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By virtue of order adjacent derivative of set-valued maps, relationships between higher-order adjacent derivative of a set-valued map and its profile map are discussed.. In3, Tanino stud

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,sxkcqu@163.com

Received 29 March 2010; Accepted 3 August 2010

Academic Editor: Jong Kim

Copyrightq 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 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 In3, 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 In4, 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 mapsi.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 assumptionssee 7

On the other hand, some interesting results have been proved for stability analysis in vector optimization problems In8, Tanino studied some qualitative results concerning the behavior of the perturbation map in convex vector optimization In9, Li investigated the continuity and the closedness of contingent derivative of the marginal map in multiobjective optimization In10, 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 in3 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 InSection 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 X and Y be two finite dimensional spaces, and let K ⊆ Y be a

pointed closed convex cone with a nonempty interior intK, where K is said to be pointed

if K ∩ −K  {0} Let F : X ⇒ Y be a set-valued map The domain and the graph of

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 : Fx  K, for all x ∈ DomF.

At first, let us recall some important definitions

Definition 2.1see 11 Let Q be a nonempty subset of Y An elements y ∈ Q is said to be

a minimal pointresp 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

MinK Q resp., WMin K Q.

Definition 2.2see 12 A base for K is a nonempty convex subset B of Kwith 0 /∈ B such that

everyk ∈ K, k / 0 has a unique representation k  αb, where b ∈ B and α > 0.

Definition 2.3see 13 The weak domination property is said to hold for a subset H of Y if

H ⊆ WMin K H  int K ∪ {0}.

Definition 2.4see 14 Let F 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 } with x n → x and y ∈ Fx, there exists a generalized sequence {y n} with

y n ∈ Fx n  such that y n → y.

ii F is said to be upper semicontinuous u.s.c at x ∈ X if for any neighborhood

NFx of Fx, there exists a neighborhood Nx of x such that Fx ⊆ NFx,

for allx ∈ Nx.

iii F is said to be closed at x ∈ X if for any generalized sequence x n , y n  ∈ GraphF,

x n , y n  → x, y, it yields x, y ∈ GraphF.

We say thatF 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 2.5see 14 F is said to be Lipschitz around x ∈ X if there exist a real number

M > 0 and a neighborhood Nx of x such that

Fx1  ⊆ Fx2  Mx1− x2B Y , ∀x1, x2 ∈ Nx, 2.1 whereB Y denotes the closed unit ball of the origin inY.

Definition 2.6 see 14 F is said to be uniformly compact near x ∈ X if there exists a

neighborhoodNx of x such thatx∈Nx Fx 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, letm be an integer number and m > 1.

Definition 3.1 see 15 Let x ∈ C ⊆ X and u1, , um−1 be elements of X The set

T C bm x, u1, , um−1  is called the mth-order adjacent set of C at x, u1, , um−1, if and only

if, for anyx ∈ T C bm x, u1, , um−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 u1  h2

n u2  · · ·  h m−1

n u m−1  h m

n x n ∈ C, ∀n. 3.1

Definition 3.2 see 15 Let x, y ∈ GraphF and u i , v i  ∈ X × Y, i  1, 2, , m − 1.

Themth-order adjacent derivative D bm Fx, y, u1, v1, , u m−1 , v m−1  of F at x, y for vectors

u1, v1, , u m−1 , v m−1  is the set-valued map from X to Y defined by

Graph

D bm Fx, y, u1, v1, , u m−1 , v m−1 TGraphFbm x, y, u1, v1, , u m−1 , v m−1. 3.2

Proposition 3.3 Let x, y ∈ GraphF and u i , v i  ∈ X × Y, i  1, 2, , m − 1 Then, for any

x ∈ DomD bm Fx, y, u1, v1, , u m−1 , v m−1 ,

D bm Fx, y, u1, v1, , u m−1 , v m−1x  K ⊆ D bm F  Kx, y, u1, v1, , u m−1 , v m−1x.

