Volume 2009, Article ID 898213,13pages doi:10.1155/2009/898213
Research Article
Optimality Conditions of Globally Efficient
Solution for Vector Equilibrium Problems with
Generalized Convexity
Qiusheng Qiu
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
Correspondence should be addressed to Qiusheng Qiu,[email protected]
Received 19 March 2009; Accepted 21 September 2009
Recommended by Yeol Je Cho
We study optimality conditions of globally efficient solution for vector equilibrium problems with generalized convexity. The necessary and sufficient conditions of globally efficient solution for the vector equilibrium problems are obtained. The Kuhn-Tucker condition of globally efficient solution for vector equilibrium problems is derived. Meanwhile, we obtain the optimality conditions for vector optimization problems and vector variational inequality problems with constraints.
Copyrightq2009 Qiusheng Qiu. 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
Throughout the paper, letX,Y,andZbe real Hausdorfftopological vector spaces,D⊂ Xa nonempty subset, and 0Ydenotes the zero element ofY. LetC⊂YandK⊂Zbe two pointed
convex conessee1such that intC /∅, intK /∅, where intCdenotes the interior ofC. Let
g :D → Zbe a mapping and letF :D×D → Y be a mapping such thatFx, x 0, for all
x∈D. For eachx∈D, we denoteFx, D y∈DFx, yand define the constraint set
Ax∈D:gx∈ −K, 1.1
which is assumed to be nonempty.
Consider the vector equilibrium problems with constraintsfor short, VEPC: finding
x∈Asuch that
Fx, y/∈ −P, ∀y∈A, VEPC
Vector equilibrium problems, which contain vector optimization problems, vector variational inequality problems, and vector complementarity problems as special case, have been studied by Ansari et al.2,3, Bianchi et al.4, Fu5, Gong6, Gong and Yao7,8, Hadjisavvas and Schaible9, Kimura and Yao10–13, Oettli14, and Zeng et al.15. But so far, most papers focused mainly on the existence of solutions and the properties of the solutions, there are few papers which deal with the optimality conditions. Giannessi et al.
16turned the vector variational inequalities with constraints into another vector variational inequalities without constraints. They gave sufficient conditions for efficient solution and weakly efficient solution of the vector variational inequalities in finite dimensional spaces. Morgan and Romaniello17gave scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities in Hilbert space by using the concept of subdifferential of the function. Gong18presented the necessary and sufficient conditions for weakly efficient solution, Henig efficient solution, and superefficient solution for the vector equilibrium problems with constraints under the condition of cone-convexity. However, the condition of cone-convexity is too strong. Some generalized convexity has been developed, such as cone-preinvexitysee19, cone-convexlikenesssee20, cone-subconvexlikenesssee21, and generalized cone-convexlikenesssee22. Among them, the generalized cone-subconvexlikeness has received more attention. Then, it is important to give the optimality conditions for the solution ofVEPCunder conditions of generalized convexity. Moreover, it appears that no work has been done on the Kuhn-Tucker condition of solution forVEPC. This paper is the effort in this direction.
In the paper, we study the optimality conditions for the vector equilibrium problems. Firstly, we present the necessary and sufficient conditions for globally efficient solution of VEPC under generalized cone-subconvexlikeness. Secondly, we prove that the Kuhn-Tucker condition for VEPC is both necessary and sufficient under the condition of cone-preinvexity. Meanwhile, we obtain the optimality conditions for vector optimization problems with constraints and vector variational inequality problems with constraints in
Section 4.
2. Preliminaries and Definitions
LetY∗,Z∗be the dual space ofY,Z, respectively, then the dual cone ofCis defined as
C∗ϕ∈Y∗:ϕc≥0, ∀c∈C. 2.1
The set of strictly positive functional inC∗is denoted byC i, that is,
C iϕ∈C∗:ϕc>0, ∀c∈C\ {0Y}. 2.2
It is well known that
iifC i/∅, thenChas a base;
iiifY is a Hausdorfflocally convex space, thenC i/∅if and only ifChas a base;
Remark 2.1. The positive cone in many common Banach spaces possesses strictly positive functionals. However, this is not always the casesee23.
LetM⊂Y be an arbitrary nonempty subset and cone M {λx:λ >0, x∈M}. The symbol clMdenotes the closure ofM, and coneMdenotes the generated cone ofM, that is, coneM {λx:λ≥0, x∈M}. WhenMis a convex, so is coneM.
