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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 582792, 17 pages
http://dx.doi.org/10.1155/2012/582792
Research Article

Well-Posedness of Generalized Vector Quasivariational Inequality Problems

School of Mathematics, Chongqing Normal University, Chongqing 400047, China

Received 28 October 2011; Accepted 14 December 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Jian-Wen Peng and Fang Liu. 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.

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.

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 [311] 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 [3137] 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 Section 2, 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 [2630].

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 𝑋1𝑋, 𝐾𝑍 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, 𝑦1𝑦2 if and only if 𝑦2𝑦1𝐶 for any 𝑦1,𝑦2𝑌. Let 𝐿(𝑋,𝑌) be the space of all the linear continuous operators from 𝑋 to 𝑌. Let 𝑇𝑋12𝐿(𝑋,𝑌) and 𝑆𝑋12𝑋1 be strict set-valued mappings (i.e., 𝑇(𝑥) and 𝑆(𝑥), forall𝑥𝑋1), and let 𝑔𝑋1𝑍 be a continuous vector-valued mapping. We denote by 𝑧, 𝑥 the value 𝑧(𝑥), where 𝑧𝐿(𝑋,𝑌), 𝑥𝑋1. Let 𝑋0={𝑥𝑋1𝑔(𝑥)𝐾} be nonempty. We consider the following generalized vector quasi-variational inequality problem with functional constraints and abstract set constraints.

Find 𝑥𝑋0 such that 𝑥𝑆(𝑥) and there exists 𝑧𝑇(𝑥) satisfying𝑧,𝑥𝑥int𝐶,𝑥𝑆𝑥.(GVQVI) Denote by 𝑋 the solution set of (GVQVI).

Let 𝑍1,𝑍2 be two normed spaces. A set-valued map 𝐹 from 𝑍1 to 2𝑍2 is

(i) closed, on 𝑍3𝑍1, if for any sequence {𝑥𝑛}𝑍3 with 𝑥𝑛𝑥 and 𝑦𝑛𝐹(𝑥𝑛) with 𝑦𝑛𝑦, one has 𝑦𝐹(𝑥);

(ii) lower semicontinuous (l.s.c. in short) at 𝑥𝑍1, if {𝑥𝑛}𝑍1,𝑥𝑛𝑥, and 𝑦𝐹(𝑥) imply that there exists a sequence {𝑦𝑛}𝑍2 satisfying 𝑦𝑛𝑦 such that 𝑦𝑛𝐹(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) at 𝑥𝑍1, if for any neighborhood 𝑉 of 𝐹(𝑥), there exists a neighborhood 𝑈 of 𝑥 such that 𝐹(𝑥)𝑉, forall𝑥𝑈. 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 map 𝐹 is closed.

Let (𝑃,𝑑) be a metric space, 𝑃1𝑃, and 𝑥𝑃. We denote by 𝑑𝑃1(𝑥)=inf{𝑑(𝑥,𝑝)𝑝𝑃1} 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 Φ𝑉, we denote the positive polar cone of Φ byΦ=𝜆𝑉𝜆(𝑥)0,𝑥Φ.(2.1)

Let 𝑒int𝐶 be fixed. Denote𝐶0=𝜆𝐶.𝜆(𝑒)=1(2.2)

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}𝐑1+={𝑟0|𝑟isarealnumber} with 𝜖𝑛0 and 𝑧𝑛𝑇(𝑥𝑛) such that 𝑑𝑋0𝑥𝑛𝜖𝑛,𝑥(2.3)𝑛𝑥𝑆𝑛,(2.4)𝑧𝑛,𝑥𝑥𝑛+𝜖𝑛𝑥𝑒int𝐶,𝑥𝑆𝑛.(2.5)
(ii) {𝑥𝑛}𝑋1 is called a type II LP approximating solution sequence if there exist {𝜖𝑛}𝐑1+ with 𝜖𝑛0 and 𝑧𝑛𝑇(𝑥𝑛) such that (2.3)–(2.5) hold, and, for any 𝑧𝑇(𝑥𝑛), there exists 𝑤(𝑛,𝑧)𝑆(𝑥𝑛) satisfying 𝑧,𝑤(𝑛,𝑧)𝑥𝑛𝜖𝑛𝑒𝐶.(2.6)
(iii) {𝑥𝑛}𝑋1 is called a generalized type I LP approximating solution sequence if there exist {𝜖𝑛}𝐑1+ with 𝜖𝑛0 and 𝑧𝑛𝑇(𝑥𝑛) such that 𝑑𝐾𝑔𝑥𝑛𝜖𝑛(2.7) and (2.4), (2.5) hold.
(iv) {𝑥𝑛}𝑋1 is called a generalized type II LP approximating solution sequence if there exist {𝜖𝑛}𝐑1+ with 𝜖𝑛0, 𝑧𝑛𝑇(𝑥𝑛) such that (2.4), (2.5), and (2.7) hold, and, for any 𝑧𝑇(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 𝑥𝑋 such that 𝑥𝑛𝑗𝑥.

