2, ,m − 1.
Suppose that F is Lipschitz at
x. Then, D
x
2
M
x
1
− x
2
B
Y
, ∀x
1
,x
2
∈ N
x
. 3.30
First, we prove that D
bm
Fx, y,u
1
,v
1
, ,u
m−1
n
→ 0, there exists a sequence {x
n
, y
n
} with x
n
, y
n
→ x, y such that
y h
n
v
1
··· h
m−1
n
v
m−1
h
m
n
y
n
∈ F
x h
n
u
1
u
1
···
h
m−1
n
u
m−1
h
m
n
x
n
∈ Nx, for any n sufficiently large. Therefore, by 3.30, we have
F
x h
n
u
1
··· h
m−1
n
u
m−1
h
m
n
x
n
3.32
So, with 3.31, there exists −b
n
∈ B
Y
such that
y h
n
v
1
··· h
m−1
n
v
m−1
h
m
n
y
n
M
x
n
− x
n
b
n
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
.
3.34
It follows from 3.34 that for any sequence {x
k
} with x
k
→ x, y ∈ D
bm
Fx, y,u
1
,v
1
, ,
u
m−1
3.35
Journal of Inequalities and Applications 9
Obviously, y
k
→ y. Hence, D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
is l.s.c. on DomD
bm
Fx, y,
u
1
,v
1
, ,u
m−1
,v
m−1
.
We will prove that D
bm
Fx, y,u
1
,v
m−1
x. From the definition
of D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
x, we have that for any sequence {h
n
}⊆R
\{0} with
h
n
→ 0, there exists a sequence {x
n
,y
n
} with x
n
,y
n
→ x, y such that
y h
. 3.36
Take any x
n
→ x. Obviously, x h
n
u
1
···h
m−1
n
u
m−1
h
m
n
x
n
, xh
n
u
1
···h
m−1
n
u
m−1
h
m
n
x
u
m−1
h
m
n
x
n
Mh
m
n
x
n
− x
n
B
Y
.
3.37
Similar to the proof of l.s.c., there exists b ∈ B
Y
such that
y M
x − x
b ∈ D
bm
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
is u.s.c. on DomD
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
,
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 F be a
set-valued map from U to Y , where U is the Banach space of perturbation parameter vectors,
Y is the objective space, and F is considered as the feasible set map in the objective space.
In the optimization problem corresponding to each parameter valued x, our aim is to find
the set of weakly minimal points of the feasible objective valued set Fx. Hence, we define
another set-valued map S from U to Y by
S
x
. 4.2
Hence, if F is K-minicomplete by S near
x, then, for any y ∈ Sx
D
bm
F K
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
D
bm
S K
x, y,u
1
,v
bm
Sx, y,u
1
,v
1
, ,u
m−1
,v
m−1
;
ii F is Lipschitz at
x;
iii F is
K-minicomplete by S near
x.
Then, for any x ∈ DomD
bm
Sx, y,u
1
,v
1
, ,u
m−1
,v
m−1
,
D
bm
S
.
4.4
Proof. We first prove that
WMin
K
D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
⊆ D
bm
S
x, y,u
1
,v
1
x
WMin
K
D
bm
F
K
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
WMin
K
D
bm
x
.
4.6
Thus, result 4.5 holds.
Now, we prove that
D
bm
S
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
⊆ WMin
K
D
bm
F
x, y,u
n
→ 0, there exists a sequence {x
n
,y
n
} with x
n
,y
n
→ x, y such that
y h
n
v
1
··· h
m−1
n
v
m−1
h
m
n
y
n
∈ S
x h
n
u
1
Suppose that y
/
∈ WMin
K
D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
x. Then, there exists y ∈
D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
x such that y− y ∈ int K. Thus, for the preceding sequence
{h
n
}, there exists a sequence { x
n
u
m−1
h
m
n
x
n
. 4.9
Obviously, x h
n
u
1
··· h
m−1
n
u
m−1
h
m
n
x
n
, x h
n
u
1
··· h
m−1
n
u
1
··· h
m−1
n
u
m−1
h
m
n
x
n
Mh
m
n
x
n
− x
n
B
Y
.
4.10
So, with 4.9, there exists −b
n
∈ B
Y
u
1
··· h
m−1
n
u
m−1
h
m
n
x
n
.
