# A Note on Scalar-Valued Gap Functions for Generalized Vector Variational Inequalities

Authors: Xiang-Kai Sun; Hong-Yong Fu; Yi Chai
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with gap functions of generalized vector variational inequalities (GVVI). By using scalarization approach, scalar-valued variational inequalities of (GVVI) are introduced. Some relationships between the solutions of (GVVI) and its scalarized versions are established. Then, by using these relationships and some mild conditions, scalar-valued gap functions for (GVVI) are established.


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## Body

## 1. Introduction

The concept of vector variational inequalities was firstly introduced by Giannessi [1] in a finite-dimensional space. Since then, extensive study of vector variational inequalities has been done by many authors in finite- or infinite-dimensional spaces under generalized monotonicity and convexity assumptions. See [2–10] and the references therein. Among solution approaches for vector variational inequalities, scalarization is one of the most analyzed topics at least from the computational point of view; see [8–10].Gap functions are very useful for solving vector variational inequalities. One advantage of the introduction of gap functions in vector variational inequalities is that vector variational inequalities can be transformed into optimization problems. Then, powerful optimization solution methods and algorithms can be applied for finding solutions of vector variational inequalities. Recently, some authors have investigated the gap functions for vector variational inequalities. Yang and Yao [11] introduced gap functions and established necessary and sufficient conditions for the existence of a solution of vector variational inequalities. Chen et al. [12] extended the theory of gap function for scalar variational inequalities to the case of vector variational inequalities. They also obtained the set-valued gap functions for vector variational inequalities. Li and Chen [13] introduced set-valued gap functions for a vector variational inequality and obtained some related properties. Li et al. [14] investigated differential and sensitivity properties of set-valued gap functions for vector variational inequalities and weak vector variational inequalities. Meng and Li  [15] also investigated the differential and sensitivity properties of set-valued gap functions for Minty vector variational inequalities and Minty weak vector variational inequalities.The purpose of this paper is to define a single variable gap function for generalized vector variational inequalities by using the scalarization approach. To this end, we first transform the generalized vector variational inequality into an equivalent scalar variational inequality by using the scalarization approach of  [9]. Then, we establish the relations between vector variational inequalities and variational inequalities. Finally, we apply the results to obtain gap functions for generalized vector variational inequalities.


