Volume 2009, Article ID 684304, 14 pagesdoi:10.1155/2009/684304 Research Article Generalized Levitin-Polyak Well-Posedness of Vector Equilibrium Problems Jian-Wen Peng,1 Yan Wang,1 and L
Trang 1Volume 2009, Article ID 684304, 14 pages
doi:10.1155/2009/684304
Research Article
Generalized Levitin-Polyak Well-Posedness of
Vector Equilibrium Problems
Jian-Wen Peng,1 Yan Wang,1 and Lai-Jun Zhao2
1 College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China
2 Management School, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Lai-Jun Zhao,zhao laijun@163.com
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
Copyrightq 2009 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 Polyak2 extended the notion to constrained scalar optimization, allowing minimizing sequences {x n } to be outside of the feasible set X0 and requiring
d x n , X0 the distance from x n 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
g is a continuous map between two metric spaces and K is a closed set For minimizing
sequences{x n }, instead of dx n , X0, here the distance dgx n , K is required to tend to zero.
This generalization is appropriate for penalty-type methodse.g., penalty function methods, augmented Lagrangian methods with iteration processes terminating when dgxn , K is
small enoughbut dx n , X0 may be large Recently, the study of generalized Levitin-Polyak well-posedness was extended to nonconvex vector optimization problems with abstract and functional constraints see 5, variational inequality problems with abstract and functional constraintssee 6, generalized variational inequality problems with abstract and functional constraints7, generalized vector variational inequality problems with abstract
Trang 2and functional constraints 8, and equilibrium problems with abstract and functional constraints9 Most recently, S J Li and M H Li 10 introduced and researched two types
of Levitin-Polyak well-posedness of vector equilibrium problems with variable domination structures Huang et al.11 introduced and researched the Levitin-Polyak well-posedness of vector quasiequilibrium problems Li et al.12 introduced 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 constraint gx ∈ 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, d X , Z, d Z , and Y be locally convex Hausdorff topological vector spaces, where
d X d Z is the metric which compatible with the topology of XZ Throughout this paper,
we suppose that K ⊂ Z and X1 ⊂ X are nonempty and closed sets, C : X → 2 Y is a
set-valued mapping such that for any x ∈ X, Cx is a pointed, closed, and convex cone in Z with nonempty interior int Cx, e : X → Y is a continuous vector-valued mapping and satisfies that for any x ∈ X, ex ∈ int Cx, f : X × X1 → Y and g : X1 → Z are two vector-valued mappings, and X0 {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 point x∗∈ X0, such that
f
x∗, y
/
∈ − int Cx∗, ∀y ∈ X0. VEP
We always assume that X0/ and g is continuous on X1and the solution set ofVEP
is denoted byΩ
LetP, d be a metric space, P1 ⊆ P, and x ∈ P We denote by dx, P1 inf{dx, p :
p ∈ P1} the distance function from the point x ∈ P to the set P1
Definition 2.1 i A sequence {x n } ⊂ X1 is called a type I Levitin-Polyak in short LP approximating solution sequence forVEP if there exists { n} ⊂ R1
with n → 0 such that
d x n , X0 ≤ n , 2.1
f
x n , y
n e x n /∈ − int Cx n , ∀y ∈ X0. 2.2
ii{x n } ⊂ X1is called type II approximating solution sequence forVEP if there exists
{ n} ⊂ R1
with n → 0 and {y n } ⊂ X0satisfying2.1, 2.2, and
f
x n , y n
− n e x n ∈ −Cx n . 2.3
Trang 3iii{x n } ⊂ X1is called a generalized type I approximating solution sequence forVEP
if there exists{ n} ⊂ R1
with n → 0 satisfying
d
and2.2
iv{x n } ⊂ X1 is called a generalized type II approximating solution sequence for
VEP if there exists { n} ⊂ R1
with n → 0 and {y n } ⊂ X0satisfying2.2, 2.3, and 2.4
Definition 2.2 The vector equilibrium problem VEP is said to be type I resp., type II, generalized type I, generalized type II LP well-posed if Ω / ∅ and for any type I resp., type
II, generalized type I, generalized type II LP approximating solution sequence {xn} of VEP, there exists a subsequence{x n j } of {x n } and x ∈ Ω such that x n j → x.