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 that 𝑔 is uniformly continuous functions on a set 𝑋1𝛿0=𝑥𝑋1𝑑𝑋0(𝑥)𝛿0,(2.8) 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) If 𝑌=𝐑1, 𝐶=𝐑1+, 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=𝐑1+, 𝑆(𝑥)=𝑋0 for all 𝑥𝑋1, 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) If 𝑆(𝑥)=𝑋0 for all 𝑥𝑋1, 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 map 𝑇 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: 𝑥𝑋andforanytypeIresp.,typeII,generalizedtypeI,generalizedtypeIILPapproximatingsolutionsequence𝑛,wehave𝑑𝑋𝑥𝑛.0(2.9)

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 Proposition 2.4 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:min𝑓(𝑥)s.t.𝑥𝑋1𝑔(𝑥)𝐾,(P)

where 𝑋1𝑋1 is nonempty and f𝑋1𝐑1{+} is proper. The feasible set of (P) is 𝑋0, where 𝑋0={𝑥𝑋1𝑔(𝑥)𝐾}. The optimal set and optimal value of (P) are denoted by 𝑋 and 𝑣, respectively. Note that if Dom(𝑓)𝑋0, whereDom(𝑓)=𝑥𝑋1𝑓(𝑥)<+,(2.10) 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 limsup𝑛𝑓𝑥𝑛𝑑𝑣,(2.11)𝑋0𝑥𝑛0.(2.12)
(ii) {𝑥𝑛}𝑋1 is called a type II LP minimizing sequence for (P) if lim𝑛𝑓𝑥𝑛=𝑣(2.13) and (2.12) hold.
(iii) {𝑥𝑛}𝑋1 is called a generalized type I LP minimizing sequence for (P) if 𝑑𝐾𝑔𝑥𝑛0(2.14) 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 𝑥𝑋 such that 𝑥𝑛𝑗𝑥.
The Auslender gap function for (GVQVI) is defined as follows: 𝑓(𝑥)=inf𝑧𝑇(𝑥)sup𝑥𝑆(𝑥)inf𝜆𝐶0𝜆𝑧,𝑥𝑥,𝑥𝑋1.(2.15)
Let 𝑋2𝑋 be defined by 𝑋2={𝑥𝑋𝑥𝑆(𝑥)}.(2.16)
In the rest of this paper, we set 𝑋1 in (P) equal to 𝑋1𝑋2. Note that if 𝑆 is closed on 𝑋1, then 𝑋1 is closed.
Recall the following widely used function (see, e.g., [38]) 𝜉𝑌𝐑1min𝑡𝐑1.𝑦𝑡𝑒𝐶(2.17)
It is known that 𝜉 is a continuous, (strictly) monotone (i.e., for any 𝑦1,𝑦2𝑌, 𝑦1𝑦2𝐶 implies that 𝜉(𝑦1)𝜉(𝑦2) and (𝑦1𝑦2int𝐶 implies that 𝜉(𝑦1)>𝜉(𝑦2)), subadditive and convex function. Moreover, it holds that 𝜉(𝑡𝑒)=𝑡,forall𝑡𝐑1 and 𝜉(𝑦)=sup𝜆𝐶0𝜆(𝑦),forall𝑦𝑌.
Now we given some properties for the function 𝑓 defined by (2.15).

