Journal of Inequalities and Applications Volume 2009, Article ID 617185,16pages doi:10.1155/2009/617185
Research Article
Generalized Vector Complementarity Problems
with Moving Cones
Lu-Chuan Ceng
1and Yen-Cherng Lin
21Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Department of Occupational Safety and Health, China Medical University, Taichung 404, Taiwan
Correspondence should be addressed to Yen-Cherng Lin,[email protected]
Received 3 October 2009; Accepted 30 November 2009
Recommended by Charles E. Chidume
We introduce and discuss a class of generalized vector complementarity problems with moving cones. We discuss the existence results for the generalized vector complementarity problem under inclusive type condition. We obtain equivalence results between the generalized vector complementarity problem, the generalized vector variational inequality problem, and other related problems. The theorems presented here improve, extend, and develop some earlier and very recent results in the literature.
Copyrightq2009 L.-C. Ceng and Y.-C. Lin. 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 and Preliminaries
It is well known that vector variational inequalities were initially studied by Giannessi1 and ever since have been widely studied in infinite-dimensional spaces see, for example,2–
8and the references therein.
Very recently, Huang et al.9considered a class of vector complementarity problems with moving cones. They established existence results of a solution for this class of vector complementarity problems under an inclusive type condition. They also obtained some equivalence results among a vector complementarity problem, a vector variational inequality problem, a vector optimization problem, a weak minimal element problem, and a vector unilateral optimization problem in ordered Banach spaces. Their results generalized the main results in4.
family of closed and convex cones satisfy an inclusion relation so long as their corresponding variables satisfy certain conditions. We also obtain some equivalence results among a generalized vector complementarity problem, a generalized vector variational inequality problem, a generalized vector optimization problem, a generalized weak minimal element problem, and a generalized vector unilateral optimization problem under some monotonicity conditions and some inclusive type conditions in ordered Banach spaces. The theorems presented in this paper improve, extend, and develop some earlier and very recent results in the literature including4,9.
LetXbe a Banach space, andAa subset ofX. The topological interior of a subsetAin
X is denoted by intA. A nonempty subsetP inX is called a convex cone ifPP ⊂ P, and
λP ⊂Pfor anyλ >0. The relations≤P and/≤PinXare defined asx≤Pyify−x∈Pandx/≤Py
ify−x /∈P, for anyx, y∈X. Similarly, we can define the relations≤intPand/≤intP if we replace
the setPby intP.P is called a pointed cone ifP is a cone andP∩−P {0}.
LetLX, Ybe the space of all continuous linear mappings fromXtoY. We denote the value ofl∈LX, Yatx∈Xbyl, x.
LetX, Y be two Banach spaces, andP :K → 2Y a set-valued mapping such that, for
eachx ∈ K,Pxis a proper closed convex and pointed cone with apex at the origin and intPx/∅, andT :X → LX, Y. Very recently, Huang et al.9introduced the following three kinds of vector complementarity problems.
(Weak) vector complementarity problem (VCP): findingx∈Ksuch that
Tx, x /≥intPx0, Tx, y≤/intPx0, ∀y∈K. 1.1
Positive vector complementarity problem (PVCP): findingx∈Ksuch that
Tx, x /≥intPx0, Tx, y≥Px0, ∀y∈K. 1.2
Strong vector complementarity problem (SVCP): findingx∈Ksuch that
Tx, x 0, Tx, y≥Px0, ∀y∈K. 1.3
We remark that ifPx P for allx ∈ K, whereP is a closed, pointed, and convex cone inY with nonempty interior intPx, thenVCP,PVCPand SVCPreduce to the problems considered in Chen and Yang4. In9, they actually only studied the first two kinds complementarity problems. For the existence results ofSVCP, we refer the reader to our recent resultsSubmitted, On theF-implicit vector complementarity problem.
