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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 492103, 4 pages
A Note on Scalar-Valued Gap Functions for Generalized Vector Variational Inequalities
1College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
2College of Automation, Chongqing University, Chongqing 400030, China
3School of Management, Southwest University of Political Science and Law, Chongqing 401120, China
Received 1 November 2013; Accepted 17 December 2013
Academic Editor: Khalil Ezzinbi
Copyright © 2013 Xiang-Kai Sun 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.
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.
The concept of vector variational inequalities was firstly introduced by Giannessi  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  introduced gap functions and established necessary and sufficient conditions for the existence of a solution of vector variational inequalities. Chen et al.  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  introduced set-valued gap functions for a vector variational inequality and obtained some related properties. Li et al.  investigated differential and sensitivity properties of set-valued gap functions for vector variational inequalities and weak vector variational inequalities. Meng and Li  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 . 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 by , the origins of all finite-dimensional spaces denoted by , the norms of all finite-dimensional spaces denoted by , and the inner products of all finite-dimensional spaces denoted by . Furthermore, let be nonempty closed convex set. Let be vector-valued functions, and let be real-valued functions. For abbreviation, we put and for any ,
In this paper, we consider the following generalized vector variational inequality The solution set of is denoted by .
Clearly, for , and collapse to the generalized variational inequality and the variational inequality respectively.
Now, by using the scalarization scheme of Lee et al. , we introduce scalar gap functions for and . So, for any , we consider the following scalar variational inequalities: The solution sets of and are denoted by and , respectively.
Lemma 1. The following properties hold. (i)If is an affine function for every , then, (ii)If and are continuous functions for every , then is a closed set.
Proof. (i) We first prove the inclusion. In fact, take any , where . Then, for any ,
Thus, there cannot exist such that
which means that .
Now, we prove the equality in (8). If , then, Moreover, since each is an affine function is a convex set. Thus, by using the separation theorem (see [16, Theorem 11.3]), there exists such that This means that , and for any , Then, . Conversely, for any , where , it is easy to see that .
(ii) Set Then, is a closed set. For any , let Since and are continuous, is a closed set. Moreover, since we get that is a closed set. The proof is complete.
Taking in Lemma 1, we can easily get the following result.
Corollary 2 (see ). The following properties hold.(i)(ii)If is a continuous function for every , then is a closed set.
3. Gap Functions for (GVVI) and (VVI)
In this section, we propose some new gap functions for . Now, we first introduce the definitions of gap functions for and .
Definition 3. A real-valued function is said to be a scalar-valued gap function of if it satisfies the following conditions:(i), for any ;(ii) if and only if is a solution of .
Definition 4. A real-valued function is said to be a scalar-valued gap function of if it satisfies the following conditions: (i), for any ;(ii) if and only if is a solution of .
Now, by using Lemma 1 and Corollary 2, we generalize the gap function introduced by Auslender  for scalar variational inequalities to the case of vector variational inequalities. The gap functions for and are defined by respectively. The symbol in the above expression denotes the unit simplex in ; that is, it is given as The use of in the above expression is to stress the fact that the vector , and we just express the normalized version. Further, use of has an advantage since if additionally is compact and each is convex for any , then, the functions and are finite.
Theorem 5. If is an affine function for every , then, the function defined by (19) is a gap function for .
Proof. (i) It is easy to prove that for all .
(ii) If there exists such that , set Then, It is easy to observe that for , the function is a convex function. Moreover, since , is a proper lower semicontinuous convex function. Then, there exists such that which follows that for all , Then, This means that solves . Thus, using Lemma 1, we get that is a solution of .
Conversely, let . By Lemma 1, there exists such that . Then, for all , that is, Then, So, . Moreover, as for all , then, and the proof is complete.
By Theorem 5, it is easy to see that the following result holds.
Corollary 6. The function defined by (20) is a gap function for .
In this paper, by using the scalarization approach of , 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.
This research was partially supported by the National Natural Science Foundation of China (Grant no. 11301570), the Basic and Advanced Research Project of CQ CSTC (Grant no. cstc2013jcyjA00003), the China Postdoctoral Science Foundation funded project (Grant no. 2013M540697), and the Research Fund of Chongqing Technology and Business University (Grant no. 2013-56-03).
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