Remark 2.2. Obviously, we have
iconeM cone M{0Y}; iiclconeM clcone M;
iiiifP ⊂Y satisfying for allλ >0,λP ⊂P, then cone M P cone M P.
Several definitions of generalized convex mapping have been introduced in literature.
1LetS0⊂Xbe a nonempty convex subset and letC⊂Ybe a convex cone. A mapping f :S0 → Y is calledC-convex, if for allx1, x2∈S0, for allλ∈0,1, we have
λfx1 1−λfx2−fλx1 1−λx2∈C. 2.3
2LetD⊂Xbe a nonempty subset and letC⊂Y be a convex cone.
iA mappingf :D → Y is calledC-convexlikesee20, if for allx1, x2 ∈D,
for allλ∈0,1, there existsx3∈Dsuch that
λfx1 1−λfx2−fx3∈C. 2.4
iifis said to beC-subconvexlikesee21, if there existsθ∈intCsuch that for allx1, x2∈D, for allλ∈0,1, for allε >0, there existsx3∈Dsuch that
εθ fx1 1−λfx2−fx3∈C. 2.5
iiifis said to be generalizedC-subconvexlikesee22, if there existsθ∈intC such that for allx1, x2 ∈D, for allλ∈0,1, for allε > 0, there existsx3 ∈D, ρ >0 such that
εθ λfx1 1−λfx2−ρfx3∈C. 2.6
A nonempty subsetS ⊂ X is called invex with respect toη, if there exists a mappingη:S×S → Xsuch that for anyx, y∈S, andt∈0,1,x tηy, x∈S.
3Let S ⊂ X be a invex set with respect toη. A mappingf : S → Y is said to be
C-preinvex with respect toηsee19, if for anyx, y∈S, andt∈0,1, we have
Remark 2.3. iFrom21, we know thatfisC-convexlike onDif and only iffD Cis a convex set andfisC-subconvexlike onDif and only iffD intCis a convex set.
iiIffD Cis a convex set, so is intclfD C. By Lemma 2.5 of24,fD intC is convex. This shows thatC-convexlikeness impliesC-subconvexlikeness. But in general the converse is not truesee21.
iiiIt is clear thatC-subconvexlikeness implies generalizedC-subconvexlikeness. But in general the converse is not truesee22.
Remark 2.4. Forηx, y x−y, the invex set is a convex set and theC-preinvex mapping is
a convex mapping. However, there are mappings which areC-preinvex but not convexsee
25.
Relationships among various types of convexity are as shown below:
C-convexity⇒C-preinvexity⇒C-convexlikeness⇒C-subconvexlikeness
⇒generalizedC-subconvexlikeness. 2.8
Yang 26 proved the following Lemma in Banach space; Chen and Rong 27 generalized the result to topological vector space.
Lemma 2.5. Assume thatintC /∅. Thenf :D → Y is generalizedC-subconvexlike if and only if
cone fD intCis convex.
Lemma 2.6. Assume thatiM⊂Yis a nonempty subset andC⊂Yis a convex cone withintC /∅.
iicone M intCis convex. ThenclconeM Cis also convex.
Proof. ByLemma 2.5andRemark 2.1iii, we deduce that cone M intCis a convex set. It
is not difficult to prove that coneM intCis a convex set.
Note that clconeM C clconeM intCand the closure of a convex set is convex, then clconeM Cis a convex set. The proof is finished.
Lemma 2.7see1. Ifψ∈K∗\ {0Z∗},z∈ −intK, thenψ, z<0.
