Volume 2010, Article ID 306403,14pages doi:10.1155/2010/306403
Research Article
Saddle Point Problems, Bilevel Problems,
and Mathematical Program with Equilibrium
Constraint on Complete Metric Spaces
Lai-Jiu Lin and Chih-Sheng Chuang
Department of Mathematics, National Changhua University of Education, Changhua 50058, Taiwan
Correspondence should be addressed to Lai-Jiu Lin,[email protected]
Received 4 November 2009; Accepted 15 January 2010
Academic Editor: Yeol Je Cho
Copyrightq2010 L.-J. Lin and C.-S. Chuang. 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.
We apply an existence theorem of variational inclusion problem on metric spaces to study opti-mization problems, set-valued vector saddle point problems, bilevel problems, and mathematical programs with equilibrium constraint on metric spaces. We study these problems without any convexity and compactness assumptions. Our results are different from any existence results of these types of problems in topological vector spaces.
1. Introduction
LetX, dXandY, dYbe two metric spaces, letZbe a real Hausdorfftopological vector space
ordered by a nonempty pointed closed convex coneKinZwith nonempty interior, and letW
be a real Banach space ordered by a nonempty pointed closed convex coneCwith nonempty interior. LetG:X×Y W,S :X×Y ×XZ, andP :X×Y Zbe multivalued maps. Throughout this paper, we use these notations unless specified otherwise. In this paper, the following vector mathematical programs with equilibrium constraint on metric spaces are considered.
MPEC-1MinCGx, yis subject tox, y∈X×Y, andSx, y, u∩K\ {0} ∅for allu∈X.
MPEC-2MinCGx, yis subject tox, y∈X×Y, andSx, y, u/⊆K\ {0}for allu∈X.
IfZR,K 0,∞, andSis a real function, then problemsMPEC-1andMPEC-2 are reduced to the following problem:
We also study the following bilevel problems.
BL-1MinCGx, yis subject to x, y ∈ X ×Y, fory ∈ Y,x is a solution of problem
Q1y: WMinKPx, y i.e., fory ∈ Y, Px, y∩WMinKPX, y/∅, forx ∈ X,
and y is a solution of problem R1x: WMaxKPx, y.i.e., for x ∈ X,Px, y∩
WMaxKPx, Y/∅.
BL-2MincGx, yis subject tox, y∈X×Y,Px, y/⊆Pa, y intKfor alla∈X, and
Px, b/⊆Px, y intKfor allb∈Y.
BL-3MincGx, y is subject to x, y ∈ X ×Y, forx ∈ X,y is a solution of problem
Q2x: MinRm
Px, y and i.e., for x ∈ X, Px, y∩ MinRmPx, Y/∅, for y ∈
Y, x is a solution of problem R2y: MaxRm
Px, y. i.e., for y ∈ Y, Px, y ∩
MaxRm
PX, y/∅.
IfP :X×Y → Zis a single valued function, thenBL-2is reduced to the following problem.
BL-4MinCGx, yis subject tox, y∈X×Y, fory ∈Y, andxis a solution of problem
and Q3y: WMinKPx, y; and for x ∈ X, y is a solution of problem R3x:
WMaxKPx, y.
Problem BL-1 has applications in real world. Let X be the set of government’s agricultural policies and let Y be the set of government’s industrial policies. For each
x ∈ X and y ∈ Y, let Gx, y be the amount of money that the government uses to promote agriculture and industrial developments, and letPx, ybe the degree of industrial development. We suppose that as the industry becomes more and more developed, the losses from the agriculture sector rise accordingly. Therefore, the solution of problem BL-1 represents the government’s best policy to promote the development of the industry so that the losses from agriculture sector will be as minimal as possible, while the amount of money that the government uses to promote the policies can be the lowest possible.
Mathematical program with equilibrium constraint and bilevel problem represent two important classes of optimization problems which have been investigated in a large number of articles and books. We find in the literatures that Luo et al.1, Stein and Still2, Stein
3, Birbil et al. 4, Liou et al. 5, Lin and Still 6, Lin 7, as well asLin and Hsu 8
have studied mathematical program with equilibrium constraint and bilevel problem on topological vector spaces. As usual in linear and nonlinear optimization, these studies mainly deal with optimality conditions and numerical methods to solve these problems and typically the existence of feasible points is tacitly assumed. Besides, the domains of the functions they consider are subsets of topological vector spaces and certain convexity assumptions on the functions are needed. In this paper, we study these problems with functions defined on metric spaces, so we do not need any convexity assumptions on the functions we consider. We study the existence theorems of solutions forMPEC-1,MPEC-2,BL-1,BL-2, andBL-3. To the best of our knowledge, there is no result of these types of problems on metric space.
In this paper, we also study the following loose set-valued vector saddle point problems.
LSP-1For eachu, v ∈ X×Y, findx, y ∈ X×Y such thatPu, y∩Px, v K/∅,
Px, y∩MinKPX, y/∅, andPx, y∩MaxKPx, Y/∅.
LSP-2For each u, v ∈ X ×Y, findx, y ∈ X ×Y such that Pu, y ⊆ Px, v K,
IfP : X ×Y → Z is a map, then problemsLSP-1andLSP-2are reduced to the following vector saddle point problemsVSP.
iFindx, y∈X×Ysuch thatPx, y∈MinKPX, y∩MaxKPx, Y.
