# Well-Posedness of Generalized Vector Quasivariational Inequality Problems

Authors: Jian-Wen Peng; Fang Liu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce several types of the Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with both abstract set constraints and functional constraints. Criteria and characterizations of these types of the Levitin-Polyak well-posednesses with or without gap functions of generalized vector quasivariational inequality problem are given. The results in this paper unify, generalize, and extend some known results in the literature.


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## Body

## 1. Introduction

The vector variational inequality in a finite-dimensional Euclidean space has been introduced in [1] and applications have been given. Chen and Cheng [2] studied the vector variational inequality in infinite-dimensional space and applied it to vector optimization problem. Since then, many authors [3–11] have intensively studied the vector variational inequality on different assumptions in infinite-dimensional spaces. Lee et al. [12, 13], Lin et al. [14], Konnov and Yao [15], Daniilidis and Hadjisavvas [16], Yang and Yao [17], and Oettli and Schläger [18] studied the generalized vector variational inequality and obtained some existence results. Chen and Li [19] and Lee et al. [20] introduced and studied the generalized vector quasi-variational inequality and established some existence theorems.On the other hand, 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 [21] in dealing with unconstrained optimization problems. Its extension to the constrained case was developed by Levitin and Polyak [22]. The study of generalized Levitin-Polyak well-posedness for convex scalar optimization problems with functional constraints originates from Konsulova and Revalski [23]. Recently, this research was extended to nonconvex optimization problems with abstract set constraints and functional constraints (see [24]), nonconvex vector optimization problem with abstract set constraints and functional constraints (see [25]), variational inequality problems with abstract set constraints and functional constraints (see [26]), generalized inequality problems with abstract set constraints and functional constraints [27], generalized quasi-inequality problems with abstract set constraints and functional constraints [28], generalized vector inequality problems with abstract set constraints and functional constraints [29], and vector quasivariational inequality problems with abstract set constraints and functional constraints [30]. For more details on well-posedness on optimizations and related problems, please also see [31–37] and the references therein. It is worthy noting that there is no study on the Levitin-Polyak well-posedness for a generalized vector quasi-variational inequality problem.In this paper, we will introduce four types of Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with an abstract set constraint and a functional constraint. In Section2, by virtue of a nonlinear scalarization function and a gap function for generalized vector quasi-varitional inequality problems, we show equivalent relations between the Levitin-Polyak well-posedness of the optimization problem and the Levitin-Polyak well-posedness of generalized vector quasi-varitional inequality problems. In Section 3, we derive some various criteria and characterizations for the (generalized) Levitin-Polyak well-posedness of the generalized vector quasi-variational inequality problems. The results in this paper unify, generalize, and extend some known results in [26–30].


## 2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations and assumptions.Let(X,∥·∥) be a normed space equipped with norm topology, and let (Z,d1) be a metric space. Let X1⊂X, K⊂Z be nonempty and closed sets. Let Y be a locally convex space ordered by a nontrivial closed and convex cone C with nonempty interior intC, that is, y1≤y2 if and only if y2-y1∈C for any y1,y2∈Y. Let L(X,Y) be the space of all the linear continuous operators from X to Y. Let T:X1→2L(X,Y) and S:X1→2X1 be strict set-valued mappings (i.e., T(x)≠∅ and S(x)≠∅, forallx∈X1), and let g:X1→Z be a  continuous vector-valued mapping. We denote by 〈z, x〉 the value z(x), where z∈L(X,Y), x∈X1. Let X0={x∈X1:g(x)∈K} be nonempty. We consider the following generalized vector quasi-variational inequality problem with functional constraints and abstract set constraints.Findx¯∈X0 such that x¯∈S(x¯) and there exists z¯∈T(x¯) satisfying(GVQVI)〈z¯,x-x¯〉∉-int⁡C,∀x∈S(x¯).
