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Journal of Function Spaces
Volume 2016 (2016), Article ID 7297854, 11 pages
http://dx.doi.org/10.1155/2016/7297854
Research Article

Continuity Results and Error Bounds on Pseudomonotone Vector Variational Inequalities via Scalarization

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received 31 August 2015; Accepted 11 January 2016

Academic Editor: Giuseppe Marino

Copyright © 2016 Xin Zuo et al. 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

Continuity (both lower and upper semicontinuities) results of the Pareto/efficient solution mapping for a parametric vector variational inequality with a polyhedral constraint set are established via scalarization approaches, within the framework of strict pseudomonotonicity assumptions. As a direct application, the continuity of the solution mapping to a parametric weak Minty vector variational inequality is also discussed. Furthermore, error bounds for the weak vector variational inequality in terms of two known regularized gap functions are also obtained, under strong pseudomonotonicity assumptions.

1. Introduction

The concept of the vector variational inequality (VVI, for short) was first introduced by Giannessi in his well-known paper [1]. This model has received extensive attentions in the last three decades. Many important results on various kinds of vector variational inequalities (VVIs, for short) have been established; for example, see [24] and the references therein.

Nowadays, VVIs as powerful tools appear in many important problems from theory to applications, such as multiobjective/vector optimization, network economics, and financial equilibrium. In such situation it is very important to understand behaviors of solutions of a VVI when the problem’s data vary. In other words, we need to know properties of solutions of parametric VVIs when the parameters vary. Therefore, one of the main topics is to investigate stability of the solution mappings for parametric VVIs and vector equilibrium problems (VEPs, for short). Usually, solution stability investigations were devoted to upper and lower semicontinuities, Lipschitz/Hölder continuity, and error bounds; see, for example, [522]. Our interest in this paper is to further discuss the continuity (both upper and lower semicontinuities) of solution mappings for parametric VVIs and error bounds for weak VVIs in terms of the known regularized gap functions.

In the available literature on the subject of solution semicontinuity for parametric VVIs and VEPs, there are two phenomena that could be observed. On the one hand, among many approaches dealing with the lower semicontinuity and continuity of solution mappings for parametric VVIs and VEPs, the scalarization method is of considerable interest and effective (see [7, 8, 1014, 19]). On the other hand, most of the semicontinuity results were devoted to the weak Pareto/efficient solutions of parametric VVIs and VEPs, while there have been only few investigations on the Pareto/efficient solutions of parametric VVIs and VEPs (see [1214]). Obviously, the latter is more difficult, as the ordering relations involved are neither closed nor open. Based on the above observations, we would study the continuity (both lower and upper semicontinuities) of Pareto/efficient solution mappings for parametric VVIs via scalarization.

It is well known that the monotonicity of mappings plays a vital role in the study of VVIs and VEPs, such as solution existence and stability analysis. In particular, we notice that almost all scalarization methods dealing with the lower semicontinuity of parametric VVIs and VEPs share a common feature: sufficient conditions are guaranteed under strict monotonicity assumptions or some variants (see [7, 8, 11, 12, 14]). Recently, Wang and Huang [10] have discussed the lower semicontinuity of the weak Pareto/efficient solutions to a parametric vector mixed variational inequality under a kind of strict pseudomonotonicity assumptions. To the best of our knowledge, there was nearly no lower semicontinuity result for parametric VVIs and VEPs with strict pseudomonotone mappings via scalarization in the literature. Therefore, we will further study the continuity (both lower and upper semicontinuities) of the Pareto/efficient solution mapping for a parametric VVI with a polyhedral constraint set discussed in our previous work [12], within the framework of strict pseudomonotonicity assumptions. The technique of proofs is adopted by scalarization, based on the useful properties proposed by Lee and Yen [23] and Lee et al. [24]. The results obtained relax strict monotonicity assumptions used in [12] to strict pseudomonotonicity ones. As a direct application, the continuity of the solution mapping to a parametric weak Minty VVI is also discussed.