There are many results on loose saddle point problems, vector saddle point, and saddle point problemssee, e.g.,9–17. But to the best of our knowledge, there are no existence theorems in the literatures for loose saddle point and vector saddle point problems for functions defined on the product of metric spaces. We study the loose saddle point problems and vector saddle point problems for functions defined on the product of metric spaces. We do not assume any compact assumptions on the spaces and convexity assumptions on the maps we consider. As for applications of our existence theorems on saddle point problems, we study bilevel problems on metric spaces. We also study mathematical programs with equilibrium constraint. Our results on mathematical programs with equilibrium constraint, bilevel problems, loose saddle point problems, and vector saddle point problems are different from any existence results of these types of problems in the literatures.
2. Preliminaries
LetXandY be topological spacesin short t.s.,T:X Y be a multivalued map.Tis said to be u.s.c.resp., l.s.c.atx∈Xif for every open setUinY withTx⊆Uresp.,Tx∩U /∅, there exists an open neighborhoodV ofxsuch thatTx ⊆ Uresp.,Tx∩U /∅for all
x ∈V;T is said to be u.s.c.resp., l.s.c.onXifT is u.s.c.resp., l.s.c.at every point ofX;
T is continuous atxifTis both u.s.c. and l.s.c. atx;T is said to be closed if GrT{x, y∈ X×Y :y∈Tx, x∈X}is a closed set inX×Y;T is said to be open if GrTis an open set in
X×Y. For a subsetAof topological spaceX, let clAdenote the closure ofA.
Lemma 2.1see18. LetXandYbe topological spaces, and letT :X Y be a multivalued map.
ThenT is l.s.c. atx ∈X if and only if for anyy ∈Txand any net{xα}α∈ΛinX converges tox, there exists a net{yα}α∈Λsuch thatyα∈Txαfor allα∈Λandyα → y.
Lemma 2.2 see19. Let X and Y be Hausdorff topological spaces, and let T : X Y be a
multivalued map.iIfT is an u.s.c. multivalued map with nonempty closed values, thenTis closed; iiifX is a compact set andT is an u.s.c. multivalued map with nonempty compact values, then
TXis compact.
Definition 2.3. LetAbe a nonempty subset of a t.v.s.Zordered by a nonempty pointed closed convex coneK. An element a ∈ A is said to be a minimalresp., maximal point of Aif
A∩a−K {a}resp.,A∩aK {a}. Here, MinKAand MaxKAdenote the sets of
minimal point ofAand maximal point ofA, respectively.
Definition 2.4. LetAbe a nonempty subset of a t.v.s.Zordered by a nonempty pointed closed convex coneK with nonempty interior. An element a ∈ Ais said to be a weakly minimal
resp., weakly maximalpoint ofAifA∩a−intK ∅resp.,A∩aintK ∅. Here, WMinKAand WMaxKAdenote the sets of weakly minimal point ofAand weakly maximal
point ofA, respectively.
Theorem 2.5see20. LetZ be a Hausdorfft.v.s. ordered by a nonempty pointed closed convex
coneK with nonempty interior. If A is a nonempty compact subset of Z, then MinKA /∅ and
Definition 2.6see21. LetK be pointed closed convex cone with nonempty interior in a Banach space Z. Then K is called normal if there exists λ > 0 such that if x, y ∈ K and
y−x∈K, thenx ≤λy. Here, it is calledλ-normal pointed closed convex cone.
Theorem 2.7see22. LetX, dXbe a complete metric space, and letZbe a Hausdorfft.v.s.. Let
F:X×XZbe a multivalued map. Assume that
i0∈Fx, xfor eachx∈X,
iifor eachx∈X,{y∈X : 0∈Fx, y}is a closed subset ofX,
iiifor eachx, y, z∈X, if0∈Fx, yand0∈Fy, z, then0∈Fx, z,
ivfor each sequence{xn}n∈NinX, if0∈Fxn, xn1, thendXxn, xn1 → 0asn → ∞.
Then there existsx∈Xsuch that0/∈Fx, yfor ally∈X\ {x}.
Theorem 2.8. LetX, dXbe a complete metric space. LetG :X W be a multivalued map with
nonempty values. Assume that
iGis closed,
iifor each sequence{xn, yn}n∈NinX×W, ifyn∈Gxnandyn−yn1∈Cfor alln∈N,
thendXxn, xn1 → 0andyn−yn1 → 0asn → ∞.
Then there existsx∈Xsuch thatGx∩MinCGX/∅.
Proof. Letk0 ∈ C\ {0}be fixed. Clearly, GrGis a complete metric space. LetF : GrG×
GrG R be defined by Fx1, y1,x2, y2 {t ∈ R : y1 − y2 tk0 ∈ C} for each
x1, y1,x2, y2 ∈ GrG. Clearly, 0 ∈ Fx, y,x, y for each x, y ∈ GrG. For each
x, y∈GrG, we know that
u, v∈GrG: 0∈Fx, y,u, vu, v∈GrG:y−v∈C. 2.1
If{un, vn}n∈Nis a sequence in{u, v∈GrG: 0∈Fx, y,u, v}andun, vn →
u, vasn → ∞, thenun, vn∈GrGandy−vn ∈Cfor alln∈N. Clearly,y−v∈Cand
u, v∈GrG. Byii, conditionivofTheorem 2.7is satisfied. ByTheorem 2.7, there exists
x, y∈GrGsuch thaty−y /∈Cfor allx, y∈GrG\ {x, y}. Clearly,y∈GX∩y−C. Ify ∈ GX∩y−Candy /y, then there existsx ∈X such thaty ∈Gxandy−y ∈C. Thenx, y∈ GrG\ {x, y}and this implies thaty−y /∈C. This leads to a contradiction. Hence,GX∩y−C {y}. This implies thatGx∩MinCGX/∅.
Example 2.9. LetW R,X −∞,0∪ {1},C 0,∞, and k0 1. LetG : X W be
defined byGx: {−x}for eachx ∈X. Then byTheorem 2.8, there existsx∈X such that
Gx∩MincGX/∅.