Denote by X¯ the solution set of (GVQVI).LetZ1,Z2 be two normed spaces. A set-valued map F from Z1 to 2Z2 is(i) closed, onZ3⊆Z1, if for any sequence {xn}⊆Z3 with xn→x and yn∈F(xn) with yn→y, one has y∈F(x);(ii) lower semicontinuous (l.s.c. in short) atx∈Z1, if {xn}⊆Z1,xn→x, and y∈F(x) imply that there exists a sequence {yn}⊆Z2 satisfying yn→y such that yn∈F(xn) for n sufficiently large. If F is l.s.c. at each point of Z1, we say that F is l.s.c. on Z1;(iii) upper semicontinuous (u.s.c. in short) atx∈Z1, if for any neighborhood V of F(x), there exists a neighborhood U of x such that F(x′)⊆V, forallx′∈U. If F is u.s.c. at each point of Z1, we say that F is u.s.c. on Z1.It is obvious that any u.s.c. nonempty closed-valued mapF is closed.Let(P,d) be a metric space, P1⊂P, and x∈P. We denote by dP1(x)=inf{d(x,p′):p′∈P1} the distance from the point x to the set P1. For a topological vector space V, we denote by V* its dual space. For any set Φ⊂V, we denote the positive polar cone of Φ by(2.1)Φ*={λ∈V*:λ(x)≥0,∀x∈Φ}.Lete∈intC be fixed. Denote(2.2)C*0={λ∈C*:λ(e)=1}.Definition 2.1.
(i) A sequence{xn}⊆X1 is called a type I Levitin-Polyak (LP in short) approximating solution sequence if there exist {ϵn}⊆R+1={r≥0|risarealnumber} with ϵn→0 and zn∈T(xn) such that
(2.3)dX0(xn)≤ϵn,(2.4)xn∈S(xn),(2.5)〈zn,x-xn〉+ϵne∉-int⁡C,∀x∈S(xn).
(ii){xn}⊆X1 is called a type II LP approximating solution sequence if there exist {ϵn}⊆R+1 with ϵn→0 and zn∈T(xn) such that (2.3)–(2.5) hold, and, for any z∈T(xn), there exists w(n,z)∈S(xn) satisfying
(2.6)〈z,w(n,z)-xn〉-ϵne∈-C.
(iii){xn}⊆X1 is called a generalized type I LP approximating solution sequence if there exist {ϵn}⊆R+1 with ϵn→0 and zn∈T(xn) such that
(2.7)dK(g(xn))≤ϵn
and (2.4), (2.5) hold.
(iv){xn}⊆X1 is called a generalized type II LP approximating solution sequence if there exist {ϵn}⊆R+1 with ϵn→0, zn∈T(xn) such that (2.4), (2.5), and (2.7) hold, and, for any z∈T(xn), there exists w(n,z)∈S(xn) such that (2.6) holds.Definition 2.2.
(GVQVI) is said to be type I (resp., type II, generalized type I, generalized type II) LP well-posed if the solution set X¯ of (GVQVI) is nonempty, and, for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence {xn}, there exists a subsequence {xnj} of {xn} and x¯∈X¯ such that xnj→x¯.Remark 2.3.
(i) It is clear that any (generalized) type II LP approximating solution sequence is a (generalized) type I LP approximating solution sequence. Thus, (generalized) type I LP well-posedness implies (generalized) type II LP well-posedness.
(ii) Each type of LP well-posedness of (GVQVI) implies that the solution set X¯ is compact.
(iii) Suppose thatg is uniformly continuous functions on a set
(2.8)X1(δ0)={x∈X1:dX0(x)≤δ0},
for some δ0>0. Then generalized type I (resp., generalized type II) LP well-posedness of (GVQVI) implies its type I (resp., type II) LP well-posedness.
(iv) IfY=R1, C=R+1, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the generalized quasi-variational inequality problem defined by Jiang et al. [28]. If Y=R1, C=R+1, S(x)=X0 for all x∈X1, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the generalized variational inequality problem defined by Huang, and Yang [27] which contains as special cases for the type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the variational inequality problem in [26].
(v) IfS(x)=X0 for all x∈X1, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the generalized vector variational inequality problem defined by Xu et al. [29].
(vi) If the set-valued mapT is replaced by a single-valued map F, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the vector quasivariational inequality problems defined by Zhang et al. [30].
Consider the following statement:(2.9)[δ2X¯≠∅andforanytypeI(resp.,typeII,generalizedtypeI,generalizedtypeIIδ2)LPapproximatingsolutionsequence{xn},wehavedX¯(xn)⟶0δ2].Proposition 2.4.
