R E S E A R C H
Open Access
Well-posed generalized vector equilibrium
problems
Xicai Deng
1,2*and Shuwen Xiang
2*Correspondence:
1Department of Mathematics and
Computer, Guizhou Normal College, Guiyang, 550018, China
2Department of Mathematics,
Guizhou University, Guiyang, 550025, China
Abstract
In this paper, we establish the bounded rationality modelMfor generalized vector equilibrium problems by using a nonlinear scalarization technique. By using the modelM, we introduce a new well-posedness concept for generalized vector equilibrium problems, which unifies its Hadamard and Levitin-Polyak well-posedness. Furthermore, sufficient conditions for the well-posedness for generalized vector equilibrium problems are given. As an application, sufficient conditions on the well-posedness for generalized equilibrium problems are obtained.
MSC: 49K40; 90C31
Keywords: well-posedness; generalized vector equilibrium problems; bounded rationality model; nonlinear scalarization function
1 Introduction
As is well known, the notion of well-posedness can be divided into two different groups: Hadamard type and Tykhonov type []. Roughly speaking, Hadamard types of well-posedness for a problem means the continuous dependence of the optimal solution from the data of the problem. Tykhonov types of well-posedness for a problem such as Tykhonov and Levitin-Polyak well-posedness are based on the convergence of approx-imating solution sequences of the problem. Some researchers have investigated the re-lations between them for different problems (see [–]). The notion of extended well-posedness has been proposed by Zolezzi [] in the context of scalar optimization. In some sense this notion unifies the ideas of Tykhonov and Hadamard well-posedness. Moreover, the notion of extended well-posedness has been generalized to vector optimization prob-lems by Huang [–].
On the other hand, the vector equilibrium problem provides a very general model for a wide range of problems, for example, the vector optimization problem, the vector vari-ational inequality problem, the vector complementarity problem and the vector saddle point problem. In the literature, existence results for various types of vector equilibrium problems have been investigated intensively; see,e.g., [, ] and the references therein. The study of well-posedness for vector equilibrium problems is another important topic in vector optimization theory. Recently, Tykhonov types well-posedness for vector opti-mization problems, vector variational inequality problems and vector equilibrium prob-lems have been intensively studied in the literature, such as [–]. Among those papers, we observe that the scalarization technique is an efficient approach to deal with Tykhonov types well-posedness for vector optimization problems. As noted in [, ], the notions of
well-posedness in the scalar case can be extended to the vector case and, for this end, one needs an appropriate scalarizarion technique. Such a technique is supposed to preserve some well-posedness properties when one passes from the vectorial to the scalar case, and simple examples show that linear scalarization is not useful from this point of view even in the convex case. An effort in this direction was made in the papers (see [, , , ]). Miglierina et al.[] investigated several types of well-posedness concepts for vectorial optimization problems by using a nonlinear scalarization procedure. Some equivalences between well-posedness of vectorial optimization problems and well-posedness of cor-responding scalar optimization problems are given. By virtue of a nonlinear scalarization function, Durea [] proved the Tykhonov well-posedness of the scalar optimization prob-lems are equivalent to the Tykhonov well-posedness of the original vectorial optimization problems. Very recently, Li and Xia [] investigated Levitin-Polyak well-posedness for vectorial optimization problems by using a nonlinear scalarization function. They also showed the equivalence relations between the Levitin-Polyak well-posedness of scalar op-timization problems and the vectorial opop-timization problems.
Motivated and inspired by the research work mentioned above, we introduce a new well-posedness concept for generalized vector equilibrium problems (in short (GVEP)), which unifies its Hadamard and Levitin-Polyak well-posedness. The concept of well-posedness for (GVEP) is investigated by using a new method which is different from the ones used in [–]. Our method is based on a nonlinear scalarization technique and the bounded rationality modelM(see [–]). Furthermore, we give some sufficient conditions on various types of well-posedness for (GVEP). Finally, we apply these results to generalized equilibrium problems (in short (GEP)).