Lemma 2.7. Let the function 𝑓 be defined by (2.15), and let the set-valued map 𝑇 be compact-valued on 𝑋1. Then(i)𝑓(𝑥)0,forall𝑥𝑋1;(ii)for any 𝑥𝑋0, 𝑓(𝑥)=0 if and only if 𝑥𝑋.

Proof. (i) Let 𝑥𝑋1. 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 𝑧𝑇(𝑥), such that sup𝑥𝑆(𝑥)inf𝜆𝐶0𝜆𝑧,𝑥𝑥𝛿𝑓(𝑥)+2𝛿<2<0.(2.18) Thus, we have inf𝜆𝐶0𝜆𝑧,𝑥𝑥<0,𝑥𝑆(𝑥),(2.19) which is impossible when 𝑥=𝑥. This proves (i).
(ii) Suppose that 𝑥𝑋0 such that 𝑓(𝑥)=0.
Then, it follows from the definition of 𝑋0 that 𝑥𝑆(𝑥). And from the definition of 𝑓(𝑥) we know that there exist 𝑧𝑛𝑇(𝑥) and 0<𝜖𝑛0 such that inf𝜆𝐶0𝜆𝑧𝑛,𝑥𝑥𝑓(𝑥)+𝜖𝑛=𝜖𝑛,𝑥𝑆(𝑥),(2.20) that is, 𝜉𝑧𝑛,𝑥𝑥𝜖𝑛,𝑥𝑆(𝑥).(2.21) By the compactness of 𝑇(𝑥), there exists a sequence {𝑧𝑛𝑗} of {𝑧𝑛} and some 𝑧𝑇(𝑥) such that 𝑧𝑛𝑗𝑧.(2.22) This fact, together with the continuity of 𝜉 and (2.21), implies that 𝜉𝑧,𝑥𝑥0,𝑥𝑆(𝑥).(2.23) It follows that 𝑥𝑋.
Conversely, assume that 𝑥𝑋. It follows from the definition of 𝑋 that 𝑥𝑆(𝑥). Suppose to the contrary that 𝑓(𝑥)>0. Then, for any 𝑧𝑇(𝑥), sup𝑥𝑆(𝑥)inf𝜆𝐶0𝜆𝑧,𝑥𝑥>0.(2.24)
Thus, there exist 𝛿>0 and 𝑥0𝑆(𝑥) such that inf𝜆𝐶0𝜆𝑧,𝑥𝑥0𝛿.(2.25) It follows that 𝜉𝑧,𝑥0𝑥𝛿<0.(2.26) As a result, we have 𝑧,𝑥0𝑥int𝐶.(2.27) This contradicts the fact that 𝑥𝑋. So, 𝑓(𝑥)=0. This completes the proof.

