Journal of Applied MathematicsVolume 2012 (2012), Article ID 582792, 17 pagesdoi:10.1155/2012/582792

Research Article

Well-Posedness of Generalized Vector Quasivariational Inequality Problems

Jian-Wen Peng and Fang Liu

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 [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 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 [26–30].

2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations and assumptions.

Let be a normed space equipped with norm topology, and let be a metric space. Let , be nonempty and closed sets. Let be a locally convex space ordered by a nontrivial closed and convex cone with nonempty interior , that is, if and only if for any . Let be the space of all the linear continuous operators from to . Let and be strict set-valued mappings (i.e., and , ), and let be a continuous vector-valued mapping. We denote by , the value , where , . Let be nonempty. We consider the following generalized vector quasi-variational inequality problem with functional constraints and abstract set constraints.

Find such that and there exists satisfying Denote by the solution set of (GVQVI).

Let be two normed spaces. A set-valued map from to is

(i) closed, on , if for any sequence with and with , one has ;

(ii) lower semicontinuous (l.s.c. in short) at , if , and imply that there exists a sequence satisfying such that for sufficiently large. If is l.s.c. at each point of , we say that is l.s.c. on ;

(iii) upper semicontinuous (u.s.c. in short) at , if for any neighborhood of , there exists a neighborhood of such that , . If is u.s.c. at each point of , we say that is u.s.c. on .

It is obvious that any u.s.c. nonempty closed-valued map is closed.

Let be a metric space, , and . We denote by the distance from the point to the set . For a topological vector space , we denote by its dual space. For any set , we denote the positive polar cone of by

Let be fixed. Denote

Definition 2.1. (i) A sequence is called a type I Levitin-Polyak (LP in short) approximating solution sequence if there exist with and such that (ii) is called a type II LP approximating solution sequence if there exist with and such that (2.3)–(2.5) hold, and, for any , there exists satisfying (iii) is called a generalized type I LP approximating solution sequence if there exist with and such that and (2.4), (2.5) hold.(iv) is called a generalized type II LP approximating solution sequence if there exist with , such that (2.4), (2.5), and (2.7) hold, and, for any , there exists 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 of (GVQVI) is nonempty, and, for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence , there exists a subsequence of and such that .

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 is compact.(iii) Suppose that is uniformly continuous functions on a set for some . 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 , , 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 , , for all , 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 for all , 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 is replaced by a single-valued map , 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:

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 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:

where is nonempty and is proper. The feasible set of (P) is , where . The optimal set and optimal value of (P) are denoted by and , respectively. Note that if , where then . In this paper, we always assume that .

Definition 2.5. (i) A sequence is called a type I LP minimizing sequence for (P) if (ii) is called a type II LP minimizing sequence for (P) if and (2.12) hold.(iii) is called a generalized type I LP minimizing sequence for (P) if and (2.11) hold.(iv) 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 of (P) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP minimizing sequence , there exists a subsequence of and such that .The Auslender gap function for (GVQVI) is defined as follows: Let be defined by In the rest of this paper, we set in (P) equal to . Note that if is closed on , then is closed.Recall the following widely used function (see, e.g., [38]) It is known that is a continuous, (strictly) monotone (i.e., for any , implies that and ( implies that ), subadditive and convex function. Moreover, it holds that and .Now we given some properties for the function defined by (2.15).

Lemma 2.7. Let the function be defined by (2.15), and let the set-valued map be compact-valued on . Then (i);(ii)for any , if and only if .

Proof. (i) Let . Suppose to the contrary that . Then, there exists a such that . By definition, for , there exists a , such that Thus, we have which is impossible when . This proves (i). (ii) Suppose that such that .Then, it follows from the definition of that . And from the definition of we know that there exist and such that that is, By the compactness of , there exists a sequence of and some such that This fact, together with the continuity of and (2.21), implies that It follows that .Conversely, assume that . It follows from the definition of that . Suppose to the contrary that . Then, for any , Thus, there exist and such that It follows that As a result, we have This contradicts the fact that . So, . This completes the proof.

Lemma 2.8. Let be defined by (2.15). Assume that the set-valued map is compact-valued and u.s.c. on and the set-valued map is l.s.c. on . Then is l.s.c. function from to . Further assume that the solution set of (GVQVI) is nonempty, then .

Proof. First we show that , . Suppose to the contrary that there exists such that . Then, there exist and with such that Thus, By the compactness of , there exist a sequence and some such that . This fact, together with (2.29) and the continuity of on , implies that which is impossible, since is a finite function on . Second, we show that is l.s.c. on . Let . Suppose that satisfies , and . It follows that, for each , there exist and such that For any , by the l.s.c. of , we have a sequence with converging to such that By the u.s.c. of at and the compactness of , we obtain a subsequence of and some such that . Taking the limit in (2.32) (with replaced by ), by the continuity of , we have It follows that . Hence, is l.s.c. on . Furthermore, if , by Lemma 2.7, we see that .

Lemma 2.9. Let the function be defined by (2.15), and let the set-valued map be compact-valued on . Then, (i) is a sequence such that there exist with and satisfying (2.4) and (2.5) if and only if and (2.11) hold with ,(ii) is a sequence such that there exist with and satisfying (2.4) and (2.5), and for any , there exists satisfying (2.6) if and only if and (2.13) hold with .

Proof. (i) Let be any sequence if there exist with and satisfying (2.4) and (2.5), then we can easily verify that It follows that (2.11) holds with .For the converse, let and (2.11) hold with . We can see that and (2.4) hold. Furthermore, by (2.11), we have that there exists such that . By the compactness of , we see that for every there exists such that It follows that for every there exists such that (2.5) holds.(ii) Let be any sequence we can verify that holds if and only if there exists with and, for any , there exists such that From the proof of (i), we know that and hold if and only if such that there exist with satisfying (2.4) and (2.5) (with replaced by ). Finally, we let and the conclusion follows.