2 Preliminaries
LetXbe a nonempty subset of the Hausdorff topological spaceHandYbe a Hausdorff topological vector space. Assume thatCdenotes a nonempty, closed, convex, and pointed cone inYwith apex at the origin andintC=∅, whereintCdenotes the topological interior ofC.
LetG:X⇒Xbe a set-valued mapping andϕ:X×X→Ybe a vector-valued mapping, the problem of interest, called generalized vector equilibrium problems (in short (GVEP)), which consist of finding an elementx∈Xsuch thatx∈G(x) and
ϕ(x,y) /∈–intC, ∀y∈G(x).
WhenY =RandC= [, +∞[, the generalized vector equilibrium problem becomes the generalized equilibrium problem (in short (GEP)): finding an elementx∈Xsuch that x∈G(x) and
ϕ(x,y)≥, ∀y∈G(x).
Now we introduce the notion of Levitin-Polyak approximating solution sequence for (GVEP).
that
dxn,G(xn)≤n,
and
ϕ(xn,y) +ne∈/–intC, ∀y∈G(xn).
Next, we introduce a nonlinear scalarization function and their related properties.
Lemma .([, , ]) For fixed e∈intC,the nonlinear scalarization function is defined by
ξe(y) =inf{r∈R:y∈re–C}, ∀y∈Y.
The nonlinear scalarization functionξehas the following properties:
(i) ξe(y)≥r⇐⇒y∈/re–intC;
(ii) ξe(re) =r;
(iii) ξe(y+y)≤ξe(y) +ξe(y),for ally,y∈Y.
Definition . Letϕ:X→Ybe a vector-valued mapping.
(i) ϕis said to beC-upper semicontinuous atxif for any open neighborhoodVof zero element inY, there is an open neighborhoodUatxinXsuch that for anyx∈U, ϕ(x)∈ϕ(x) +V–C;
(ii) ϕis said to beC-upper semicontinuous onXifϕisC-upper semicontinuous on eachx∈X;
(iii) ϕis said to beC-lower semicontinuous atxif for any open neighborhoodV of zero element inY, there is an open neighborhoodUatxinXsuch that for anyx∈U, ϕ(x)∈ϕ(x) +V+C;
(iv) ϕis said to beC-lower semicontinuous onXifϕisC-lower semicontinuous on eachx∈X.
Remark . In Definition ., whenY=R,C= [, +∞[, beingC-upper semicontinuous reduces to being upper semicontinuous and beingC-lower semicontinuous reduces to being lower semicontinuous.
Lemma . Ifϕ:X×X→Y is C-upper semicontinuous on X×X,thenξe◦ϕ:X×X→ is upper semicontinuous on X×X.
Proof In order to show thatξe◦ϕ:X×X→ is upper semicontinuous onX×X, we must check, for anyr∈ , the setL={(x,y)∈X×X:ξe(ϕ(x,y))≥r}is closed.
Let (xn,yn) ∈L and (xn,yn)→(x,y), we have ξe(ϕ(xn,yn))≥r, that is to say, by Lemma .(i),ϕ(xn,yn) /∈re–intC. Next, we only need to prove thatξe(ϕ(x,y))≥r, that is,ϕ(x,y) /∈re–intC. By way of contradiction, assume thatϕ(x,y)∈re–intC, then there exists an open neighborhoodVof zero element inYsuch that
SinceϕisC-upper semicontinuous at (x,y)∈X×X, we have
ϕ(xn,yn)∈ϕ(x,y) +V–C⊂re–intC–C⊂re–intC.
It contradictsϕ(xn,yn) /∈re–intC. SoLis closed. It showsξe◦ϕ:X×X→ is upper
semicontinuous onX×X.
Finally, we recall some useful definitions and lemmas.
LetF:X⇒Ybe a set-valued mapping.Fis said to be upper semicontinuous atx∈Xif for any open setU⊃F(x), there is an open neighborhoodO(x) ofxsuch thatU⊃F(x) for eachx∈O(x);Fis said to be lower semicontinuous atxif for any open setU∩F(x)=∅, there is an open neighborhoodO(x) ofxsuch thatU∩F(x)=∅, for eachx∈O(x);Fis said to be an usco mapping ifFis upper semicontinuous andF(x) is nonempty compact for eachx∈X;Fis said to be closed ifGraph(F) is closed, whereGraph(F) ={(x,y)∈X×Y: x∈X,y∈F(x)}is the graph ofF.