3.3

Proof The proof follows on the lines of Proposition 2.1 in 3 by replacing contingent derivative bymth-order adjacent derivative.

Note that the converse inclusion of3.3 may not hold The following example explains the case where we only takem  2, 3.

Example 3.4 Let X  Y  R and K  R, letF : X ⇒ Y be defined by

Fx 

⎧

⎨

⎩

{0} ifx ≤ 0,

−1, x3

ifx > 0. 3.4

Letx, y  0, 0 ∈ GraphF and u1, v1  u2, v2  1, 0 For any x > 0, we have

D b2 Fx, y, u1, v1x  {0}, D b2 F  Kx, y, u1, v1x  R,

D b3 Fx, y, u1, v1, u2, v2x  {1}, D b3 F  Kx, y, u1, v1, u2, v2x  R. 3.5

Thus, for anyx > 0, we have

D b2 F  Kx, y, u1, v1x /⊆ D b2 Fx, y, u1, v1x  K,

D b3 F  Kx, y, u1, v1, u2, v2x /⊆ D b3 Fx, y, u1, v1, u2, v2x  K. 3.6

Proposition 3.5 Let x, y ∈ GraphF and u i , v i  ∈ X × Y, i  1, 2, , m − 1 Assume that K

has a compact base Then, for any x ∈ DomD bm Fx, y, u1, v1, , u m−1 , v m−1 ,

WMinK D bm

K

x, y, u1, v1, , u m−1 , v m−1x ⊆ D bm Fx, y, u1, v1, , u m−1 , v m−1x.

3.7

K is a closed convex cone contained in int K ∪ {0}.

Proof If WMin K D bm Kx, y, u1, v1, , u m−1 , v m−1 x  ∅, the inclusion holds trivially.

Thus, we suppose that WMinK D bm Kx, y, u1, v1, , u m−1 , v m−1 x / ∅ Let y0 ∈ WMinK D bm Kx, y, u1, v1, , u m−1 , v m−1 x Then,

y0 ∈ D bm

Kx, y, u1, v1, , u

m−1 , v m−1x. 3.8

K ⊆ int K ∪ {0},

WMinK D bm

Kx, y, u1, v1, , u

m−1 , v m−1x

⊆ MinK D bm

Kx, y, u1, v1, , u

m−1 , v m−1x,

3.9

then it follows that

y0∈ MinK D bm

Kx, y, u1, v1, , u

m−1 , v m−1x. 3.10

From3.8 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, y0 and

n K such that

y  h n v1  · · ·  h m−1

n v m−1  h m

n y n n ∈ Fx  h n u1  · · ·  h m−1

n u m−1  h m

n x n. 3.11

K has a compact base It is clear

that K, where B is a compact base for K In this proposition, we

n K, there exist α n > 0 and b n

n  α n b n B is compact, we may assume without loss of generality that b n

Now, we show thatα n /h m

n → 0 Suppose that α n /h m

n  0 Then, for some ε > 0, we

may assume, without loss of generality, thatα n /h m

n ≥ ε, for all n Let k n  εh m

n /α n n K.

Then, we have

By3.11 and 3.12, we obtain that

y  h n v1  · · ·  h m−1

n v m−1  h m

n y n − k n ∈ Fx  h n u1  · · ·  h m−1

n u m−1  h m

n x n K. 3.13

From3.13 and k n /h m

n  ε/α n n  εb n → εb / 0, we have

y0 − εb ∈ D bm

K

x, y, u1, v1, , u m−1 , v m−1

x, 3.14

which contradicts3.10 Therefore, α n /h m

n → 0 and y n n /h m

n → y0 Thus, it follows from

3.11 that y0∈ D bm Fx, y, u1, v1, , u m−1 , v m−1 x, and the proof is complete.

Remark 3.6 The inclusion of

WMinK D bm F  Kx, y, u1, v1, , u m−1 , v m−1x ⊆ D bm Fx, y, u1, v1, , u m−1 , v m−1x

3.15

may not hold under the assumptions ofProposition 3.5 The following example explains the case where we only takem  2, 3.