If (GVQVI) is type I (resp., type II, generalized type I, generalized type II) LP well-posed, then (2.9) holds. Conversely if (2.9) holds and X¯ is compact, then (1) is type I (resp., type II, generalized type I, generalized type II) LP well-posed.The proof of Proposition2.4 is elementary and thus omitted.To see the various LP well-posednesses of (1) are adaptations of the corresponding LP well-posednesses in minimizing problems by using the Auslender gap function, we consider the following general constrained optimization problem:(P)min⁡f(x)s.t.x∈X1′g(x)∈K,whereX1′⊆X1 is nonempty and f:X1→R1∪{+∞} is proper. The feasible set of (P) is X0′, where X0′={x∈X1′:g(x)∈K}. The optimal set and optimal value of (P) are denoted by X′¯ and v¯, respectively. Note that if Dom(f)∩X0′≠∅, where(2.10)Dom⁡(f)={x∈X1:f(x)<+∞},
then v¯<+∞. In this paper, we always assume that v¯>-∞.Definition 2.5.
(i) A sequence{xn}⊆X1′ is called a type I LP minimizing sequence for (P) if
(2.11)limsup⁡n→∞f(xn)≤v¯,(2.12)dX0′(xn)⟶0.
(ii){xn}⊆X1′ is called a type II LP minimizing sequence for (P) if
(2.13)lim⁡n→∞f(xn)=v¯
and (2.12) hold.
(iii){xn}⊆X1′ is called a generalized type I LP minimizing sequence for (P) if
(2.14)dK(g(xn))⟶0
and (2.11) hold.
(iv){xn}⊆X1′ is called a generalized type II LP minimizing sequence for (P) if (2.13) and (2.14) hold.Definition 2.6.
(P) is said to be type I (resp., type II, generalized type I, generalized type II) LP well-posed if the solution set X¯′ of (P) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP minimizing sequence {xn}, there exists a subsequence {xnj} of {xn} and x¯∈X¯′ such that xnj→x¯.
The Auslender gap function for (GVQVI) is defined as follows:
(2.15)f(x)=inf⁡z∈T(x)sup⁡x′∈S(x)inf⁡λ∈C*0λ(〈z,x-x′〉),∀x∈X1.
LetX2⊆X be defined by
(2.16)X2={x∈X∣x∈S(x)}.
In the rest of this paper, we setX1′ in (P) equal to X1∩X2. Note that if S is closed on X1, then X1′ is closed.
Recall the following widely used function (see, e.g., [38])
(2.17)ξ:Y⟶R1:min⁡{t∈R1:y-te∈-C}.
It is known thatξ is a continuous, (strictly) monotone (i.e., for any y1,y2∈Y, y1-y2∈C implies that ξ(y1)≥ξ(y2) and (y1-y2∈intC implies that ξ(y1)>ξ(y2)), subadditive and convex function. Moreover, it holds that ξ(te)=t,forallt∈R1 and ξ(y)=supλ∈C*0λ(y),forally∈Y.
Now we given some properties for the functionf defined by (2.15).Lemma 2.7.
Let the functionf be defined by (2.15), and let the set-valued map T be compact-valued on X1. Then(i)
f(x)≥0,forallx∈X1′;(ii)
for anyx∈X0′, f(x)=0 if and only if x∈X¯.Proof.
(i) Letx∈X1′. Suppose to the contrary that f(x)<0. Then, there exists a δ>0 such that f(x)<-δ. By definition, for δ/2>0, there exists a z∈T(x), such that
(2.18)sup⁡x′∈S(x)inf⁡λ∈C*0λ(〈z,x-x′〉)≤f(x)+δ2<-δ2<0.
Thus, we have
(2.19)inf⁡λ∈C*0λ(〈z,x-x′〉)<0,∀x′∈S(x),
which is impossible when x′=x. This proves (i).
(ii) Suppose thatx∈X0′ such that f(x)=0.
Then, it follows from the definition ofX0′ that x∈S(x). And from the definition of f(x) we know that there exist zn∈T(x) and 0<ϵn→0 such that
(2.20)inf⁡λ∈C*0λ(〈zn,x-x′〉)≤f(x)+ϵn=ϵn,∀x′∈S(x),
that is,
(2.21)ξ(〈zn,x′-x〉)≥-ϵn,∀x′∈S(x).