Example 2.10. Let W X R, C 0,∞, and k0 1. Let G : X W be defined by Gx:{−x}for eachx∈X. Clearly,Gis closed. But conditioniiofTheorem 2.8does not hold. Indeed, let{xn, yn}{n,−n}n∈N. Thenyn∈Gxnfor alln∈N, anddXxn, xn1 1
andyn−yn11 for alln∈N. Furthermore, there is nox∈Xsuch thatGx∩MincGX/∅.
Theorem 2.11. Let X be a metric space. LetG : X W be a multivalued map with nonempty values. Assume that
iGXis a nonempty closed subset ofW,
iifor each sequence{yn}n∈NinGX, ifyn−yn1∈C, thenyn−yn1 → 0asn → ∞.
Then there existsx∈Xsuch thatGx∩MincGX/∅.
Proof. Letk0 ∈ C\ {0}be fixed. Clearly,GXis a complete metric space. LetF : GX× GX Rbe defined byFy1, y2 {t ∈ R : y2−y1−tk0 ∈ −C}for eachy1, y2 ∈ GX.
Clearly, 0∈Fy, yfor eachy∈GX. Besides, for eachy1 ∈GX, letAy1:{y∈GX: 0∈ Fy1, y}. ThenAy1 {y∈GX:y−y1 ∈ −C}is a closed set. Next, for eachy1, y2, y3∈GX,
if 0∈Fy1, y2and 0∈Fy2, y3, then 0∈Fy1, y3. Byii, for each sequence{yn}n∈NinGX
with 0∈Fyn, yn1, we haveyn−yn1 → 0 asn → ∞. ByTheorem 2.7, there existsx∈X
such thaty∈Gx∩MincGX/∅.
Example 2.12. LetEW R,X {−1} ∪0,∞,C 0,∞, andk0 1. LetG:X Wbe
defined by
Gx:
⎧ ⎨ ⎩
{1,2} ifx−1,
{2x,4} ifx∈0,∞.
2.2
Thus byTheorem 2.11, there existsx∈ Xsuch thatGx∩MincGX/∅. Note thatX is not
closed, not open, and not convex.
Example 2.13. LetX{1,2},WR2, andCR2. LetG:XWbe defined by
Gx:
⎧ ⎨ ⎩
y1, y2
∈R2:y
11, 0≤y2≤2
ifx1,
y1, y2
∈R2:y
12, 0≤y2≤1
ifx2.
2.3
ByTheorem 2.11, there existsx∈Xsuch thatGx∩MincGX/∅. Indeed,x1.
Remark 2.14. Example 2.10 also shows that condition ii of Theorem 2.11 is essential in
Theorem 2.11. Next, the following result is a special case of Theorem 2.11. Note that it is different fromTheorem 2.5since we do not assume thatAis a compact set, but we assume thatWis a Banach space.
Corollary 2.15. LetAbe a nonempty closed subset ofW. Suppose that for each sequence{yn}n∈Nin
Aifyn−yn1∈C, thenyn−yn1 → 0asn → ∞. ThenMincA /∅.
Proof. LetXbe a singleton subset ofW. ThenXis a complete metric space. LetG:X W
be defined by Gx A for each x ∈ X. By Theorem 2.11, there existsx ∈ X such that
Gx∩MincGX/∅and this implies that MincA /∅.
Example 2.16. LetW R2andACR2
. It is easy to see that all conditions ofCorollary 2.15
are satisfied. Hence, MincA /∅. Indeed, MincA{0,0}. Note thatAis not a compact subset
Corollary 2.17. LetWbe a Banach space, and letCbe aλ-normal pointed closed convexconewith nonempty interiorλ∈0,1. IfAis a nonempty bounded closed subset ofW, thenMincA /∅.
Proof. Take any sequence{yn}n∈NinAwithyn−yn1 ∈C. This implies thatyn−k−yn1−
yn−k−1−yn1∈C, k1,2, . . . ,n−2.By assumption, we get
yn−yn1 ≤λ yn−1−yn1 ≤ · · · ≤λn−1 y
1−yn1 . 2.4
SinceAis bounded, there existsM > 0 such that yn−yn1 ≤ λn−1Mfor alln ∈ N.This
implies thatyn−yn1 → 0 asn → ∞andCorollary 2.17follows fromCorollary 2.15.
Remark 2.18. Theorem 2.11andCorollary 2.15are equivalent.
Remark 2.19. From the above results and examples, we observe that ifA⊆Rmis a closed set
and there existsa ∈ Rsuch thatA ⊆ aRm
, then MinRm
A /∅. Indeed, for each sequence
{yn}n∈N {y1n, y2n, . . . , ymn}n∈N in A with yn −yn1 ∈ Rm, it is easy to see that, for each j 1,2, . . . , m,{ynj}n∈Nis a decreasing sequence inRand bounded from below. Thenyn−
yn1 → 0 asn → ∞and MinRm
A /∅.
3. Saddle Point Problems
Theorem 3.1. LetX, dXandY, dYbe two complete metric spaces. Assume that
ifor eachx, y∈X×Y,{u, v∈X×Y :Px, v∩Pu, y K/∅}is a closed set, iifor eachx, y,u, v,a, b∈X×Y, ifPx, v∩Pu, yK/∅andPu, b∩Pa, v
K/∅, thenPx, b∩Pa, y K/∅,
iiifor each sequence {xn, yn}n∈N in X ×Y, if Pxn, yn1∩Pxn1, yn K/∅, then
dXxn, xn1 → 0anddYyn, yn1 → 0asn → ∞.