Motivated and inspired by the above three kinds of vector complementarity problems, in this paper we introduce three kinds of generalized vector complementarity problems. Let
X, Ybe two Banach spaces, andP :K → 2Y a set-valued mapping such that, for eachx∈K,
Px is a proper closed convex and pointed cone with apex at the origin and intPx/∅, letA : LX, Y → LX, Ybe a single-valued mapping, andT : X → 2LX,Ya set-valued
mapping, where 2LX,Y is a collection of all nonempty subsets ofLX, Y. We consider the
(Weak) generalized vector complementarity problem (GVCP): finding x ∈ K andu ∈ Tx
such that
Au, x /≥intPx0, Au, y≤/intPx0, ∀y∈K. 1.4
Generalized positive vector complementarity problem (GPVCP): findingx∈Kandu∈Tx
such that
Au, x />intPx0, Au, y ≥Px0, ∀y∈K. 1.5
Generalized strong vector complementarity problem (GSVCP): findingx ∈K andu∈ Tx
such that
Au, x 0, Au, y≥Px0, ∀y∈K. 1.6
We remark that ifAI the identity mapping ofLX, Y, andT :KX → 2LX,Yis
a single-valued mapping, then three kinds of generalized vector complementarity problems reduce to three kinds of vector complementarity problems in Huang et al.9, respectively.
2. Existence of a Solution for GVCP
Huang et al. 9 established some equivalence results between the positive vector complementarity problem and the vector extremum problem and also sufficient conditions for the existence of a solution of the vector extremum problem. In this section, we extend their results to the cases involving the set-valued mappings.
LetXbe an arbitrary real Hausdorfftopological vector spaces, andY a Banach space.
LX, Y denotes the space of all continuous linear mappings from X to Y. Let K be a nonempty set ofX, andP :K → 2Y a set-valued mapping such that, for eachx∈K,Pxis
a proper closed convex and pointed cone with apex at the origin and intPx/∅. LetAbe a subset ofY. For eachx∈K, a pointz∈Ais called a minimal pointofAwith respect to the conePxifA∩z−Px {z}; MinPxAis the set of all minimal points ofAwith respect to the conePx; a pointz∈Ais called a weakly minimal pointofAwith respect to the conePx
ifA∩z−intPx ∅; MinPwxAis the set of all weakly minimal points ofAwith respect to
the conePx, we refer the reader to10for more detail.
LetA : LX, Y → LX, Ybe a single-valued mapping and letT :X → 2LX,Y be
a set-valued mapping. Now, we consider the following generalized vector complementarity problemGVCP. Findx∈Kandu∈Txsuch that
Au, x /≥intPx0, Au, y≤/intPx0, ∀y∈K. 2.1
A feasible set ofGVCPis
We consider the following generalized vector optimization problemGVOP:
MinPAu, x subject tox, u∈F. 2.3
A pointx, u ∈ Fis called a weakly minimal solution ofGVOPwith respect to the cone
Px, ifAu, x is a weakly minimal point ofGVOPwith respect to the conePx, that is,
Au, x ∈MinPwx{Au, x :x, u∈F}. We denote the set of all weakly minimal solutions of GVOPwith respect to the conePxbyΩPwxand the set of all weakly minimal solutions of GVOPbyΩw, that is,
Ωw x∈K
ΩPx
w . 2.4
Theorem 2.1. IfΩw/∅and, for somex∈K, there existsx, u∈ΩwPxsuch thatAu, x /≥intPx0,
then the generalized vector complementarity problem (GVCP) is solvable.
Proof. Letx, u∈ΩPwxandAu, x /≥intPx0. Thenx∈K, u∈Tx, and
Au, x /≥intPx0, Au, y ≤/intPx0, ∀y∈K. 2.5
It follows thatxis a solution ofGVCP. This completes the proof.
We remark that ifA I the identity mapping of LX, Yand T is a single-valued mapping fromK XtoLX, Y, thenTheorem 2.1coincides with Theorem 2.1 in Huang et al.9.
Definition 2.2. LetT :K → 2LX,Y,P:K → 2Y be two set-valued mappings with intPx/∅
for everyx∈K,A:LX, Y → LX, Ya single-valued mapping, andFa subset ofK×TK. We say thatPis inclusive with respect toFif for anyx, u,y, v∈F,
Au, x ≤intPyAv, yimplies thatPx⊂Py. 2.6
It is easy to see that, ifPx Pfor allx∈K, wherePis a closed, pointed, and convex cone inY, thenPis inclusive with respect toF.
Example 2.3. LetX Y R2, K 0,1×0,1, A I be the identity mapping ofLX, Y. For eachx x1, x2∈K, define
Px z1, z2∈R2: 0≤z2≤1x1z1
, 2.7
and, for eachx∈K,
Tx
⎧ ⎪ ⎨ ⎪ ⎩ ⎡ ⎢ ⎣3
2
1x1 0
0 3 2
1x2
⎤ ⎥ ⎦
⎫ ⎪ ⎬ ⎪
Also, define
u, x
3x1
2x1 1x1,
3x2 2x2 1x2
, ∀x, u∈F. 2.9
Then it is easy to see thatPis inclusive with respect toF. Indeed, for anyx, u,y, v∈Fwith
x x1, x2, y y1, y2, u Tx, andv Ty, ifu, x ≤intPyv, y , thenv, y − u, x ∈
intPyand sox1< y1. Therefore,Px⊂PyandPis inclusive with respect toF.