Lemma .[] Let X and Y be two metric spaces.Suppose that F :Y⇒X is a usco mapping.Then for any yn→y and any xn∈F(yn),there is a subsequence{xnk} ⊂ {xn}such that xnk→x∈F(y).
Lemma .[] If F:Y⇒X is closed and X is compact,then F is upper semicontinuous on Y.
Let (X,d) be a metric space. Denote byK(X) all nonempty compact subsets ofX. For arbitraryC,C⊂X, define
h(C,C) =max
h(C,C),h(C,C)
,
where
h(C,C) =sup
d(b,C) :b∈C
and
d(b,C) =inf
d(b,c),c∈C
.
It is obvious thathis a Hausdorff metric onK(X).
Lemma . [] Let (X,d) be a metric space and h be Hausdorff metric on X. Then (K(X),h)is complete if and only if(X,d)is complete.
3 Bounded rationality model and definition of well-posedness for (GVEP) Let (X,d) be a metric space. The problem spaceof (GVEP) is given by
=
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
λ= (ϕ,G) :
ϕ:X×X→YisC-upper semicontinuous onX×X,
∀x∈X,ϕ(x,x) =,sup(x,y)∈X×Xϕ(x,y)< +∞,
G:X⇒Xis continuous with nonempty compact value onX,
∃x∈Xsuch thatx∈G(x) andϕ(x,y) /∈–intC,∀y∈G(x).
For anyλ= (ϕ,G),λ= (ϕ,G)∈, define
ρ(λ,λ) := sup (x,y)∈X×X
ϕ(x,y) –ϕ(x,y)+sup x∈Xh
G(x),G(x)
,
wherehdenotes a Hausdorff distance onX. Clearly, (,ρ) is a metric space.
Next, the bounded rationality modelM={,X,f,}for (GVEP) is defined as follows:
(i) andXare two metric spaces;
(ii) the feasible set of the problemλ∈is defined by
f(λ) :=x∈X:x∈G(x);
(iii) the solution set of the problemλ∈is defined by
E(λ) :=x∈G(x) :ϕ(x,y) /∈–intC,∀y∈G(x);
(iv) the rationality function of the problemλ∈is defined by
(λ,x) := sup
y∈G(x)
–ξe
ϕ(x,y).
Lemma .
() x∈f(λ),(λ,x)≥. () For allλ∈,E(λ)=∅.
() For allλ∈and≥,(λ,x) =supy∈G(x){–ξe(ϕ(x,y))} ≤if and only if ϕ(x,y) +e∈/–intC,∀y∈G(x).Particularly,x∈E(λ)if and only if(λ,x) = .
Proof () Ifx∈f(λ), thenx∈G(x). By Lemma .(i), we have
(λ,x)≥–ξe
ϕ(x,x)= .
() Obvious.
() Ifϕ(x,y) +e∈/–intC,∀y∈G(x), by Lemma .(i),ξe(ϕ(x,y))≥–,∀y∈G(x). Thus, we have(λ,x) =supy∈G(x){–ξe(ϕ(x,y))} ≤.
Conversely, if (λ,x) =supy∈G(x){–ξe(ϕ(x,y))} ≤, thenξe(ϕ(x,y))≥–,∀y∈G(x). By Lemma .(i), we getϕ(x,y) +e∈/–intC,∀y∈G(x).
Remark . By Definition . and Lemma ., for alln> withn→, the set of LP approximating solution for the problemλis defined as
E(λ,n) :=
x∈X:dx,f(λ)≤n,(λ,x)≤n
;
the set of solutions for the problemλis defined as
E(λ) =E(λ, ) :=x∈X:x∈f(λ),(λ,x) = .
Definition .
