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

Authors: Xin Zuo; Hong-Zhi Wei; Chun-Rong Chen
Journal: Journal of Function Spaces
 (2016)

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## 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.


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## Body

## 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 [2–4] 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, [5–22]. 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, 10–14, 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 [12–14]). 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, 20–22]). 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 [20–22]). 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 [20–22].The rest of the paper is organized as follows. In Section2, 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 ξ-pseudomonotonicity, strict ξ-pseudomonotonicity, and strong ξ-pseudomonotonicity are presented. In Section 3, we discuss sufficient conditions that guarantee the continuity of solution mappings S(·) for (PVVI) and SMw(·) 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 ϕα and gα, under strong pseudomonotonicity assumptions. The last section gives some concluding remarks.


## 2. Preliminaries

LetK⊂Rn be a nonempty, closed, and convex set. Let Fi:Rn→Rn (i=1,…,p) be vector-valued functions. For abbreviation we write F≔(F1,…,Fp) and F(x)(v)≔(〈F1(x),v〉,…,〈Fp(x),v〉) for every x∈K and v∈Rn. The scalar product and the Euclidean norm in an Euclidean space are denoted by 〈·,·〉 and ·, respectively. For a set A in an Euclidean space, int⁡A and cl A denote the interior and the closure of A, respectively. C≔R+p is the nonnegative orthant of Rp. Let Λ and Ω be nonempty subsets of Euclidean spaces, and set Δ≔{ξ=(ξ1,…,ξp)∈R+p:ξ=1} and Δ+≔{ξ∈int⁡R+p:ξ=1}.Consider the vector variational inequality (VVI) (resp., the weak vector variational inequality (WVVI)), which consists in findingx¯∈K such that(1)Fx¯x-x¯∉-C∖0Rpresp.,Fx¯x-x¯∉-int⁡C,∀x∈K.The 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 mappingF is perturbed by the parameter μ∈Ω, we can consider the following parametric vector variational inequality (PVVI) (resp., parametric weak vector variational inequality (PWVVI)) of finding x¯∈K such that(2)Fx¯,μx-x¯∉-C∖0Rpresp.,Fx¯,μx-x¯∉-int⁡C,∀x∈K,where Fi:K×Ω→Rn (i=1,…,p) are vector-valued functions.For eachμ∈Ω, we denote the solution mappings of (PVVI) and (PWVVI) by S(μ) and Sw(μ), respectively; that is,(3)Sμ≔x¯∈K:Fx¯,μx-x¯∉-C∖0Rp,∀x∈K,Swμ≔x¯∈K:Fx¯,μx-x¯∉-int⁡C,∀x∈K.For everyξ∈C∖{0Rp}, we consider the variational inequality (VI)ξ of finding x¯∈K such that(4)fξx¯,x-x¯≥0,∀x∈K,where fξ(x)≔∑i=1p‍ξiFi(x), with the corresponding parametric variational inequality (PVI)ξ of finding x¯∈K such that(5)fξx¯,μ,x-x¯≥0,∀x∈K,where fξ(x,μ)≔∑i=1p‍ξiFi(x,μ).Denote the solution set of(VI)ξ by sol(VI)ξ and the solution mapping of (PVI)ξ by Sξ(μ): that is,(6)Sξμ≔x¯∈K:fξx¯,μ,x-x¯≥0,∀x∈K.Fori=1,…,p, we denote the variational inequality associated with Fi as (VI)i, that is, to find x¯∈K such that(7)Fix¯,x-x¯≥0,∀x∈K.The solution set of (VI)i (i=1,…,p) is denoted by sol(VI)i.Lemma 1 (see [19, 23, 24]).
It holds that(8)⋃ξ∈Δ+solVIξ⊆solVVI⊆solWVVI=⋃ξ∈ΔsolVIξ.And, sol(WVVI) is a closed set provided that F is a continuous mapping. If K is a polyhedral convex set, that is, K is the intersection of finitely many closed half-spaces of Rn, then the first inclusion in the above formula holds as equality.Remark 2.
LetΣ≔{ξ=(ξ1,…,ξp)∈R+p:∑i=1p‍ξi=1} be the unit simplex in C. The relative interior of Σ is described by the formula Σ+≔{ξ∈int⁡R+p:∑i=1p‍ξi=1}. If we replace “Δ” by “Σ” in Lemma 1, then the corresponding result still holds (cf. [19, Theorem 2.1]).Lemma 3 (see [24, 25]).
Suppose that there existα>0 such that(9)Fix′-Fix,x′-x≥αx′-x2,∀x,x′∈K,∀i∈1,…,p,and l>0 such that(10)Fix′-Fix≤lx′-x,∀x,x′∈K,∀i∈1,…,p.Then the solution sets sol(VVI) and sol(WVVI) are nonempty bounded and compact, respectively, and(11)⋃ξ∈Δ+solVIξ⊆solVVI⊆solWVVI=⋃ξ∈ΔsolVIξ=cl⋃ξ∈Δ+solVIξ.Moreover, for every ξ∈Δ, the variational inequality (VI)ξ has a unique solution in K.Recalling from [26], we say that the function H:Rn→Rn is pseudomonotone on K⊂Rn iff(12)Hx,y-x≥0⟹Hy,y-x≥0,∀x,y∈K.It is called strictly pseudomonotone on K iff(13)Hx,y-x≥0⟹Hy,y-x>0,∀x,y∈K,x≠y.It is called strongly pseudomonotone on K iff there exists a constant c>0 such that(14)Hx,y-x≥0⟹Hy,y-x≥cx-y2,∀x,y∈K.Definition 4.