Additionally, as we know, error bounds for VVIs and VEPs have played important roles in stability analysis. Using error bounds, one can obtain an upper estimate of the distance between an arbitrary feasible point and the solution set of VVIs or VEPs. Gap functions have turned out to be very useful in deriving the error bounds (cf. [18, 2022]). About error bounds for VVIs and VEPs, there are also two phenomena that should be noticed. On the one hand, most of the error bound results were devoted to scalar variational inequalities, while there still have been only few investigations for VVIs and VEPs. On the other hand, nearly all error bound results for VVIs and VEPs are obtained under strong monotonicity assumptions (see [2022]). Whence, we would further deduce error bounds for weak VVIs in terms of the known regularized gap functions. Our models are discussed within the framework of strong pseudomonotonicity assumptions, which are properly weaker than strong monotonicity ones used in most papers. Thus, the conclusions obtained improve main results of [2022].

The rest of the paper is organized as follows. In Section 2, we introduce the weak vector variational inequality (WVVI), the parametric VVI problem (PVVI), and the parametric weak Minty VVI problem (PWMVVI) and recall some necessary concepts and properties. In particular, the concepts of -pseudomonotonicity, strict -pseudomonotonicity, and strong -pseudomonotonicity are presented. In Section 3, we discuss sufficient conditions that guarantee the continuity of solution mappings for (PVVI) and for (PWMVVI) by using scalarization approaches, under strict pseudomonotonicity assumptions. In Section 4, we deduce error bounds for (WVVI) in terms of regularized gap functions and , under strong pseudomonotonicity assumptions. The last section gives some concluding remarks.

2. Preliminaries

Let be a nonempty, closed, and convex set. Let () be vector-valued functions. For abbreviation we write and for every and . The scalar product and the Euclidean norm in an Euclidean space are denoted by and , respectively. For a set in an Euclidean space, and denote the interior and the closure of , respectively. is the nonnegative orthant of . Let and be nonempty subsets of Euclidean spaces, and set and .

Consider the vector variational inequality (VVI) (resp., the weak vector variational inequality (WVVI)), which consists in finding such thatThe solution sets of (VVI) and (WVVI) are denoted by sol(VVI) and sol(WVVI), respectively. The elements of the first set (resp., the second set) are called the Pareto/efficient solutions (resp., the weak Pareto/efficient solutions) of (VVI).

When the mapping is perturbed by the parameter , we can consider the following parametric vector variational inequality (PVVI) (resp., parametric weak vector variational inequality (PWVVI)) of finding such thatwhere () are vector-valued functions.

For each , we denote the solution mappings of (PVVI) and (PWVVI) by and , respectively; that is,

For every , we consider the variational inequality of finding such thatwhere , with the corresponding parametric variational inequality of finding such thatwhere .

Denote the solution set of by and the solution mapping of by : that is,

For , we denote the variational inequality associated with as , that is, to find such thatThe solution set of () is denoted by .

Lemma 1 (see [19, 23, 24]). It holds thatAnd, is a closed set provided that is a continuous mapping. If is a polyhedral convex set, that is, is the intersection of finitely many closed half-spaces of , then the first inclusion in the above formula holds as equality.

Remark 2. Let be the unit simplex in . The relative interior of is described by the formula . If we replace “” by “” in Lemma 1, then the corresponding result still holds (cf. [19, Theorem ]).

Lemma 3 (see [24, 25]). Suppose that there exist such thatand such thatThen the solution sets and are nonempty bounded and compact, respectively, andMoreover, for every , the variational inequality has a unique solution in .

Recalling from [26], we say that the function is pseudomonotone on iffIt is called strictly pseudomonotone on iffIt is called strongly pseudomonotone on iff there exists a constant such that

Definition 4. The mapping is said to be -pseudomonotone (resp., strictly -pseudomonotone and strongly -pseudomonotone) on iff , is pseudomonotone (resp., strictly pseudomonotone and strongly pseudomonotone) on .

Clearly, the strong -pseudomonotonicity implies the strict -pseudomonotonicity, which, in turn, implies the -pseudomonotonicity. Definition 4 is motivated by [10, Definition ] and [25, Definitions and ]. Similar to [25], the vector variational inequality (VVI) is said to be pseudomonotone (resp., strictly pseudomonotone and strongly pseudomonotone) iff is -pseudomonotone (resp., strictly -pseudomonotone, strongly -pseudomonotone) on . Next, we give an example to illustrate Definition 4.

