Volume 2009, Article ID 684304,14pages doi:10.1155/2009/684304
Research Article
Generalized Levitin-Polyak Well-Posedness of
Vector Equilibrium Problems
Jian-Wen Peng,
1Yan Wang,
1and Lai-Jun Zhao
21College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China 2Management School, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Lai-Jun Zhao,zhao [email protected]
Received 1 July 2009; Revised 19 October 2009; Accepted 18 November 2009
Recommended by Nanjing Jing Huang
We study generalized Levitin-Polyak well-posedness of vector equilibrium problems with fun-ctional constraints as well as an abstract set constraint. We will introduce several types of generalized Levitin-Polyak well-posedness of vector equilibrium problems and give various cri-teria and characterizations for these types of generalized Levitin-Polyak well-posedness.
Copyrightq2009 Jian-Wen Peng et al. 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.
1. Introduction
It is well known that the well-posedness is very important for both optimization theory and numerical methods of optimization problems, which guarantees that, for approximating solution sequences, there is a subsequence which converges to a solution. The study of well-posedness originates from Tykhonov 1 in dealing with unconstrained optimization problems. Levitin and Polyak2extended the notion to constrainedscalaroptimization, allowing minimizing sequences {xn} to be outside of the feasible set X0 and requiring
dxn, X0 the distance from xn to X0 to tend to zero. The Levitin and Polyak well-posedness is generalized in 3, 4 for problems with explicit constraint gx ∈ K, where
and functional constraints 8, and equilibrium problems with abstract and functional constraints9. Most recently, S. J. Li and M. H. Li10introduced and researched two types of Levitin-Polyak well-posedness of vector equilibrium problems with variable domination structures. Huang et al.11introduced and researched the Levitin-Polyak well-posedness of vector quasiequilibrium problems. Li et al.12introduced and researched the Levitin-Polyak well-posedness for two types of generalized vector quasiequilibrium problems. However, there is no study on the generalized Levitin-Polyak well-posedness for vector equilibrium problems and vector quasiequilibrium problems with explicit constraintgx∈K.
Motivated and inspired by the above works, in this paper, we introduce two types of generalized Levitin-Polyak well-posedness of vector equilibrium problems with functional constraints as well as an abstract set constraint and investigate criteria and characterizations for these two types of generalized Levitin-Polyak well-posedness. The results in this paper generalize and extend some known results in literature.
2. Preliminaries
Let X, dX, Z, dZ, and Y be locally convex Hausdorff topological vector spaces, where
dXdZ is the metric which compatible with the topology ofXZ. Throughout this paper, we suppose thatK ⊂ Z and X1 ⊂ X are nonempty and closed sets, C : X → 2Y is a set-valued mapping such that for anyx ∈ X,Cxis a pointed, closed, and convex cone inZ
with nonempty interior intCx, e : X → Y is a continuous vector-valued mapping and satisfies that for any x ∈ X,ex ∈ intCx,f : X ×X1 → Y and g : X1 → Z are two vector-valued mappings, andX0 {x ∈ X1 : gx ∈ K}. We consider the following vector equilibrium problem with variable domination structures, functional constraints, as well as an abstract set constraint: finding a pointx∗∈X0, such that
fx∗, y/∈ −intCx∗, ∀y∈X0. VEP
We always assume thatX0/ andgis continuous onX1and the solution set ofVEP is denoted byΩ.
LetP, dbe a metric space,P1 ⊆P,andx ∈P. We denote bydx, P1 inf{dx, p :
p∈P1}the distance function from the pointx∈Pto the setP1.
Definition 2.1. i A sequence {xn} ⊂ X1 is called a type I Levitin-Polyak in short LP approximating solution sequence forVEPif there exists{n} ⊂R1withn → 0 such that
dxn, X0≤n, 2.1
fxn, y
nexn/∈ −intCxn, ∀y∈X0. 2.2
ii{xn} ⊂X1is called type II approximating solution sequence forVEPif there exists
{n} ⊂R1withn → 0 and{yn} ⊂X0satisfying2.1,2.2, and
fxn, yn
iii{xn} ⊂X1is called a generalized type I approximating solution sequence forVEP if there exists{n} ⊂R1 withn → 0 satisfying
dgxn, K
≤n 2.4
and2.2.
iv{xn} ⊂ X1 is called a generalized type II approximating solution sequence for
VEPif there exists{n} ⊂R1 withn → 0 and{yn} ⊂X0satisfying2.2,2.3, and2.4.