(i) If∀xn∈E(λ,n),n> withn→, there must exist a subsequence{xnk} ⊂ {xn} such thatxnk→x∈E(λ), then the problemλ∈is said to be generalized Levitin-Polyak well-posed (in short GLP-wp);
(ii) IfE(λ) ={x}(a singleton),∀xn∈E(λ,n),n> withn→, there must have xn→x, then the problemλ∈is said to be Levitin-Polyak well-posed (in short
LP-wp).
Referring to [], Hadamard well-posedness for (GVEP) is defined as follows.
Definition .
(i) If∀λn∈,λn→λ,∀xn∈E(λn), there must exist a subsequence{xnk} ⊂ {xn}such thatxnk→x∈E(λ), then the problemλ∈is said to be generalized Hadamard well-posed (in short GH-wp);
(ii) IfE(λ) ={x}(a singleton),∀λn∈,λn→λ,∀xn∈E(λn), we must havexn→x, then
the problemλ∈is said to be Hadamard well-posed (in short H-wp).
Next, we establish a new well-posedness concept for (GVEP), which unifies its Hada-mard and Levitin-Polyak well-posedness.
Definition .
(i) If∀λn∈,λn→λ,∀xn∈E(λn,n),n> withn→, there must exist a
subsequence{xnk} ⊂ {xn}such thatxnk→x∈E(λ), then the problemλ∈is said to be generalized well-posed (in short G-wp);
(ii) IfE(λ) ={x}(a singleton),∀λn∈,λn→λ,∀xn∈E(λn,n),n> withn→,
there must havexn→x, thenλ∈is said to be well-posed (in short wp).
By Definitions ., . and ., it is easy to check the following.
Lemma .
() If the problemλ∈is G-wp,thenλmust be GLP-wp. () If the problemλ∈is wp,thenλmust be LP-wp.
Lemma .
() If the problemλ∈is G-wp,thenλmust be GH-wp. () If the problemλ∈is wp,thenλmust be H-wp.
4 Sufficient conditions for well-posedness of (GVEP)
Assume that the bounded rationality modelM={,X,f,} for (GVEP) is given. Now, let (X,d) be a compact metric space, (Y, · ) be a Banach space, andCbe a nonempty, closed, convex, and pointed cone inYwith apex at the origin andintC=∅.
In order to show sufficient conditions for well-posedness of (VEP), we first give the fol-lowing lemmas.
Proof Let{λn= (ϕn,Gn)}be any Cauchy sequence in, then for any> , there is a posi-tive integerNsuch that for anyn,m≥N,
ρ(λn,λm) = sup (x,y)∈X×X
ϕn(x,y) –ϕm(x,y)+sup x∈Xh
Gn(x),Gm(x)
<.
Then, for any fixed (x,y)∈X×Y, {ϕn(x,y)} is a Cauchy sequence inY, and{Gn(x)}is a Cauchy sequence inK(X). By Lemma ., (K(X),h) is a complete spaces and (Y, · ) is also complete spaces. It follows that there existϕ(x,y)∈Y andG(x)∈K(X) such that
limm→∞ϕm(x,y) =ϕ(x,y) andlimm→∞Gm(x) =G(x). Thus, for alln≥N, we have
sup
(x,y)∈X×X
ϕn(x,y) –ϕ(x,y)+sup x∈Xh
Gn(x),G(x)
≤.
Next, we will prove thatλ= (ϕ,G)∈, thus (,ρ) is a complete metric space.
(i) For any open convex neighborhoodVof zero element inY, there is a positive integer nsuch that for allx,y∈X,
ϕ(x,y)∈ϕn(x,y) + V
, ()
and
ϕn(x,y)∈ϕ(x,y) + V
. ()
Sinceλn = (ϕn,Gn)∈,ϕn isC-upper semicontinuous onX×X, thus there are an open neighborhood ofUatxand an open neighborhood ofUatysuch that
ϕn
x,y∈ϕn(x,y) + V
–C, ∀x ∈U
,y∈U. ()
By (), (), and (), we have
ϕx,y∈ϕn
x,y+V
⊂ϕn(x,y) +
V–C⊂ϕ(x,y) +V–C.
It showsϕ:X×X→YisC-upper semicontinuous onX×X.