Example 5. Let . Define as and for every , respectively. For every and ,Thus, we have that ,Hence, is strongly -monotone on with modulus . Clearly, a strongly -monotone mapping is strongly -pseudomonotone. So, is strongly -pseudomonotone on , of course, it is strictly -pseudomonotone and -pseudomonotone.

Remark 6. If is -pseudomonotone (resp., strictly -pseudomonotone and strongly -pseudomonotone) on , then it is obvious that, for every , is pseudomonotone (resp., strictly pseudomonotone and strongly pseudomonotone) on . However, the following examples show the converses are not true.

Example 7. Let . Define as and , respectively. Clearly, are pseudomonotone on . However, we can show that is not -pseudomonotone on . For , , . Take and , in . Then , but , which implies that is not -pseudomonotone on .

Example 8. Let . Define as and , respectively. Clearly, are strictly pseudomonotone on . However, we can show that is not strictly -pseudomonotone on . For , , . Take and distinct points in . Then , but , which implies that is not strictly -pseudomonotone on .

Example 9. Let . Define as and , respectively. It is easy to see that are strongly pseudomonotone on with constants , respectively. Taking , for example, , if , we have . Noting that for all , then, we deduce that . For , , . Take and distinct points in . Then , but for any , which implies that is not strongly -pseudomonotone on .

Remark 10. If we use “” to replace “” in Definition 4, then analogous concepts can be introduced, and similar discussions hold as above.

Associated with (WVVI), we consider the following weak Minty vector variational inequality (WMVVI) of finding such thatWhen is perturbed by the parameter , we consider the parametric weak Minty vector variational inequality (PWMVVI) of finding such thatThe solution set of (WMVVI) is denoted as , and the solution mapping of (PWMVVI) is denoted by : that is,

The following result is a direct corollary deduced from [27, Theorem ] and Lemma 1.

Lemma 11. If each is continuous and pseudomonotone, then

In what follows, the notation denotes the open ball with center and radius .

Definition 12 (see [28]). Let be a set-valued mapping and let be given. (i)is called lower semicontinuous (l.s.c) at iff for any open set satisfying , there exists such that, for every , .(ii) is called upper semicontinuous (u.s.c) at iff for any open set satisfying , there exists such that, for every , .We say is l.s.c (resp., u.s.c) on , iff it is l.s.c (resp., u.s.c) at each . is said to be continuous on iff it is both l.s.c and u.s.c on .

Remark that is l.s.c at iff for any sequence with and any , there exists a sequence such that .

If has compact values (i.e., is a compact set for each ), then is u.s.c at iff for any sequences with and with , there exist and a subsequence of , such that .

The following lemma plays an important role in the proof of the lower semicontinuity of the solution mappings and .

Lemma 13 (see [28, page ]). The union of a family of l.s.c set-valued mappings from a topological space into a topological space is also an l.s.c set-valued mapping from into , where is an index set.

3. Continuity Results

Throughout this section, we make the following assumption (A): , is nonempty and bounded; : that is, , .

For example, based on Lemma 3, assumption (A) is fulfilled if there exist constants and such that ,Assumption (A) is also fulfilled if is a compact convex set and for any , , , are continuous on (see [24, Theorem ]).

Now we study the lower semicontinuity of with a strictly -pseudomonotone mapping , but not strictly monotone functions , . The latter was considered in our previous work [12].

If , , are strictly monotone on , that is, , , then it is clear that , is strictly -monotone on ; that is,Obviously, it implies that is strictly -pseudomonotone on . However, the following example shows that the converse is not true. That is, the strict -pseudomonotonicity of is properly weaker than the strict monotonicity of ().

Example 14. Let . Define as and , respectively. For , , . We show that is strictly -pseudomonotone on . For any and ,, suppose that . As and , we have . Thus, , because . Hence, is strictly -pseudomonotone on . However, are not both strictly monotone on . It is clear that is not strictly monotone on , since , , . Moreover, is also not strictly -monotone on . In fact, taking and in , we get .