Example 3.7 Let X  R and Y  R2, letK  R2

andF : X ⇒ Y be defined by Fx y ∈ R2 :y x3, x3

Suppose thatx, y  0, 0, 0 ∈ GraphF, u1, v1  u2, v2  1, 0, 0 Then, for any

x ∈ X,

D b2 Fx, y, u1, v1x  {0, 0},

D b3 Fx, y, u1, v1, u2, v2x  {1, 1},

D b2 F  Kx, y, u1, v1x 

y1, y2∈ R2:y1 ≥ 0, y2≥ 0,

D b3 F  Kx, y, u1, v1, u2, v2x 

y1, y2∈ R2:y1 ≥ 1, y2≥ 1.

3.17

Naturally, we have

WMinK D b2 F  Kx, y, u1, v1x 

y1, y2∈ R2:y1y2  0, y1≥ 0, y2≥ 0,

WMinK D b3 F  Kx, y, u1, v1, u2, v2x 

y1, y2∈ R2:y1 ≥ 1, y2 1

∪

y1, y2∈ R2:y1  1, y2≥ 1.

3.18 Thus, for anyx ∈ X,

WMinK D b2 F  Kx, y, u1, v1x /⊆ D b2 Fx, y, u1, v1x,

WMinK D b3 F  Kx, y, u1, v1, u2, v2x /⊆ D b3 Fx, y, u1, v1, u2, v2x. 3.19

Proposition 3.8 Let x, y ∈ GraphF, and u i , v i  ∈ X × Y, i  1, 2, , m − 1, and let K

has a compact base Suppose that Px : D bm Kx, y, u1, v1, , u m−1 , v m−1 x fulfills the

weak domination property for any x ∈ DomD bm Fx, y, u1, v1, , u m−1 , v m−1  Then, for any

x ∈ DomD bm Fx, y, u1, v1, , u m−1 , v m−1 ,

WMinK D bm

K

x, y, u1, v1, , u m−1 , v m−1

x

 WMinK D bm Fx, y, u1, v1, , u m−1 , v m−1x,

3.20

K is a closed convex cone contained in int K ∪ {0}.

Proof Let y0∈ WMinK D bm Kx, y, u1, v1, , u m−1 , v m−1 x Then,

y0 ∈ D bm

K

x, y, u1, v1, , u m−1 , v m−1

ByProposition 3.5, we also havey0 ∈ D bm Fx, y, u1, v1, , u m−1 , v m−1 x.

Suppose thaty0 /∈ WMin K D bm Fx, y, u1, v1, , u m−1 , v m−1 x Then, there exists y∈

D bm Fx, y, u1, v1, , u m−1 , v m−1 x such that

Fromy∈ D bm Fx, y, u1, v1, , u m−1 , v m−1 x andProposition 3.3, we have

y∈ D bm

Kx, y, u1, v1, , u

m−1 , v m−1x. 3.23

So, by3.21, 3.22, and 3.23, y0/∈ WMin K D bm Kx, y, u1, v1, , u m−1 , v m−1 x, which

leads to a contradiction Thus,y0∈ WMinK D bm Fx, y, u1, v1, , u m−1 , v m−1 x.

Conversely, lety0∈ WMinK D bm Fx, y, u1, v1, , u m−1 , v m−1 x Then,

y0 ∈ D bm Fx, y, u1, v1, , u m−1 , v m−1

x ⊆ D bm

K

x, y, u1, v1, , u m−1 , v m−1

x.