Lemma 2.8. Let 𝑓 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 that 𝑓(𝑥)>, forall𝑥𝑋1. Suppose to the contrary that there exists 𝑥0𝑋1 such that 𝑓(𝑥0)=. Then, there exist 𝑧𝑛𝑇(𝑥0) and {𝑀𝑛}𝑅1+ with 𝑀𝑛+ such that sup𝑥𝑥𝑆0inf𝜆𝐶0𝜆𝑧𝑛,𝑥0𝑥𝑀𝑛.(2.28) Thus, 𝜉𝑧𝑛,𝑥𝑥0𝑀𝑛,𝑥𝑥𝑆0.(2.29) By the compactness of 𝑇(𝑥0), there exist a sequence {𝑧𝑛𝑗}{𝑧𝑛} and some 𝑧0𝑇(𝑥0) such that 𝑧𝑛𝑗𝑧0. This fact, together with (2.29) and the continuity of 𝜉 on 𝑌, implies that 𝜉𝑧0,𝑥𝑥0+,𝑥𝑥𝑆0(2.30) which is impossible, since 𝜉 is a finite function on 𝑌.
Second, we show that 𝑓 is l.s.c. on 𝑋1. Let 𝑎𝐑1. Suppose that {𝑥𝑛}𝑋1 satisfies 𝑓(𝑥𝑛)𝑎,forall𝑛, and 𝑥𝑛𝑥0𝑋1. It follows that, for each 𝑛, there exist 𝑧𝑛𝑇(𝑥𝑛) and 0<𝛿𝑛0 such that 𝜉𝑧𝑛,𝑦𝑥𝑛𝑎+𝛿𝑛𝑥,𝑦𝑆𝑛.(2.31)
For any 𝑥𝑆(𝑥0), by the l.s.c. of 𝑆, we have a sequence {𝑦𝑛} with {𝑦𝑛}𝑆(𝑥𝑛) converging to 𝑥 such that 𝜉𝑧𝑛,𝑦𝑛𝑥𝑛𝑎+𝛿𝑛.(2.32)
By the u.s.c. of 𝑇 at 𝑥0 and the compactness of 𝑇(𝑥0), we obtain a subsequence {𝑧𝑛𝑗} of {𝑧𝑛} and some 𝑧0𝑇(𝑥0) such that 𝑧𝑛𝑗𝑧0. Taking the limit in (2.32) (with 𝑛 replaced by 𝑛𝑗), by the continuity of 𝜉, we have 𝑧𝜉0,𝑥𝑥0𝑎,𝑥𝑥𝑆0.(2.33)
It follows that 𝑓(𝑥0)=inf𝑧𝑇(𝑥0)sup𝑥𝑆(𝑥0)𝜉(𝑧,𝑥𝑥0)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 function 𝑓 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}𝐑1+ with 𝜖𝑛0 and 𝑧𝑛𝑇(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}𝐑1+ with 𝜖𝑛0 and 𝑧𝑛𝑇(𝑥𝑛) satisfying (2.4) and (2.5), and for any 𝑧𝑇(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}𝐑1+ with 𝜖𝑛0 and 𝑧𝑛𝑇(𝑥𝑛) satisfying (2.4) and (2.5), then we can easily verify that 𝑥𝑛𝑋1𝑥,𝑓𝑛𝜖𝑛.(2.34) 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}𝐑1+with𝜖𝑛0 such that 𝑓(𝑥𝑛)𝜖𝑛. By the compactness of 𝑇(𝑥𝑛), we see that for every 𝑛 there exists 𝑧𝑛𝑇(𝑥𝑛) such that 𝜉𝑧𝑛,𝑥𝑥𝑛𝜖𝑛,𝑥𝑥𝑆𝑛.(2.35)
It follows that for every 𝑛 there exists 𝑧𝑛𝑇(𝑥𝑛) such that (2.5) holds.
(ii) Let {𝑥𝑛}𝑋1 be any sequence we can verify that liminf𝑛𝑓𝑥𝑛0(2.36)
holds if and only if there exists {𝛼𝑛}𝐑1+ with 𝛼𝑛0 and, for any 𝑧𝑇(𝑥𝑛), there exists 𝑤(𝑛,𝑧)𝑆(𝑥𝑛) such that 𝑧,𝑤(𝑛,𝑧)𝑥𝑛𝛼𝑛𝑒𝐶.(2.37)
From the proof of (i), we know that limsup𝑛𝑓(𝑥𝑛)0 and {𝑥𝑛}𝑋1 hold if and only if {𝑥𝑛}𝑋1 such that there exist {𝛽𝑛}𝐑1+ with 𝛽𝑛0𝑧𝑛𝑇(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 that 𝑋 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. Let 𝑓(𝑥) be defined by (2.15). Since 𝑋, it follows from Lemma 2.7 that 𝑥𝑋 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 Lemma 2.9, 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)Since 𝑋0𝑋0,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 function 𝑐=𝑐(𝑡,𝑠,𝑟) defined for 𝑡,𝑠,𝑟0 sufficiently small, such that𝑠𝑐(𝑡,𝑠,𝑟)0,𝑡,𝑠,𝑟,𝑐(0,0,0)=0,𝑛0,𝑡𝑛0,𝑟𝑛𝑡=0,𝑐𝑛,𝑠𝑛,𝑟𝑛0implythat𝑡𝑛0,(3.1)