Then, for eachu, v ∈ X×Y, there existsx, y ∈X×Y such thatPu, y∩Px, v K/∅,
Px, y∩MinKPX, y/∅, andPx, y∩MaxKPx, Y/∅.
Proof. LetdX×Yx, y,u, v dXx, u dYy, v. ThenX×Y, dX×Yis a complete metric
space. Take anyu, v ∈ X×Y. Define the set M : {x, y ∈ X×Y : Pu, y∩Px, v K/∅}. Clearly, M, dX×Y is a complete metric space. LetG : M×M Z be defined
byGx, y, u, v Px, v−Pu, y−K. By Theorem 2.7, there exists x, y ∈ Msuch that
Px, y∩Px, y K ∅for allx, y∈M\ {x, y}. Byiiand the definition ofM, it is easy to see thatPx, y∩Px, y K ∅for allx, y∈X×Y\ {x, y}. Hence,
Px, y∩Px, yK∅ ∀y∈Y \y,
Px, y∩Px, y−K∅ ∀x∈X\ {x}. 3.1
SincePx, yis a nonempty compact set, byTheorem 2.5, there existz1, z2∈Px, ysuch that z1∈MinKPx, yandz2 ∈MaxKPx, y. Next, it is easy to see thatPx, Y∩z2K {z2}
andPX, y∩z1−K {z1}. Therefore,z2 ∈Px, y∩MaxKPx, Y/∅andz1 ∈Px, y∩
Remark 3.2. IfPis an u.s.c. multivalued map with nonempty compact values, then condition
iofTheorem 3.1holds.
Example 3.3. LetX 0,1,Y 1,2,Z R2, andK R2
. LetP : X×Y Zbe defined
byPx, y {x,−y}for eachx, y∈X×Y. ByTheorem 3.1, for eachu, v∈X×Y, there existsx, y ∈ X ×Y such that Pu, y∩Px, v K/∅,Px, y∩MinKPX, y/∅, and
Px, y∩MaxKPx, Y/∅. Indeed, for eachu, v∈X×Y,x, y 0,1.
InTheorem 3.1, ifY is singleton, then we have the following result.
Corollary 3.4. LetF :XZbe a multivalued map with nonempty compact values. Assume that
ifor eachx∈X,{y∈X :Fx∩Fy K/∅}is a closed set,
iifor eachx, y, z ∈ X, ifFx∩Fy K/∅andFy∩Fz K/∅, thenFx∩
Fz K/∅,
iiifor each sequence{xn}n∈NinX, ifFxn∩Fxn1 K/∅, thendXxn, xn1 → 0as n → ∞.
Then, for eachu∈X, there existsx∈Xsuch thatFu∩Fx K/∅andFx∩MinKFX/∅.
Theorem 3.5. Assume that
ifor eachx, y∈X×Y,{u, v∈X×Y :Px, v⊆Pu, y K}is closed,
iifor eachx, y,u, v,a, b∈X×Y, ifPx, v⊆Pu, y KandPu, b⊆Pa, v K, thenPx, b⊆Pa, y K,
iiifor each sequence {xn, yn}n∈N in X × Y, if Pxn, yn1 ⊆ Pxn1, yn K, then
dXxn, xn1 → 0anddYyn, yn1 → 0asn → ∞.
Then, for eachu, v ∈ X ×Y, there existsx, y ∈ X×Y such thatPu, y ⊆ Px, v K, and
Px, y/⊆Px, y Kfor allx, y∈X×Y \ {x, y}.
Proof. LetdX×Yx, y,u, v dXx, u dYy, v. Take anyu, v ∈ X×Y. Define the set
M{x, y∈X×Y :Pu, y⊆Px, v K}. Clearly,M, dX×Yis a complete metric space.
LetG:M×MZbe defined byGx, y, u, v Z\ {Px, v−Z\Pu, y K}. For each
x, y∈M, byi,{u, v∈M: 0∈Gx, y, u, v}{u, v∈M:Px, v⊆Pu, y K}is a closed set. Then byTheorem 2.7, there existsx, y∈Msuch thatPx, y/⊆Px, y Kfor all
x, y∈M\ {x, y}.
Now, we want to show thatPx, y/⊆Px, y K for all x, y ∈ X×Y \ {x, y}. In fact, we only need to consider thatx, y/∈M. Suppose thatPx, y ⊆ Px, y K. By ii andx, y∈M,x, y∈Mand this is a contradiction. Therefore,Px, y/⊆Px, y Kfor all
x, y∈X×Y\ {x, y}.
Remark 3.6. Condition i ofTheorem 3.5can be replaced by y Px, ybeing l.s.c. and
xPx, ybeing an u.s.c. multivalued map with nonempty compact values.
Example 3.7. LetX{1,2},Y 1,2,ZR2, andKR2
, and letP:X×Y Zbe defined
by
Px, y:
⎧ ⎨ ⎩
1, z∈R2:y≤z≤2 if x1,
2, z∈R2:y≤z≤2 if x2. 3.2
Example 3.8. LetX 0,∞,Y −∞,0,ZR2, andKR2
. LetP :X×Y Zbe defined
byPx, y x,∞×y,∞for eachx, y∈X×Y. ByTheorem 3.5, for eachu, v∈X×Y, there existsx, y ∈ X ×Y such that Pu, y ⊆ Px, v K, andPx, y/⊆Px, y K for all
x, y∈X×Y\ {x, y}. Indeed,x, y 0,0for eachu, v∈X×Y.
4. Bilevel Problems
Proposition 4.1. LetQ:X×Y X×Y be defined by
Qu, v A1u, v∩B1∩C1, where
A1u, v:
x, y∈X×Y :Pu, y∩Px, v K/∅,
B1:
x, y∈X×Y :Px, y∩WMinKP
X, y/∅,
C1:
x, y∈X×Y :Px, y∩WMaxKPx, Y/∅
.