3.24

Suppose thaty0 /∈ WMin K D bm Kx, y, u1, v1, , u m−1 , v m−1 x Then, there exists y∈

D bm Kx, y, u1, v1, , u m−1 , v m−1 x such that

y0 − y k ∈ int K. 3.25

SincePx fulfills the weak domination property for any x ∈ DomD bm Fx, y, u1, v1, ,

u m−1 , v m−1 , there exists k∈ int K ∪ {0} such that

y− k∈ WMinK D bm

K

x, y, u1, v1, , u m−1 , v m−1

x. 3.26 From3.25 and 3.26, we have

y0 − k − k∈ WMinK D bm

Kx, y, u1, v1, , u

m−1 , v m−1x. 3.27

It follows fromProposition 3.5and3.27 that

y0 − k − k∈ D bm Fx, y, u1, v1, , u m−1 , v m−1

which contradicts y0 ∈ WMinK D bm Fx, y, u1, v1, , u m−1 , v m−1 x Thus, y0 ∈ WMinK D bm Kx, y, u1, v1, , u m−1 , v m−1 x, and the proof is complete.

Obviously,Example 3.4can also show that the weak domination property ofPx is

essential forProposition 3.8

Remark 3.9 FromExample 3.7, the equality of

WMinK D bm F  Kx, y, u1, v1, , u m−1 , v m−1x

 WMinK D bm Fx, y, u1, v1, , u m−1 , v m−1x 3.29

may still not hold under the assumptions ofProposition 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 bm Fx, y, u1, v1, , u m−1 , v m−1  is continuous on

DomDbm Fx, y, u1, v1, , u m−1 , v m−1 .

Proof Since F is Lipschitz at x, there exist a real number M > 0 and a neighborhood Nx of

x such that

Fx1  ⊆ Fx2  Mx1− x2B Y , ∀x1, x2 ∈ Nx. 3.30

First, we prove thatD bm Fx, y, u1, v1, , u m−1 , v m−1  is l.s.c at x ∈ DomD bm Fx, y, u1, v1, , u m−1 , v m−1  Indeed, for any y ∈ D bm Fx, y, u1, v1, , u m−1 , v m−1 x From the

definition ofmth-order adjacent derivative, we have that for any sequence {h n } ⊆ R\ {0} withh n → 0, there exists a sequence {x n , y n } with x n , y n  → x, y such that

y  h n v1  · · ·  h m−1

n v m−1  h m

n y n ∈ Fx  h n u1  · · ·  h m−1

n u m−1  h m

n x n



. 3.31 Take anyx ∈ X and x n → x Obviously, x  h n u1  · · ·  h m−1

n u m−1  h m

n x n,x  h n u1 · · · 

h m−1

n u m−1  h m

n x n ∈ Nx, for any n sufficiently large Therefore, by 3.30, we have

Fx  h n u1  · · ·  h m−1

n u m−1  h m

n x n



⊆ Fx  h n u1  · · ·  h m−1

n u m−1  h m

n x n



 Mh m

n x n − x n B Y

3.32

So, with3.31, there exists −b n ∈ B Y such that

y  h n v1  · · ·  h m−1

n v m−1  h m

n

y n  Mx n − x n b n

∈ Fx  h n u1  · · ·  h m−1

n u m−1  h m

n x n



.

3.33

We may assume, without loss of generality, thatb n → b ∈ B Y Thus, by3.33,

y  Mx − xb ∈ D bm Fx, y, u1, v1, , u m−1 , v m−1x. 3.34

It follows from3.34 that for any sequence {x k } with x k → x, y ∈ D bm Fx, y, u1, v1, ,

u m−1 , v m−1 x, there exists a sequence {y k} with

y k: y  Mx − x k b ∈ D bm Fx, y, u1, v1, , u m−1 , v m−1

x k . 3.35

Obviously,y k → y Hence, D bm Fx, y, u1, v1, , u m−1 , v m−1  is l.s.c on DomD bm Fx, y, u1, v1, , u m−1 , v m−1 .