The mappingF≔(F1,…,Fp) is said to be ξ-pseudomonotone (resp., strictly ξ-pseudomonotone and strongly ξ-pseudomonotone) on K iff ∀ξ=(ξ1,…,ξp)∈Δ, fξ is pseudomonotone (resp., strictly pseudomonotone and strongly pseudomonotone) on K.Clearly, the strongξ-pseudomonotonicity implies the strict ξ-pseudomonotonicity, which, in turn, implies the ξ-pseudomonotonicity. Definition 4 is motivated by [10, Definition 2.3] and [25, Definitions 1 and 2]. Similar to [25], the vector variational inequality (VVI) is said to be pseudomonotone (resp., strictly pseudomonotone and strongly pseudomonotone) iff F is ξ-pseudomonotone (resp., strictly ξ-pseudomonotone, strongly ξ-pseudomonotone) on K. Next, we give an example to illustrate Definition 4.Example 5.
LetK=R2. Define F1,F2:K→R2 as F1(x)=(x1-1,x2) and F2(x)=1/2x1,x2+1 for every x=(x1,x2)∈R2, respectively. For every ξ=(ξ1,ξ2)∈Δ and x∈K,(15)fξx=∑i=12ξiFix=ξ1+ξ22x1-ξ1,ξ1+ξ2x2+ξ2.Thus, we have that ∀x=(x1,x2),x′=x1′,x2′∈K,(16)fξx′-fξx,x′-x=ξ1+ξ22x1′-x12+ξ1+ξ2x2′-x22≥ξ1+ξ22x1′-x12+x2′-x22=ξ1+ξ22x′-x2≥ξ12+ξ222x′-x2=12x′-x2.Hence, F≔(F1,F2) is strongly ξ-monotone on K with modulus c=1/2. Clearly, a strongly ξ-monotone mapping is strongly ξ-pseudomonotone. So, F is strongly ξ-pseudomonotone on K, of course, it is strictly ξ-pseudomonotone and ξ-pseudomonotone.Remark 6.
IfF is ξ-pseudomonotone (resp., strictly ξ-pseudomonotone and strongly ξ-pseudomonotone) on K, then it is obvious that, for every i=1,…,p, Fi is pseudomonotone (resp., strictly pseudomonotone and strongly pseudomonotone) on K. However, the following examples show the converses are not true.Example 7.
LetK=[1,+∞[. Define F1,F2:K→R as F1(x)=-x and F2(x)=1, respectively. Clearly, F1,F2 are pseudomonotone on K. However, we can show that F≔(F1,F2) is not ξ-pseudomonotone on K. For ξ=(ξ1,ξ2)∈Δ, fξ(x)=∑i=12‍ξiFi(x)=-ξ1x+ξ2, ∀x∈K. Take ξ=1/2,3/2∈Δ and x1=1, x2=2 in K. Then 〈fξ(x1),x2-x1〉=-1/2+3/2·1=(3-1)/2>0, but 〈fξ(x2),x2-x1〉=-1+3/2·1=(3-2)/2<0, which implies that F is not ξ-pseudomonotone on K.Example 8.
LetK=[-1,1]. Define F1,F2:K→R as F1(x)=|x|+1 and F2(x)=-1, respectively. Clearly, F1,F2 are strictly pseudomonotone on K. However, we can show that F≔(F1,F2) is not strictly ξ-pseudomonotone on K. For ξ=(ξ1,ξ2)∈Δ, fξ(x)=ξ1|x|+ξ1-ξ2, ∀x∈K. Take ξ=2/2,2/2∈Δ and distinct points x1=-1,x2=0 in K. Then 〈fξ(x1),x2-x1〉=2/2>0, but 〈fξ(x2),x2-x1〉=0, which implies that F is not strictly ξ-pseudomonotone on K.Example 9.
LetK=[1,2]. Define F1,F2:K→R as F1(x)=x and F2(x)=-x, respectively. It is easy to see that F1,F2 are strongly pseudomonotone on K with constants 1, respectively. Taking F1, for example, ∀x,y∈K, if 〈F1(x),y-x〉=x(y-x)≥0, we have y-x≥0. Noting that |x-y|≤1 for all x,y∈K, then, we deduce that 〈F1(y),y-x〉=y(y-x)≥y-x=|x-y|≥|x-y|2. For ξ=(ξ1,ξ2)∈Δ, fξ(x)=(ξ1-ξ2)x, ∀x∈K. Take ξ=2/2,2/2∈Δ and distinct points x1,x2 in K. Then 〈fξ(x1),x2-x1〉=0, but 〈fξ(x2),x2-x1〉=0<c|x1-x2|2 for any c>0, which implies that F is not strongly ξ-pseudomonotone on K.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 findingx¯∈K such that(17)Fxx¯-x∉int⁡C,∀x∈K.When F is perturbed by the parameter μ∈Ω, we consider the parametric weak Minty vector variational inequality (PWMVVI) of finding x¯∈K such that(18)Fx,μx¯-x∉int⁡C,∀x∈K.The solution set of (WMVVI) is denoted as sol(WMVVI), and the solution mapping of (PWMVVI) is denoted by SMw(μ): that is,(19)SMwμ≔x¯∈K:Fx,μx¯-x∉int⁡C,∀x∈K.The following result is a direct corollary deduced from [27, Theorem 4.2] and Lemma 1.Lemma 11.