Proposition 2.10. Assume that and is compact-valued on . 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 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 defined by (2.15).

Proof. Let be defined by (2.15). Since , it follows from Lemma 2.7 that is a solution of (GVQVI) if and only if is an optimal solution of (5) with .(i)Similar to the proof of Lemma 2.9, it is also routine to check that a sequence 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 defined by (2.15).(ii)Since for any . 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 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 defined for sufficiently small, such that

Theorem 3.1. Let the set-valued map be compact-valued on . If (GVQVI) is type II LP well-posed, the set-valued map is closed-valued, then there exist a function satisfying (3.1) such that where is defined by (2.15). Conversely, suppose that is nonempty and compact and (3.2) holds for some satisfying (3.1). Then (GVQVI) is type II LP well-posed.

Proof. Define Since , it is obvious that . Moreover, if , and , then there exists a sequence with , , such that Since is closed-valued, for any . This fact, combined with (3.4) and (3.5) and Lemma 2.9 (ii) implies that is a type II LP approximating solution sequence of (GVQVI). By Proposition 2.4, we have that .Conversely, let be a type II LP approximating solution sequence of (GVQVI). Then, by (3.2), we have Let Then and . Moreover, by Lemma 2.9, we have that . Then, . These facts together with the properties of the function imply that . By Proposition 2.4, we see that (GVQVI) is type II LP well-posed.

Theorem 3.2. Let the set-valued map be compact-valued on . If (GVQVI) is generalized type II LP well-posed, the set-valued map is closed, then there exist a function satisfying (3.1) such that where is defined by (2.15). Conversely, suppose that is nonempty and compact and (3.8) holds for some 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 Next we give the Furi-Vignoli-type characterizations [39] for the (generalized) type I LP well-posedness of (GVQVI).Let be a Banach space. Recall that the Kuratowski measure of noncompactness for a subset of is defined as where is the diameter of defined by Given two nonempty subsets A and B of a Banach space , the Hausdorff distance between A and B is defined by For any , two types of approximating solution sets for (GVQVI) are defined, respectively, by

Theorem 3.3. Assume that is u.s.c. and compact-valued on and is l.s.c. and closed on . Then (a) (GVQVI) is type I LP well-posed if and only if (b) (GVQVI) is generalized type I LP well-posed if and only if

Proof. (a) First we show that, for every , is closed. In fact, let and . Then (2.4) and the following formula hold: Since , by the closedness of and (2.4), we have . From (3.16), we get For any , by the lower semi-continuity of and (3.18), we can find with such that By the u.s.c. of at and the compactness of , there exist a subsequence and some such that This fact, together with the continuity of and (3.19), implies that It follows that Hence, .Second, we show that . It is obvious that . Now suppose that with and . Then From (3.23), we have From (3.25), we have that is . Hence, .Now we assume that (GVQVI) is type I LP well-posed. By Remark 2.3, we know that the solution is nonempty and compact. For every positive real number , since , one gets For every , the following relations hold: where since is compact. Hence, in order to prove that , we only need to prove that Suppose that this is not true, then there exist , , and sequence , , such that for sufficiently large.Since is type I LP approximating sequence for (GVQVI), it contains a subsequence conversing to a point of , which contradicts (3.31).For the converse, we know that, for every , the set is closed, , and . The theorem on Page. 412 in [40, 41] can be applied, and one concludes that the set is nonempty, compact, and If is type I LP approximating sequence for (GVQVI), then there exists a sequence of positive real numbers decreasing to 0 such that , for every . Since is compact and 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 be a topological space, and let be nonempty. Suppose that is an extended real-valued function. is said to be level-compact on if, for any , the subset is compact.(ii) Let be a finite-dimensional normed space, and let be nonempty. A function is said to be level-bounded on if is bounded or 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 of (GVQVI) is nonempty and set-valued map is l.s.c. and closed on , the set-valued map T is u.s.c. and compact-valued on . Suppose that one of the following conditions holds: (i) there exists such that is compact, where (ii) the function defined by (2.15) is level-compact on ; (iii) is finite-dimensional and where is defined by (2.15); (iv) there exists such that is level-compact on 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 , the set is bounded since is finite-dimensional space and the function defined by (2.15) is l.s.c. on and thus is closed. Suppose to the contrary that there exists and such that and . From , we have .Thus, which contradicts (3.36).Therefore, we only need to we show that if (iv) holds, then (GVQVI) is type I LP well-posed. Let be a type I LP approximating solution sequence for (GVQVI). Then, there exist with and such that (2.3), (2.4), and (2.5) hold. From (2.3) and (2.4), we can assume without loss of generality that . By Lemma 2.9, we can assume without loss of generality that . By the level-compactness of on , we can find a subsequence of and such that . Taking the limit in (2.3) (with replaced by ), we have . Since is closed and (2.4) holds, we also have .Furthermore, from the . of at and the compactness of , we deduce that there exist a subsequence of and some such that . From this fact, together with (2.5), we have Thus, .The next proposition can be proved similarly.

Proposition 3.6. Suppose that the solution set of (GVQVI) is nonempty and set-valued map is l.s.c. and closed on , the set-valued map T is u.s.c. and compact-valued on . Suppose that one of the following conditions holds: (i) there exists such that is compact, where (ii) the function defined by (2.15) is level-compact on ; (iii) is finite-dimension and where is defined by (2.15), (iv) there exists such that is level-compact on defined by (3.40). Then (GVQVI) is generalized type II LP well-posed.

Remark 3.7. If 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.

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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