We will prove thatD bm Fx, y, u1, v1, , u m−1 , v m−1  is u.s.c on x ∈ DomD bm Fx, y, u1, v1, , u m−1 , v m−1  In fact, for any ε > 0, we consider the neighborhood x  ε/MB X of

x Let x ∈ x  ε/MB X andy ∈ D bm Fx, y, u1, v1, , u m−1 , v m−1 x From the definition

ofD bm Fx, y, u1, v1, , 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 n v1  · · ·  h m−1

n v m−1  h m

n y n ∈ Fx  h n u1  · · ·  h m−1

n u m−1  h m

n x n. 3.36

Take anyx n → x Obviously, xh n u1 · · ·h m−1

n u m−1 h m

n x n,xh n u1 · · ·h m−1

n u m−1 h m

n x n∈

Nx, for any n sufficiently large Therefore, by 3.30, we have

Fx  h n u1  · · ·  h m−1

n u m−1  h m

n x n



⊆ Fx  h n u1  · · ·  h m−1

n 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 existsb ∈ B Y such that

y  Mx − xb ∈ D bm Fx, y, u1, v1, , u m−1 , v m−1x. 3.38

Thus, y ∈ D bm Fx, y, u1, v1, , u m−1 , v m−1 x  εB Y Hence,

D bm Fx, y, u1, v1, , u m−1 , v m−1  is u.s.c on DomD bm Fx, y, u1, v1, , 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 LetF be a

set-valued map fromU 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 valuedx, our aim is to find

the set of weakly minimal points of the feasible objective valued setFx Hence, we define

another set-valued mapS from U to Y by

Sx  WMin K Fx, for any x ∈ U. 4.1

The set-valued map S is called the weak perturbation map Throughout this section, we

K is a closed convex cone contained in int K ∪ {0}.

Definition 4.1see 11 F is said to be K-minicomplete by S near x if Fx ⊆ Sx  K, for

anyx ∈ Nx, where Nx is a neighborhood of x.

Remark 4.2 Since Sx ⊆ Fx, the K-minicompleteness of F by S near x implies that

Sx  K  Fx  K, for any x ∈ Nx. 4.2 Hence, ifF is K-minicomplete by S near x, then, for any y ∈ Sx

D bm F  Kx, y, u1, v1, , u m−1 , v m−1 D bm S  Kx, y, u1, v1, , u m−1 , v m−1.

4.3 The following lemma palys a crucial role in this paper

Lemma 4.3 Let x, y ∈ GraphS and u i , v i  ∈ U × Y, i  1, 2, , m − 1, and let K have a

compact base Suppose that the following conditions are satisfied:

i Px : D bm Kx, y, u1, v1, , u m−1 , v m−1 x fulfills the weak domination

property for any x ∈ DomD bm Sx, y, u1, v1, , u m−1 , v m−1 ;

ii F is Lipschitz at x;

K-minicomplete by S near x.

Then, for any x ∈ DomD bm Sx, y, u1, v1, , u m−1 , v m−1 ,

D bm Sx, y, u1, v1, , u m−1 , v m−1

x  WMin K D bm Fx, y, u1, v1, , u m−1 , v m−1

x.

4.4

Proof We first prove that

WMinK D bm Fx, y, u1, v1, , u m−1 , v m−1x ⊆ D bm Sx, y, u1, v1, , u m−1 , v m−1x.

4.5

In fact, fromProposition 3.5,Proposition 3.8 K-minicompleteness of F by S near x,

we have

WMinK D bm Fx, y, u1, v1, , u m−1 , v m−1

x

 WMinK D bm

Kx, y, u1, v1, , u

m−1 , v m−1x

 WMinK D bm

Kx, y, u1, v1, , u

m−1 , v m−1x

⊆ D bm Sx, y, u1, v1, , u m−1 , v m−1

x.

4.6

Thus, result4.5 holds

Now, we prove that

D bm Sx, y, u1, v1, , u m−1 , v m−1x ⊆ WMin K D bm Fx, y, u1, v1, , u m−1 , v m−1x.

4.7

From3.8 and the definition of mth-order adjacent derivative, we have that for any sequence... the weak domination property ofPx is

essential forProposition 3.8