## 2. Generalized Vector Variational Inequalities

Throughout this paper, let the set of nonnegative real numbers be denoted byR+, the origins of all finite-dimensional spaces denoted by 0, the norms of all finite-dimensional spaces denoted by ∥·∥, and the inner products of all finite-dimensional spaces denoted by 〈,〉. Furthermore, let K⊆Rn be nonempty closed convex set. Let Fi:Rn→Rn (i=1,2,…,m) be vector-valued functions, and let gi:Rn→R (i=1,2,…,m) be real-valued functions. For abbreviation, we put
(1)F:=(F1,F2,…,Fm),g:=(g1,g2,…,gm),
and for any x,v∈Rn,
(2)〈F(x),v〉:=(〈F1(x),v〉,〈F2(x),v〉,…,〈Fm(x),v〉).In this paper, we consider the following generalized vector variational inequality(GVVI):(3)(GVVI){Findx0∈Ksuchthat〈F(x0),x-x0〉+g(x)-g(x0)∉-intR+m,〈F(x0),x-x0〉+g(x)DD-foranyx∈K.
The solution set of (GVVI) is denoted by sol (GVVI).Ifg=0, then (GVVI) collapses to the following vector variational inequality (VVI), introduced and studied by [2, 3]:
(4)(VVI){Findx0∈Ksuchthat〈F(x0),x-x0〉∉-intR+m,〈F(x0),22〉foranyx∈K.
The solution set of (VVI) is denoted by sol(VVI).Clearly, form=1, (GVVI) and (VVI) collapse to the generalized variational inequality (GVI)(5)(GVI){Findx0∈Ksuch  that〈F1(x0),x-x0〉+g1(x)-g1(x0)≥0,〈F1(x0),x-x0〉ffffffffffforanyx∈K,
and the variational inequality (VI)(6)(VI){Findx0∈Ksuchthat〈F1(x0),x-x0〉≥0,〈(x0)〉foranyx∈K,
respectively.Now, by using the scalarization scheme of Lee et al. [9], we introduce scalar gap functions for (GVVI) and (VVI). So, for any ξ∈R+m∖{0}, we consider the following scalar variational inequalities:
(7)(GVI)ξ{Findx0∈Ksuchthat〈∑i=1m‍ξiFi(x0),x-x0〉+∑i=1m‍ξigi(x)-∑i=1m‍ξigi(x0)≥0forallx∈K,(VI)ξ{Findx0∈Ksuchthat〈∑i=1m‍ξiFi(x0),x-x0〉≥0forallx∈K.
The solution sets of (GVI)ξ and (VI)ξ are denoted by sol(GVI)ξ and sol(VI)ξ, respectively.Lemma 1.
The following properties hold.(i)
Ifgi is an affine function for every i, then,
(8)⋃ξ∈intR+m‍sol(GVI)ξ⊂sol(GVVI)=⋃ξ∈R+m∖{0}‍sol(GVI)ξ.(ii)
IfFi and gi are continuous functions for every i, then sol(GVVI) is a closed set.Proof.
(i) We first prove the inclusion. In fact, take anyx0∈sol(GVI)ξ, where ξ∈intR+m. Then, for any x∈K,
(9)〈∑i=1m‍ξiFi(x0),x-x0〉+∑i=1m‍ξigi(x)-∑i=1m‍ξigi(x0)=∑i=1m‍ξi(〈Fi(x0),x-x0〉+gi(x)-gi(x0))=〈ξ,〈F(x0),x-x0〉+g(x)-g(x0)〉≥0.
Thus, there cannot exist x∈K such that
(10)〈F(x0),x-x0〉+g(x)-g(x0)∈-intR+m,
which means that x0∈sol(GVVI).
Now, we prove the equality in (8). If x0∈sol(GVVI), then,
(11){〈F(x0),x-x0〉+g(x)-g(x0):x∈K}∩(-intR+m)=∅.
Moreover, since each gi is an affine function
(12){〈F(x0),x-x0〉+g(x)-g(x0):x∈K}
is a convex set. Thus, by using the separation theorem (see [16, Theorem 11.3]), there exists ξ∈Rm∖{0} such that
(13)infx∈K{ξT(〈F(x0),x-x0〉+g(x)-g(x0))}≥supv∈-intR+mξTv.
This means that ξ∈Rm∖{0}, and for any x∈K,
(14)ξT(〈F(x0),x-x0〉+g(x)-g(x0))≥0.
Then, x0∈sol(GVI)ξ. Conversely, for any x0∈sol(GVI)ξ, where ξ∈R+m∖{0}, it is easy to see that x0∈sol(GVVI).
(ii) Set(15)Δ:=Rm∖(-intR+m).
Then, Δ is a closed set. For any x∈K, let
(16)K(x):={y∈K:〈F(y),x-y〉+g(x)-g(y)∈Δ}.
Since F and g are continuous, K(x) is a closed set. Moreover, since
(17)sol(GVVI)=⋂x∈K‍K(x),
we get that sol(GVVI) is a closed set. The proof is complete.Takingg=0 in Lemma 1, we can easily get the following result.Corollary 2 (see [9]).
The following properties hold.(i)
(18)⋃ξ∈intR+m‍sol(VI)ξ⊂sol(VVI)=⋃ξ∈R+m∖{0}‍sol(VI)ξ.(ii)
IfFi is a continuous function for every i, then sol(VVI) is a closed set.