Theorem 2.4. Suppose thatPis inclusive with respect toF. If there exist at most a finite number of solutions for (GVCP), then (GVCP) is solvable if and only ifΩw/∅, and there existsx, u∈ΩPwx
such thatAu, x /≥intPx0.
Proof. Letη1be a solution ofGVCP. Then there existsu1∈Tη1such that
Au1, η1 /≥intPη10,
Au1, y
/
≤intPη10, ∀y∈K. 2.10
Ifη1, u1∈ΩPwη1, then
Au1, η1
/
≥intPη10, 2.11
and hence the conclusion holds. If η1, u1∈/ΩPwη1, by the definition of a weakly minimal
solution, there existsη2, u2∈Fsuch that
Au2, y
/
≤intPη20, ∀y∈K,
Au2, η2
≤intPη1
Au1, η1
/
≥intPη10.
2.12
This implies that
Au2, η2
/
≥intPη10. 2.13
SinceAu2, η2 ≤intPη1Au1, η1 , andPis inclusive with respect toF, it follows thatPη2⊂
Pη1and this implies that
Au2, η2
/
≥intPη20. 2.14
Thus,η2is a solution ofGVCPandη2/η1. Continuing this process, there existsηn, un∈F
such thatηnis a solution ofGVCPandηn, un∈ΩPwηn, sinceGVCPhas at most a finite
number of solutions. Thus,Aun, ηn ∈MinwPηn{Au, ηn :ηn, u∈F}and
Aun, ηn
/
≥intPηn0. 2.15
Remark 2.5. 1IfAI the identity mapping ofLX, Y,T is a single-valued mapping from
X toLX, Y, andPx P for allx ∈X, whereP is a closed, pointed, and convex cone in
Y, thenPxsatisfies the inclusive condition with respect toFandTheorem 2.4reduces to Theorem 3.2 of Chen and Yang4.
2IfA I the identity mapping ofLX, YandT is a single-valued mapping from
KXtoLX, Y, thenTheorem 2.4coincides with Theorem 2.2 of Huang et al.9.
We next consider the generalized positive vector complementarity problemGPVCP. Findingx∈Kandu∈Txsuch that
Au, x /≥intPx0, Au, y≥Px0, ∀y∈K. 2.16
Let
Gx, u∈K×TK:u∈Tx, Au, y≥Px0, ∀y∈K. 2.17
Consider the following generalized vector optimization problemGVOP0to be
MinPAu, x subject tox, u∈G. 2.18
We denote the set of all minimal points ofGVOP0 with respect to the conePxbyΓPx, that is,ΓPx MinPx{Au, x : x, u ∈ G}, and denote the set of all minimal points of
GVOP0by
Γ x∈K
ΓPx.
2.19
Using a similar argument ofTheorem 2.1, we have the following results of solvability forGPVCP.
Theorem 2.6. IfΓ/∅and there existsx, u∈ΓPxsuch thatAu, x /≥
intPx0, then (GPVCP) is
solvable.
Theorem 2.7. Suppose thatP is inclusive with respect toG. If there exist at most a finite number of solutions of (GPVCP), then (GPVCP) is solvable if and only ifΓ/∅, and there existsx, u ∈ΓPx
such thatAu, x /≥intPx0.
One remarks that IfA I the identity mapping ofLX, YandT is a single-valued mapping fromKXtoLX, Y, then Theorems2.6and2.7coincide with Theorems 2.3 and 2.4 of Huang et al.9, respectively.
3. Equivalences between Generalized Vector Complementarity
3.1. Problems and Generalized Weak Minimal Element Problems
LetX, Ybe two Banach spaces, andP :K → 2Y a set-valued mapping such that, for eachx∈
letA : LX, Y → LX, Ybe a single-valued mapping, andT : X → 2LX,Ya set-valued
mapping, where 2LX,Yis a collection of all nonempty subsets ofLX, Y, andf :X → Y a
given operator.