Theorem 3.1. Let the set-valued map 𝑇 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 ||𝑓||𝑑(𝑥)𝑐𝑋(𝑥),𝑑𝑋0(𝑥),𝑑𝑆(𝑥)(𝑥),𝑥𝑋1,(3.2) 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 𝑐||𝑓||(𝑡,𝑠,𝑟)=inf(𝑥)𝑥𝑋1,𝑑𝑋(𝑥)=𝑡,𝑑𝑋0(𝑥)=𝑠,𝑑𝑆(𝑥).(𝑥)=𝑟(3.3) Since 𝑋, it is obvious that 𝑐(0,0,0)=0. Moreover, if 𝑠𝑛0,𝑡𝑛0,𝑟𝑛=0, and 𝑐(𝑡𝑛,𝑠𝑛,𝑟𝑛)0, then there exists a sequence {𝑥𝑛}𝑋1 with 𝑑𝑋(𝑥𝑛)=𝑡𝑛, 𝑑𝑆(𝑥𝑛)(𝑥𝑛)=𝑟𝑛=0, 𝑑𝑋0𝑥𝑛=𝑠𝑛0,(3.4) such that ||𝑓𝑥𝑛||0.(3.5)
Since 𝑆 is closed-valued, 𝑥𝑛𝑆(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 𝑡𝑛0.
Conversely, let {𝑥𝑛} be a type II LP approximating solution sequence of (GVQVI). Then, by (3.2), we have ||𝑓𝑥𝑛||𝑑𝑐𝑋𝑥𝑛,𝑑𝑋0𝑥𝑛,𝑑𝑆(𝑥𝑛)𝑥𝑛.(3.6)
Let 𝑡𝑛=𝑑𝑋𝑥𝑛,𝑠𝑛=𝑑𝑋0𝑥𝑛,𝑟𝑛=𝑑𝑆(𝑥𝑛)𝑥𝑛.(3.7)
Then 𝑠𝑛0 and 𝑟𝑛=0,forall𝑛𝑁. 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 𝑡𝑛0. By Proposition 2.4, we see that (GVQVI) is type II LP well-posed.

Theorem 3.2. Let the set-valued map 𝑇 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 ||𝑓||𝑑(𝑥)𝑐𝑋(𝑥),𝑑𝐾(𝑔(𝑥)),𝑑𝑆(𝑥)(𝑥),𝑥𝑋1,(3.8) 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 Theorem 3.1. The only difference lies in the proof of the first part of Theorem 3.1. Here we define 𝑐||𝑓||(𝑡,𝑠,𝑟)=inf(𝑥)𝑥𝑋1,𝑑𝑋(𝑥)=𝑡,𝑑𝐾(𝑔(𝑥))=𝑠,𝑑𝑆(𝑥).(𝑥)=𝑟(3.9)
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 𝐻𝜇(𝐻)=inf𝜖>0𝐻𝑖𝐻,diam𝑖<𝜖,𝑖=1,,𝑛,(3.10)
where diam(𝐻𝑖) is the diameter of 𝐻𝑖 defined by 𝐻diam𝑖=sup𝑥1𝑥2𝑥1,𝑥2𝐻𝑖.(3.11)
Given two nonempty subsets A and B of a Banach space (𝑋,), the Hausdorff distance between A and B is defined by 𝑑(𝐴,𝐵)=maxsup𝐵𝑑(𝑎)𝑎𝐴,sup𝐴(𝑏)𝑏𝐵.(3.12)
For any 𝜖0, two types of approximating solution sets for (GVQVI) are defined, respectively, by Ω1(𝜖)=𝑥𝑋1𝑥𝑆(𝑥),𝑑𝑋0(𝑥)𝜖,𝑧𝑇(𝑥),s.t.𝑧,𝑥𝑥+𝜖𝑒int𝐶,𝑥,Ω𝑆(𝑥)2(𝜖)=𝑥𝑋1𝑥𝑆(𝑥),𝑑𝐾(𝑔(𝑥))𝜖,𝑧𝑇(𝑥),s.t.𝑧,𝑥𝑥+𝜖𝑒int𝐶,𝑥.𝑆(𝑥)(3.13)

Theorem 3.3. Assume that 𝑇 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 lim𝜖0𝜇Ω1(𝜖)=0,(3.14)
(b) (GVQVI) is generalized type I LP well-posed if and only if lim𝜖0𝜇Ω2(𝜖)=0.(3.15)