If eachFi:K→Rn(i=1,…,p) is continuous and pseudomonotone, then(20)solWMVVI=solWVVI=⋃ξ∈ΔsolVIξ.In what follows, the notationB(λ¯,δ) denotes the open ball with center λ¯∈Λ and radius δ>0.Definition 12 (see [28]).
LetG:Λ⇉Rn be a set-valued mapping and let λ¯∈Λ be given. (i)
Gis called lower semicontinuous (l.s.c) at λ¯ iff for any open set V satisfying V∩G(λ¯)≠∅, there exists δ>0 such that, for every λ∈B(λ¯,δ), V∩G(λ)≠∅.(ii)
G is called upper semicontinuous (u.s.c) at λ¯ iff for any open set V satisfying G(λ¯)⊆V, there exists δ>0 such that, for every λ∈B(λ¯,δ), G(λ)⊆V.We say G is l.s.c (resp., u.s.c) on Λ, iff it is l.s.c (resp., u.s.c) at each λ∈Λ. G is said to be continuous on Λ iff it is both l.s.c and u.s.c on Λ.Remark thatG is l.s.c at λ¯ iff for any sequence {λn}⊂Λ with λn→λ¯ and any x¯∈G(λ¯), there exists a sequence xn∈G(λn) such that xn→x¯.IfG has compact values (i.e., G(λ) is a compact set for each λ∈Λ), then G is u.s.c at λ¯ iff for any sequences {λn}⊂Λ with λn→λ¯ and {xn} with xn∈G(λn), there exist x¯∈G(λ¯) and a subsequence {xnk} of {xn}, such that xnk→x¯.The following lemma plays an important role in the proof of the lower semicontinuity of the solution mappingsS(·) and SMw(·).Lemma 13 (see [28, page 114]).
The unionΓ=⋃i∈I‍Γi of a family of l.s.c set-valued mappings Γi from a topological space X into a topological space Y is also an l.s.c set-valued mapping from X into Y, where I is an index set.


## 3. Continuity Results

Throughout this section, we make the following assumption (A):∀μ∈Ω, S(μ) is nonempty and bounded; {ξ∈Δ+:Sξ(μ)≠∅}=Δ+: that is, ∀ξ∈Δ+, Sξ(μ)≠∅.For example, based on Lemma3, assumption (A) is fulfilled if there exist constants α>0 and l>0 such that ∀μ,μ′∈Ω,(21)Fix′,μ-Fix,μ,x′-x≥αx′-x2,∀x,x′∈K,∀i∈1,…,p,Fix′,μ′-Fix,μ≤lx′-x+μ′-μ,∀x,x′∈K,∀i∈1,…,p.Assumption (A) is also fulfilled if K is a compact convex set and for any μ∈Ω, Fi(·,μ), i=1,…,p, are continuous on K (see [24, Theorem 2.2]).Now we study the lower semicontinuity ofS(·) with a strictly ξ-pseudomonotone mapping F, but not strictly monotone functions Fi, i=1,…,p. The latter was considered in our previous work [12].IfFi:K→Rn, i=1,…,p, are strictly monotone on K, that is, 〈Fix′-Fix,x′-x〉>0, ∀x,x′∈K,x≠x′,i=1,…,p, then it is clear that ∀ξ=(ξ1,…,ξp)∈Δ, F≔(F1,…,Fp) is strictly ξ-monotone on K; that is,(22)fξx′-fξx,x′-x=∑i=1pξiFix′-Fix,x′-x>0,∀x,x′∈K,x≠x′.Obviously, it implies that F is strictly ξ-pseudomonotone on K. However, the following example shows that the converse is not true. That is, the strict ξ-pseudomonotonicity of F is properly weaker than the strict monotonicity of Fi (i∈{1,…,p}).Example 14.
LetK=[1,+∞[. Define F1,F2:K→R as F1(x)=x and F2(x)=1, respectively. For ξ=(ξ1,ξ2)∈Δ, fξ(x)=∑i=12‍ξiFi(x)=ξ1x+ξ2, ∀x∈K. We show that F≔(F1,F2) is strictly ξ-pseudomonotone on K. For any ξ=(ξ1,ξ2)∈Δ and x1,x2∈K,x1≠x2, suppose that 〈fξ(x1),x2-x1〉=(ξ1x1+ξ2)(x2-x1)≥0. As ξ1x1+ξ2>0 and x1≠x2, we have x2-x1>0. Thus, 〈fξ(x2),x2-x1〉=(ξ1x2+ξ2)(x2-x1)>0, because ξ1x2+ξ2>0. Hence, F is strictly ξ-pseudomonotone on K. However, F1,F2 are not both strictly monotone on K. It is clear that F2 is not strictly monotone on K, since 〈F2(x2)-F2(x1),x2-x1〉=0, ∀x1,x2∈K, x1≠x2. Moreover, F is also not strictly ξ-monotone on K. In fact, taking ξ=(0,1)∈Δ and x1≠x2 in K, we get 〈fξ(x2)-fξ(x1),x2-x1〉=ξ1(x2-x1)2=0.Lemma 15.
Letξ∈Δ+. Suppose that assumption (A) holds and the following conditions are satisfied: (i)
Fi(·,·), i=1,…,p, are continuous on K×Ω, where K⊂Rn is a nonempty, closed, and convex set.(ii)
For anyμ∈Ω, F(·,μ) is strictly ξ-pseudomonotone on K: that is,(23)fξx,μ,x′-x≥0⟹fξx′,μ,x′-x>0,∀x,x′∈K,x≠x′.Then, Sξ(·) is l.s.c on Ω.Proof.