Assume thatintC /∅, a vectorx ∈ Ais called a weakly efficient solution of VEPC, ifx satisfies
Fx, y/∈ −intC, ∀y∈A. 2.9
Definition 2.8see6. LetC⊂Ybe a convex cone. Also,x∈Ais said to be a globally efficient
solution ofVEPC, if there exists a pointed convex coneH ⊂Y withC\ {0Y} ⊂ intHsuch
that
Fx, A∩−H\ {0Y} ∅. 2.10
Remark 2.9. Obviously,x∈Ais a globally efficient solution ofVEPC, thenxis also a weakly
3. Optimality Conditions
Theorem 3.1. Assume that i x ∈ Aand there exists x0 ∈ D such that gx0 ∈ −intK;ii hy Fx, y, gyis a generalizedC×K-subconvexlike onD. Thenx∈Ais a globally efficient
solution of VEPCif and only if there existsϕ∈C iandψ∈K∗such that
ϕ, Fx, x ψ, gxmin
y∈D
ϕ, Fx, y ψ, gy, 3.1
ψ, gx0. 3.2
Proof. Assume thatx∈Ais a globally efficient solution ofVEPC, then there exists a pointed
convex coneH⊂Y withC\ {0Y} ⊂intHsuch that
Fx, A∩ −H{0Y}. 3.3
SinceHis a pointed convex cone withC\ {0Y} ⊂intH, then
Fx, A C∩ −intH∅. 3.4
Note thathy Fx, y, gy, for ally∈Dand above formula, it is not difficult to prove
hD C×K∩−intH×−intK ∅. 3.5
Since intH and intK are two open sets andC, K are two pointed convex cones, by
3.5, we have
clconehD C×K∩−intH×−intK ∅. 3.6
Moreover, since hy Fx, y, gy is a generalized C × K-subconvexlike on D, by
Lemma 2.5, cone hD intH × intK is convex. This follows from Lemma 2.6 that clconehD C×K is convex. By the standard separation theorem see1, page 76, there existsϕ, ψ∈Y∗×Z∗\ {0Y∗,0Z∗}such that
ϕ, ψ,clconehD C×K>ϕ,−intC ψ,−intK. 3.7
Since clconehD C×Kis a cone, it follows from3.7that
ϕ, ψ,clconehD C×K≥0. 3.8
Note that0Y,0Z∈C×K, thushD⊂clconehD C×K. By3.8, we obtain immediately
It implies that
ϕ, Fx, y ψ, gy≥0, ∀y∈D. 3.10
On the other hand, by0Y,0Z∈clconehD C×Kand3.7, we get
ϕ,−intH ψ,−intK<0. 3.11
Since for allh∈intH, for allλ >0, we haveλh∈intH, by3.11, we get
ϕ, h> 1 λ
ψ,−k0
, ∀h∈intH, ∀λ >0, k0∈intK. 3.12
Lettingλ → ∞, we have
ϕ, h ≥0, ∀h∈intH. 3.13
Firstly, we prove that
ϕ∈H∗\ {0Y∗}, ψ ∈K∗. 3.14
SinceHis convex and intHis nonempty, thenH⊂clH clintH. Note thatϕ∈Y∗ and3.13, and we haveϕ∈H∗. With similar proof ofϕ∈H∗, we can prove thatψ∈K∗.
We need to show thatϕ /0Y∗.
In fact, ifϕ0Y∗, thenψ ∈K∗\ {0Z∗}. By3.10, we have
ψ, gy≥0, ∀y∈D. 3.15
On the other hand, sinceψ ∈K∗,gx0∈ −intK, byLemma 2.7, we haveψ, gx0<
0, which is a contradiction with3.15. Secondly, we show thatϕ∈C i.
For anyc∈C\ {0Y}, sinceC\ {0Y} ⊂intH, then there exists a balanced neighborhood
Uof zero element such that
c U⊂H. 3.16
Note thatϕ /0Y∗, and there exists−u∈Usuch thatϕ, u>0.
Sinceϕ∈H∗, then
ϕ, c ≥ ϕ, u>0. 3.17
By the arbitrariness ofc∈C\ {0Y}, we haveϕ∈C i.
Takingyxin3.10, we get
ψ, gx≥0. 3.18
Moreover, sincex∈A{x∈D:gx∈ −K}, ψ ∈K∗, then
ψ, gx≤0. 3.19
Thus3.2holds.
SinceFx, x 0 andψ, gx0, by3.10, we have
ϕ, Fx, x ψ, gxmin
y∈D
ϕ, Fx, y ψ, gy. 3.20
Then3.1holds.