Definition 2.2. The vector equilibrium problem VEP is said to be type I resp., type II, generalized type I, generalized type IILP well-posed ifΩ/∅and for any type Iresp., type II, generalized type I, generalized type IILP approximating solution sequence{xn}ofVEP, there exists a subsequence{xnj}of{xn}andx∈Ωsuch thatxnj → x.
Remark 2.3. i IfY Rand Cx R1 {r ∈ R : r ≥ 0} for allx ∈ X, then the type I
resp., type II, generalized type I, generalized type IILP well-posedness ofVEPdefined inDefinition 2.2reduces to the type Iresp., type II, generalized type I, generalized type II LP well-posedness of the scalar equilibrium problem with abstract and functional constraints introduced by Long et al.9. Moreover, ifX∗is the topological dual space ofX,F:X1 → X∗ is a mapping,Fx, zdenotes the value of the functionalFxatz, andfx, y Fx, y− xfor all x, y ∈ X1, then the type I resp., type II, generalized type I, generalized type II LP well-posedness of VEP defined inDefinition 2.2 reduces to the type I resp., type II, generalized type I, generalized type IILP well-posedness for the variational inequality with abstract and functional constraints introduced by Huang et al.6. IfK Z, thenX1 X0 and the type Iresp., type IILP well-posedness ofVEPdefined inDefinition 2.2reduces to the type Iresp., type IILP well-posedness of the vector equilibrium problem introduced by S. J. Li and M. H. Li10.
ii It is clear that anygeneralizedtype II LP approximating solution sequence of
VEP is a generalized type I LP approximating solution sequence of VEP. Thus the
generalizedtype I LP well-posedness ofVEPimplies thegeneralizedtype II LP well-posedness ofVEP.
iii Each type of LP well-posedness of VEP implies that the solution set Ω is nonempty and compact.
ivLetgbe a uniformly continuous functions on the set
Sδ0
x∈X1:d
gx, K≤δ0
2.5
for someδ0 > 0. Then generalized type I resp., type IILP well-posedness implies type I
resp., type IILP well-posedness.
3. Criteria and Characterizations for Generalized LP
Well-Posedness of
VEP
In this section, we present necessary and/or sufficient conditions for the various types of
3.1. Criteria and Characterizations without Using Gap Functions
In this subsection, we give some criteria and characterizations for thegeneralizedLP well-posedness ofVEPwithout using any gap functions ofVEP.
Now we introduce the Kuratowski measure of noncompactness for a nonempty subset
AofXsee13defined by
αA inf
>0 :A⊂
n
i1
Ai, for everyAi, diamAi<
, 3.1
where diamAiis the diameter ofAidefined by
diamAi sup{dx1, x2:x1, x2 ∈Ai}. 3.2
Given two nonempty subsetsAandBofX, the excess of setAto setBis defined by
eA, B sup{da, B:a∈A}, 3.3
and the Hausdorffdistance betweenAandBis defined by
HA, B max{eA, B, eB, A}. 3.4
For any > 0, four types of approximating solution sets for VEP are defined, respectively, by
T1:{x∈X1:dgx, K≤andfx, y ex/∈ −intCx, for ally∈X0},
T2:{x∈X1:dx, X0≤andfx, y ex/∈ −intCx, for all y∈X0},
T3:{x∈X1 :dgx, K≤ andfx, y ex/∈ −intCx, for all y∈X0and
fx, y−ex∈ −Cx, for somey∈X0},
T4 : {x ∈ X1 : dx, X0 ≤ and fx, y ex/∈ −intCx, for ally ∈ X0and
fx, y−ex∈ −Cx, for somey∈X0}.