Suppose to the contrary that there existsμ0∈Ω such that Sξ(·) is not l.s.c at μ0. Then there exist {μn} with μn→μ0 and x0∈Sξ(μ0), such that, for any xn∈Sξ(μn), xn↛x0.
Sincex0∈K and K⊂Rn is a closed set, there exists x¯n∈K such that x¯n→x0. Fix any yn∈Sξ(μn). From Lemma 1, Sξ(μn)⊆S(μn). Hence, the sequence {yn}⊂K is bounded by the boundedness of S(μn). Without loss of generality, we can assume that there is a y0∈Rn such that yn→y0. As the set K is closed, y0∈K. It follows from x0∈Sξ(μ0) and y0∈K that(24)fξx0,μ0,y0-x0≥0.Moreover, since yn∈Sξ(μn) and x¯n∈K, we get 〈fξ(yn,μn),x¯n-yn〉≥0. By the continuity of Fi(i=1,…,p), taking limit on above inequality, we get that(25)fξy0,μ0,x0-y0≥0.
Assume thaty0≠x0. Then by the strict ξ-pseudomonotonicity of F and (24), we have 〈fξ(y0,μ0),x0-y0〉<0, which contradicts (25). Therefore, y0=x0. This is impossible by the wrong assumption. The proof is complete.Theorem 16.
Suppose that all conditions of Lemma15 are satisfied and K is a polyhedral convex set. Then S(·) is continuous on Ω.Proof.
“l.s.c”: SinceK is a polyhedral convex set, by virtue of Lemma 1, for each μ∈Ω,(26)Sμ=⋃ξ∈Δ+Sξμ.It follows from Lemma 15 that, for each ξ∈Δ+, Sξ(·) is l.s.c on Ω. Thus, in view of Lemma 13, we immediately obtain that S(·) is l.s.c on Ω.
“u.s.c”: We prove thatS(·) is u.s.c on Ω. Suppose that there exists some μ0∈Ω such that S(·) is not u.s.c at μ0. Then there exist an open set V satisfying S(μ0)⊆V and sequences μn→μ0 and xn∈S(μn), such that xn∉V, ∀n.
Notice that because the strictξ-pseudomonotonicity of F is imposed, it is easy to verify that, for every ξ∈Δ+ and μ∈Ω, Sξ(μ) is a singleton; namely, Sξ(·) is single-valued.
By Lemma1, xn∈S(μn)=⋃ξ∈Δ+‍Sξ(μn); thus, there exists ξ′∈Δ+ such that {xn}=Sξ′(μn). Let {x0}=Sξ′(μ0). Since Sξ′(·) is single-valued, so it is continuous at μ0 by Lemma 15; thus, xn→x0. Note that x0∈⋃ξ∈Δ+‍Sξ(μ0)=S(μ0)⊆V. It follows from xn∉V and the openness of V that x0∉V, which yields a contradiction. Thus, we have proved the upper semicontinuity of S(·).Remark 17.
Theorem16 improves [12, Theorem 3.2], by weakening the strict monotonicity of Fi (i∈{1,…,p}) to the strict ξ-pseudomonotonicity of F.Example 18.
LetK=Ω=[1,2]. Define F1,F2:K×Ω→R as F1(x,μ)=μx and F2(x,μ)=μ, respectively. Clearly, all conditions of Theorem 16 are satisfied (cf. Example 14), and hence it derives the continuity of the solution mapping S (in fact, S(μ)={1}, ∀μ∈Ω). However, Theorem 3.2 of [12] is not applicable, because the strict monotonicity of F2(·,μ) is violated.We further give an example to illustrate Theorem16 when S is set-valued. Based on the union property S(μ)=⋃ξ∈Δ+‍Sξ(μ), for any μ∈Ω, S(μ) in Theorem 16 need not be a singleton in general, although for each ξ∈Δ+ the problem (PVI)ξ has a unique solution by the strict ξ-pseudomonotonicity of F. This is because as we change the parameter ξ the solution of (PVI)ξ changes as well and all these solutions are in fact solutions of (PVVI).Example 19.
LetK=[0,1] and Ω=[1,2]. Define F1,F2:K×Ω→R as F1(x,μ)=μx and F2(x,μ)=-μ, respectively. For ξ=(ξ1,ξ2)∈Δ+, fξ(x,μ)=∑i=12‍ξiFi(x,μ)=(ξ1x-ξ2)μ, ∀x∈K,μ∈Ω. It is easy to check that ∀ξ∈Δ+,x,x′∈K,x≠x′, 〈fξ(x′,μ)-fξ(x,μ),x′-x〉=ξ1μ(x′-x)2>0, which implies that condition (ii) of Lemma 15 holds. Thus, all conditions of Theorem 16 are satisfied. Direct computations show that ∀ξ∈Δ+, μ∈Ω,(27)Sξμ=ξ2ξ1,ifξ2≤ξ1,1,ifξ2>ξ1,Sμ=0,1.Clearly, S(μ)=⋃ξ∈Δ+‍Sξ(μ) and S(·) is continuous on Ω.In Lemma15, the strict ξ-pseudomonotonicity condition is strict that the solution set Sξ(μ) 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 Sξ(μ) may be a general set but not a singleton; that is, the solution mapping Sξ(·) is set-valued in general.Lemma 20.
Letξ∈Δ+. Suppose that assumption (A) holds and the following conditions are satisfied: (i)
Fi(·,·), i=1,…,p, are continuous on K×Ω, where K⊂Rn is a nonempty, closed, and convex set.(ii)
There exists a constantα>0 such that for each μ∈Ω and x∈K∖Sξ(μ), there exists y∈Sξ(μ) satisfying Fix,μ,y-x≤-αx-y2,i=1,…,p.Then, Sξ(·) is l.s.c on Ω.Proof.
