Volume 2011, Article ID 857520,13pages doi:10.1155/2011/857520
Research Article
Second-Order Contingent Derivative of the
Perturbation Map in Multiobjective Optimization
Q. L. Wang
1and S. J. Li
21College of Sciences, Chongqing Jiaotong University, Chongqing 400074, China
2College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China
Correspondence should be addressed to Q. L. Wang,[email protected]
Received 14 October 2010; Accepted 24 January 2011
Academic Editor: Jerzy Jezierski
Copyrightq2011 Q. L. Wang 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.
Some relationships between the second-order contingent derivative of a set-valued map and its profile map are obtained. By virtue of the second-order contingent derivatives of set-valued maps, some results concerning sensitivity analysis are obtained in multiobjective optimization. Several examples are provided to show the results obtained.
1. Introduction
In this paper, we consider a family of parametrized multiobjective optimization problems
PVOP ⎧ ⎨ ⎩
min fu, x f1u, x, f2u, x, . . . , fmu, x
,
s.t. u∈Xx⊆Rp. 1.1
Here,uis ap-dimensional decision variable,xis ann-dimensional parameter vector,Xis a nonempty set-valued map fromRntoRp, which specifies a feasible decision set, andf is an objective map fromRp×RntoRm, wherem,n,pare positive integers. The norms of all finite dimensional spaces are denoted by · .Cis a closed convex pointed cone with nonempty interior inRm. The coneCinduces a partial order≤
ConRm, that is, the relation≤Cis defined by
We use the following notion. For anyy, y∈Rm,
y <Cy←→y−y∈intC. 1.3
Based on these notations, we can define the following two sets for a setMinRm:
iy0 ∈Mis aC-minimal point ofMwith respect toCif there exists noy∈M, such
thaty≤Cy0,y /y0,
iiy0∈Mis a weaklyC-minimal point ofMwith respect toCif there exists noy∈M,
such thaty <Cy0.
The sets ofC-minimal point and weaklyC-minimal point ofMare denoted by MinCMand WMinCM, respectively.
LetGbe a set-valued map fromRntoRmdefined by
Gx y∈Rm|yfu, x, for someu∈Xx. 1.4
Gxis considered as the feasible set map. In the vector optimization problem corresponding to each parameter valuedx, our aim is to find the set ofC-minimal point of the feasible set mapGx. The set-valued mapWfromRntoRmis defined by
Wx MinCGx, 1.5
for anyx∈Rn, and call it the perturbation map forPVOP.
Sensitivity and stability analysis is not only theoretically interesting but also practically important in optimization theory. Usually, by sensitivity we mean the quantitative analysis, that is, the study of derivatives of the perturbation function. On the other hand, by stability we mean the qualitative analysis, that is, the study of various continuity properties of the perturbationor marginalfunctionor mapof a family of parametrized vector optimization problems.
Some interesting results have been proved for sensitivity and stability in optimization
see 1–16. Tanino 5 obtained some results concerning sensitivity analysis in vector optimization by using the concept of contingent derivatives of set-valued maps introduced in 17, and Shi 8 and Kuk et al. 7, 11 extended some of Tanino’s results. As for vector optimization with convexity assumptions, Tanino 6 studied some quantitative and qualitative results concerning the behavior of the perturbation map, and Shi 9
studied some quantitative results concerning the behavior of the perturbation map. Li10
discussed the continuity of contingent derivatives for set-valued maps and also discussed the sensitivity, continuity, and closeness of the contingent derivative of the marginal map. By virtue of lower Studniarski derivatives, Sun and Li 14 obtained some quantitative results concerning the behavior of the weak perturbation map in parametrized vector optimization.
order tangent sets. To the best of our knowledge, second-order contingent derivatives of perturbation map in multiobjective optimization have not been studied until now. Motivated by the work reported in 5–11, 14, we discuss some second-order quantitative results concerning the behavior of the perturbation map forPVOP.
The rest of the paper is organized as follows. In Section2, we collect some important concepts in this paper. In Section3, we discuss some relationships between the second-order contingent derivative of a set-valued map and its profile map. In Section4, by the second-order contingent derivative, we discuss the quantitative information on the behavior of the perturbation map forPVOP.