Theorem 3.1. LetXbe complete.
i VEPis generalized type I LP well-posed if and only if the solution setΩis nonempty and compact and
eT1,Ω−→0 as−→0. 3.5
ii VEPis type I LP well-posed if and only if the solution setΩis nonempty and compact and
eT2,Ω−→0 as−→0. 3.6
iii VEPis generalized type II LP well-posed if and only if the solution setΩis nonempty and compact and
iv VEPis type II LP well-posed if and only if the solution setΩis nonempty and compact and
eT4,Ω−→0 as−→0. 3.8
Proof. The proofs ofii,iii, andivare similar with that ofiand they are omitted here. LetVEPbe generalized type I LP well-posed. ThenΩis nonempty and compact. Now we show that3.5holds. Suppose to the contrary that there existl >0,n>0 withn → 0 and
zn∈T1nsuch that
dzn,Ω≥l. 3.9
Since{zn} ⊂T1nwe know that{zn}is generalized type I LP approximating solution forVEP. By the generalized type I LP well-posedness ofVEP, there exists a subsequence
{znj}of{zn}converging to some element ofΩ. This contradicts3.9. Hence3.5holds.
Conversely, suppose thatΩis nonempty and compact and3.5holds. Let{xn}be a generalized type I LP approximating solution forVEP. Then there exists a sequence{n} with{n} ⊆R1andn → 0 such that
dgxn, K
≤n,
fxn, y
nexn/∈ −intCxn, ∀y∈X0.
3.10
Thus,{xn} ⊂T1. It follows from3.5that there exists a sequence{zn} ⊆Ωsuch that
dxn, zn dxn,Ω≤eT1,Ω−→0. 3.11
SinceΩis compact, there exists a subsequence {znk} of{zn} converging tox0 ∈ Ω.
And so the corresponding subsequence{xnk}of{xn}converging to x0. ThereforeVEPis
generalized type I LP well-posed. This completes the proof.
Theorem 3.2. LetXbe complete. Assume that
ifor anyy∈X1, the vector-valued functionx→fx, yis continuous;
iithe mappingW :X → 2Y defined byWx Y\ −intCxis closed.
ThenVEPis generalized type I LP well-posed if and only if
T1/, ∀ >0, lim
→0αT1 0. 3.12
Proof. First we show that for every >0,T1is closed. In fact, let{xn} ⊂T1andxn → x. Then
dgxn, K
≤,
fxn, y
exn/∈ −intCxn, ∀y∈X0.
From3.13, we get
dgx, K≤,
fxn, y
exn∈Wxn, ∀y∈X0.
3.14
By assumptionsi,ii, we havefx, y ex/∈ −intCx, for ally∈X0.Hencex∈T1. Second, we show that
Ω >0T1. 3.15
It is obvious that
Ω⊂ >0T1. 3.16
Now suppose thatn>0 withn → 0 andx∗∈
∞
n1T1n. Then
dgx∗, K≤n, ∀n∈N, 3.17
fx∗, y nex∗/∈ −intCx∗, ∀y∈X0. 3.18
SinceK is closed,g is continuous, and3.17holds, we havex∗ ∈X0. By3.18and closedness ofWx∗, we getfx∗, y∈ Wx∗, for ally ∈ X0,that is,x∗ ∈Ω.Hence3.15 holds.
Now we assume that 3.12 holds. Clearly, T1· is increasing with > 0. By the Kuratowski theoremsee14, we have
HT1,Ω−→0, as−→0. 3.19
Let {xn} be any generalized type I LP approximating solution sequence for VEP. Then there existsn >0 withn → 0 such that3.13holds. Thus,xn∈T1n. It follows from
3.19thatdxn,Ω → 0. So there exsistun∈Ω, such that
dxn, un−→0. 3.20
SinceΩis compact, there exists a subsequence{unj} of{un}and a solutionx∗ ∈ Ω
satisfying
unj −→x∗. 3.21
From3.20and3.21, we getdxnj, x∗ → 0.
Conversely, letVEP be generalized type I LP well-posed. Observe that for every
>0,
Hence,
αT1≤2HT1,Ω αΩ 2eT1,Ω, 3.23
whereαΩ 0 sinceΩis compact. FromTheorem 3.1i, we know thateT1,Ω → 0 as
→ 0. It follows from3.23that3.12holds. This completes the proof.
Similar toTheorem 3.2, we can prove the following result.