Suppose to the contrary that there existsμ0∈Ω such that Sξ(·) is not l.s.c at μ0. Then there exist {μn} with μn→μ0 and x0∈Sξ(μ0), such that, for any xn∈Sξ(μn), xn↛x0.
Sincex0∈K and K⊂Rn is a closed set, there exists x¯n∈K such that x¯n→x0. Whence, it is clear that x¯n∈K∖Sξ(μn). Thus, by assumption (ii), there exists yn∈Sξ(μn) such that(28)fξx¯n,μn,yn-x¯n=∑i=1pξiFix¯n,μn,yn-x¯n≤-∑i=1pξiαyn-x¯n2≤-∑i=1pξi2αyn-x¯n2=-αyn-x¯n2.Similarly as in the proof of Lemma 15, the sequence {yn} is bounded; thus, without loss of generality, we can assume that yn→y0. As the set K is closed, y0∈K. Taking the limit as n→∞ in above inequality, we have(29)fξx0,μ0,y0-x0≤-αy0-x02.
Now we claim thaty0=x0. Otherwise, by (29), we obtain that 〈fξ(x0,μ0),y0-x0〉<0, which contradicts (24), because x0∈Sξ(μ0) and y0∈K. Hence, y0=x0. However, it is impossible by the wrong assumption. The proof is complete.Remark 21.
Condition (ii) of Lemma20 is a modification of the strong pseudomonotonicity of Fi, which may be called the partially strong pseudomonotonicity of Fi, because our assumption is not imposed on all x,y∈K. In addition, if “〈Fi(x,μ),y-x〉≤-αx-y2, i=1,…,p” is replaced by “〈fξ(x,μ),y-x〉≤-αx-y2”, then the conclusion of Lemma 20 still holds. Whence, condition (ii) of Lemma 20 is also from a modification of the strong ξ-pseudomonotonicity of F. 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 3.1] and [18, Theorem 4.2].We give the following trivial example to illustrate Lemma20, where Sξ is set-valued.Example 22.
LetK={0}×[0,1] and Ω=[1,2]. Define F1,F2:K×Ω→R2 as F1(x,μ)=(μx2,0) and F2(x,μ)=(μx22,0) for every x=(x1,x2)∈R2, respectively. For ξ=(ξ1,ξ2)∈Δ+, x=(x1,x2)∈K and μ∈Ω, fξ(x,μ)=∑i=12‍ξiFi(x,μ)=(μx2(ξ1+ξ2x2),0). By a direct calculation, we obtain that Sξ(μ)={0}×[0,1]=K, ∀ξ∈Δ+, μ∈Ω. Thus, the partially strong pseudomonotonicity of Fi (i=1,2) holds trivially, as K∖Sξ(μ)=∅. Whence, all conditions of Lemma 20 are satisfied, and hence it derives the lower semicontinuity of the set-valued solution mapping Sξ.Theorem 23.
Suppose that all conditions of Lemma20 are satisfied and K is a polyhedral convex set. Then S(·) is continuous on Ω.Proof.
“l.s.c”: By virtue of Lemmas1, 13, and 20, the lower semicontinuity is valid.
“u.s.c”: We prove thatS(·) is u.s.c on Ω. Suppose that there exists some μ0∈Ω such that S(·) is not u.s.c at μ0. Then there exist an open set V satisfying S(μ0)⊆V and sequences μn→μ0 and xn∈S(μn), such that xn∉V, ∀n.
It is easy to check that, for everyξ∈Δ+ and μ∈Ω, because of the closedness of K and the continuity of Fi, Sξ(μ) is a closed set in Rn. On the other hand, Sξ(μ)⊆S(μ) together with the boundedness of S(μ) yields that Sξ(μ) is also a bounded set in Rn. Thus, for every ξ∈Δ+ and μ∈Ω, Sξ(μ) is a compact set in Rn.
By Lemma1, xn∈S(μn)=⋃ξ∈Δ+‍Sξ(μn); thus, there exists ξ′∈Δ+ such that xn∈Sξ′(μn). It follows from [12, Lemma 3.3] that Sξ′(·) is u.s.c at μ0 with compact values. Whence, for {μn} and {xn}, there exist x0∈Sξ′(μ0) and a subsequence {xnk} of {xn} such that xnk→x0. Note that x0∈⋃ξ∈Δ+‍Sξ(μ0)=S(μ0)⊆V. It follows from xnk∉V and the openness of V that x0∉V, which yields a contradiction. Thus, we have proved the upper semicontinuity of S(·).Remark 24.
Theorem23 modifies [12, Theorem 3.5], by changing the partially strong monotonicity of Fi (i∈{1,…,p}) to the partially strong pseudomonotonicity of Fi.Example 25.