2. Preliminaries
In this section, we state several important concepts. LetF:Rn → 2Rm
be nonempty set-valued maps. The efficient domain and graph ofF are defined by
domF {x∈Rn|Fx/∅},
gphF x, y∈Rn×Rm|y∈Fx, x∈Rn,
2.1
respectively. The profile mapF ofFis defined byFx Fx C, for everyx∈domF,
whereCis the order cone ofRm.
Definition 2.1see18. A base forCis a nonempty convex subsetQofCwith 0Rm ∈/ clQ,
such that everyc ∈C,c /0Rm, has a unique representation of the formαb, whereb∈Qand
α >0.
Definition 2.2see19. Fis said to be locally Lipschitz atx0∈Rnif there exist a real number
γ >0 and a neighborhoodUx0ofx0, such that
Fx1⊆Fx2 γx1−x2BRm, ∀x1, x2∈Ux0, 2.2
whereBRm denotes the closed unit ball of the origin inRm.
3. Second-Order Contingent Derivatives for Set-Valued Maps
Definition 3.1see20. LetAbe a nonempty subsetX,x0 ∈clA, andu∈X, whereclA
denotes the closure ofA.
iThe second-order contingent setTA2x0, uofAatx0, uis defined as
TA2x0, u x∈X| ∃hn−→0, xn−→x,s.t. x0hnuh2nxn∈A
. 3.1
iiThe second-order adjacent setTA2x0, uofAatx0, uis defined as
TA2x0, u x∈X| ∀hn−→0,∃xn−→x,s.t. x0hnuh2nxn∈A
. 3.2
Definition 3.2see20. LetX,Y be normed spaces andF : X → 2Y be a set-valued map, and letx0, y0∈gphFandu, v∈X×Y.
iThe set-valued mapD2Fx
0, y0, u, vfromXtoY defined by
gphD2Fx0, y0, u, v
Tgph2Fx0, y0, u, v
, 3.3
is called second-order contingent derivative ofFatx0, y0, u, v. iiThe set-valued mapD2Fx
0, y0, u, vfromXtoY defined by
gphD2Fx0, y0, u, v
Tgph2Fx0, y0, u, v
, 3.4
is called second-order adjacent derivative ofFatx0, y0, u, v.
Definition 3.3see21. TheC-domination property is said to be held for a subsetHofY if H⊂MinCHC.
Proposition 3.4. Letx0, y0∈gphFandu, v∈X×Y, then
D2Fx0, y0, u, v
x C⊆D2FCx0, y0, u, v
x, 3.5
for anyx∈X.
Proof. The conclusion can be directly obtained similarly as the proof of5, Proposition 2.1.
It follows from Proposition3.4that
domD2Fx0, y0, u, v
⊆domD2Fx0, y0, u, v
Note that the inclusion of
D2Fx0, y0, u, v
x⊆D2Fx0, y0, u, v
x C, 3.7
may not hold. The following example explains the case.
Example 3.5. LetX R,Y R, andCR. Consider a set-valued mapF :X → 2Y defined
by
Fx ⎧ ⎨ ⎩
y|y≥x2 if x≤0,
x2,−1 if x >0. 3.8
Letx0, y0 0,0∈gphFandu, v 1,0, then, for anyx∈X,
D2Fx0, y0, u, v
x R, D2Fx0, y0, u, v
x {1}. 3.9
Thus, one has
D2Fx0, y0, u, v
x/⊆D2Fx0, y0, u, v
x C, x∈X, 3.10
which shows that the inclusion of3.7does not hold here.