Remark 2.3 i If Y R and Cx R1
{r ∈ R : r ≥ 0} for all x ∈ X, then the type I
resp., type II, generalized type I, generalized type II LP well-posedness of VEP defined
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, if X∗is the topological dual space of X, F : X1 → X∗
is a mapping, Fx, z denotes the value of the functional Fx at z, and fx, y Fx, y −
x for 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 II LP well-posedness for the variational inequality with abstract and functional constraints introduced by Huang et al.6 If K Z, then X1 X0
and the type Iresp., type II LP well-posedness of VEP defined inDefinition 2.2reduces to the type Iresp., type II LP well-posedness of the vector equilibrium problem introduced by
S J Li and M H Li10
ii It is clear that any generalized type II LP approximating solution sequence of
VEP is a generalized type I LP approximating solution sequence of VEP Thus the
generalized type I LP well-posedness of VEP implies the generalized type II LP well-posedness ofVEP
iii Each type of LP well-posedness of VEP implies that the solution set Ω is nonempty and compact
iv Let g be a uniformly continuous functions on the set
S δ0 x ∈ X1: d
g x, K≤ δ0
2.5
for some δ0 > 0 Then generalized type I resp., type II LP well-posedness implies type I
resp., type II LP 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
generalized LP well-posedness of VEP defined inSection 2
Trang 43.1 Criteria and Characterizations without Using Gap Functions
In this subsection, we give some criteria and characterizations for thegeneralized LP well-posedness ofVEP without using any gap functions of VEP
Now we introduce the Kuratowski measure of noncompactness for a nonempty subset
A of Xsee 13 defined by
α A inf
> 0 : A⊂n
i1
A i , for every A i , diamA i <
where diamA i is the diameter of A idefined by
diamA i sup{dx1, x2 : x1, x2 ∈ A i }. 3.2
Given two nonempty subsets A and B of X, the excess of set A to set B is defined by
e A, B sup{da, B : a ∈ A}, 3.3 and the Hausdorff distance between A and B is defined by
H A, 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 ≤ and fx, y ex /∈ − int Cx, for all y ∈ X0},
T2 : {x ∈ X1: dx, X0 ≤ and fx, y ex /∈ − int Cx, for all y ∈ X0},
T3 : {x ∈ X1 : dgx, K ≤ and fx, y ex /∈ − int Cx, for all y ∈ X0and
f x, y − ex ∈ −Cx, for some y ∈ X0},
T4 : {x ∈ X1 : dx, X0 ≤ and fx, y ex /∈ − int Cx, for all y ∈ X0and
f x, y − ex ∈ −Cx, for some y ∈ X0}
Theorem 3.1 Let X be complete.
i VEP is generalized type I LP well-posed if and only if the solution set Ω is nonempty and compact and
e T1, Ω −→ 0 as −→ 0. 3.5
ii VEP is type I LP well-posed if and only if the solution set Ω is nonempty and compact and
e T2, Ω −→ 0 as −→ 0. 3.6
iii VEP is generalized type II LP well-posed if and only if the solution set Ω is nonempty and compact and
e T3, Ω −→ 0 as −→ 0. 3.7
Trang 5iv VEP is type II LP well-posed if and only if the solution set Ω is nonempty and compact and
e T4, Ω −→ 0 as −→ 0. 3.8
Proof The proofs ofii, iii, and iv are similar with that of i and they are omitted here LetVEP be generalized type I LP well-posed Then Ω is nonempty and compact Now we show that3.5 holds Suppose to the contrary that there exist l > 0, n > 0 with n → 0 and
z n ∈ T1 n such that
Since{z n } ⊂ T1 n we know that {z n} is generalized type I LP approximating solution forVEP By the generalized type I LP well-posedness of VEP, there exists a subsequence
{z n j } of {z n} converging to some element of Ω This contradicts 3.9 Hence 3.5 holds Conversely, suppose thatΩ is nonempty and compact and 3.5 holds Let {x n} be a generalized type I LP approximating solution forVEP Then there exists a sequence { n} with{ n} ⊆ R1
and n → 0 such that
d
g x n , K≤ n ,
f
x n , y
n e x n /∈ − int Cx n , ∀y ∈ X0. 3.10 Thus,{x n } ⊂ T1 It follows from 3.5 that there exists a sequence {z n} ⊆ Ω such that
d x n , z n dx n , Ω ≤ eT1, Ω −→ 0. 3.11
SinceΩ is compact, there exists a subsequence {z n k } of {z n } converging to x0 ∈ Ω And so the corresponding subsequence{x n k } of {x n } converging to x0 ThereforeVEP is generalized type I LP well-posed This completes the proof
Theorem 3.2 Let X be complete Assume that
i for any y ∈ X1, the vector-valued function x
ii the mapping W : X → 2 Y defined by W x Y \ − int Cx is closed.