Theorem 3.3. LetXbe complete. Assume that
ifor anyy∈X1, the vector-valued functionx→fx, yis continuous;
iithe mappingW :X → 2Y defined byWx Y\ −intCxis closed;
iiithe set-valued mappingC:X1 → 2Y is closed;
ivfor anyx∗ ∈Ω,fx∗, y∈ −∂C, for somey ∈X0. ThenVEPis generalized type II LP
well-posed if and only if
T3/, ∀ >0, lim
→0αT3 0. 3.24
Definition 3.4. VEP is said to be generalized type Iresp., generalized type IIwell-set if
Ω/∅and for any generalized type Iresp., generalized type IILP approximating solution sequence{xn}forVEP, we have
dxn,Ω−→0, asn−→ ∞. 3.25
From the definitions of the generalized LP well-posedness forVEPand those of the generalized well-set forVEP, we can easily obtain the following proposition.
Proposition 3.5. The relations between generalized LP well-posedness and generalized well set are
i VEPis generalized type I LP well-posed if and only ifVEPis generalized type I well-set andΩis compact.
ii VEPis generalized type II LP well-posed if and only if VEP is generalized type II well-set andΩis compact.
By combining the proof ofTheorem 3.3in10and that ofTheorem 3.1, we can prove that the following results show that the relations between the generalized LP well-posedness forVEPand the solution setΩofVEP.
Theorem 3.6. LetXbe finite dimensional. Assume that
ifor anyy∈X1, the vector-valued functionx→fx, yis continuous;
iithe mappingW :X → 2Y defined byWx Y\ −intCxis closed;
iiithere exists0>0such thatT10(resp.,T30) is bounded.
Corollary 3.7. SupposeΩ/. And assume that
ifor anyy∈X1the vector-valued functionx→fx, yis continuous;
iithe mappingW :X → 2Y defined byWx Y\ −intCxis closed;
iiithere exists0>0such thatT10(resp.,T30) is compact.
IfΩis nonempty, thenVEPis generalized type Iresp., generalized type IILP well-posed.
3.2. Criteria and Characterizations Using Gap Functions
In this subsection, we give some criteria and characterizations for thegeneralizedLP well-posedness of VEP using the gap functions ofVEP introduced by S. J. Li and M. H. Li
10.
Chen et al.15introduced a nonlinear scalarization functionξe:X×Z → Rdefined by
ξe
x, yinfλ∈R:y∈λex−Cx. 3.26
Definition 3.810. A mappingg :X → Ris said to be a gap function onX0forVEPif
igx≥0, for allx∈X0;
iigx∗ 0 andx∗∈X0if and only ifx∗∈Ω.
S. J. Li and M. H. Li10introduced a mappingφ:X → Rdefined as follows:
φx sup y∈X0
−ξe
x, fx, y. 3.27
Lemma 3.9 see10. If for any x ∈ X0,fx, x ∈ −∂Cx, where ∂Cx is the topological
boundary ofCx, then the mappingφdefined by3.27is a gap function onX0forVEP.
Now we consider the following general constrained optimization problems introduced and researched by Huang and Yang [4]:
Pminφx
s.t. x∈X1, gx∈K.
3.28
We useargminφandv∗denote the optimal set and value of (P), respectively.
The following example illustrates that it is useful to consider sequences that satisfy
Example 3.10. Letα >0,XR1,ZR1,Cx R2,andex 1,1for eachx∈X,KR1 −,
X1R1, gx
⎧ ⎪ ⎨ ⎪ ⎩
x, ifx∈0,1,
1
x2, ifx≥1,
fx, y
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
xα−yα,−xα−y−1, if x∈0,1, ∀y∈X 1,
1
xα − 1
yα,− 1
xα −y−1
, if x >1, ∀y∈X1,
−1,−1, if x <0, ∀y∈X1.
3.29
Then, it is easy to verify thatX0{x∈X1:gx∈K}andVEPis equivalent to the optimization problemPwith
φx
⎧ ⎪ ⎨ ⎪ ⎩
−xα, ifx∈0,1,
− 1
xα, ifx≥1.
3.30
Huang and Yang4showed thatxn 2n1/αis the unique solution to the following penalty problemP Pαn:
P Pαnmin x∈X1
φx nmax0, gxα, n∈N, 3.31
anddgxn, K → 0 anddxn, X0 → ∞.