Lemma 15. Let . Suppose that assumption holds and the following conditions are satisfied: (i), , are continuous on , where is a nonempty, closed, and convex set.(ii)For any , is strictly -pseudomonotone on : that is,Then, is l.s.c on .

Proof. Suppose to the contrary that there exists such that is not l.s.c at . Then there exist with and , such that, for any , .
Since and is a closed set, there exists such that . Fix any . From Lemma 1, . Hence, the sequence is bounded by the boundedness of . Without loss of generality, we can assume that there is a such that . As the set is closed, . It follows from and thatMoreover, since and , we get . By the continuity of , taking limit on above inequality, we get thatAssume that . Then by the strict -pseudomonotonicity of and (24), we have , which contradicts (25). Therefore, . This is impossible by the wrong assumption. The proof is complete.

Theorem 16. Suppose that all conditions of Lemma 15 are satisfied and is a polyhedral convex set. Then is continuous on .

Proof. “l.s.c”: Since is a polyhedral convex set, by virtue of Lemma 1, for each ,It follows from Lemma 15 that, for each , is l.s.c on . Thus, in view of Lemma 13, we immediately obtain that is l.s.c on .
“u.s.c”: We prove that is u.s.c on . Suppose that there exists some such that is not u.s.c at . Then there exist an open set satisfying and sequences and , such that , .
Notice that because the strict -pseudomonotonicity of is imposed, it is easy to verify that, for every and , is a singleton; namely, is single-valued.
By Lemma 1, ; thus, there exists such that . Let . Since is single-valued, so it is continuous at by Lemma 15; thus, . Note that . It follows from and the openness of that , which yields a contradiction. Thus, we have proved the upper semicontinuity of .

Remark 17. Theorem 16 improves [12, Theorem ], by weakening the strict monotonicity of () to the strict -pseudomonotonicity of .

Example 18. Let . Define as and , respectively. Clearly, all conditions of Theorem 16 are satisfied (cf. Example 14), and hence it derives the continuity of the solution mapping (in fact, , ). However, Theorem of [12] is not applicable, because the strict monotonicity of is violated.

We further give an example to illustrate Theorem 16 when is set-valued. Based on the union property , for any , in Theorem 16 need not be a singleton in general, although for each the problem has a unique solution by the strict -pseudomonotonicity of . This is because as we change the parameter the solution of changes as well and all these solutions are in fact solutions of (PVVI).

Example 19. Let and . Define as and , respectively. For , , . It is easy to check that , , which implies that condition (ii) of Lemma 15 holds. Thus, all conditions of Theorem 16 are satisfied. Direct computations show that , ,Clearly, and is continuous on .

In Lemma 15, the strict -pseudomonotonicity condition is strict that the solution set is confined to be a singleton. In this paper, like done in our previous work [12], we introduce the following assumption (ii) of Lemma 20 to weaken this condition. In the case, the solution set may be a general set but not a singleton; that is, the solution mapping is set-valued in general.

Lemma 20. Let . Suppose that assumption holds and the following conditions are satisfied: (i), , are continuous on , where is a nonempty, closed, and convex set.(ii)There exists a constant such that for each and , there exists satisfying .Then, is l.s.c on .

Proof. Suppose to the contrary that there exists such that is not l.s.c at . Then there exist with and , such that, for any , .
Since and is a closed set, there exists such that . Whence, it is clear that . Thus, by assumption (ii), there exists such thatSimilarly as in the proof of Lemma 15, the sequence is bounded; thus, without loss of generality, we can assume that . As the set is closed, . Taking the limit as in above inequality, we haveNow we claim that . Otherwise, by (29), we obtain that , which contradicts (24), because and . Hence, . However, it is impossible by the wrong assumption. The proof is complete.

Remark 21. Condition (ii) of Lemma 20 is a modification of the strong pseudomonotonicity of , which may be called the partially strong pseudomonotonicity of , because our assumption is not imposed on all . In addition, if “, ” is replaced by “”, then the conclusion of Lemma 20 still holds. Whence, condition (ii) of Lemma 20 is also from a modification of the strong -pseudomonotonicity of . We notice that this kind of monotonicity has been used to deal with solution stability (e.g., Hölder continuity, error bound) of vector variational inequalities and vector equilibrium problems; see, for example, [17, Theorem ] and [18, Theorem ].