By the compactness of T(x), there exists a sequence {znj} of {zn} and some z∈T(x) such that
(2.22)znj⟶z.
This fact, together with the continuity of ξ and (2.21), implies that
(2.23)ξ(〈z,x′-x〉)≥0,∀x′∈S(x).
It follows that x∈X¯.
Conversely, assume thatx∈X¯. It follows from the definition of X¯ that x∈S(x). Suppose to the contrary that f(x)>0. Then, for any z∈T(x),
(2.24)sup⁡x′∈S(x)inf⁡λ∈C*0λ(〈z,x-x′〉)>0.
Thus, there existδ>0 and x0∈S(x) such that
(2.25)inf⁡λ∈C*0λ(〈z,x-x0〉)≥δ.
It follows that
(2.26)ξ(〈z,x0-x〉)≤-δ<0.
As a result, we have
(2.27)〈z,x0-x〉∈-int⁡C.
This contradicts the fact that x∈X¯. So, f(x)=0. This completes the proof.Lemma 2.8.
Letf be defined by (2.15). Assume that the set-valued map T is compact-valued and u.s.c. on X1 and the set-valued map S is l.s.c. on X1. Then f is l.s.c. function from X1 to R1∪{+∞}. Further assume that the solution set X¯ of (GVQVI) is nonempty, then Dom(f)≠∅.Proof.
First we show thatf(x)>-∞, forallx∈X1. Suppose to the contrary that there exists x0∈X1 such that f(x0)=-∞. Then, there exist zn∈T(x0) and {Mn}⊂R+1 with Mn→+∞ such that
(2.28)sup⁡x′∈S(x0)inf⁡λ∈C*0λ(〈zn,x0-x′〉)≤-Mn.
Thus,
(2.29)ξ(〈zn,x′-x0〉)≥Mn,∀x′∈S(x0).
By the compactness of T(x0), there exist a sequence {znj}⊂{zn} and some z0∈T(x0) such that znj→z0. This fact, together with (2.29) and the continuity of ξ on Y, implies that
(2.30)ξ(〈z0,x′-x0〉)≥+∞,∀x′∈S(x0)
which is impossible, since ξ is a finite function on Y.
Second, we show thatf is l.s.c. on X1. Let a∈R1. Suppose that {xn}⊂X1 satisfies f(xn)≤a,foralln, and xn→x0∈X1. It follows that, for each n, there exist zn∈T(xn) and 0<δn→0 such that
(2.31)-ξ(〈zn,y-xn〉)≤a+δn,∀y∈S(xn).
For anyx′∈S(x0), by the l.s.c. of S, we have a sequence {yn} with {yn}∈S(xn) converging to x′ such that
(2.32)-ξ(〈zn,yn-xn〉)≤a+δn.
By the u.s.c. ofT at x0 and the compactness of T(x0), we obtain a subsequence {znj} of {zn} and some z0∈T(x0) such that znj→z0. Taking the limit in (2.32) (with n replaced by nj), by the continuity of ξ, we have
(2.33)-ξ(〈z0,x′-x0〉)≤a,∀x′∈S(x0).
It follows thatf(x0)=infz∈T(x0)supx′∈S(x0)-ξ(〈z,x′-x0〉)≤a. Hence, f is l.s.c. on X1. Furthermore, if X¯≠∅, by Lemma 2.7, we see that Dom(f)≠∅.Lemma 2.9.
Let the functionf be defined by (2.15), and let the set-valued map T be compact-valued on X1. Then,(i)
{xn}⊆X1 is a sequence such that there exist {ϵn}⊆R+1 with ϵn→0 and zn∈T(xn) satisfying (2.4) and (2.5) if and only if {xn}⊆X1′ and (2.11) hold with v¯=0,(ii)
{xn}⊆X1 is a sequence such that there exist {ϵn}⊆R+1 with ϵn→0 and zn∈T(xn) satisfying (2.4) and (2.5), and for any z∈T(xn), there exists w(n,z)∈S(xn) satisfying (2.6) if and only if {xn}⊆X1′ and (2.13) hold with v¯=0.Proof.
(i) Let{xn}⊆X1 be any sequence if there exist {ϵn}⊆R+1 with ϵn→0 and zn∈T(xn) satisfying (2.4) and (2.5), then we can easily verify that
(2.34){xn}⊆X1′,f(xn)≤ϵn.