Now, we recall the definitions about generalized well-posedness forPintroduced by Huang and Yang4 or7as follows
Definition 3.11. A sequence{xn} ⊂ X1 is called a generalized type Iresp., generalized type IILP approximating solution sequence forPif the following3.32and3.33 resp.,3.32 and3.34hold:
dgxn, K
−→0, asn−→ ∞, 3.32
lim sup
n→ ∞ φxn≤v
∗, 3.33
lim
n→ ∞φxn v
∗. 3.34
Definition 3.12. Pis said to be generalized type Iresp., generalized type IILP well-posed if
iargminφ /;
iifor every generalized type Iresp., generalized type IILP approximating solution sequence{xn}forP, there exists a subsequence{xnj}of{xn}converging to some element
The following result shows the equivalent relations between the generalized LP well-posedness ofVEPand the generalized LP well-posedness ofP.
Theorem 3.13. Suppose thatfx, x∈ −∂Cx, for allx∈X0. Then
i VEPis generalized type I well-posed if and only if (P) is generalized type I well-posed; ii VEPis generalized type II well-posed if and only if (P) is generalized type II well-posed.
Proof. iByLemma 3.9, we know thatφis a gap function onX0,x ∈ Ωif and only if x ∈ argminφwithv∗φx 0.
Assume that{xn}is any generalized type I LP approximating solution sequence for
VEP. Then there existsn>0 withn → 0 such that
dgxn, K
≤n, 3.35
fxn, y
nexn/∈ −intCxn, ∀y∈X0. 3.36
It follows from3.35and3.36that
dgxn, K
−→0, asn−→ ∞, 3.37
ξe
xn, f
xn, y
≥ −n, ∀y∈X0. 3.38
Hence, we obtain
φxn sup y∈X0
−ξe
xn, f
xn, y
≤n. 3.39
Thus,
lim sup
n→ ∞ φxn≤0 sincen−→0. 3.40
The above formula and3.37imply that{xn}is a generalized type I LP approximating solution sequence forP.
Conversely, assume that {xn} is any generalized type I LP approximating solution sequence forP. Thendgxn, K → 0 and lim supn→ ∞φxn≤0.
Thus, there existsn>0 withn → 0 satisfying3.35and
φxn sup y∈X0
−ξe
xn, f
xn, y
≤n. 3.41
From3.41, we have
ξe
xn, f
xn, y
Equivalently,3.36 holds. Hence, {xn} is a generalized type I LP approximating solution sequence forVEP.
iiThe proof is similar toiand is omitted. This completes the proof.
Now we consider a real-valued functionc ct, s defined fort, s ≥ 0 sufficiently small, such that
ct, s≥0, ∀t, s, c0,0 0,
sn−→0, tn≥0, ctn, sn−→0, implytn−→0.
3.43
Lemma 3.14see4, Theorem 2.2. Suppose thatfx, x∈ −∂Cxfor anyx∈X0.
iIf (P) is generalized type II LP well-posed, then there exists a functioncsatisfying3.43
such that
φx−v∗≥cdx,argminφ, dgx, K, ∀x∈X1. 3.44
iiAssume thatargminφis nonempty and compact, and3.44holds for somecsatisfying 3.43. Then (P) is generalized type II LP well-posed.
The following theorem follows immediately from Lemma 3.14and Theorem 3.13 with φx defined by3.27andv∗0.
Theorem 3.15. Suppose thatfx, x∈ −∂Cxfor anyx∈X0.
i If VEP is generalized type II LP well-posed, then there exists a function c satisfying 3.43such that
φx≥c
dx,Ω, dgx, K, ∀x∈X1. 3.45
iiAssume thatΩis nonempty and compact, and3.45holds for somecsatisfying3.43. ThenVEPis generalized type II LP well-posed.
Definition 3.16see4,7. iLet Z be a topological space and letZ1 ⊂ Zbe a nonempty subset. Suppose thatG:Z → R∪ { ∞}is an extend real-valued function. Then the function
Gis said to be level-compact onZ1if for anys∈R1the subset{z∈Z1 :Gz≤s}is compact.
iiLetZbe a finite dimensional normed space andZ1 ⊂Zbe nonempty. A function
h:Z → R1∪ { ∞}is said to be level-bounded onZ
1ifZ1is bounded or
lim z∈Z1,z → ∞
hz ∞. 3.46
Proposition 3.17. Assume that for any y ∈ X1, the vector-valued function x → fx, y is
continuous and the mappingW : X → 2Y defined byWx Y \ −intCxis closed, and Ω
is nonempty. Then,VEPis generalized type I LP well-posed if one of the following conditions holds: ithere existsδ1>0such thatSδ1is compact, where
Sδ1
x∈X1:d
gx, K≤δ1
iithe functionφdefined by3.27is level-compact onX1;
iiiXis a finite-dimensional normed space and
lim x∈X1,x → ∞
maxφx, dgx, K ∞; 3.48
ivthere existsδ1>0such thatφis level-compact onSδ1defined by3.47.