4.1
IfPis a continuous multivalued map with nonempty compact values, thenQis closed, andB1∩C1is
a closed set.
Proof. Here, we only need to show that Q is closed. If {uα, vα, xα, yα}α∈Λ ⊆ GrQ and uα, vα, xα, yα → u, v, x, y, for eachα∈Λ, we have then
1Puα, yα∩Pxα, vα K/∅,
2Pxα, yα∩WMinKPX, yα/∅,
3Pxα, yα∩WMaxKPxα, Y/∅.
By1, for eachα∈Λ, there existsaα∈Puα, yα∩Pxα, vαKandbα∈Pxα, vαsuch that
aα−bα∈K. LetL1:{xα:α∈Λ} ∪ {x},L2:{yα:α∈Λ} ∪ {y},L3:{vα:α∈Λ} ∪ {v}, and
L4:{uα:α∈Λ}∪{u}. ThenL1,L2,L3, andL4are compact sets. ByLemma 2.2,PL4×L2and PL1×L3are compact sets. Hence, we may assume thataα → aandbα → b. ByLemma 2.2,
a∈Pu, yandb∈Px, v. Clearly,a−b∈KandPu, y∩Px, v K/∅. By2, for each
α∈Λ, there existsdα ∈Pxα, yα∩WMinKPX, yα. ByLemma 2.2,PL1×L2is a compact
set. Hence, we may assume thatdα → d. ByLemma 2.2,d∈Px, y⊆PX, y.
For eachα∈Λ,PX, yα∩dα−intK ∅.Take anyp ∈PX, y. Then there exists
r ∈ X such that p ∈ Pr, y. There exists a net {pα}α∈Λ in Z such that pα ∈ Pr, yα ⊆
PX, yαfor allα∈Λ,andpα → p,pα−dα/∈ −intKfor all α∈Λ.Therefore,p−d /∈ −intK
for allp∈PX, y. That is,PX, y∩d−intK ∅. Hence,d∈Px, y∩WMinKPX, y/∅.
Similarly, we can prove thatPx, y∩WMaxKPx, Y/∅. So,u, v, x, y ∈ GrQand Qis
closed.
Lemma 4.2. Let X and Y be topological spaces, and letG : X Y be a multivalued map. Let
G−1:GXXbe defined byG−1y:{x∈X :y∈Gx}for eachy∈GX. IfG−1is an u.s.c. multivalued map with nonempty compact values andAis a nonempty closed subset ofX, thenGA
is a closed subset ofY.
Proof. Ify∈clGA, then there exist a net{yα}α∈Λsuch thatyα → yand a net{xα}α∈Λin Asuch thatyα ∈ Gxαfor allα∈ Λ. Hence,xα ∈G−1yαfor allα ∈Λ. LetL {yα :α ∈
SinceAis a closed set,x ∈ A. ByLemma 2.2,x ∈ G−1yandy ∈Gx⊆ GA. Therefore,
GAis a closed subset ofY.
Theorem 4.3. InTheorem 3.1, letG:X×Y Wbe a multivalued map with nonempty values, and
further assume that
aG−1is an u.s.c. multivalued map with nonempty compact values, bPis a continuous multivalued map with nonempty compact values,
cfor each sequence{wn}n∈NinGX×Y, ifwn−wn1 ∈C, thenwn−wn1 → 0as n → ∞.
Then there is a solution of problem (BL-1).
Proof. LetB1 andC1 be defined as in Proposition 4.1. ByTheorem 3.1and Proposition 4.1, B1∩C1is a nonempty closed subset ofX×Y. ByLemma 4.2,GB1∩C1is a nonempty closed
subset ofW. ByTheorem 2.11, there exists x, y ∈ B1∩C1 such thatGx, y∩MincGB1∩ C1/∅.
Example 4.4. InExample 3.3, letW R2,CR2
, andGx, y: x, y R2for eachx, y∈ X×Y. Clearly,GX×Y 0,1 R2
is a closed subset ofR2. Besides, we have
G−1u, v: 0,min{u,1}×1,min{v,2}, 4.2
andG−1is an u.s.c. multivalued map with nonempty compact values. ByTheorem 4.3, there is a solution of problemBL-1. Indeed, the solution set is{0,1}.
The following theorem is similar toTheorem 4.3. Note that the conditions of Theorems
4.3and4.5are different.
Theorem 4.5. InTheorem 3.1, letG:X×Y Wbe a multivalued map, and further assume that
aGis an u.s.c. multivalued map with nonempty compact values,
bfor each x, y, z ∈ X×Y, ifGx∩Gy K/∅ and Gy∩Gz K/∅, then
Gx∩Gz K/∅,
cPis a continuous multivalued map with nonempty compact values,
dfor each{xn}n∈NinX×Y, ifGxn∩Gxn1 K/∅, thendX×Yxn, xn1 → 0as n → ∞.
Then there is a solution of problem (BL-1).
Proof. LetB1 andC1 be defined as in Proposition 4.1. ByTheorem 3.1and Proposition 4.1, B1∩C1 is a nonempty closed subset ofX ×Y. Hence,B1 ∩C1, dX×Yis a complete metric
space. Now, for eachx, y∈B1∩C1, letAx,y:{u, v∈B1∩C1:Gx, y∩Gu, v K/∅}.
Ifu, v ∈ clAx,y, then there exists a net{uα, vα}α∈Λ inAx,y such that uα, vα → u, v.