Conversely, ifx∈Ais not a globally efficient solution ofVEPC, then for any pointed convex coneH⊂Y withC\ {0Y} ⊂intH, we have
Fx, A∩−H\ {0Y}/∅. 3.21
Byϕ∈C i, let
H0
y∈Y :ϕ, y>0∪ {0Y}. 3.22
Obviously,H0 is a pointed convex cone andC\ {0Y} ⊂ intH0. By3.21, then there exists y0∈Asuch that
Fx, y0
∈Fx, A∩−H\ {0Y}. 3.23
By the definition ofH0, we get
ϕ, Fx, y0
<0. 3.24
Moreover, sincey0∈A{x∈D:gx∈ −K}andψ∈K∗, then
ψ, gy0
≤0. 3.25
This together with3.24implies that
ϕ, Fx, y0
ψ, gy0
On the other hand, sinceFx, x 0, by3.1and3.2, we get
0ϕ, Fx, x ψ, gx
min
y∈D
ϕ, Fx, y ψ, gy
≤ϕ, Fx, y0
ψ, gy0
,
3.27
which contradicts3.26. The proof is finished.
Corollary 3.2. Assume thatiD⊂Xis invex with respect toη;iix∈Aand there existsx0∈D
such thatgx0 ∈ −intK;iiiFx,·isC-preinvex onD with respect toη, andg : D → Y is K-preinvex onDwith respect toη. Thenx∈Ais a globally efficient solution ofVEPCif and only
if there existϕ∈C iandψ∈K∗such that3.1and3.2hold.
Proof. SinceFx,·isC-preinvex onDwith respect toη, g :D → Y isK-preinvex onDwith
respect toη. Thenhy Fx, y, gyisC×K-preinvex onDwith respect toη. Thus by
Theorem 3.1, the conclusion ofCorollary 3.2holds.
Remark 3.3. Corollary 3.2 generalizes and improves the recent results of Gong see 18,
Theorem 3.3. Especially, Corollary 3.2 generalizes and improves in the following several aspects.
1The condition that the subsetDis convex is extended to invex.
2Fx, yisC-convex inyis extended toC-preinvex iny.
3gyisK-convex is extended toK-preinvex.
Next, we introduce Gateaux derivative of mapping.
Letx ∈Xand letf :X → Y be a mapping.f is called Gateaux differentiable atxif for anyx∈X, there exists limit
f
xx limt→0
fx tx−fx
t . 3.28
Mappingfx :x → fxxis called Gateaux derivative offatx.
The following theorem shows that the Kuhn-Tucker condition for VEPC is both necessary and sufficient.
Theorem 3.4. Assume thatiC⊂Y,K⊂Zare closed,D⊂Xis invex with respect toη;iix∈A
and there existsx0 ∈D such thatgx0 ∈ −intK;iiiFx,·isC-preinvex onDwith respect to ηand Gateaux differentiable atx, andg :D → Y is Gateaux differentiable atxandK-preinvex on
Dwith respect toη; . Thenx∈Ais a globally efficient solution of VEPCif and only if there exists
ϕ∈C iandψ ∈K∗such that
ϕ, F
x
x, ηy, x ψ, g
x
ηy, x≥0, ∀y∈D, 3.29
Proof. Assume thatx ∈ Ais a globally efficient solution ofVEPC, byCorollary 3.2, there existsϕ∈C iandψ∈K∗such that
ψ, gx0, 3.31
ϕ, Fx, y−Fx, x ψ, gy−gx≥0, ∀y∈D. 3.32
SinceDis invex with respect toη, then for anyy∈D,
x tηy, x∈D, ∀t∈0,1. 3.33
By3.32, for anyt∈0,1, we have
ϕ,F
x, x tηy, x−Fx, x
t
ψ,g
x tηy, x−gx
t
≥0, ∀y∈D. 3.34
SinceFx,·is Gateaux differentiable atx, and g : D → Y is Gateaux differentiable atx, lettingt → 0 in3.34, we have
ϕ, F
x
x, ηy, x ψ, g
x
ηy, x≥0, ∀y∈D. 3.35
Conversely, ifxis not a globally efficient solution ofVEPC, a similar proof of3.24 inTheorem 3.1, there existsx1∈Asuch that
ϕ, Fx, x1
<0. 3.36
SinceFx, x 0, thus we have
ϕ, Fx, x1−Fx, x
<0. 3.37
Moreover, sinceFx,·isC-preinvex onDwith respect toη, then for anyλ∈0,1,x, x1∈D,
we have
λFx, x1 1−λFx, x−F
x, x ληx1, x
∈C. 3.38
This together withCbeing cone yields that
Fx, x1−Fx, x−
Fx, x ληx1, x
−Fx, x
λ ∈C. 3.39
SinceCis closed, takingλ → 0 in the above formula, we have
Fx, x1−Fx, x−Fx
x, ηx1, x
Note thatϕ∈C∗, then we have
ϕ, Fx, x1−Fx, x
≥ϕ, F
x
x, ηx1, x
. 3.41
This together with3.37yields that
ϕ, F
x
x, ηx1, x
<0. 3.42
Moreover, sincex1∈A,ψ ∈K∗andψ, gx0, then we have
ψ, gx1−gx
≤0. 3.43
With similar proof of3.41, we get
ψ, g
x
ηx1, x
≤ψ, gx1−gx
≤0. 3.44
This together with3.42implies that
ϕ, F
x
x, ηx1, x
ψ, g
x
ηx1, x
<0, 3.45
which contradicts3.29. The proof is finished.