4.11
Since y
n
− y
n
Mx
n
− x
n
b
n
→ y − y and y − y ∈ int K, y
n
− y
n
Mx
v
m−1
h
m
n
y
n
M
x
n
−x
n
b
n
∈ int K,
4.12
which contradicts 4.8. T hen, y ∈ WMin
K
D
bm
Fx, y,u
1
,v
1
, ,u
∈ R
2
: y
1
≥ 0,y
2
≥ 0
∪
y
1
,y
2
∈ R
2
: y
2
>
y
1
. 4.13
0
. 4.14
Suppose that
x, y0, 0, 0 ∈ GraphS, u
1
,v
1
u
2
,v
2
1, 0, 0. Then, F is
Lipschitz at
x, and for any x ∈ U,
D
b2
F
K
x, y,u
1
,v
1
x
2
: y
1
≥ 0,y
2
≥ 0
∪
y
1
,y
2
∈ R
2
: y
2
≥
y
1
4.15
fulfills the weak domination property. We also have
D
y
1
,y
2
∈ R
2
: y
1
≥ 0,y
2
≥ 0
∪
y
1
,y
2
∈ R
2
: y
2
≥
y
1
x
S
x
,
WMin
K
D
b2
F
x, y,u
1
,v
1
x
WMin
K
D
b3
F
x, y,u
y
1
,y
2
∈ R
2
: y
2
−y
1
,y
1
< 0
.
4.17
Thus, for any x ∈ X,
D
b2
S
x, y,u
1
,v
1
x
x
/
WMin
K
D
b3
F
x, y,u
1
,v
1
,u
2
,v
2
x
.
4.18
Theorem 4.5. Let
x, y ∈ GraphS and u
i
,v
i
∈ U × Y,i 1, 2, ,m − 1. Then,
D
x, we have that
Graph
D
bm
S
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
T
bm
Graph
S
x, y,u
1
,v
1
, ,u
m−1
1
,v
1
, ,u
m−1
,v
m−1
, and the proof is complete.
Theorem 4.6. Let x, y ∈ GraphS and u
i
,v
i
∈ U×Y, i 1, 2, ,m−1.IfY is a compact space,
then D
bm
Sx, y,u
1
,v
1
, ,u
m−1
,v
m−1
is u.s.c. on DomD
bm
Sx, y,u
1
,v
1
, ,u
1
,v
1
, ,u
m−1
,v
m−1
. Suppose t hat D
bm
Fx, y,
u
1
,v
1
, ,u
m−1
,v
m−1
x is a compact set and the assumptions of Lemma 4.3 are satisfied. Then,
D
bm
Sx, y,u
1
,v
1
, ,u
m−1
,v
m−1
is u.s.c. at x.
, ,u
m−1
,v
m−1
x is a compact set, it
follows from Theorem 8 of Chapter 3 in 14 that
D
bm
S
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
WMin
K
D
bm
F
x, y,u
1
,v
1
, ,u
, ,u
m−1
,v
m−1
4.20
is u.s.c. at x, and the proof is complete.
Now, we give an example to illustrate Theorem 4.7, where we also take m 2, 3.
Example 4.8. Let U 0, 1, Y R
2
,andK R
2
,andletF : U ⇒ Y be defined by
F
x
y
1
,y
2
∈ R
2
:0≤ y
1
∪
y
1
,y
2
∈ R
2
: y
1
0, 0 ≤ y
2
≤ x
3
. 4.22
Suppose that
x, y0, 0, 0 ∈ GraphS, x 1/3, u
1
,v
1
u
2
,v
2
1, 0, 0 and
x
K,
D
b3
F
K
x, y,u
1
,v
1
,u
2
,v
2
x
y
x
{
0, 0
}
,
D
b2
S
x, y,u
1
,v
1
x
{
0, 0
}
,
D
b3
,
D
b3
S
x, y,u
1
,v
1
,u
2
,v
2
x
y
1
,y
2
∈ R
2
:0≤ y
1
≤ 1,y
Sx, y,u
1
,v
1
,u
2
,v
2
x are u.s.c at x.
Theorem 4.9. Let x ∈ DomD
bm
Sx, y,u
1
,v
1
, ,u
m−1
,v
m−1
. Suppose t hat D
bm
Fx, y,
u
1
,v
1
, ,u
m−1
,v
m−1
D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
.
4.25
By Proposition 3.10, we have that D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
m−1
is l.s.c. at x. Then, there
exists a sequence {y
1
, ,u
m−1
,v
m−1
x
⊆ Min
K
D
bm
F
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
.
4.26
F
x
,
y
,u
1
,v
1
, ,u
m−1
,v
m−1
x
n
,
4.27
then it follows that
y
n
− y
n
∈
K.
4.28
x, y,u
1
,v
1
, ,u
m−1
,v
m−1
x
. 4.29
Journal of Inequalities and Applications 15
From 4.28 and
K is closed, we have y − y ∈
K ⊆ int K ∪{0}. Then, it follows from 4.25 and
4.29 that y y.Thus,WMin
K
D
bm
Fx, y,u
1
,v
1
, ,u
m−1
,v
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auser, Boston, Mass, USA, 1990.
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