Proposition 3.6. Letx0, y0∈gphFandu, v∈X×Y. Suppose thatChas a compact baseQ,
then for anyx∈X,
MinCD2F
x0, y0, u, v
x⊆D2Fx0, y0, u, v
x. 3.11
Proof. Letx∈X. If MinCD2Fx0, y0, u, vx ∅, then3.11holds trivially. So, we assume
that MinCD2Fx0, y0, u, vx/∅, and let
y∈MinCD2F
x0, y0, u, v
x. 3.12
Sincey ∈D2Fx
0, y0, u, vx, there exist sequences{hn}withhn → 0,{xn, yn} withxn, yn → x, y, and{cn}withcn∈C, such that
y0hnvh2n
yn−cn
∈Fx0hnuh2nxn
, for anyn. 3.13
We now showαn → 0. Suppose thatαn 0, then for someε > 0, we may assume without loss of generality that αn ≥ ε, for alln, by taking a subsequence if necessary. Let cn ε/αncn, then, for anyn,cn−cn∈Cand
y0hnvh2n
yn−cn
∈F
x0hnuh2nxn
. 3.14
Sincecn ε/αncn εbn, for alln, cn → εb /0Y. Thus,yn−cn → y−εb. It follows from
3.14that
y−εb∈D2Fx0, y0, u, v
x, 3.15
which contradicts3.12, sinceεb∈C. Thus,αn → 0 andyn−cn → y. Then, it follows from
3.13thaty∈D2Fx0, y0, u, vx. So,
MinCD2Fx0, y0, u, v
x⊆D2Fx0, y0, u, v
x, 3.16
and the proof of the proposition is complete. Note that the inclusion of
WMinCD2Fx0, y0, u, v
x⊆D2Fx0, y0, u, v
x, 3.17
may not hold under the assumptions of Proposition3.6. The following example explains the case.
Example 3.7. LetX R,Y R2, andC R2
. Obviously,Chas a compact base. Consider a
set-valued mapF:X → 2Y defined by
Fx y1, y2
|y1 ≥x, y2x2
. 3.18
Letx0, y0 0,0,0∈gphFandu, v 1,1,0. For anyx∈X,
D2Fx0, y0, u, v
x y1, y2
|y1≥x, y2≥1
,
D2Fx0, y0, u, v
x y1,1
|y1≥x
.
3.19
Then, for anyx∈X,WMinCD2Fx0, y0, u, vx {y1,1|y1 ≥x} ∪ {x, y2|y2≥1}. So,
the inclusion of3.17does not hold here.
Proposition 3.8. Let x0, y0 ∈ gphF and u, v ∈ X × Y. Suppose that C has a compact
base Qand Px : D2Fx
0, y0, u, vxsatisfies theC-domination property for allx ∈ K :
domD2Fx
0, y0, u, v, then for anyx∈K,
MinCD2F
x0, y0, u, v
x MinCD2F
x0, y0, u, v
Proof. From Proposition3.4, one has
D2Fx0, y0, u, v
x C⊆D2Fx0, y0, u, v
x, for anyx∈K. 3.21
It follows from theC-domination property ofD2F
x0, y0, u, vxand Proposition3.6that
D2Fx0, y0, u, v
x⊆MinCD2F
x0, y0, u, v
x C
⊆D2Fx0, y0, u, v
x C, for anyx∈K,
3.22
and then
D2Fx0, y0, u, v
x CD2Fx0, y0, u, v
x, for anyx∈K. 3.23
Thus, for anyx∈K,
MinCD2Fx0, y0, u, v
x MinCD2F
x0, y0, u, v
x, 3.24
and the proof of the proposition is complete.
The following example shows that the C-domination property of Px in Proposi-tion3.8is essential.
Example 3.9 Px does not satisfy the C-domination property. LetX R,Y R2, and
CR2
, and letF:X → 2Y be defined by
Fx ⎧ ⎨ ⎩
{0,0} ifx≤0,
0,0,−x,−√x ifx >0, 3.25
then
Fx
⎧ ⎨ ⎩
R2 if x≤0,
y1, y2
|y1≥ −x, y2≥ −
√
x if x >0. 3.26
Letx0, y0 0,0,0∈gphF,u, v 1,0,0, then, for anyx∈X,
D2Fx0, y0, u, v
x {0,0}, Px D2Fx0, y0, u, v
x R2. 3.27
Obviously,Pxdoes not satisfy theC-domination property and
MinCD2F
x0, y0, u, v
x/MinCD2F
x0, y0, u, v
4. Second-Order Contingent Derivative of the Perturbation Maps
The purpose of this section is to investigate the quantitative information on the behavior of the perturbation map forPVOPby using second-order contingent derivative. Hereafter in this paper, letx0∈E,y0∈Wx0, andu, v∈Rn×Rm, and letCbe the order cone ofRm.
Definition 4.1. We say thatGisC-minicomplete byWnearx0if
Gx⊆Wx C, ∀x∈Vx0, 4.1
whereVx0is some neighborhood ofx0.