Then for eachα∈Λ, there existswα ∈Gx, y∩Guα, vα K. SinceGx, yis a compact
set, we may assume thatwα → w∈Gx, y. There existstα∈Guα, vαsuch thatwα−tα∈K.
LetL {uα, vα : α∈ Λ} ∪ {u, v}. Clearly,LandGLare compact sets. Hence, we may
Gx, y∩Gu, v K/∅. Therefore,u, v∈Ax,yandAx,yis a closed set. ByCorollary 3.4,
there existsx, y∈B1∩C1such thatGx, y∩MincGB1∩C1/∅.
Furthermore, we have the following result which is different from Theorems4.3and
4.5. In Theorem 4.6, W is a Hausdorff t.v.s., and X and Y are compact metric spaces. In Theorems4.3and4.5,Wis a Banach space, andXandY are complete metric spaces.
Theorem 4.6. InTheorem 3.1, letG:X×Y Wbe a multivalued map, and further assume that
aGis an u.s.c. multivalued map with nonempty compact values, bPis a continuous multivalued map with nonempty compact values, cXandY are compact.
Then there is a solution of problem (BL-1).
Proposition 4.7. LetQ:X×Y X×Y be defined by
Qu, v:A2u, v∩B2∩C2, where
A2u, v:
x, y∈X×Y :Pu, y⊆Px, v K,
B2:
x, y∈X×Y :Px, y/⊆Pa, yintK ∀a∈X,
C2:
x, y∈X×Y :Px, b/⊆Px, yintK ∀b∈Y.
4.3
IfPis a continuous multivalued map with nonempty compact values, thenQis closed, andB2∩C2is
a closed set.
Proof. Here, we only need to show that Q is closed. If {uα, vα, xα, yα}α∈Λ ⊆ GrQ and uα, vα, xα, yα → u, v, x, y, then we have
1Puα, yα⊆Pxα, vα K,
2Pxα, yα/⊆Pa, yα intKfor alla∈X,
3Pxα, b/⊆Pxα, yα intKfor allb∈Y.
Take anyz∈Pu, y; there exists a net{zα}α∈Λsuch thatzα ∈Puα, yαfor allα∈Λ
andzα → z. There existswα∈Pxα, vαsuch thatzα−wα∈K. LetL1 :{xα:α∈Λ} ∪ {x},
L2 : {yα : α ∈ Λ} ∪ {y},L3 : {vα : α ∈ Λ} ∪ {v}, andL4 : {uα : α ∈ Λ} ∪ {u}. Then
L1,L2,L3, andL4are compact sets andPL1×L3is a compact set. Hence, we may assume
thatwα → w. ByLemma 2.2,w ∈Px, v. Clearly,z ∈wK. Hence,z∈Px, v K, and
Pu, y⊆Px, v K.
Take anya∈X; there existssα∈Pxα, yαsuch thatsα/∈Pa, yα intK.PL1×L2
is a compact set, and we may assume thatsα → s; ByLemma 2.2, s ∈ Px, y. Take any
t ∈ Pa, y, there exists a net {tα}α∈Λ such that tα ∈ Pa, yα for all α ∈ Λ and tα → t,
sα−tα/∈intKfor allα∈Λ. Clearly,s−t /∈intKfor allt∈Pa, y. Hence,Px, y/⊆Pa, y
intKfor alla∈X. Similarly,Px, b/⊆Px, yintKfor allb∈Y. Therefore,Qis closed.
Theorem 4.8. InTheorem 3.5, letG : X×Y W be a multivalued map with nonempty values. Further assume that conditions (a)–(c) ofTheorem 4.3(resp., conditions (a)–(d) ofTheorem 4.5) are satisfied. Then there is a solution of problem (BL-2).
Proposition 4.9. LetP :X×Y → Rmbe a map withPx, y P
1x, y, P2x, y, . . . , Pmx, y.
LetQ:X×Y X×Ybe defined by
Qu, v:A3u, v∩B3∩C3, where
A3u, v:
x, y∈X×Y :Pu, y−Px, v∈K,
B3:
x, y∈X×Y :Px, y∈MaxRm
Px, Y
,
C3:
x, y∈X×Y :Px, y∈MinRm
P
X, y.
4.4
Suppose that, for eachi1,2, . . . , m,Pi :X×Y → Ris continuous,x → Pix, yis one to one,
andy → Pix, yis one to one. ThenQis closed, andB3∩C3is a closed set.
Proof. Let{un, vn, xn, yn}n∈N ⊆ GrQandun, vn, xn, yn → u, v, x, yasn → ∞. Then
Pun, yn−Pxn, vn ∈ Rm,Pxn, b−Pxn, yn/∈Rm for allb ∈ Y \ {yn}, and Pxn, yn−
Pa, yn/∈Rm for alla∈X\ {xn}. SincePis continuous,Pu, y−Px, v∈Rm. Furthermore,
for eachn∈N, we have
Pxn, yn
−Pa, yn
/ ∈Rm
\ {0} ∀a∈X. 4.5
Take anya∈X\{x}. There existsn0∈Nsuch thatxn/afor alln≥n0. Furthermore, for
eachn∈Nwithn≥n0, there existsjjn∈ {1,2, . . . , m}such thatPjxn, yn−Pja, yn<0.
Indeed, if not, there exists n ∈ N with n ≥ n0 such that Pjxn, yn−Pja, yn ≥ 0 for all
j ∈ {1,2, . . . , m}. Sincen≥n0andx → Pjx, yis one to one,Pjxn, yn−Pja, yn>0 for all
j∈ {1,2, . . . , m}. Hence,Pxn, yn−Pa, yn∈intRm. This leads to a contradiction.