Consider Example18. To illustrate that Theorem 23 is valid, we only need to verify condition (ii) of Lemma 20. It is clear that Sξ(μ)={1}, ∀ξ∈Δ+,μ∈Ω. Choose α=1. For any μ∈Ω=[1,2] and x∈K∖Sξ(μ)=]1,2], taking y=1∈Sξ(μ), we get that(30)F1x,μ,y-x+αx-y2=μx1-x+x-12=-x-1μ-1x+1<0,F2x,μ,y-x+αx-y2=μ1-x+x-12=-x-1μ+1-x≤0.That is, condition (ii) of Lemma 20 holds. Hence, all conditions of Theorem 23 hold and it is valid. However, Theorem 3.5 of [12] is not applicable, because the partially strong monotonicity of F2(·,μ) is violated. In fact, for every α>0 and μ∈Ω,x∈K∖Sξ(μ)=]1,2], taking any y∈Sξ(μ)={1}, we have 〈F2(y,μ)-F2(x,μ),y-x〉=0(y-x)=0<α|y-x|2.In the sequel, we will show an application to the parametric weak Minty vector variational inequality (PWMVVI). Similarly, we make the following assumption(A′): ∀μ∈Ω, SMw(μ) is nonempty and bounded; {ξ∈Δ:Sξ(μ)≠∅}=Δ; that is, ∀ξ∈Δ, Sξ(μ)≠∅.Theorem 26.
Suppose that assumption(A′) holds and the following conditions are satisfied: (i)
Fi(·,·), i=1,…,p, are continuous on K×Ω, where K⊂Rn is a nonempty, closed, and convex set.(ii)
For anyμ∈Ω, F(·,μ) is strictly ξ-pseudomonotone on K; that is, ∀ξ∈Δ, (31)fξx,μ,x′-x≥0⟹fξx′,μ,x′-x>0,∀x,x′∈K,x≠x′.Then, SMw(·) is continuous on Ω.Proof.
“l.s.c”: Since for anyμ∈Ω, F(·,μ) is strictly ξ-pseudomonotone on K, it is clear that Fi(·,μ), i=1,…,p, are strictly pseudomonotone on K, thus pseudomonotone on K. By virtue of Lemma 11, for each μ∈Ω,(32)SMwμ=⋃ξ∈ΔSξμ.Similar to the proof of Lemma 15 we know, for each ξ∈Δ, Sξ(·) is l.s.c on Ω. Thus, in view of Lemma 13, we immediately obtain that SMw(·) is l.s.c on Ω.
“u.s.c”: The proof is similar to that of Theorem16.The following result can be deduced by the similar proof of Theorem23.Theorem 27.
Suppose that assumption(A′) holds and the following conditions are satisfied: (i)
Fi(·,·), i=1,…,p, are continuous on K×Ω, where K⊂Rn is a nonempty, closed, and convex set, and Fi(·,μ), i=1,…,p, are pseudomonotone on K for any μ∈Ω.(ii)
For anyξ∈Δ, there exists a constant α>0 such that, for each μ∈Ω and x∈K∖Sξ(μ), there exists y∈Sξ(μ) satisfying Fix,μ,y-x≤-αx-y2,i=1,…,p.Then, SMw(·) is continuous on Ω.


## 4. Gap Functions and Error Bounds

Throughout this section, assume thatsol(VI)ξ≠∅ for all ξ∈Σ. The existence, for instance, can be guaranteed by the compactness and convexity of K, and the continuity of F (e.g., [24, Theorem 2.2]).We will now introduce a regularized gap function for (WVVI). This gap function was studied by Charitha and Dutta [20].Forα>0, we define the function ϕα as(33)ϕαx≔minξ∈Σ⁡maxy∈K⁡∑i=1pξiFix,x-y-α2y-x2.For fixed x∈Rn and ξ∈C∖{0Rp} consider the following problem:(34)fαx,ξ≔maxy∈K⁡∑i=1pξiFix,x-y-α2y-x2,which is equivalently written as(35)fαx,ξ=-miny∈K⁡∑i=1pξiFix,y-x+α2y-x2.Lemma 28 (see [20, Lemma 2.4]).
For anyx∈Rn and any ξ∈C∖{0Rp}, let ϕα(x) and fα(x,ξ) be defined by (33) and (34), respectively. Then, fα(x,ξ) is continuous on Rn×Σ and ϕα is well-defined.Note thatϕα is finite without the assumption that K is compact (cf. [20, Remark 2.5]).Lemma 29 (see [20, Theorem 2.7]).
Letϕα(x) be defined by (33). Then ϕα(x)≥0 for all x∈K. Furthermore, ϕα(x∗)=0,x∗∈K if and only if x∗ solves (WVVI). That is, ϕα is a gap function for (WVVI).We will now present an error bound for(WVVI) with a strongly ξ-pseudomonotone mapping F, but not strongly monotone functions Fi(i=1,…,p) done in [20–22]. In our setting we will devise error bounds in terms of the regularized gap function ϕα(x). In what follows by the notation dist(x,A) we mean the distance between the point x and the set A.IfFi:K→Rn,i=1,…,p, are strongly monotone with μi>0 on K, that is, Fiy-Fix,y-x≥μiy-x2,∀x,y∈K,i=1,…,p, and set μ≔min1≤i≤pμi, then it is clear that ∀ξ=(ξ1,…,ξp)∈Σ, F is strongly ξ-monotone on K with μ>0: that is,(36)fξy-fξx,y-x=∑i=1pξiFiy-Fix,y-x≥∑i=1pξiμiy-x2≥μy-x2,∀x,y∈K.Obviously, it implies that F is strongly ξ-pseudomonotone on K. However, the following example shows the converse is not true. That is, the strong ξ-pseudomonotonicity of F is properly weaker than the strong monotonicity of Fi (i=1,…,p).Example 30.