Define the feasible set associated withT andA
Fx∈K: there isu∈Tx such thatAu, y/≤intPx0, ∀y∈K. 3.1
We now consider the following five problems.
iThe generalized vector optimization problem GVOPl: for a given l ∈ LX, Y, findingx∈Fsuch that
lx∈MinPwxl
F. 3.2
iiThe generalized weak minimal element problem GWMEP: finding x ∈ F such thatx∈MinKwF.
iiiThe generalized vector complementarity problemGVCP: findingx∈Fsuch that
Au, x /≥intPx0 whereu∈Txis associated withxin the definition ofF.
ivThe generalized vector variational inequality problemGVVIP: findingx∈Kand
u∈Txsuch that
Au, y−x/≤intPx0, ∀y∈K. 3.3
vThe generalized vector unilateral optimization problemGVUOP: findingx∈ K
such thatfx∈MinwPxfK.
We remark that if A I the identity mapping of LX, Y and T is a single-valued mapping fromX toLX, Y, then theGVOPl,GWMEP,GVCP,GVVIP, and
GVUOPreduce to Huang, et al.’s problemsVOPl,WMEP,VCP,VVIP, andVUOP, respectively; see9for more details.
Definition 3.1 see4. A linear operator l : X → Y is called weakly positive if, for any
x, y∈X, x/≥intCyimplies thatlx/≥intPxly.
Definition 3.2. Let X and Y be two Banach spaces and l a linear operator from X to Y. If the image of any bounded set inX is a self-sequentially compact set in Y, then l is called completely continuous.
A mappingf:X → Yis said to be convex if
fλx 1−λy≤Pxλfx 1−λfy 3.4
Definition 3.3. LetA: LX, Y → LX, Yandf :X → Y be two mappings.f is said to be
A-subdifferentiable atx0 ∈Xif there existsu0∈LX, Ysuch that
fx−fx0≥Px0Au0, x−x0 , ∀x∈X. 3.5
IffisA-subdifferentiable atx0∈X, then we define theA-subdifferential offatx0as follows:
∂Afx0:
u∈LX, Y:fx−fx0≥Px0Au, x−x0 , ∀x∈X
. 3.6
IffisA-subdifferentiable at eachx∈X, then we say thatfisA-subdifferentiable onX.
Remark 3.4. We note that as the mentions in9, ifXandYare two Banach spaces, a mapping
f:X → Yis Fr´echet differentiable atx0 ∈Xif there exists a linear bounded operatorDfx0 such that
lim
x→0
fx0x−fx0−
Dfx0, x
x 0, 3.7
where Dfx0is said to be the Fr´echet derivative of f at x0. The mappingf is said to be Fr´echet differentiable onXiff is Fr´echet differentiable at each point ofX. Iff :X → Y is convex and Fr´echet differentiable onX, then
fy−fx≥PxDfx, y−x, ∀x, y∈X. 3.8
Iff is Fr´echet differentiable at x0 ∈ X, thenf isI-subdifferentiable atx0 ∈ X and
Dfx0∈∂Ifx0.
IffisA-subdifferentiable onX, then for eachx, y∈Xwe have
fy−fx≥PxAu, y−x, ∀u∈∂Afx. 3.9
Definition 3.5. LetXbe a Banach space,K⊂Xa proper closed convex and pointed cone with apex at the origin and intK /∅. The norm · inXis called strictly monotonically increasing onK9if, for eachy∈K,
x∈y−intK∩K only impliesx<y. 3.10
For the example of the strictly monotonically increasing property, we refer the reader to9, Example 3.1.
Theorem 3.6. LetX, Y be two Banach spaces,K⊂Xa proper closed convex and pointed cone with apex at the origin andintK /∅, andP : K → 2Y a set-valued mapping with closed, convex, and
pointed cones values such thatintPx/∅for allx∈K. Suppose that
3there existsx∈Fsuch thatAuis one to one and completely continuous, whereu∈Txis associated withxin the definition ofF;
4X is a topological dual space of a real normed space and the norm · in X is strictly monotonically increasing onK.
If (GVVIP) is solvable, then (GVOP)l, (GWMEP), (GVCP), and (GVUOP) are also solvable.