Remark 4.2. LetCbe a convex cone. SinceWx ⊆Gx, theC-minicompleteness ofGbyW nearx0implies that
Wx CGx C, ∀x∈Vx0. 4.2
Hence, ifGisC-minicomplete byWnearx0, then
D2WCx0, y, u, v
D2GCx0, y, u, v
, ∀y∈Wx0. 4.3
Theorem 4.3. Suppose that the following conditions are satisfied:
iGis locally Lipschitz atx0;
iiD2Gx0, y0, u, v D2Gx0, y0, u, v; iiiGisC-minicomplete byWnearx0;
ivthere exists a neighborhoodUx0ofx0, such that for anyx ∈ Ux0,Wxis a single
point set,
then, for allx∈Rn,
D2Wx0, y0, u, v
x⊆MinCD2G
x0, y0, u, v
x. 4.4
Proof. Letx∈Rn. IfD2Wx0, y0, u, vx ∅, then4.4holds trivially. Thus, we assume that
D2Wx
0, y0, u, vx/∅. Lety∈D2Wx0, y0, u, vx, then there exist sequences{hn}with hn → 0and{xn, yn}withxn, yn → x, y, such that
y0hnvh2nyn∈W
x0hnuh2nxn
⊆Gx0hnuh2nxn
, ∀n.
4.5
So,y∈D2Gx
0, y0, u, vx.
Suppose that y /∈ MinCD2Gx0, y0, u, vx, then there exists y ∈ D2Gx0, y0,
u, vx, such that
SinceD2Gx
0, y0, u, v D2Gx0, y0, u, v, for the preceding sequence{hn}, there exists a sequence{xn, yn}withxn, yn → x, y, such that
y0hnvh2nyn∈G
x0hnuh2nxn
, ∀n. 4.7
It follows from the locally Lipschitz continuity of G that there exist γ > 0 and a neighborhoodVx0ofx0, such that
Gx1⊆Gx2 γx1−x2BRm, ∀x1, x2∈Vx0, 4.8
whereBRm is the closed ball ofRm.
From assumptioniii, there exists a neighborhoodV1x0ofx0, such that
Gx⊆Wx C, ∀x∈V1x0. 4.9
Naturally, there existsN >0, such that
x0hnuh2nxn, x0hnuh2nxn∈Ux0∩Vx0∩V1x0, ∀n > N. 4.10
Therefore, it follows from4.7and4.8that for anyn > N, there existsbn ∈BRm, such that
y0hnvh2n
yn−γxn−xnbn
∈Gx0hnuh2nxn
. 4.11
Thus, from4.5,4.9, and assumptioniv, one has
y0hnvh2n
yn−γxn−xnbn
−y0hnvh2nyn
h2n
yn−γxn−xnbn−yn
∈C, ∀n > N,
4.12
and then it follows fromyn−γxn−xnbn−yn → y−yandCis a closed convex cone that
y−y∈C, 4.13
which contradicts4.6. Thus,y∈MinCD2Gx0, y0, u, vxand the proof of the theorem is
complete.
The following two examples show that the assumptionivin Theorem4.3is essential.
Example 4.4Wx is not a single-point set nearx0. LetC {y1, y2 ∈R2 | y1 ≥ y2}and
G:R → 2R2be defined by
Gx C∪ y1, y2
|y1≥x2x, y2≥x2
then
Wx {0,0} ∪ y1, y2
|y1 x2x, y2 > x2x
. 4.15
Letx00,y0 0,0, andu, v 1,1,1, thenWxis not a single-point set nearx0, and
it is easy to check that other assumptions of Theorem4.3are satisfied. For anyx∈R, one has
D2Gx0, y0, u, v
x y1, y2
|y1∈R, y1≥y2
∪y1, y2
|y1≥1x, y2∈R
,
D2Wx0, y0, u, v
x 1x, y2
|y2≥1x
,
4.16
and then
MinCD2G
x0, y0, u, v
x 1x, y2
|y2>1x
. 4.17
Thus, for anyx∈R, the inclusion of4.4does not hold here.
Example 4.5Wxis not a single-point set nearx0. LetC {y1, y2 ∈ R2 | y1 0}and
G:R → 2R2
be defined by
Gx ⎧ ⎨ ⎩
C ifx0, C∪ y1, y2
|y1x, y2≥ −
1|x|
ifx /0, 4.18
then
Wx ⎧ ⎨ ⎩
{0,0} ifx0,
0,0,x,−1|x| ifx /0. 4.19
Letx0 0,y0 0,0, andu, v 0,0,0, thenWxis not a single-point set near
x0, and it is easy to check that other assumptions of Theorem4.3are satisfied.