It follows that (2.11) holds with v¯=0.
For the converse, let{xn}⊆X1′ and (2.11) hold with v¯=0. We can see that {xn}⊆X1 and (2.4) hold. Furthermore, by (2.11), we have that there exists {ϵn}⊆R+1withϵn→0 such that f(xn)≤ϵn. By the compactness of T(xn), we see that for every n there exists zn∈T(xn) such that
(2.35)ξ(〈zn,x′-xn〉)≥-ϵn,∀x′∈S(xn).
It follows that for everyn there exists zn∈T(xn) such that (2.5) holds.
(ii) Let{xn}⊆X1 be any sequence we can verify that
(2.36)liminf⁡n→∞f(xn)≥0
holds if and only if there exists{αn}⊆R+1 with αn→0 and, for any z∈T(xn), there exists w(n,z)∈S(xn) such that
(2.37)〈z,w(n,z)-xn〉-αne∈-C.
From the proof of (i), we know thatlimsupn→∞f(xn)≤0 and {xn}⊆X1′ hold if and only if {xn}⊆X1 such that there exist {βn}⊆R+1 with βn→0zn∈T(xn) satisfying (2.4) and (2.5) (with ϵn replaced by βn). Finally, we let ϵn=max{αn,βn} and the conclusion follows.Proposition 2.10.
Assume thatX¯≠∅ and T is compact-valued on X1. Then(i)
(GVQVI) is generalized type I (resp., generalized type II) LP well-posed if and only if (P) is generalized type I (resp., generalized type II) LP well-posed with f(x) defined by (2.15).(ii)
If (GVQVI) is type I (resp., type II) LP well-posed, then (P) is type I (resp., type II) LP well-posed with f(x) defined by (2.15).Proof.
Letf(x) be defined by (2.15). Since X¯≠Ø, it follows from Lemma 2.7 that x¯∈X¯ is a solution of (GVQVI) if and only if x¯ is an optimal solution of (5) with v¯=f(x¯)=0.(i)
Similar to the proof of Lemma2.9, it is also routine to check that a sequence {xn} is a generalized type I (resp., generalized type II) LP approximating solution sequence if and only if it is a generalized type I (resp., generalized type II) LP minimizing sequence of (P). So (GVQVI) is generalized type I (resp., generalized type II) LP well-posed if and only if (P) is generalized type I (resp., generalized type II) LP well-posed with f(x) defined by (2.15).(ii)
SinceX0′⊆X0,dX0(x)≤dX0′(x) for any x. This fact together with Lemma 2.9 implies that a type I (resp., type II) LP minimizing sequence of (P) is a type I (resp., type II) LP approximating solution sequence. So the type I (resp., type II) LP well-posedness of (GVQVI) implies the type I (resp., type II) LP well-posedness of (P) with f(x) defined by (2.15).


## 3. Criteria and Characterizations for Generalized LP Well-Posedness of (GVQVI)

In this section, we shall present some necessary and/or sufficient conditions for the various types of (generalized) LP well-posedness of (GVQVI) defined in Section 2.Now consider a real-valued functionc=c(t,s,r) defined for t,s,r≥0 sufficiently small, such that(3.1)c(t,s,r)≥0,∀t,s,r,c(0,0,0)=0,sn⟶0,tn≥0,rn=0,c(tn,sn,rn)⟶0implythattn⟶0,Theorem 3.1.
Let the set-valued mapT be compact-valued on X1. If (GVQVI) is type II LP well-posed, the set-valued map S is closed-valued, then there exist a function c satisfying (3.1) such that
(3.2)|f(x)|≥c(dX¯(x),dX0(x),dS(x)(x)),∀x∈X1,
where f(x) is defined by (2.15). Conversely, suppose that X¯ is nonempty and compact and (3.2) holds for some c satisfying (3.1). Then (GVQVI) is type II LP well-posed.Proof.
Define(3.3)c(t,s,r)=inf⁡{|f(x)|:x∈X1,dX¯(x)=t,dX0(x)=s,dS(x)(x)=r}.
Since X¯≠Ø, it is obvious that c(0,0,0)=0. Moreover, if sn→0,tn≥0,rn=0, and c(tn,sn,rn)→0, then there exists a sequence {xn}⊆X1 with dX¯(xn)=tn, dS(xn)(xn)=rn=0,
(3.4)dX0(xn)=sn⟶0,
such that
(3.5)|f(xn)|⟶0.