Corollary 3.7 see9, Theorem 3.1. Let X, Y be two Banach spaces,K ⊂ X a proper closed convex and pointed cone with apex at the origin andintK /∅, and{Px:x∈X}that a family of closed, pointed, and convex cones inY such thatintPx/∅for allx∈X. Suppose that
1T Df is the Fr´echet derivative of a convex operatorf:X → Y;
2lis a weakly positive linear operator;
3there exists x ∈ F such that Tx is one to one and completely continuous, where
F{x∈K:Tx, y /≤intPx0, ∀y∈K};
4X is a topological dual space of a real normed space and the norm · in X is strictly monotonically increasing onK.
If (VVIP) is solvable, then (VOP)l, (WMEP), (VCP), and (VUOP) are also solvable.
Proof. SinceAIthe identity mapping ofLX, YandT is a single-valued mapping fromX
toLX, Y, we have
Fx∈K: there isu∈Tx such thatAu, y/≤intPx0, ∀y∈K
x∈K:Tx, y /≤intPx0, ∀y∈K.
3.11
UtilizingTheorem 3.6, we immediately obtain the desired conclusion.
Remark 3.8. IfPx Pfor allx∈X, wherePis a closed, pointed, and convex cone inY, then
Corollary 3.7coincides with Theorem 3.1 of Chen and Yang4.
We need the following propositions to proveTheorem 3.6.
Proposition 3.9. LetA:LX, Y → LX, Yandf :X → Y be two mappings, and letT ∂Af
be theA-subdifferential of f. Thenxsolves (GVUOP) which implies thatxsolves (GVVIP). If in addition, f is a convex mapping, then conversely, x solves (GVVIP) which implies that x solves (GVUOP).
Proof. Let x be a solution of GVUOP. Then x ∈ K and fx ∈ MinPwxfK, that is,
fx/≥intPxfyfor ally∈K. SinceKis a convex cone,
fx/≥intPxfxtw−x, 0< t <1, w∈K. 3.12
Also, sincefisA-subdifferentiable onX, it follows that for allu∈Tx∂Afx
This implies that
Au, tw−x /≤intPx0, 0< t <1, w∈K, 3.14
and hence
Au, w−x /≤intPx0, ∀w∈K. 3.15
Thus,xsolvesGVVIP.
Conversely, letxsolveGVVIP. Then there existsu∈Tx∂Afxsuch that
Au, w −x /≤intPx0, ∀w∈K. 3.16
SincefisA-subdifferentiable onX, we have for allu∈Tx∂Afx
fw−fx≥PxAu, w−x , ∀w∈K, 3.17
and hence
fw−fx≥PxAu, w −x /≤intPx0, ∀w∈K. 3.18
This implies that
fw/≤intPxfx, ∀w∈K. 3.19
Consequently,xsolvesGVUOP. This completes the proof.
Proposition 3.10. Ifxsolves (GVVIP), thenxalso solves (GVCP). Conversely, ifAu, x ≤Px0,
for allx∈K, u∈Tx, thenxsolves (GVCP) which implies thatxsolves (GVVIP).
Proof. Letxbe a solution ofGVVIP. Then there existsu∈Txsuch that
Au, y−x/≤intPx0, ∀y∈K. 3.20
Lettingy0, we getAu, x /≥intPx0. Forywxwith anyw∈K, we have
Au, w /≤intPx0, ∀w∈K. 3.21
Thus,xis a solution of theGVCP.
Conversely, letxsolve theGVCP. Then there existsu∈Txsuch that
This implies
Au, x /≥intPxAu, y, ∀y∈K, 3.23
and so
Au, y−x /≥intPx0, ∀y∈K. 3.24
This completes the proof.
Proposition 3.11. Letlbe a weakly positive linear operator. Thenxsolves (GWMEP) which implies thatxsolves (GVOP)l.
Proof. Letxbe a solution ofGWMEP. Thenx ∈F andx ∈ MinKwF, that is, for anyz ∈ F,
x/≥intKz. Sincelis a weakly positive linear operator, it follows thatlx/≥intPxlzand so
lx∈MinPwxl
F, 3.25
hencexsolvesGVOPl. This completes the proof.
Definition 3.12see9. LetXbe a Banach space,K⊂Xa proper closed convex and pointed cone with apex at the origin and intK /∅,Ea nonempty subset ofX.
1If, for somex∈X, Ex {x} −K∩E /∅, thenExis called a section of the setE. 2Eis called weakly closed if{xn} ⊂ E, x ∈ X, x∗, xn → x∗, x for allx∗ ∈ X∗,
thenx∈E.
3Eis called bounded below if there exists a pointpinXsuch thatE⊂pK.