(ii) It is easy to check thatϕ(x,x) =,∀x∈X,sup(x,y)∈X×Xϕ(x,y)< +∞,G:X⇒Xis continuous onXand∀x∈X,G(x) is a nonempty compact set.
(iii) Sinceλn= (ϕn,Gn)∈, there existsxn∈Xsuch thatxn∈Gn(xn) andϕn(xn,y) /∈ –intC, ∀y∈Gn(xn). Firstly, we may suppose that xn→x, sinceX is a compact metric space. For alln≥N,
hGn(xn),G(x)
≤hGn(xn),G(xn)
+hG(xn),G(x)
≤+hG(xn),G(x).
()
Note thatGis continuous onX, we have
By () and (), we get
dx,G(x)≤d(x,xn) +dxn,Gn(xn)+hGn(xn),G(x)
=d(x,xn) +h
Gn(xn),G(xn)
→.
Hence,x∈G(x).
Finally, we only need to prove thatϕ(x,y) /∈–intC,∀y∈G(x). By way of contradiction, assume that there existsy∈G(x) such thatϕ(x,y)∈–intC. It shows that there exists an open convex neighborhoodVof zero element inY such thatϕ(x,y) +V⊂–intC.
Sincesup(x,y)∈X×Xϕn(x,y)–ϕ(x,y) →, there is a positive integerNsuch that∀n≥N,
ϕn(x,y)∈ϕ(x,y) +V
, ∀x,y∈X. ()
By virtue of h(Gn(xn),G(x))→, there existsyn∈Gn(xn) such thatyn→y. Note that
ϕ:X×X→Y isC-upper semicontinuous onX×X, then there exists a positive integer Nsuch that∀n≥N,
ϕ(xn,yn)∈ϕ(x,y) + V
–C. ()
LetN=max{N,N},∀n≥N, by () and (), we have
ϕn(xn,yn)∈ϕ(xn,yn) + V
⊂ϕ(x,y) +V–C⊂–intC–C⊂–intC.
This is a contradiction toϕn(xn,y) /∈–intC,∀y∈Gn(xn).
Lemma . f:⇒X is an usco mapping.
Proof SinceXis a compact metric space, by Lemma ., it suffices to show thatGraph(f) is closed, whereGraph(f) ={(λ,x)∈×X:x∈f(λ)}. That is to say,∀λn∈,λn→λ,
∀xn∈f(λn),xn→x, we need to show thatx∈f(λ).
In fact, for eachn= , , , . . . , sincexn∈f(λn), then there existsxn∈Xsuch thatxn∈ Gn(xn). Letsupx∈Xh(Gn(x),G(x)) =nwithn→, there must beh(Gn(xn),G(xn))≤n. Sincexn∈Gn(xn), there existsxn∈G(xn) such thatd(xn,xn)≤n. By
dxn,x≤dxn,xn
+d(xn,x)→ ()
we getxn→x. Note that set-value mappingGis continuous onX, then
hG(xn),G(x)→. ()
By () and (), we get
dx,G(x)≤dx,xn+dxn,G(xn)+hG(xn),G(x)→. ()
SinceG(x) is a nonempty compact subset ofX, by (), we havex∈G(x). It shows that
Lemma . For all(λ,x)∈×X,(λ,x)is lower semicontinuous at(λ,x).
Proof By Lemma ., it is only need to show that∀> ,∀λn= (ϕn,Gn)∈,λn→λ= (ϕ,G)∈,∀xn∈X,xn→x∈X, there exists a positive integerNsuch that∀n≥N,
(λn,xn) >(λ,x) –. ()
By definition of the least upper bound, there existsy∈G(x) such that
–ξe
ϕ(x,y)
>(λ,x) –
. ()
Note thatsupx∈Xh(Gn(x),G(x))→ andh(G(xn),G(x))→, we have
hGn(xn),G(x)≤hGn(xn),G(xn)+hG(xn),G(x)→. ()
From (), there existsyn∈Gn(xn) such thatd(yn,y)→.