LetK=[1,2]. Define F1,F2:K→R as F1(x)=x and F2(x)=1, respectively. For ξ=(ξ1,ξ2)∈Σ, fξ(x)=∑i=12‍ξiFi(x)=ξ1x+ξ2,∀x∈K. We show that F≔(F1,F2) is strongly ξ-pseudomonotone on K. For any ξ=(ξ1,ξ2)∈Σ and x1,x2∈K, suppose that 〈fξ(x1),x2-x1〉=(ξ1x1+ξ2)(x2-x1)≥0. As ξ1x1+ξ2>0, we have x2-x1≥0. Thus, 〈fξ(x2),x2-x1〉=(ξ1x2+ξ2)(x2-x1)≥x2-x1≥(x2-x1)2, because ξ1+ξ2=1; then ξ1x2+ξ2=1+ξ1(x2-1)≥1, and x2-x1∈[0,1]. Hence, F is strongly ξ-pseudomonotone on K with the modulus of strong pseudomonotonicity μ=1. However, F1,F2 are not both strongly monotone on K. It is clear that F2 is not strongly monotone on K, since 〈F2(x2)-F2(x1),x2-x1〉=0, ∀x1,x2∈K. Moreover, F is also not strongly ξ-monotone on K. In fact, taking ξ=(0,1)∈Σ and x1,x2 in K, we get 〈fξ(x2)-fξ(x1),x2-x1〉=ξ1(x2-x1)2=0.Theorem 31.
Letξ∈Σ. Suppose that F is strongly ξ-pseudomonotone on K with the modulus of strong pseudomonotonicity μ>0, and let α>0 be chosen so that α<2μ. Then for any x∈K we have(37)distx,solWVVI≤1μ-α/2ϕαx.Proof.
We can write the functionϕα(x) in the way: ϕα(x)=minξ∈Σfα(x,ξ). From Lemma 28 we know that fα is continuous on Rn×Σ, so the function fα(x,·) is continuous on Σ. Noting that Σ is compact, hence there exists ξ∗∈Σ (ξ∗ will depend on the chosen x) such that ϕα(x)=fα(x,ξ∗). Whence, using the definition of fα(x,ξ∗) we have, for all y∈K,(38)ϕαx≥∑i=1pξi∗Fix,x-y-α2y-x2.Since sol(VI)ξ∗≠∅, letting x∗∈sol(VI)ξ∗, further from Remark 2 and Lemma 1 we know that x∗ also solves (WVVI). We set y=x∗ in (38):(39)ϕαx≥∑i=1pξi∗Fix,x-x∗-α2x-x∗2.Since x∗ solves (VI)ξ∗, we have ∑i=1p‍ξi∗Fi(x∗),x-x∗≥0. Then by the strong ξ-pseudomonotonicity of F with μ>0, we get(40)∑i=1pξi∗Fix,x-x∗≥μx-x∗2.Thus, combining with (39), we obtain ϕα(x)≥μ-α/2x-x∗2. We have noted that α<2μ; hence x-x∗≤1/μ-α/2ϕα(x), which implies(41)distx,solWVVI≤1μ-α/2ϕαx,∀x∈K.The proof is complete.Remark 32.
(a) Based on the union propertysol(WVVI)=⋃ξ∈Σ‍sol(VI)ξ, sol(WVVI) in Theorem 31 need not be a singleton in general, although for each ξ∈Σ the scalar variational inequality (VI)ξ admits a unique solution by the strong ξ-pseudomonotonicity of F. This is because as we change the parameter ξ the solution set sol(VI)ξ changes as well and all these solutions are in fact solutions of (WVVI).
(b) Theorem31 improves [20, Theorem 2.9], since we use the strong ξ-pseudomonotonicity of F but not strong monotonicity of Fi(i=1,…,p).We give the following example to illustrate Theorem31.Example 33.
LetK=[1,2]. Define F1,F2:K→R as F1(x)=x and F2(x)=1, respectively. For ξ=(ξ1,ξ2)∈Σ, ξ1F1(x)+ξ2F2(x)=ξ1x+ξ2, ∀x∈K. By virtue of Example 30, F is strongly ξ-pseudomonotone on K with μ=1, and we let α=1. Moreover, sol(WVVI)=sol(VI)ξ={1},∀ξ∈Σ. Thus, all conditions of Theorem 31 are satisfied. Direct computations show that the regularized gap function for (WVVI) has the following representation:(42)ϕαx=minξ∈Σ⁡maxy∈K⁡ξ1x+ξ2x-y-12y-x2=minξ∈Σ⁡ξ1x+ξ2x-1-12x-12=minξ∈Σ⁡ξ1x-1+1x-1-12x-12=x-1-12x-12=12x-13-x,∀x∈K.Hence, 1/μ-α/2ϕα(x)=(x-1)(3-x). Clearly, dist(x,sol(WVVI))=|x-1|≤1/μ-α/2ϕα(x), ∀x∈K. However, Theorem 2.9 of [20] is not applicable, because the strong monotonicity of F2 is violated.By Remarks10 and 6 we know the strong pseudomonotonicity of Fi(i=1,…,p) is properly weaker than the strong ξ-pseudomonotonicity of F. Next, we make some discussions on the error bounds of (WVVI) when Fi(i=1,…,p) are strongly pseudomonotone on K.It is clear that⋂i=1p‍sol(VI)i⊆sol(VI)ξ, ∀ξ∈Σ and we assume that sol(VI)ξ≠∅ for all ξ∈Σ, but it is possible that ⋂i=1p‍sol(VI)i=∅. Now we make the following assumption(A~): ⋂i=1p‍sol(VI)i≠∅.Theorem 34.