Lemma 3.13see11. LetXbe a Banach space,K ⊂ Xa proper closed convex and pointed cone with apex at the origin andintK /∅,Ea nonempty subset ofX andX the topological dual space of a real normed spaceZ, · Z. Suppose there existsx∈Xsuch that the sectionExis weakly closed
and bounded below and the norm · inXis strictly monotonically increasing, then the setEhas at least one weakly minimal point.
Lemma 3.14. If (GVVIP) is solvable, then the feasible setFis nonempty.
Proof. Letxbe a solution ofGVVIP. Then there existsu∈Txsuch that
Au, y−x/≤intPx0, ∀y∈K. 3.26
Takingyzxwith anyz∈K, we know thaty∈Cand
Au, z /≤intPx0, ∀z∈K. 3.27
LetXbe a Banach space,K⊂Xa proper closed convex and pointed cone with apex at the origin and intK /∅. For anyx, y∈X,x, y xK∩y−Kis called an order interval.
Lemma 3.15see4. LetXbe a Banach space,K ⊂ X a proper closed convex and pointed cone with apex at the origin andintK /∅. If the norm · inXis strictly monotonically increasing, then the order intervals inXare bounded.
Proposition 3.16. Suppose that (GVVIP) is solvable and
1there existsxinFsuch thatAuis one to one and completely continuous, whereu∈Txis associated withxin the definition ofF;
2Xis the topological dual space of a real normed spaceZ, · Zand the norm · inXis
strictly monotonically increasing.
Then (GWMEP) has at least one solution.
Proof. By the assumption andLemma 3.14,F/∅. Letx∈F be a point such thatAuis one to one and completely continuous, whereu∈Txis associated withxin the definition ofF, and let{yn} ⊂Fwithyn → yweakly. Since
Fx {x} −C∩F⊂{x} −C∩C 0, x, 3.28
byLemma 3.15,0, xis bounded and so isFx. SinceAuis completely continuous,Au,Fx
is a self-sequentially compact set and so{Au, yn } ⊂ Au,Fx implies that there exists a
subsequence{Au, ynk }which converges toz∈ Au,Fx . We get a pointy0 ∈Fxsuch that
Au, ynk
−→Au, y0 strongly
. 3.29
On the other hand, sinceyn → yweaklyandAuis completely continuous,
Au, yn −→
Au, y strongly. 3.30
By the uniqueness of the limit, we getAu, y Au, y0 . SinceAuis one to one,yy0, and soy∈Fx. SinceFxis weakly closed, it follows fromLemma 3.13thatFhas a weakly minimal
pointpsuch thatp/≥intPpxfor allx∈F. Therefore,GWMEPhas at least one solution. This
completes the proof.
Definition 3.17. LetX, Y be two Banach spaces,K ⊂X a proper closed convex and pointed cone with apex at the origin and intK /∅, andP :K → 2Y a set-valued mapping with closed, convex and pointed cones values such that intPx/∅ for all x ∈ K. Let A : LX, Y → LX, Ybe a single-valued mapping andT :X → 2LX,Y a set-valued mapping.T is called
A-positive if
We now consider the generalized positive vector complementarity problemGPVCP. Findingx∈Kandu∈Txsuch that
Au, x /≥intPx0, Au, y≥Px0, ∀y∈K. 3.32
The feasible set related toGPVCPis defined as
F0
x∈K: there isu∈Txsuch thatAu, y≥Px0, ∀y∈K. 3.33
Let us consider the following problems.
The generalized vector optimization problemGVOPl0: findingx∈F0such thatlx∈ MinPwlF0.
The generalized weak minimal element problemGWMEP0: findingx∈F0such that
x∈MinKwF0.
The generalized positive vector complementarity problemGPVCP: findingx ∈ F0 such that
Au, x /≥intPx0, 3.34
whereu∈Txis associated withxin the definition ofF0.
The generalized vector variational inequality problemGVVIP: findingx ∈ K and
u∈Txsuch that
Au, y−x/≤intPx0, ∀y∈K. 3.35
The generalized vector unilateral optimization problem GVUOP: for a given mappingf:X → Y, findingx∈Ksuch thatfx∈MinPwfK.
