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

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.


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 [2][3][4] 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, [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].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,[10][11][12][13][14]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 [12][13][14]).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,[20][21][22]).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 [20][21][22]).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 [20][21][22].
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.
Remark 10.If we use "Σ" to replace "Δ" 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 that When  is perturbed by the parameter  ∈ Ω, we consider the parametric weak Minty vector variational inequality (PWMVVI) of finding  ∈  such that The solution set of (WMVVI) is denoted as sol(WMVVI), and the solution mapping of (PWMVVI) is denoted by The following result is a direct corollary deduced from [27, Theorem 4.2] and Lemma 1.
We say  is l.s.c (resp., u.s.c) on Λ, iff it is l.s.c (resp., u.s.c) at each  ∈ Λ.  is said to be continuous on Λ iff it is both l.s.c and u.s.c on Λ.
Remark that  is l.s.c at  iff for any sequence {  } ⊂ Λ with   →  and any  ∈ (), there exists a sequence   ∈ (  ) 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 114]).The union Γ = ⋃ ∈ Γ  of a family of l.s.c set-valued mappings Γ  from a topological space  into a topological space  is also an l.s.c set-valued mapping from  into , where  is an index set.
Theorem 16.Suppose that all conditions of Lemma 15 are satisfied and  is a polyhedral convex set.Then (⋅) is continuous on Ω.
Notice that because the strict -pseudomonotonicity of  is imposed, it is easy to verify that, for every  ∈ Δ + and  ∈ Ω,   () is a singleton; namely,   (⋅) is single-valued.
as  1 (, ) =  and  2 (, ) = , respectively.Clearly, all conditions of Theorem 16 are satisfied (cf.Example 14), and hence it derives the continuity of the solution mapping  (in fact, () = {1}, ∀ ∈ Ω).However, Theorem 3.2 of [12] is not applicable, because the strict monotonicity of  2 (⋅, ) is violated.We further give an example to illustrate Theorem 16 when  is set-valued.Based on the union property () = ⋃ ∈Δ +   (), for any  ∈ Ω, () in Theorem 16 need not be a singleton in general, although for each  ∈ Δ + the problem (PVI)  has a unique solution by the strict pseudomonotonicity of .This is because as we change the parameter  the solution of (PVI)  changes as well and all these solutions are in fact solutions of (PVVI).
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.
We give the following trivial example to illustrate Lemma 20, where   is set-valued.
Theorem 23.Suppose that all conditions of Lemma 20 are satisfied and  is a polyhedral convex set.Then (⋅) is continuous on Ω.
It is easy to check that, for every  ∈ Δ + and  ∈ Ω, because of the closedness of  and the continuity of   ,   () is a closed set in R  .On the other hand,   () ⊆ () together with the boundedness of () yields that   () is also a bounded set in R  .Thus, for every  ∈ Δ + and  ∈ Ω,   () is a compact set in R  .
We will now present an error bound for (WVVI) with a strongly -pseudomonotone mapping , but not strongly monotone functions   ( = 1, . . ., ) done in [20][21][22].In our setting we will devise error bounds in terms of the regularized gap function   (𝑥).In what follows by the notation dist(, ) we mean the distance between the point  and the set .
It is clear that ⋂ The proof is complete.
Remark 35.The assumption ( Ã) in Theorem 34 has been used to deal with error bounds of (WVVI); for example, see [21,Theorem 4.3] and [22,Theorem 3.3].In fact, by using the scalar regularized gap function mentioned in [21,22], we can also obtain a similar error bound for (WVVI) under the assumptions in Theorem 34 (see Theorem 38).However, under the strong pseudomonotonicity of   and assumption ( Ã), we see that, for every  = 1, . . ., , sol(VI)  is a singleton and all (VI)  ( = 1, . . ., ) must have the same solution, and thus sol(WVVI) should be a singleton.
Clearly, all conditions of Theorem 34 are satisfied with  1 =  2 = 1 and  = 1, and hence it derives the error bound of (WVVI) 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 (WVVI), which is independent with the scalarization parameter .
For  > 0, we define the function  Charitha et al. [21] and Sun and Chai [22] have explained that   () is finite for every  and thus is well-defined.