SinceS is closed-valued, xn∈S(xn) for any n. This fact, combined with (3.4) and (3.5) and Lemma 2.9 (ii) implies that {xn} is a type II LP approximating solution sequence of (GVQVI). By Proposition 2.4, we have that tn→0.
Conversely, let{xn} be a type II LP approximating solution sequence of (GVQVI). Then, by (3.2), we have
(3.6)|f(xn)|≥c(dX¯(xn),dX0(xn),dS(xn)(xn)).
Let(3.7)tn=dX¯(xn),sn=dX0(xn),rn=dS(xn)(xn).
Thensn→0 and rn=0,foralln∈N. Moreover, by Lemma 2.9, we have that |f(x)|→0. Then, c(tn,sn,rn)→0. These facts together with the properties of the function c imply that tn→0. By Proposition 2.4, we see that (GVQVI) is type II LP well-posed.Theorem 3.2.
Let the set-valued mapT be compact-valued on X1. If (GVQVI) is generalized type II LP well-posed, the set-valued map S is closed, then there exist a function c satisfying (3.1) such that
(3.8)|f(x)|≥c(dX¯(x),dK(g(x)),dS(x)(x)),∀x∈X1,
where f(x) is defined by (2.15). Conversely, suppose that X¯ is nonempty and compact and (3.8) holds for some c satisfying (3.4) and (3.5). Then, (GVQVI) is generalized type II LP well-posed.Proof.
The proof is almost the same as that of Theorem3.1. The only difference lies in the proof of the first part of Theorem 3.1. Here we define
(3.9)c(t,s,r)=inf⁡{|f(x)|:x∈X1,dX¯(x)=t,dK(g(x))=s,dS(x)(x)=r}.
Next we give the Furi-Vignoli-type characterizations [39] for the (generalized) type I LP well-posedness of (GVQVI).
Let(X,∥·∥) be a Banach space. Recall that the Kuratowski measure of noncompactness for a subset H of X is defined as
(3.10)μ(H)=inf⁡{ϵ>0:H⊆⋃Hi,diam⁡(Hi)<ϵ,i=1,…,n},
wherediam(Hi) is the diameter of Hi defined by
(3.11)diam⁡(Hi)=sup⁡{∥x1-x2∥:x1,x2∈Hi}.
Given two nonempty subsets A and B of a Banach space(X,∥·∥), the Hausdorff distance between A and B is defined by
(3.12)h(A,B)=max⁡{sup⁡{dB(a):a∈A},sup⁡{dA(b):b∈B}}.
For anyϵ≥0, two types of approximating solution sets for (GVQVI) are defined, respectively, by
(3.13)Ω1(ϵ)={x∈X1:x∈S(x),dX0(x)≤ϵ,∃z∈T(x),s.t.〈z,x′-x〉+ϵe∉-int⁡C,∀x′∈S(x)},Ω2(ϵ)={x∈X1:x∈S(x),dK(g(x))≤ϵ,∃z∈T(x),s.t.〈z,x′-x〉+ϵe∉-int⁡C,∀x′∈S(x)}.Theorem 3.3.
Assume thatT is u.s.c. and compact-valued on X1and S is l.s.c. and closed on X1. Then
(a)  (GVQVI) is type I LP well-posed if and only if
(3.14)lim⁡ϵ→0μ(Ω1(ϵ))=0,
(b) (GVQVI) is generalized type I LP well-posed if and only if
(3.15)lim⁡ϵ→0μ(Ω2(ϵ))=0.Proof.
(a) First we show that, for everyϵ>0, Ω1(ϵ) is closed. In fact, let xn∈Ω1(ϵ) and xn→x0. Then (2.4) and the following formula hold:
(3.16)dX0(xn)≤ϵ,∃zn∈T(xn),s.t.〈zn,x′-xn〉+ϵe∉-int⁡C,∀x′∈S(xn).
Since xn→x0, by the closedness of S and (2.4), we have x0∈S(x0). From (3.16), we get
(3.17)dX0(x0)≤ϵ,(3.18)∃zn∈T(xn),s.t.ξ(〈zn,x′-xn〉)≥-ϵ,∀x′∈S(xn).