Letξ∈Σ. Suppose that assumption (A~) holds and Fi(i=1,…,p) are strongly pseudomonotone with the modulus of strong pseudomonotonicity μi>0 on K. Moreover, let μ=min1≤i≤pμi and let α>0 be chosen so that α<2μ. Then for any x∈K we have(43)distx,solWVVI≤1μ-α/2ϕαx.Proof.
Similar to the proof of Theorem31, there exists ξ∗∈Σ such that (38) holds.
Since assumption(A~) holds, we let x∗∈⋂i=1p‍sol(VI)i. Obviously x∗∈sol(WVVI). As x∗ solves (VI)i(i=1,…,p), we have Fi(x∗),x-x∗≥0,i=1,…,p. Then by the strong pseudomonotonicity of Fi(i=1,…,p), we get Fi(x),x-x∗≥μix-x∗2. Because ξ∗∈Σ, we obtain(44)∑i=1pξi∗Fix,x-x∗≥∑i=1pξi∗μix-x∗2≥μx-x∗2.Thus, combining with (38) by setting y=x∗, we obtain(45)ϕαx≥μ-α2x-x∗2.Noting that α<2μ, we have x-x∗≤1/μ-α/2ϕα(x), which implies(46)distx,solWVVI≤1μ-α/2ϕαx,∀x∈K.The proof is complete.Remark 35.
The assumption(A~) 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 ofFi and assumption (A~), we see that, for every i=1,…,p, sol(VI)i is a singleton and all (VI)i (i=1,…,p) must have the same solution, and thus sol(WVVI) should be a singleton.Example 36.
Consider Example33. Direct computations show that(47)solWVVI=solVIξ=solVI1=solVI2=1,∀ξ∈Σ.Clearly, all conditions of Theorem 34 are satisfied with μ1=μ2=1 and α=1, and hence it derives the error bound of (WVVI) in terms of ϕα.Besides the scalar regularized gap functionϕα(x) mentioned above, Charitha et al. [21] and Sun and Chai [22] also have constructed another scalar regularized gap function gα(x) for (WVVI), which is independent with the scalarization parameter ξ.Forα>0, we define the function gα as(48)gαx≔supy∈K⁡min1≤i≤p⁡Fix,x-y-α2y-x2.Charitha et al. [21] and Sun and Chai [22] have explained that gα(x) is finite for every x and thus is well-defined.Lemma 37 (see [21, Theorem 4.2] and [22, Corollary 3.2]).
The functiongα is a gap function for (WVVI).Now we use the regularized gap functiongα(x) to develop an error bound for (WVVI) under assumption (A~) and strongly pseudomonotone component functions Fi(i=1,…,p).Theorem 38.
Suppose that assumption(A~) holds and Fi(i=1,…,p) are strongly pseudomonotone with the modulus of strong pseudomonotonicity μi>0 on K. Moreover, let μ=min1≤i≤pμi and α>0 be chosen so that α<2μ. Then for any x∈K we have(49)distx,solWVVI≤1μ-α/2gαx.Proof.
From our notations and the presentation ofgα(x), we have, for all y∈K,(50)gαx≥min1≤i≤p⁡Fix,x-y-α2y-x2.As assumption (A~) holds, we let x∗∈⋂i=1p‍sol(VI)i. Obviously x∗∈sol(WVVI). We set y=x∗ in (50):(51)gαx≥min1≤i≤p⁡Fix,x-x∗-α2x∗-x2.Without loss of generality, we assume that(52)F1x,x-x∗=min1≤i≤p⁡Fix,x-x∗.Since x∗∈⋂i=1p‍sol(VI)i, x∗ solves (VI)1; that is, 〈F1(x∗),x-x∗〉≥0. Then, by the strong pseudomonotonicity of F1, we have F1(x),x-x∗≥μ1x-x∗2≥μx-x∗2. Thus, combining with (51) and (52), we obtain(53)gαx≥μ-α2x-x∗2.Noting that α<2μ, we have x-x∗≤1/μ-α/2gα(x), which implies(54)distx,solWVVI≤1μ-α/2gαx,∀x∈K.The proof is complete.Remark 39.
Theorem38 improves [21, Theorem 4.3] and [22, Corollary 3.3], since we use the strong pseudomonotonicity of Fi but not strong monotonicity of Fi(i=1,…,p).Example 40.
Consider Example36. All conditions of Theorem 38 are satisfied with μ1=μ2=1 and α=1, and hence it derives the error bound of (WVVI) in terms of gα. Actually, we can verify that(55)gαx=supy∈K⁡min⁡xx-y,x-y-12y-x2=supy∈K⁡x-y-12x-y2,ifx-y>0supy∈K⁡xx-y-12x-y2,ifx-y≤0=supy∈K⁡12x-13-x,if1≤y<x0,ifx≤y≤2=12x-13-x,∀x∈K.However, Theorem 4.3 of [21] (or Corollary 3.3 of [22]) is not applicable, because the strong monotonicity of F2 is violated.


## 5. Conclusions

In the paper, firstly, sufficient conditions for the continuity (both lower and upper semicontinuities) of solution mappingsS(·) to (PVVI) and SMw(·) to (PWMVVI) have been established, when the mappings on models are strictly ξ-pseudomonotone and partially strong pseudomonotone, respectively. Secondly, error bounds for (WVVI) in terms of regularized gap functions ϕα and gα have been obtained, when the mappings on models are strongly ξ-pseudomonotone and strongly pseudomonotone, respectively. All discussions have been carried out via scalarization approaches. The results obtained improve or modify corresponding ones in [12] and [20–22], respectively. Moreover, numerous examples have been provided to illustrate main conclusions.



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