For anyx∈R, one has
D2Gx0, y0, u, v
x D2Gx0, y0, u, v
x C∪y1, y2
|y1 x, y2∈R
,
D2Wx0, y0, u, v
x {0,0},
4.20
and then
MinCD2G
x0, y0, u, v
0 ∅. 4.21
Now, we give an example to illustrate Theorem4.3.
Example 4.6. LetCR2andG:R → 2R2be defined by
Gx y1, y2
∈R2|x≤y1≤xx2, x−x2 ≤y2≤x
, ∀x∈R, 4.22
then
Wx x, x−x2, ∀x∈R. 4.23
Letx0, y0 0,0,0 ∈ gphG,u, v 1,1,1. By directly calculating, for all
x∈R, one has
D2Gx0, y0, u, v
x D2Gx0, y0, u, v
x
y1, y2
|x≤y1≤x1, x−1≤y1≤x
,
D2Wx0, y0, u, v
x {x, x−1}.
4.24
Then, it is easy to check that assumptions of Theorem4.3are satisfied, and the inclusion of
4.4holds.
Theorem 4.7. IfPx:D2G
x0, y0, u, vxfulfills theC-domination property for allx∈Ω:
domD2Gx
0, y0, u, vandGisC-minicomplete byWnearx0, then
MinCD2G
x0, y0, u, v
x⊆D2Wx0, y0, u, v
x, for anyx∈Ω. 4.25
Proof. SinceC⊂Rn,Chas a compact base. Then, it follows from Propositions3.6and3.8and Remark4.2that for anyx∈Ω, one has
MinCD2G
x0, y0, u, v
x MinCD2Gx0, y0, u, v
x
MinCD2W
x0, y0, u, v
x
⊆D2Wx0, y0, u, v
x.
4.26
Then, the conclusion is obtained and the proof is complete.
Remark 4.8. If the C-domination property of Px is not satisfied in Theorem 4.7, then Theorem4.7may not hold. The following example explains the case.
Example 4.9Pxdoes not satisfy theC-domination property forx ∈ Ω. LetC R2
and
G:R → R2be defined by
Gx ⎧ ⎨ ⎩
{0,0} ifx≤0,
then,
Gx
⎧ ⎨ ⎩
R2
ifx≤0,
y1, y2
|y1 ≥ −x, y2≥ −√x
ifx >0. 4.28
Letx0, y0 0,0,0∈gphF,u, v 1,0,0, then, for anyx∈Ω R,
Wx ⎧ ⎨ ⎩
{0,0} ifx≤0,
y1, y2
|y1−x, y2−√x
ifx >0, 4.29
for anyx∈Ω,
D2Gx0, y0, u, v
x {0,0}, Px D2Gx0, y0, u, v
x R2,
D2Wx0, y0, u, v
x ∅.
4.30
Hence,Px does not satisfy the C-domination property, and MinCD2Gx0, y0, u, vx
{0,0}. Then,
MinCD2G
x0, y0, u, v
x/⊆D2Wx0, y0, u, v
x. 4.31
Theorem 4.10. Suppose that the following conditions are satisfied:
iGis locally Lipschitz atx0;
iiD2Gx0, y0, u, v D2Gx0, y0, u, v; iiiGisC-minicomplete byWnearx0;
ivthere exists a neighborhoodUx0ofx0, such that for anyx∈ Ux0,Wxis a
single-point set;
vfor anyx ∈Ω :domD2Gx0, y0, u, v,D2Gx0, y0, u, vxfulfills theC
-domi-nation property;
then
D2Wx0, y0, u, v
x MinCD2G
x0, y0, u, v
x, ∀x∈Ω. 4.32
Proof. It follows from Theorems4.3 and 4.7that4.32holds. The proof of the theorem is complete.
Acknowledgments
This research was partially supported by the National Natural Science Foundation of China
no. 10871216 and no. 11071267, Natural Science Foundation Project of CQ CSTC and Science and Technology Research Project of Chong Qing Municipal Education Commission
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