For any v∈S(x0), by the lower semi-continuity of S and (3.18), we can find vn∈S(xn) with vn→v such that
(3.19)ξ(〈zn,vn-xn〉)≥-ϵ.
By the u.s.c. ofT at x0 and the compactness of T(x0), there exist a subsequence {znj}⊂{zn} and some z0∈T(x0) such that
(3.20)znj⟶z0.
This fact, together with the continuity ofξ and (3.19), implies that
(3.21)ξ(〈z0,v-x0〉)≥-ϵ∀v∈S(x0).
It follows that(3.22)〈z0,v-x0〉+ϵe∉-int⁡C∀v∈S(x0).
Hence, x0∈Ω1(ϵ).
Second, we show thatX¯=⋂ϵ>0Ω1(ϵ). It is obvious that X¯⊆⋂ϵ>0Ω1(ϵ). Now suppose that ϵn>0 with ϵn→0 and x*∈⋂ϵ>0Ω1(ϵn). Then
(3.23)dX0(x*)≤ϵn,∀n,(3.24)x*∈S(x*),(3.25)∃z∈T(x*),s.t.〈z,x′-x*〉+ϵne∉-int⁡C,∀x′∈S(x*).
From (3.23), we have
(3.26)x*∈X0.
From (3.25), we have
(3.27)〈z,x′-x*〉∉-int⁡C,∀x′∈S(x*),
that isx*∈X¯. Hence, X¯=⋂ϵ>0Ω1(ϵ).
Now we assume that (GVQVI) is type I LP well-posed. By Remark 2.3, we know that the solution X¯ is nonempty and compact. For every positive real number ϵ, since X¯∈Ω1(ϵ), one gets
(3.28)Ω1(ϵ)≠∅,h(Ω1(ϵ),X¯)=max⁡{sup⁡u∈Ω1(ϵ)dX¯(u),sup⁡v∈X¯dΩ1(ϵ)(v)}=sup⁡u∈Ω1(ϵ)dX¯(u).
For everyn∈N, the following relations hold:
(3.29)μ(Ω1(ϵ))≤2h(Ω1(ϵ),X¯)+μ(X¯)=2h(Ω1(ϵ),X¯),
where μ(X¯)=0 since X¯ is compact. Hence, in order to prove that limϵ→0μ(Ω1(ϵ))=0, we only need to prove that
(3.30)lim⁡ϵ→0h(Ω1(ϵ),X¯)=lim⁡ϵ→0sup⁡u∈Ω1(ϵ)dX¯(u)=0.
Suppose that this is not true, then there existβ>0, ϵn→0, and sequence {un}, un∈Ω1(ϵn), such that
(3.31)dX¯(un)>β,
for n sufficiently large.
Since{un} is type I LP approximating sequence for (GVQVI), it contains a subsequence {unk} conversing to a point of X¯, which contradicts (3.31).
For the converse, we know that, for everyϵ>0, the set Ω1(ϵ) is closed, X¯=⋂ϵ>0Ω1(ϵ), and limϵ→0μ(Ω1(ϵ))=0. The theorem on Page. 412 in [40, 41] can be applied, and one concludes that the set X¯ is nonempty, compact, and
(3.32)lim⁡ϵ→0h(Ω1(ϵ),X¯)=0.
If{xn} is type I LP approximating sequence for (GVQVI), then there exists a sequence {ϵn} of positive real numbers decreasing to 0 such that xn∈Ω1(ϵn), for every n∈N. Since X¯ is compact and
(3.33)lim⁡n→+∞dX¯(xn)≤lim⁡n→+∞h(Ω1(ϵn),X¯)=0,
by Proposition 2.4, (GVQVI) is type I LP well-posed.
(b) The proof is Similar to that of (a), and it is omitted here. This completes the proof.Definition 3.4.
(i) LetZ be a topological space, and let Z1⊆Z be nonempty. Suppose that h:Z→R1∪{+∞} is an extended real-valued function. h is said to be level-compact on Z1 if, for any s∈R1, the subset {z∈Z1:h(z)≤s} is compact.
(ii) LetX be a finite-dimensional normed space, and let Z1⊂Z be nonempty. A function h:Z→R1∪{+∞} is said to be level-bounded on Z1 if Z1 is bounded or
(3.34)lim⁡z∈Z1,∥z∥→+∞h(z)=+∞.