## 3. Gap Functions for (GVVI) and (VVI)

In this section, we propose some new gap functions for(GVVI). Now, we first introduce the definitions of gap functions for (GVVI) and (VVI).Definition 3.
A real-valued functionψ:X→R is said to be a scalar-valued gap function of (GVVI) if it satisfies the following conditions:(i)
ψ(x)≥0, for any x∈K;(ii)
ψ(x0)=0 if and only if x0∈K is a solution of (GVVI).Definition 4.
A real-valued functionφ:X→R is said to be a scalar-valued gap function of (VVI) if it satisfies the following conditions: (i)
φ(x)≥0, for any x∈K;(ii)
φ(x0)=0 if and only if x0∈K is a solution of (VVI).Now, by using Lemma1 and Corollary 2, we generalize the gap function introduced by Auslender [17] for scalar variational inequalities to the case of vector variational inequalities. The gap functions for (GVVI) and (VVI) are defined by
(19)ψ(x)=infξ∈Sm{supy∈K{〈∑i=1m‍ξiFi(x),x-y〉-∑i=1m‍ξigi(y)}+∑i=1m‍ξigi(x)},(20)φ(x)=infξ∈Smsupy∈K〈∑i=1m‍ξiFi(x),x-y〉,
respectively. The symbol Sm in the above expression denotes the unit simplex in R+m; that is, it is given as
(21)Sm={x∈ℝ+m:∑i=1m‍xi=1}.
The use of Sm in the above expression is to stress the fact that the vector ξ≠0, and we just express the normalized version. Further, use of Sm has an advantage since if additionally K is compact and each gi is convex for any i=1,2,…,m, then, the functions ψ and φ are finite.Theorem 5.
Ifgi is an affine function for every i, then, the function ψ defined by (19) is a gap function for (GVVI).Proof.
(i) It is easy to prove thatψ(x)≥0 for all x∈K.
(ii) If there existsx-∈K such that ψ(x-)=0, set
(22)θ(x-,ξ)=supy∈K{〈∑i=1m‍ξiFi(x-),x--y〉-∑i=1m‍ξigi(y)}.
Then,
(23)ψ(x-)=infξ∈Sm{θ(x-,ξ)+∑i=1m‍ξigi(x-)}.
It is easy to observe that for x-∈K, the function θ(x-,·) is a convex function. Moreover, since ψ(x-)=0, θ(x-,·) is a proper lower semicontinuous convex function. Then, there exists ξ*∈Sm such that
(24)θ(x-,ξ*)+∑i=1m‍ξigi(x-)=ψ(x-)=0,
which follows that for all y∈K,
(25)〈∑i=1m‍ξiFi(x-),x--y〉+∑i=1m‍ξigi(x-)-∑i=1m‍ξigi(y)≤0.
Then,
(26)〈∑i=1m‍ξiFi(x-),y-x-〉+∑i=1m‍ξigi(y)-∑i=1m‍ξigi(x-)≥0.
This means that x- solves (GVI)ξ. Thus, using Lemma 1, we get that x- is a solution of (GVVI).
Conversely, letx-∈sol(GVVI). By Lemma 1, there exists ξ′∈Sm such that x-∈sol(GVI)ξ′. Then, for all y∈K,
(27)〈∑i=1m‍ξi′Fi(x-),y-x-〉+∑i=1m‍ξi′gi(y)-∑i=1m‍ξi′gi(x-)≥0;
that is,
(28)〈∑i=1m‍ξi′Fi(x-),x--y〉+∑i=1m‍ξi′gi(x-)-∑i=1m‍ξi′gi(y)≤0.
Then,
(29)θ(x-,ξ′)+∑i=1m‍ξi′gi(x-)≤0.
So, ψ(x-)≤0. Moreover, as ψ(x)≥0 for all x∈K, then, ψ(x-)=0 and the proof is complete.By Theorem5, it is easy to see that the following result holds.Corollary 6.
The functionφ defined by (20) is a gap function for (VVI).


## 4. Conclusions

In this paper, by using the scalarization approach of [9], we transform a generalized vector variational inequality into an equivalent scalar variational inequality. Then, we establish some relationships between the solutions of vector variational inequalities and variational inequalities. By using these relationships and some mild conditions, we obtain gap functions for the generalized vector variational inequalities and vector variational inequalities.



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