We give the following trivial example to illustrate Lemma 20, where is set-valued.

Example 22. Let and . Define as and for every , respectively. For , and , . By a direct calculation, we obtain that , , . Thus, the partially strong pseudomonotonicity of () holds trivially, as . Whence, all conditions of Lemma 20 are satisfied, and hence it derives the lower semicontinuity of the set-valued solution mapping .

Theorem 23. Suppose that all conditions of Lemma 20 are satisfied and is a polyhedral convex set. Then is continuous on .

Proof. “l.s.c”: By virtue of Lemmas 1, 13, and 20, the lower semicontinuity is valid.
“u.s.c”: We prove that is u.s.c on . Suppose that there exists some such that is not u.s.c at . Then there exist an open set satisfying and sequences and , such that , .
It is easy to check that, for every and , because of the closedness of and the continuity of , is a closed set in . On the other hand, together with the boundedness of yields that is also a bounded set in . Thus, for every and , is a compact set in .
By Lemma 1, ; thus, there exists such that . It follows from [12, Lemma ] that is u.s.c at with compact values. Whence, for and , there exist and a subsequence of such that . Note that . It follows from and the openness of that , which yields a contradiction. Thus, we have proved the upper semicontinuity of .

Remark 24. Theorem 23 modifies [12, Theorem ], by changing the partially strong monotonicity of () to the partially strong pseudomonotonicity of .

Example 25. Consider Example 18. To illustrate that Theorem 23 is valid, we only need to verify condition (ii) of Lemma 20. It is clear that , . Choose . For any and , taking , we get thatThat is, condition (ii) of Lemma 20 holds. Hence, all conditions of Theorem 23 hold and it is valid. However, Theorem of [12] is not applicable, because the partially strong monotonicity of is violated. In fact, for every and , taking any , we have .

In the sequel, we will show an application to the parametric weak Minty vector variational inequality (PWMVVI). Similarly, we make the following assumption : , is nonempty and bounded; ; that is, , .

Theorem 26. Suppose that assumption holds and the following conditions are satisfied: (i), , are continuous on , where is a nonempty, closed, and convex set.(ii)For any , is strictly -pseudomonotone on ; that is, , Then, is continuous on .

Proof. “l.s.c”: Since for any , is strictly -pseudomonotone on , it is clear that , , are strictly pseudomonotone on , thus pseudomonotone on . By virtue of Lemma 11, for each ,Similar to the proof of Lemma 15 we know, for each , is l.s.c on . Thus, in view of Lemma 13, we immediately obtain that is l.s.c on .
“u.s.c”: The proof is similar to that of Theorem 16.

The following result can be deduced by the similar proof of Theorem 23.

Theorem 27. Suppose that assumption holds and the following conditions are satisfied: (i), , are continuous on , where is a nonempty, closed, and convex set, and , , are pseudomonotone on for any .(ii)For any , there exists a constant such that, for each and , there exists satisfying .Then, is continuous on .

4. Gap Functions and Error Bounds

Throughout this section, assume that for all . The existence, for instance, can be guaranteed by the compactness and convexity of , and the continuity of (e.g., [24, Theorem ]).

We will now introduce a regularized gap function for (WVVI). This gap function was studied by Charitha and Dutta [20].

For , we define the function asFor fixed and consider the following problem:which is equivalently written as

Lemma 28 (see [20, Lemma ]). For any and any , let and be defined by (33) and (34), respectively. Then, is continuous on and is well-defined.

Note that is finite without the assumption that is compact (cf. [20, Remark ]).

Lemma 29 (see [20, Theorem ]). Let be defined by (33). Then for all . Furthermore, , if and only if solves . That is, is a gap function for .

We will now present an error bound for with a strongly -pseudomonotone mapping , but not strongly monotone functions done in [2022]. In our setting we will devise error bounds in terms of the regularized gap function . In what follows by the notation we mean the distance between the point and the set .