Therefore, there existk ∈ {1,2, . . . , m},and{xnt}t∈N,{ynt}t∈Nof{xn}n∈N,and{yn}n∈N,
respectively, such that Pkxnt, ynt −Pka, ynt < 0 for allt ∈ N. Suppose that Px, y −
Pa, y ∈ Rm
.ThenPjx, y−Pja, y ≥ 0 for allj ∈ {1,2, . . . , m}, 0 ≤ Pkx, y−Pka, y
limt→ ∞Pkxnt, ynt−Pka, ynt≤0.Hence,Pkx, y Pka, y. This leads to a contradiction
sincex → Pkx, yis one to one. Therefore,Px, y−Pa, y/∈Rm for alla∈X\{x}.Similarly,
we havePx, b−Px, y/∈Rm for allb∈Y \ {y}.Therefore,Qis closed.
Theorem 4.10. LetP :X×Y → Rmbe a map withPx, y P
1x, y, P2x, y,. . . , Pmx, y.
LetG:X×Y Wbe a multivalued map with nonempty values. Further assume that
ifor eachi1,2, . . . , m,Pi :X×Y → Ris continuous,x → Pix, yis one to one, and
y → Pix, yis one to one,
iifor eachx, y,u, v,a, b∈X×Y, ifPx, v−Pu, y∈Rm
andPu, b−Pa, v∈Rm, thenPx, b−Pa, y∈Rm
,
iiifor each sequence {xn, yn}n∈N in X × Y with Pxn, yn1 − Pxn1, yn ∈ Rm,
ivG−1is an u.s.c. multivalued map with nonempty compact values,
vfor each sequence{wn}n∈NinGX×Y, ifwn−wn1 ∈C, thenwn−wn1 → 0as
n → ∞.
Then there is a solution of problem (BL-3).
Proof. Applying Proposition 4.9 and following the similar argument as in the proof of
Theorem 4.3, we can get the proof ofTheorem 4.10.
Remark 4.11. The conditions ofTheorem 4.10andTheorem 3.5in Liou et al.5are different. Note that Liou et al. 5 assumed that the feasible set Spx is nonempty, and let the
considered multivalued mapF be proper, lower semicontinuous, and weakly coercive on GrSp.
5. Equilibrium Problems and Mathematical Program with
Equilibrium Constraint on Complete Metric Spaces
Theorem 5.1. LetXandY be a complete metric spaces, and letH:X×Y×XZbe a multivalued
map. Assume that
ifor eachx, y∈X×Y,0∈Hx, y, x,
iifor eachx, y∈X×Y,{u∈X: 0∈Hx, y, u}is a closed subset ofX,
iiifor each x, u, a ∈ X and y, v ∈ Y, if 0 ∈ Hx, y, u and 0 ∈ Hu, v, a, then 0 ∈
Hx, y, a,
ivfor each sequence{xn, yn}n∈Nin X×Y with0 ∈ Hxn, yn, xn1,dXxn, xn1 → 0
anddYyn, yn1 → 0asn → ∞.
Then there existsx, y∈X×Ysuch that0/∈Hx, y, ufor allu∈X\ {x}.
Proof. LetG:X×Y×X×Y Zbe defined byGx, y, u, v Hx, y, ufor eachx, y, u, v∈ X×Y×X×Y. ThenTheorem 5.1follows fromTheorem 2.7.
Theorem 5.2. LetXandY be a complete metric spaces, and letS:X×Y×X Zbe a multivalued
map. Assume that
iZ\K\ {0}is a closed set; and for eachx, y∈X×Y,Sx, y, x {0}, iifor eachx, y∈X×Y,{u∈X:Sx, y, u∩K /∅}is a closed subset ofX,
iiifor each x, u, a ∈ X and y, v ∈ Y, ifSx, y, u∩K /∅ and Su, v, a∩K /∅, then
Sx, y, a∩K /∅,
ivfor each sequence{xn, yn}n∈NinX×YwithSxn, yn, xn1∩K /∅,dXxn, xn1 → 0
anddYyn, yn1 → 0asn → ∞,
vGis closed; and for eachu∈X,x, ySx, y, uis l.s.c.,
vifor each sequence {xn, yn, wn}n∈N in GrG with wn − wn1 ∈ C, dX×Yxn, yn,
xn1, yn1 → 0andwn−wn1 → 0asn → ∞.
Proof. ByTheorem 5.1, there existsx, y∈X×Ysuch thatSx, y, u∩K∅for allu∈X\{x}. SinceSx, y, x {0},Sx, y, u∩K\ {0} ∅for allu∈X.
LetL1{x, y∈X×Y :Sx, y, u∩K\{0} ∅for allu∈X}. Clearly,x, y∈L1/∅.
Ifx, y∈clL1, then there exists a net{xα, yα}α∈ΓinL1such thatxα, yα → x, y. Then,
for each u ∈ X,Sxα, yα, u∩K \ {0} ∅ for allα ∈ Γ. That is,Sxα, yα, u ⊆ Z\K\
{0}for allα ∈ Γ.Take any u ∈ X and any z ∈ Sx, y, u; there exists a net{zα}α∈Γ such
thatzα ∈ Sxα, yα, u ⊆ Z\K\ {0}for allα∈ Γandzα → z. Clearly,z ∈ Z\K\ {0}.
Hence,Sx, y, u∩K\ {0} ∅ for allu ∈ X. Then L1 is a closed subset of a complete
metric spaceX×Y. Furthermore,L1, dX×Yis a complete metric space. LetG1 G|L1. Then
GrG1 {x, y, w∈X×Y×W :x, y∈L1, w∈Gx, y}is a closed subset ofX×Y×W.
ByTheorem 2.8, there is a solution of problemMPEC-1.