4. Application
As interesting applications of the results ofSection 3, we obtain the optimality conditions for vector optimization problems and vector variational inequality problems.
LetLX, Ybe the space of all bounded linear mapping fromX toY. We denote by
h, xthe value ofh∈LX, Yatx.
Equation VEPC includes as a special case a vector variational inequality with constraintsfor short,VVICinvolving
Fx, yTx, y−x, 4.1
whereTis a mapping fromDtoLX, Y.
Definition 4.1see18. IfFx, y Tx, y−x,x, y∈A, and ifx∈Ais a globally efficient
solution ofVEPC, thenx∈Ais called a globally efficient solution ofVVIC.
Theorem 4.2. Assume thatiC⊂Y,K ⊂Zare closed,D ⊂Xis a nonempty convex subset;ii
x∈Aand there existsx0∈Dsuch thatgx0∈ −intK;iiig :D → Y is Gateaux differentiable
atxandK-convex onD. Thenx ∈ Ais a globally efficient solution of (VVIC) if and only if there existsϕ∈C iandψ ∈K∗such that
ϕ,T x, y−x ψ, g
x
y−x≥0, ∀y∈D,
Proof. Let
Fx, yTx, y−x, x, y∈D,
ηy, xy−x, x, y∈D, 4.3
thenDis invex with respect toη, Fx,·is Gateaux differentiable atxandC-preinvex with respect to η, and g : D → Y is Gateaux differentiable at x and K-preinvex on D with respect toη. Thus the conditions ofTheorem 3.4are satisfied. Note thatϕ, Fxx, ηy, x
ϕ,T x, y−x, byTheorem 3.4, then the conclusion ofTheorem 4.2holds.
Another special case of VEPCis the vector optimization problem with constraints
for short, VOP:
minfx
subject tox∈A VOP
involving
Fx, yfy−fx, x, y∈D, 4.4
wheref:D → Y is a mapping.
Definition 4.3see18. IfFx, y fy−fx,x, y∈A, and ifx∈Ais a globally efficient solution ofVEPC, thenx∈Ais called a globally efficient solution ofVOP.
Theorem 4.4. Assume thatiC⊂Y,K⊂Zare closed,D⊂Xis invex with respect toη;iix∈A
and there existsx0∈Dsuch thatgx0∈ −intK;iiif:D → Yis Gateaux differentiable atxand C-preinvex onDwith respect toη, andg:D → Y is Gateaux differentiable atxandK-preinvex on
Dwith respect toη. Thenx∈ Ais a globally efficient solution of VOPif and only if there exists
ϕ∈C iandψ ∈K∗such that
ϕ, f
x
ηy, x ψ, g
x
ηy, x≥0, ∀y∈D,
ψ, gx0. 4.5
Proof. Let
Fx, yfy−fx, x, y∈D, 4.6
thenFx,·is Gateaux differentiable atxandC-preinvex with respect toη, andg :D → Y is Gateaux differentiable atxand K-preinvex onD with respect toη. Thus the conditions ofTheorem 3.4are satisfied. Note thatϕ, Fxx, ηy, xϕ, fxηy, x, byTheorem 3.4, then the conclusion ofTheorem 4.4holds.
Theorem 4.5. Assume that i x ∈ Aand there exists x0 ∈ D such that gx0 ∈ −intK;ii
hy fy−fx, gyis generalizedC×K-subconvexlike onD. Thenx ∈ Ais a globally
efficient solution ofVOPif and only if there existsϕ∈C iandψ ∈K∗such that
ϕ, fx ψ, gxmin
y∈D
ϕ, fy ψ, gy,
ψ, gx0.
4.7
Acknowledgment
This work was supported by the National Natural Science Foundation of China no. 10771228.
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