Now we establish some sufficient conditions for type I (resp., generalized I type) LP well-posedness of (GVQVI).Proposition 3.5.
Suppose that the solution setX¯ of (GVQVI) is nonempty and set-valued map S is l.s.c. and closed on X1, the set-valued map T is u.s.c. and compact-valued on X1. Suppose that one of the following conditions holds:
(i) there exists0<δ1≤δ0 such that X1(δ1) is compact, where
(3.35)X1(δ1)={x∈X1∩X2:dX0(x)≤δ1};
(ii) the functionf defined by (2.15) is level-compact on X1∩X2;
(iii)X is finite-dimensional and
(3.36)lim⁡x∈X1∩X2,∥x∥→+∞max⁡{f(x),dX0(x)}=+∞,
wheref is defined by (2.15);
(iv) there exists0<δ1≤δ0 such that f is level-compact on X1(δ1) defined by (3.35). Then (GVQVI) is type I LP well-posed.Proof.
First, we show that each of (i), (ii), and (iii) implies (iv). Clearly, either of (i) and (ii) implies (iv). Now we show that (iii) implies (iv). Indeed, we need only to show that, for anyt∈R1, the set
(3.37)A={x∈X1(δ1):f(x)≤t}
is bounded since X is finite-dimensional space and the function f defined by (2.15) is l.s.c. on X1 and thus A is closed. Suppose to the contrary that there exists t∈R1 and {xn′}⊆X1(δ1) such that ∥xn′∥→+∞ and f(xn′)≤t. From {xn′}⊆X1(δ1), we have dX0(xn′)≤δ1.
Thus,(3.38)max⁡{f(xn′),dX0(xn′)}≤max⁡{t,δ1},
which contradicts (3.36).
Therefore, we only need to we show that if (iv) holds, then (GVQVI) is type I LP well-posed. Let {xn} be a type I LP approximating solution sequence for (GVQVI). Then, there exist {ϵn}⊆R+1 with ϵn→0 and zn∈T(xn) such that (2.3), (2.4), and (2.5) hold. From (2.3) and (2.4), we can assume without loss of generality that {xn}⊆X1(δ1). By Lemma 2.9, we can assume without loss of generality that {xn}⊆{x∈X1(δ1):f(x)≤1}. By the level-compactness of f on X1(δ1), we can find a subsequence {xnj} of {xn} and x¯∈X1(δ1) such that xnj→x¯. Taking the limit in (2.3) (with xn replaced by xnj), we have x¯∈X0. Since S is closed and (2.4) holds, we also have x¯∈S(x¯).
Furthermore, from theu.s.c. of T at x¯ and the compactness of T(x¯), we deduce that there exist a subsequence {znj} of {zn} and some z¯∈T(x¯) such that znj→z¯. From this fact, together with (2.5), we have
(3.39)〈z¯,x′-x¯)∉-int⁡C,∀x′∈S(x¯).
Thus, x¯∈X¯.
The next proposition can be proved similarly.Proposition 3.6.
Suppose that the solution setX¯ of (GVQVI) is nonempty and set-valued map S is l.s.c. and closed on X1, the set-valued map T is u.s.c. and compact-valued on X1. Suppose that one of the following conditions holds:
(i) there exists0<δ1≤δ0 such that X2(δ1) is compact, where
(3.40)X2(δ1)={x∈X1∩X2:dK(g(x))≤δ1};
(ii) the functionf defined by (2.15) is level-compact on X1∩X2;
(iii)X is finite-dimension and
(3.41)lim⁡x∈X1∩X2,∥x∥→+∞max⁡{f(x),dKg(x)}=+∞,
wheref is defined by (2.15),
(iv) there exists0<δ1≤δ0 such that f is level-compact on X2(δ1) defined by (3.40). Then (GVQVI) is generalized type II LP well-posed.Remark 3.7.
IfX is finite-dimensional, then the “level-compactness” condition in Propositions 3.1 and 3.6 can be replaced by “level boundedness” condition.Remark 3.8.
It is easy to see that the results in this paper unify, generalize and extend the main results in [26–30] and the references therein.



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*Source: 582792-2012-02-29.xml*