Proof. Let{xn} ⊆ X1be a generalized type I LP approximating solution sequence forVEP. Then there exists a sequence{n} ⊆ R1 with n > 0 such that3.35and3.36hold. From
3.20, without loss of generality, we assume that {xn} ⊂ Sδ1. Since Sδ1 is compact, there exists a subsequence {xnj} of {xn} and x0 ∈ Sδ1 such that xnj → x0. This fact
combined with 3.35 yields that x0 ∈ X0. Furthermore, it follows from 3.36 and the continuity of f with respect to the first argument and the closedness of W that we have
fx0, y/∈ −intCx0, for ally∈X0. Sox0∈Ω. This implies thatVEPis generalized type I LP well-posed.
It is easy to see that conditioniiimplies conditioniv. Now we show that condition
iiiimplies conditioniv. It follows from10, Proposition 4.2that the functionφdefined by3.27is lower semicontinuous, and thus for anyt ∈ R1, the set{x ∈ Sδ
1 : φx ≤ t} is closed. SinceX is a finite dimensional space, we need only to show that for any t ∈ R1, the set{x ∈ Sδ1 : φx ≤ t}is bounded. Suppose to the contrary that there existst ∈ R1 and{xn} ⊂ Sδ1and φxn ≤ tsuch that||xn|| → ∞. It follows from{xn} ⊂ Sδ1that
dgxn, K≤δ1and so
maxφxn, dgxn, K≤max{t, δ1}. 3.49
Which contradicts with3.48.
Therefore, we only need to prove that if conditionivholds, thenVEPis generalized type I LP well-posed. Suppose that conditionivholds and{xn}is a generalized type I LP approximating solution sequence forVEP. Then there exists{n} ⊂R1 withn>0 such that
3.35and3.36hold. By3.35, we can assume without loss of generality that
{xn} ⊂Sδ1. 3.50
It follows from3.36thatξexn, fxn, y≥ −n, for ally∈X0.Thus,
φxn≤n, ∀n. 3.51
From3.51, without loss of generality, we assume that{xn} ⊆ {x ∈ Sδ1 : φx ≤ b}for someb >0. Sinceφis level-compact onSδ1, the subset{x∈Sδ1:φx≤b}is compact. It follows that there exists a subsequence{xnj}of{xn}andx∈Sδ1such thatxnj → x. This
together with3.35yieldsx∈X0. Furthermore by the continuity offwith respect to the first argument, the closedness ofW, and3.36we havex0∈Ω. This completes the proof.
Proposition 3.18. Assume that for any y ∈ X1, the vector-valued function x → fx, y is
continuous and the mappingW : X → 2Y defined byWx Y \ −intCxis closed, and Ω
is nonempty. Then,VEPis type I LP well-posed if one of the following conditions holds: ithere existsδ1>0such thatS1δ1is compact where
S1δ1 {x∈X1:dx, X0≤δ1}; 3.52
iithe functionφdefined by3.27is level-compact onX1;
iiiXis a finite-dimensional normed space and
lim x∈X1,x → ∞
maxφx, dx, X0
∞; 3.53
ivthere existsδ1>0such thatφis level-compact onS1δ1defined by3.52.
Proposition 3.19. Assume thatX is a finite dimensional space, for anyy ∈ X1, the vector-valued
functionx→fx, yis continuous and the mappingW:X → 2Y defined byWx Y\−intCx
is closed, andΩis nonempty. Suppose that there exists δ1 > 0 such that the functionφxdefined
by3.27is level-bounded on the setSδ1defined by3.47. Then VEPis generalized type I LP
well-posed.
Proof. Let{xn}be a generalized type I LP approximating solution sequence forVEP. Then there exists{n}withn>0 such that3.35and3.36hold.