Definition 3.18. A set-valued mapping T : X → 2LX,Y is said to beA-strictly monotone
whereA:LX, Y → LX, Yis single-valued, if
Au−Av, x−y≥intPx0, ∀x, y∈X, x /y, u∈Tx, v∈Ty. 3.36
Definition 3.19see9. We say thatPxsatisfies an inclusive condition if, for anyx, y∈X,
x≤intCy only implies that Px⊂P
y. 3.37
Example 3.20. LetX −∞,∞, C 0,∞, Y R2, and
Px
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
z1, z2∈R2: 0≤z2≤2z1
, x∈−∞,2,
z1, z2∈R2: 0≤z2≤xz1
, x∈2,5,
z1, z2∈R2: 0≤z2≤5z1
, x∈5,∞
3.38
for allx∈X. Then it is easy to check thatPxsatisfies the inclusive condition.
Proposition 3.21. Let T be A-strictly monotone and x a solution of (GPVCP). If P satisfies the inclusive condition, thenxis a weakly minimal point ofF0(i.e.,xsolves (GWMEP)0).
Proof. It is easy to see thatx∈F0 ⊂K. Ifx∈bdK wherebdKdenotes the boundary of
K, thenxsolvesGWMEP0. Otherwise, there existsx∈F0such thatx≥intKxand so
xx−xx∈intKK⊂intK, 3.39
which is a contradiction. Ifx∈intK, by theA-strict monotonicity ofT,
Au, x−y≥intPxAv, x−y, ∀y∈F0, y /x, v∈Ty. 3.40
Supposex≥intKy. SinceT isA-positive,Av, x−y ≥Py0 and
Au, x−y≥intPxAv, x−y≥Py0. 3.41
By the assumption, we getPy⊂Pxand so
Au, x−y∈Av, x−yintPx⊂PyintPx⊂Px intPx intPx. 3.42
It follows that
Au, x−y≥intPx0, 3.43
and thus
0/≤intPxAu, x ≥Px
Au, yk 3.44
for somek∈intPx. This implies
Au, yk/≥intPx0. 3.45
Sincek∈intPxandx∈F0,
and so
Au, yk≥intPx0, 3.47
which leads to a contradiction. Therefore,x≥intKydoes not hold, that is,x/≥intKyfor ally∈F0. It follows thatxsolvesGWMEP0. This completes the proof.
Proposition 3.22. Ifxsolves (GPVCP), thenxalso solves (GVVIP).
Proof. Suppose thatxsolvesGPVCP. Thenx∈Kand there existsu∈Txsuch that
Au, x /≥intPx0,
Au, y≥Px0, ∀y∈K. 3.48
IfAu, y−x ≤intPx0, then
Au, x −Au, y−xAu, y∈intPx Px⊂intPx, 3.49
and so
Au, x ≥intPx0, 3.50
which is a contradiction. It follows that
Au, y−x/≤intPx0, 3.51
andxsolvesGVVIP. This completes the proof.
Similarly, we can obtain other equivalence conditions. We have the following theorem.
Theorem 3.23. LetX, Ybe two Banach spaces,K⊂Xa proper closed convex and pointed cone with apex at the origin andintK /∅, and{Px:x∈X}a family of closed, pointed, and convex cones in Y such thatintPx/∅for allx∈X. Suppose thatPsatisfies the inclusive condition and
1T ∂Afis theA-subdifferential of the convex operatorf:X → Y; 2lis a weakly positive linear operator;
3T isA-strictly monotone.
If (GPVCP) is solvable, then (GVOP)l0, (GWMEP)0, (GPVCP), (GVVIP), and (GVUOP)
have at least a common solution.
inclusive condition and
1T Df is the Fr´echet derivative of the convex operatorf:X → Y;
2lis a weakly positive linear operator;
3T is strictly monotone.
If (PVCP) is solvable, then (VOP)l0, (WMEP)0, (PVCP), (VVIP), and (VUOP) have at least
a common solution.
Proof. Note thatA I the identity mapping of LX, Yand T is a single-valued mapping fromXtoLX, Y. FromTheorem 3.23, we immediately obtain the desired conclusion.
Remark 3.25. IfPx P for allx∈ X, whereP is a closed, pointed, and convex cone inY, thenCorollary 3.24coincides with Theorem 4.1 of Chen and Yang4.
Acknowledgments
This research of the first author was partially supported by the National Science Foundation of China 10771141, Ph.D. Program Foundation of Ministry of Education of China
20070270004, and Science and Technology Commission of Shanghai Municipality Grant
075105118. The second author would like to thank the support of NSC97-2115-M-039-001-Grant from the National Science Council of Taiwan.
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