Sincesup(x,y)∈X×Xϕn(x,y) –ϕ(x,y) →, that is to say, there exists a positive integerN such that∀n≥N,
ϕn(xn,yn) –ϕ(xn,yn)∈re, ()
wherer∈]–,[,e∈intC,reis an open neighborhood of zero element inY. Thus, by () and Lemma .(ii), we get
– ≤ξe
ϕn(xn,yn) –ϕ(xn,yn)
≤
. ()
By () and Lemma .(iii), we get
–ξe
ϕn(xn,yn)= –ξe
ϕn(xn,yn) –ϕ(xn,yn) +ϕ(xn,yn)
≥–ξe
ϕ(xn,yn)–ξe
ϕn(xn,yn) –ϕ(xn,yn)
≥–ξe
ϕ(xn,yn)– .
()
By Lemma ., ξe◦ϕ is upper semicontinuous onX×X. Then there exists a positive integerNsuch that∀n≥N,
–ξe
ϕ(xn,yn)> –ξe
ϕ(x,y)
–
. ()
LetN=max{N,N},∀n≥N, by (), (), and (), we get the inequality ():
(λn,xn) = sup y∈Gn(xn)
–ξe
ϕn(xn,y)
≥–ξe
ϕn(xn,yn)
> –ξe
ϕ(xn,yn)– > –ξe
ϕ(x,y)
–
Theorem .
() For allλ∈,the problemsλis G-wp.
() For allλ∈,ifE(λ) ={x}(a singleton),then the problemλis wp.
Proof ()∀λn∈,λn→λ,∀xn∈E(λn,n),n> withn→, then
dxn,f(λn)≤n ()
and
(λn,xn)≤n. ()
From (), there exists un∈f(λn) such that d(un,xn)→ as n→ ∞. It follows by Lemma . and Lemma . that there exists{unk} ⊂ {un}such thatunk→x∈f(λ). By
d(xnk,x)≤d(xnk,unk) +d(unk,x)→,
we get
xnk→x∈f(λ). ()
By Lemma . and (), we have
≤(λ,x)≤lim inf
nk→∞
(λnk,xnk)≤lim infn k→∞
nk= . ()
That is,
(λ,x) = .
By () and (), we havex∈E(λ). It shows thatλis G-wp.
() By way of contradiction. If the sequence{xn}does not convergex, then there exists an open neighborhoodOatxand a subsequence{xnk}of{xn}such thatxnk ∈/ O. Since
E(λ) ={x} (a singleton), by the proof of (), we getxnk →x. This is a contradiction to
xnk∈/O.
Similarly, by Lemmas . and ., it is easy to check the following.
Theorem .
() For allλ∈,the problemsλmust be GLP-wp and GH-wp.
() For allλ∈,ifE(λ) ={x}(a singleton),then the problemλmust be LP-wp and H-wp.
Finally, we apply these results to (GEP). LetY=R,C= [, +∞[, the problem space of (GEP) is defined as
=
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
λ= (ϕ,G) :
ϕ:X×X→Ris upper semicontinuous onX×X,
∀x∈X,ϕ(x,x) = ,sup(x,y)∈X×X|ϕ(x,y)|< +∞,
G:X⇒Xis continuous with nonempty compact value onX,
∃x∈Xsuch thatx∈G(x) andϕ(x,y)≥,∀y∈G(x).
For anyλ= (ϕ,G),λ= (ϕ,G)∈, define
(λ,λ) := sup (x,y)∈X×X
ϕ(x,y) –ϕ(x,y)+sup x∈Xh
G(x),G(x)
.
It is easy to check that (, ) is a complete metric space. Hence, we have the following.
Corollary .
() For allλ∈,the problemλis G-wp.
() For allλ∈,ifE(λ) ={x}(a singleton),then the problemλis wp.
Corollary .
() For allλ∈,the problemλmust be GLP-wp and GH-wp.
() For allλ∈,ifE(λ) ={x}(a singleton),then the problemλmust be LP-wp and H-wp.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
This research was supported by NSFC (Grant Number: 11161008), the Guizhou Provincial Science and Technology Foundation (20132235), (20122289). The authors thank the anonymous referees for their constructive comments which help us to revise the paper.
Received: 10 December 2013 Accepted: 17 March 2014 Published:28 Mar 2014
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