If , are strongly monotone with on , that is, , and set , then it is clear that , is strongly -monotone on with : that is,Obviously, it implies that is strongly -pseudomonotone on . However, the following example shows the converse is not true. That is, the strong -pseudomonotonicity of is properly weaker than the strong monotonicity of ().

Example 30. Let . Define as and , respectively. For , . We show that is strongly -pseudomonotone on . For any and , suppose that . As , we have . Thus, , because ; then , and . Hence, is strongly -pseudomonotone on with the modulus of strong pseudomonotonicity . However, are not both strongly monotone on . It is clear that is not strongly monotone on , since , . Moreover, is also not strongly -monotone on . In fact, taking and in , we get .

Theorem 31. Let . Suppose that is strongly -pseudomonotone on with the modulus of strong pseudomonotonicity , and let be chosen so that . Then for any we have

Proof. We can write the function in the way: . From Lemma 28 we know that is continuous on , so the function is continuous on . Noting that is compact, hence there exists ( will depend on the chosen ) such that . Whence, using the definition of we have, for all ,Since , letting , further from Remark 2 and Lemma 1 we know that also solves . We set in (38):Since solves , we have . Then by the strong -pseudomonotonicity of with , we getThus, combining with (39), we obtain . We have noted that ; hence , which impliesThe proof is complete.

Remark 32. (a) Based on the union property , in Theorem 31 need not be a singleton in general, although for each the scalar variational inequality admits a unique solution by the strong -pseudomonotonicity of . This is because as we change the parameter the solution set changes as well and all these solutions are in fact solutions of .
(b) Theorem 31 improves [20, Theorem ], since we use the strong -pseudomonotonicity of but not strong monotonicity of ().

We give the following example to illustrate Theorem 31.

Example 33. Let . Define as and , respectively. For , , . By virtue of Example 30, is strongly -pseudomonotone on with , and we let . Moreover, . Thus, all conditions of Theorem 31 are satisfied. Direct computations show that the regularized gap function for has the following representation:Hence, . Clearly, , . However, Theorem of [20] is not applicable, because the strong monotonicity of is violated.

By Remarks 10 and 6 we know the strong pseudomonotonicity of is properly weaker than the strong -pseudomonotonicity of . Next, we make some discussions on the error bounds of when are strongly pseudomonotone on .

It is clear that , and we assume that for all , but it is possible that . Now we make the following assumption: .

Theorem 34. Let . Suppose that assumption holds and are strongly pseudomonotone with the modulus of strong pseudomonotonicity on . Moreover, let and let be chosen so that . Then for any we have

Proof. Similar to the proof of Theorem 31, there exists such that (38) holds.
Since assumption holds, we let . Obviously . As solves , we have . Then by the strong pseudomonotonicity of , we get . Because , we obtainThus, combining with (38) by setting , we obtainNoting that , we have , which impliesThe proof is complete.

Remark 35. The assumption in Theorem 34 has been used to deal with error bounds of ; for example, see [21, Theorem ] and [22, Theorem ]. In fact, by using the scalar regularized gap function mentioned in [21, 22], we can also obtain a similar error bound for under the assumptions in Theorem 34 (see Theorem 38).
However, under the strong pseudomonotonicity of and assumption , we see that, for every , is a singleton and all () must have the same solution, and thus should be a singleton.

Example 36. Consider Example 33. Direct computations show thatClearly, all conditions of Theorem 34 are satisfied with and , and hence it derives the error bound of in terms of .

Besides the scalar regularized gap function mentioned above, Charitha et al. [21] and Sun and Chai [22] also have constructed another scalar regularized gap function for , which is independent with the scalarization parameter .

For , we define the function as

Charitha et al. [21] and Sun and Chai [22] have explained that is finite for every and thus is well-defined.

Lemma 37 (see [21, Theorem ] and [22, Corollary ]). The function is a gap function for .

Now we use the regularized gap function to develop an error bound for under assumption and strongly pseudomonotone component functions .

Theorem 38. Suppose that assumption holds and are strongly pseudomonotone with the modulus of strong pseudomonotonicity on . Moreover, let and be chosen so that . Then for any we have

Proof. From our notations and the presentation of