ThenVEP is generalized type I LP well-posed if and only if
T1 / , ∀ > 0, lim
→ 0α T1 0. 3.12
Proof First we show that for every > 0, T1 is closed In fact, let {x n } ⊂ T1 and x n → x.
Then
d
g x n , K≤ ,
f
x n , y
ex n /∈ − int Cx n , ∀y ∈ X0. 3.13
Trang 6From3.13, we get
d
g x, K≤ ,
f
x n , y
ex n ∈ Wx n , ∀y ∈ X0. 3.14
By assumptionsi, ii, we have fx, y ex /∈ − int Cx, for all y ∈ X0 Hence x ∈ T1.
Second, we show that
It is obvious that
Now suppose that n > 0 with n → 0 and x∗∈ ∞
n1T1 n Then
d
g x∗, K≤ n , ∀n ∈ N, 3.17
f
x∗, y
n e x∗ /∈ − int Cx∗, ∀y ∈ X0. 3.18
Since K is closed, g is continuous, and3.17 holds, we have x∗ ∈ X0 By3.18 and
closedness of Wx∗, we get fx∗, y ∈ Wx∗, for all y ∈ X0, that is, x∗ ∈ Ω Hence 3.15 holds
Now we assume that 3.12 holds Clearly, T1· is increasing with > 0 By the
Kuratowski theoremsee 14, we have
H T1, Ω −→ 0, as −→ 0. 3.19
Let {x n} be any generalized type I LP approximating solution sequence for VEP
Then there exists n > 0 with n → 0 such that 3.13 holds Thus, x n ∈ T1 n It follows from
3.19 that dx n , Ω → 0 So there exsist u n∈ Ω, such that
SinceΩ is compact, there exists a subsequence {u n j } of {u n } and a solution x∗ ∈ Ω satisfying
From3.20 and 3.21, we get dx n j , x∗ → 0
Conversely, letVEP be generalized type I LP well-posed Observe that for every
> 0,
H T1, Ω max{eT1, Ω, eΩ, T1} eT1, Ω. 3.22
Trang 7α T1 ≤ 2HT1, Ω αΩ 2eT1, Ω, 3.23
where αΩ 0 since Ω is compact FromTheorem 3.1i, we know that eT1, Ω → 0 as
→ 0 It follows from 3.23 that 3.12 holds This completes the proof
Similar toTheorem 3.2, we can prove the following result
Theorem 3.3 Let X be complete Assume that
i for any y ∈ X1, the vector-valued function x
ii the mapping W : X → 2 Y defined by W x Y \ − int Cx is closed;
iii the set-valued mapping C : X1 → 2Y is closed;
iv for any x∗ ∈ Ω, fx∗, y ∈ −∂C, for some y ∈ X0 ThenVEP is generalized type II LP well-posed if and only if
T3 / , ∀ > 0, lim
→ 0α T3 0. 3.24
Definition 3.4. VEP is said to be generalized type I resp., generalized type II well-set if
Ω / ∅ and for any generalized type I resp., generalized type II LP approximating solution
sequence{x n} for VEP, we have
d x n , Ω −→ 0, as n −→ ∞. 3.25
From the definitions of the generalized LP well-posedness forVEP and 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 VEP is generalized type I LP well-posed if and only if VEP is generalized type I well-set and Ω is compact.
ii VEP is generalized type II LP well-posed if and only if VEP is generalized type II well-set and Ω is compact.
By combining the proof ofTheorem 3.3in10 and that ofTheorem 3.1, we can prove that the following results show that the relations between the generalized LP well-posedness forVEP and the solution set Ω of VEP
Theorem 3.6 Let X be finite dimensional Assume that
i for any y ∈ X1, the vector-valued function x
ii the mapping W : X → 2 Y defined by W x Y \ − int Cx is closed;
iii there exists 0> 0 such that T10 (resp., T30) is bounded.
IfΩ is nonempty, then VEP is generalized type I resp., generalized type II LP well-posed
Trang 8Corollary 3.7 Suppose Ω / And assume that
i for any y ∈ X1the vector-valued function x
ii the mapping W : X → 2 Y defined by W x Y \ − int Cx is closed;
iii there exists 0> 0 such that T10 (resp., T30) is compact.
IfΩ is nonempty, then VEP is generalized type I resp., generalized type II LP well-posed
3.2 Criteria and Characterizations Using Gap Functions
In this subsection, we give some criteria and characterizations for thegeneralized LP well-posedness of VEP using the gap functions of VEP introduced by S J Li and M H Li
10
Chen et al.15 introduced a nonlinear scalarization function ξ e : X × Z → R defined
by
ξ e
x, y
infλ ∈ R : y ∈ λex − Cx. 3.26
Definition 3.810 A mapping g : X → R is said to be a gap function on X0forVEP if
i gx ≥ 0, for all x ∈ X0;
ii gx∗ 0 and x∗∈ X0if and only if x∗∈ Ω
S J Li and M H Li10 introduced a mapping φ : X → R defined as follows:
φ x sup
y ∈X0
−ξ e
x, f
x, y
boundary of C x, then the mapping φ defined by 3.27 is a gap function on X0forVEP.