Proof. (a) First we show that, for every 𝜖>0, Ω1(𝜖) is closed. In fact, let 𝑥𝑛Ω1(𝜖) and 𝑥𝑛𝑥0. Then (2.4) and the following formula hold: 𝑑𝑋0𝑥𝑛𝜖,𝑧𝑛𝑥𝑇𝑛𝑧,s.t.𝑛,𝑥𝑥𝑛+𝜖𝑒int𝐶,𝑥𝑥𝑆𝑛.(3.16) Since 𝑥𝑛𝑥0, by the closedness of 𝑆 and (2.4), we have 𝑥0𝑆(𝑥0). From (3.16), we get 𝑑𝑋0𝑥0𝜖,(3.17)𝑧𝑛𝑥𝑇𝑛𝑧,s.t.𝜉𝑛,𝑥𝑥𝑛𝜖,𝑥𝑥𝑆𝑛.(3.18) For any 𝑣𝑆(𝑥0), by the lower semi-continuity of 𝑆 and (3.18), we can find 𝑣𝑛𝑆(𝑥𝑛) with 𝑣𝑛𝑣 such that 𝜉𝑧𝑛,𝑣𝑛𝑥𝑛𝜖.(3.19)
By the u.s.c. of 𝑇 at 𝑥0 and the compactness of 𝑇(𝑥0), there exist a subsequence {𝑧𝑛𝑗}{𝑧𝑛} and some 𝑧0𝑇(𝑥0) such that 𝑧𝑛𝑗𝑧0.(3.20)
This fact, together with the continuity of 𝜉 and (3.19), implies that 𝜉𝑧0,𝑣𝑥0𝑥𝜖𝑣𝑆0.(3.21)
It follows that 𝑧0,𝑣𝑥0𝑥+𝜖𝑒int𝐶𝑣𝑆0.(3.22) Hence, 𝑥0Ω1(𝜖).
Second, we show that 𝑋=𝜖>0Ω1(𝜖). It is obvious that 𝑋𝜖>0Ω1(𝜖). Now suppose that 𝜖𝑛>0 with 𝜖𝑛0 and 𝑥𝜖>0Ω1(𝜖𝑛). Then 𝑑𝑋0𝑥𝜖𝑛𝑥,𝑛,(3.23)𝑥𝑆,𝑥(3.24)𝑧𝑇,s.t.𝑧,𝑥𝑥+𝜖𝑛𝑒int𝐶,𝑥𝑥𝑆.(3.25) From (3.23), we have 𝑥𝑋0.(3.26) From (3.25), we have 𝑧,𝑥𝑥int𝐶,𝑥𝑥𝑆,(3.27)
that is 𝑥𝑋. 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 𝑋Ω1(𝜖), one gets Ω1Ω(𝜖),1(𝜖),𝑋=maxsup𝑢Ω1(𝜖)𝑑𝑋(𝑢),sup𝑣𝑋𝑑Ω1(𝜖)(𝑣)=sup𝑢Ω1(𝜖)𝑑𝑋(𝑢).(3.28)
For every 𝑛𝑁, the following relations hold: 𝜇Ω1Ω(𝜖)21(𝜖),𝑋+𝜇𝑋Ω=21(𝜖),𝑋,(3.29) where 𝜇(𝑋)=0 since 𝑋 is compact. Hence, in order to prove that lim𝜖0𝜇(Ω1(𝜖))=0, we only need to prove that lim𝜖0Ω1(𝜖),𝑋=lim𝜖0sup𝑢Ω1(𝜖)𝑑𝑋(𝑢)=0.(3.30)
Suppose that this is not true, then there exist 𝛽>0, 𝜖𝑛0, and sequence {𝑢𝑛}, 𝑢𝑛Ω1(𝜖𝑛), such that 𝑑𝑋𝑢𝑛>𝛽,(3.31) 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 every 𝜖>0, the set Ω1(𝜖) is closed, 𝑋=𝜖>0Ω1(𝜖), and lim𝜖0𝜇(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 lim𝜖0Ω1(𝜖),𝑋=0.(3.32)
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 𝑥𝑛Ω1(𝜖𝑛), for every 𝑛𝑁. Since 𝑋 is compact and lim𝑛+𝑑𝑋𝑥𝑛lim𝑛+Ω1𝜖𝑛,𝑋=0,(3.33) 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) Let 𝑍 be a topological space, and let 𝑍1𝑍 be nonempty. Suppose that 𝑍𝑅1{+} is an extended real-valued function. is said to be level-compact on 𝑍1 if, for any 𝑠𝐑1, the subset {𝑧𝑍1(𝑧)𝑠} is compact.
(ii) Let 𝑋 be a finite-dimensional normed space, and let 𝑍1𝑍 be nonempty. A function 𝑍𝐑1{+} is said to be level-bounded on 𝑍1 if 𝑍1 is bounded or lim𝑧𝑍1,𝑧+(𝑧)=+.(3.34)