From 3.35, without loss of generality, we assume that{xn} ⊂ Sδ1. Let us show by contradiction that{xn}is bounded. Otherwise we assume without loss of generality that
||xn|| → ∞. By the level-boundedness ofφ, we have
lim
x → ∞φx ∞. 3.54
It follows from 3.36 and the proof in Proposition 3.17 that 3.51 holds. which contradicts with3.54.
Now we assume without loss of generality thatxn → x. Furthermore by the continuity offwith respect to the first argument, the closedness ofW, and3.36we havex0∈Ω. This completes the proof.
Similarly, we can prove the followingProposition 3.20.
Proposition 3.20. Assume thatX is a finite dimensional space, for anyy ∈ X1, the vector-valued
functionx→fx, yis continuous and the mappingW:X → 2Y defined byWx Y\−intCx
is closed, andΩis nonempty. Suppose that there existsδ1>0such that the functionφxdefined by
3.27is level-bounded on the setS1δ1defined by3.52. ThenVEPis type I LP well-posed.
ofVEPto the generalized well-posedness ofVEP. It is easy to see that the results in this paper generalize and extende the main results in6in several aspects.
Remark 3.22. The generalized Levitin-Polyak well-posedness for vectorquasiequilibrium problems and generalized vector-quasiequilibrium problems with explicit constraintgx∈ Kis still an open question and we will do the research in the near future.
Acknowledgments
This study was supported by Grants from the National Natural Science Foundation of ChinaProject nos. 70673012, 70741028, and 90924030and the China National Social Science FoundationProject no. 08CJY026.
References
1 A. N. Tikhonov, “On the stability of the functional optimization problem,”USSRJ Computational Mathematics and Mathematical Physics, vol. 6, no. 4, pp. 28–33, 1966.
2 E. S. Levitin and B. T. Polyak, “Convergence of minimizing sequences in conditional extremum problem,”Soviet Mathematics Doklady, vol. 7, pp. 764–767, 1966.
3 A. S. Konsulova and J. P. Revalski, “Constrained convex optimization problems—well-posedness and stability,”Numerical Functional Analysis and Optimization, vol. 15, no. 7-8, pp. 889–907, 1994.
4 X. X. Huang and X. Q. Yang, “Generalized Levitin-Polyak well-posedness in constrained optimiza-tion,”SIAM Journal on Optimization, vol. 17, no. 1, pp. 243–258, 2006.
5 X. X. Huang and X. Q. Yang, “Levitin-Polyak well-posedness of constrained vector optimization problems,”Journal of Global Optimization, vol. 37, no. 2, pp. 287–304, 2007.
6 X. X. Huang, X. Q. Yang, and D. L. Zhu, “Levitin-Polyak well-posedness of variational inequality problems with functional constraints,”Journal of Global Optimization, vol. 44, no. 2, pp. 159–174, 2009.
7 X. X. Huang and X. Q. Yang, “Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints,”Journal of Industrial and Management Optimization, vol. 3, no. 4, pp. 671–684, 2007.
8 Z. Xu, D. L. Zhu, and X. X. Huang, “Levitin-Polyak well-posedness in generalized vector variational inequality problem with functional constraints,”Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 505–524, 2008.
9 X. J. Long, N.-J. Huang, and K. L. Teo, “Levitin-Polyak well-posedness for equilibrium problems with functional constraints,”Journal of Inequalities and Applications, vol. 2008, Article ID 657329, 14 pages, 2008.
10 S. J. Li and M. H. Li, “Levitin-Polyak well-posedness of vector equilibrium problems,”Mathematical Methods of Operations Research, vol. 69, no. 1, pp. 125–140, 2009.
11 N.-J. Huang, X.-J. Long, and C.-W. Zhao, “Well-posedness for vector quasi-equilibrium problems with applications,”Journal of Industrial and Management Optimization, vol. 5, no. 2, pp. 341–349, 2009.
12 M. H. Li, S. J. Li, and W. Y. Zhang, “Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems,”Journal of Industrial and Management Optimization, vol. 5, no. 4, pp. 683–696, 2009.
13 M. Furi and A. Vignoli, “About well-posed optimization problems for functionals in metric spaces,” Journal of Optimization Theory and Applications, vol. 5, pp. 225–229, 1970.
14 C. Kuratowski,Topologie, Panstwove, Warszawa, Poland, 1952.