Now we consider the following general constrained optimization problems introduced and researched by Huang and Yang [ 4 ]:
P min φx
We use argmin φ and v∗denote the optimal set and value of (P ), respectively.
The following example illustrates that it is useful to consider sequences that satisfy
d gx n , K → 0 instead of dx n , X0 → ∞ for VEP
Trang 9Example 3.10 Let α > 0, X R1, Z R1, Cx R2
, and e x 1, 1 for each x ∈ X, K R1
−,
X1 R1
, g x
⎧
⎪
⎪
x, if x ∈ 0, 1,
1
x2, if x ≥ 1,
f
x, y
⎧
⎪
⎪
⎪
⎪
x α − y α , −x α − y − 1, if x ∈ 0, 1, ∀y ∈ X1,
1
x α − 1
y α ,− 1
x α − y − 1
, if x > 1, ∀y ∈ X1,
−1, −1, if x < 0, ∀y ∈ X1.
3.29
Then, it is easy to verify that X0 {x ∈ X1: gx ∈ K} and VEP is equivalent to the optimization problemP with
φ x
⎧
⎪
⎪
−x α , if x ∈ 0, 1,
− 1
Huang and Yang4 showed that x n 2n 1/αis the unique solution to the following penalty problemPP α n:
PP α nmin
x ∈X1
φ x nmax
0, gxα
and dgx n , K → 0 and dx n , X0 → ∞.
Now, we recall the definitions about generalized well-posedness forP introduced by
Huang and Yang4 or 7 as follows
Definition 3.11 A sequence {x n } ⊂ X1 is called a generalized type Iresp., generalized type II LP approximating solution sequence for P if the following 3.32 and 3.33 resp., 3.32 and3.34 hold:
d
g x n , K−→ 0, as n −→ ∞, 3.32 lim sup
n→ ∞ φ x n ≤ v∗, 3.33 lim
n→ ∞φ x n v∗. 3.34
Definition 3.12 P is said to be generalized type I resp., generalized type II LP well-posed
if
i argmin φ / ;
ii for every generalized type I resp., generalized type II LP approximating solution sequence{x n } for P, there exists a subsequence {x n j } of {x n} converging to some element
of argmin φ.
Trang 10The following result shows the equivalent relations between the generalized LP well-posedness ofVEP and the generalized LP well-posedness of P.
i VEP is generalized type I well-posed if and only if (P) is generalized type I well-posed;
ii VEP is generalized type II well-posed if and only if (P) is generalized type II well-posed Proof. i ByLemma 3.9, we know that φ is a gap function on X0, x ∈ Ω if and only if x ∈ argmin φ with v∗ φx 0.
Assume that{x n} is any generalized type I LP approximating solution sequence for
VEP Then there exists n > 0 with n → 0 such that
d
g x n , K≤ n , 3.35
f
x n , y
n e x n /∈ − int Cx n , ∀y ∈ X0. 3.36
It follows from3.35 and 3.36 that
d
g x n , K−→ 0, as n −→ ∞, 3.37
ξ e
x n , f
x n , y
≥ − n , ∀y ∈ X0. 3.38 Hence, we obtain
φ x n sup
y ∈X0
−ξ e
x n , f
x n , y
Thus,
lim sup
n→ ∞ φ x n ≤ 0 since n −→ 0. 3.40
The above formula and3.37 imply that {x n} is a generalized type I LP approximating solution sequence forP.
Conversely, assume that {x n} is any generalized type I LP approximating solution sequence forP Then dgx n , K → 0 and lim sup n→ ∞φ x n ≤ 0
Thus, there exists n > 0 with n → 0 satisfying 3.35 and
φ x n sup
y ∈X0
−ξ e
x n , f
x n , y
From3.41, we have
ξ e
x n , f
x n , y
≥ − n , ∀y ∈ X0. 3.42
... nonempty, then VEP is generalized type I resp., generalized type II LP well-posed Trang 8Corollary... j } of {x n} converging to some element
of argmin φ.
Trang 10The... between the generalized LP well-posedness of VEP and the generalized LP well-posedness of P.
i VEP is generalized type I well-posed if and only if (P) is generalized type