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 set 𝑋 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 exists 0<𝛿1𝛿0 such that 𝑋1(𝛿1) is compact, where 𝑋1𝛿1=𝑥𝑋1𝑋2𝑑𝑋0(𝑥)𝛿1;(3.35)
(ii) the function 𝑓 defined by (2.15) is level-compact on 𝑋1𝑋2;
(iii) 𝑋 is finite-dimensional and lim𝑥𝑋1𝑋2,𝑥+𝑓max(𝑥),𝑑𝑋0(𝑥)=+,(3.36)
where 𝑓 is defined by (2.15);
(iv) there exists 0<𝛿1𝛿0 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 any 𝑡𝐑1, the set 𝐴=𝑥𝑋1𝛿1𝑓(𝑥)𝑡(3.37) 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 𝑡𝐑1 and {𝑥𝑛}𝑋1(𝛿1) such that 𝑥𝑛+ and 𝑓(𝑥𝑛)𝑡. From {𝑥𝑛}𝑋1(𝛿1), we have 𝑑𝑋0(𝑥𝑛)𝛿1.
Thus, 𝑓𝑥max𝑛,𝑑𝑋0𝑥𝑛max𝑡,𝛿1,(3.38) 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}𝐑1+ with 𝜖𝑛0 and 𝑧𝑛𝑇(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}{𝑥𝑋1(𝛿1)𝑓(𝑥)1}. By the level-compactness of 𝑓 on 𝑋1(𝛿1), we can find a subsequence {𝑥𝑛𝑗} of {𝑥𝑛} and 𝑥𝑋1(𝛿1) such that 𝑥𝑛𝑗𝑥. Taking the limit in (2.3) (with 𝑥𝑛 replaced by 𝑥𝑛𝑗), we have 𝑥𝑋0. Since 𝑆 is closed and (2.4) holds, we also have 𝑥𝑆(𝑥).
Furthermore, from the u.s.c. of 𝑇 at 𝑥 and the compactness of 𝑇(𝑥), we deduce that there exist a subsequence {𝑧𝑛𝑗} of {𝑧𝑛} and some 𝑧𝑇(𝑥) such that 𝑧𝑛𝑗𝑧. From this fact, together with (2.5), we have 𝑧,𝑥𝑥int𝐶,𝑥𝑆𝑥.(3.39) Thus, 𝑥𝑋.
The next proposition can be proved similarly.

Proposition 3.6. Suppose that the solution set 𝑋 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 exists 0<𝛿1𝛿0 such that 𝑋2(𝛿1) is compact, where 𝑋2𝛿1=𝑥𝑋1𝑋2𝑑𝐾(𝑔(𝑥))𝛿1;(3.40)
(ii) the function 𝑓 defined by (2.15) is level-compact on 𝑋1𝑋2;
(iii) 𝑋 is finite-dimension and lim𝑥𝑋1X2,𝑥+𝑓max(𝑥),𝑑𝐾𝑔(𝑥)=+,(3.41)
where 𝑓 is defined by (2.15),
(iv) there exists 0<𝛿1𝛿0 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. If 𝑋 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 [2630] and the references therein.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 11171363 and Grant no. 10831009), the Natural Science Foundation of Chongqing (Grant No. CSTC, 2009BB8240)  and the special fund of Chongqing Key Laboratory (CSTC 2011KLORSE01).

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