Theorem 5.3. LetXandY be a complete metric spaces, and letS:X×Y×X Zbe a multivalued
map. Assume that
ifor eachx, y∈X×Y,Sx, y, x {0},
iifor eachx, y∈X×Y,{u∈X:Sx, y, u⊆K}is a closed subset ofX,
iiifor eachx, u, a∈Xandy, v∈Y, ifSx, y, u⊆Kand Su, v, a⊆K, thenSx, y, a⊆ K,
ivfor each sequence{xn, yn}n∈NinX ×Y withSxn, yn, xn1 ⊆ K,dXxn, xn1 → 0
anddYyn, yn1 → 0asn → ∞.,
vZ\K\ {0}is a closed set,
vifor eachu∈X,x, ySx, y, uis an u.s.c. multivalued map with nonempty compact values,
viiGis closed,
viiifor each sequence {xn, yn, wn}n∈N in GrG with wn − wn1 ∈ C, dX×Yxn, yn,
xn1, yn1 → 0andwn−wn1 → 0asn → ∞.
Then there is a solution of problem (MPEC-2).
Proof. LetHbe defined byHx, y, u Z\Sx, y, u−Z\Kfor eachx, y, u∈X×Y×X. ByTheorem 5.1and conditioni, there existsx, y∈X×Y such thatSx, y, u/⊆K\ {0}for allu∈X.
Let L3 {x, y ∈ X × Y : Sx, y, u/⊆K \ {0} for all u ∈ X}. If x, y ∈ clL3,
then there exists a sequence {xn, yn}n∈N in L3 such that xn, yn → x, y. Let A
{xn, yn}n∈N ∪ {x, y}. Then A is a compact set. For each n ∈ N, since xn, yn ∈ L3, Sxn, yn, u/⊆K \ {0}for allu ∈ X.For each u ∈ X and n ∈ N, there exists zn ∈ Z such
thatzn ∈Sxn, yn, uandzn/∈K\ {0}. Then{zn}n∈N⊆SA, u. Byvi, we may assume that
zn → z∈Sx, y, u. SinceZ\K\{0}is a closed set,z∈Z\K\{0}. Hence,x, y∈L3and L3 is closed. Following the similar argument as in the last part of the proof ofTheorem 5.2,
we get the proof ofTheorem 5.3.
References
2 O. Stein and G. Still, “On generalized semi-infinite optimization and bilevel optimization,”European Journal of Operational Research, vol. 142, no. 3, pp. 444–462, 2002.
3 O. Stein,Bilevel Strategies in Semi-Infinite Programming, vol. 71 ofNonconvex Optimization and Its Applications, Kluwer Academic Publishers, Boston, Mass, USA, 2003.
4 S¸. ˙I. Birbil, G. Bouza, J. B. G. Frenk, and G. Still, “Equilibrium constrained optimization problems,” European Journal of Operational Research, vol. 169, no. 3, pp. 1108–1127, 2006.
5 Y. C. Liou, X. Q. Yang, and J. C. Yao, “Mathematical programs with vector optimization constraints,” Journal of Optimization Theory and Applications, vol. 126, no. 2, pp. 345–355, 2005.
6 L.-J. Lin and G. Still, “Mathematical programs with equilibrium constraints: the existence of feasible point,”Optimization, vol. 55, no. 3, pp. 205–219, 2006.
7 L.-J. Lin, “Mathematical programming with system of equilibrium constraints,”Journal of Global Optimization, vol. 37, no. 2, pp. 275–286, 2007.
8 L.-J. Lin and H.-W. Hsu, “Existences theorems of systems of vector quasi-equilibrium problems and mathematical programs with equilibrium constraint,”Journal of Global Optimization, vol. 37, no. 2, pp. 195–213, 2007.
9 D. T. Luc and C. Vargas, “A saddlepoint theorem for set-valued maps,”Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 1, pp. 1–7, 1992.
10 T. Tanaka, “Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions,”Journal of Optimization Theory and Applications, vol. 81, no. 2, pp. 355–377, 1994.
11 I.-S. Kim and Y.-T. Kim, “Loose saddle points of set-valued maps in topological vector spaces,”Applied Mathematics Letters, vol. 12, no. 8, pp. 21–26, 1999.
12 L.-J. Lin, “System of coincidence theorems with applications,”Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 408–418, 2003.
13 Q. H. Ansari, L. J. Lin, and L. B. Su, “Systems of simultaneous generalized vector quasiequilibrium problems and their applications,”Journal of Optimization Theory and Applications, vol. 127, no. 1, pp. 27–44, 2005.
14 L.-J. Lin, “Existence theorems of simultaneous equilibrium problems and generalized vector quasi-saddle points,”Journal of Global Optimization, vol. 32, no. 4, pp. 613–632, 2005.
15 J. Y. Fu, “Stampacchia generalized vector quasiequilibrium problems and vector saddle points,” Journal of Optimization Theory and Applications, vol. 128, no. 3, pp. 605–619, 2006.
16 K. Kimura and T. Tanaka, “Existence theorem of cone saddle-points applying a nonlinear scalarization,”Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 563–571, 2006.
17 X.-H. Gong, “The strong minimax theorem and strong saddle points of vector-valued functions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2228–2241, 2008.
18 N. X. Tan, “Quasivariational inequalities in topological linear locally convex Hausdorff spaces,” Mathematische Nachrichten, vol. 122, pp. 231–245, 1985.
19 J. P. Aubin and A. Cellina,Differential Inclusion, Springer, Berlin, Germany, 1994.
20 D. T. Luc, “An existence theorem in vector optimization,”Mathematics of Operations Research, vol. 14, no. 4, pp. 693–699, 1989.
21 L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.
