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

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.

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 by 𝑅₊, 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⊆𝑅^𝑛 be nonempty closed convex set. Let 𝐹_𝑖∶𝑅^𝑛→𝑅^𝑛 (𝑖=1,2,…,𝑚) be vector-valued functions, and let 𝑔_𝑖∶𝑅^𝑛→𝑅 (𝑖=1,2,…,𝑚) be real-valued functions. For abbreviation, we put 𝐹∶=𝐹₁,𝐹₂,…,𝐹_𝑚,𝑔∶=𝑔₁,𝑔₂,…,𝑔_𝑚, (1) and for any 𝑥,𝑣∈𝑅^𝑛, ⟨𝐹(𝑥),𝑣⟩∶=𝐹₁(𝑥),𝑣,𝐹₂(𝑥),𝑣,…,𝐹_𝑚(𝑥),𝑣. (2)

In this paper, we consider the following generalized vector variational inequality (GVVI)∶(GVVI)Find𝑥₀𝑥∈𝐾suchthat⟨𝐹₀,𝑥−𝑥₀⟩+𝑔(𝑥)−𝑔𝑥₀∉−int𝑅^𝑚₊,forany𝑥∈𝐾. (3) The solution set of (GVVI) is denoted by sol (GVVI).

If 𝑔=0, then (GVVI) collapses to the following vector variational inequality (VVI), introduced and studied by [2, 3]: (VVI)Find𝑥₀𝑥∈𝐾suchthat⟨𝐹₀,𝑥−𝑥₀⟩∉−int𝑅^𝑚₊,forany𝑥∈𝐾. (4) The solution set of (VVI) is denoted by sol(VVI).

Clearly, for 𝑚=1, (GVVI) and (VVI) collapse to the generalized variational inequality (GVI)(GVI)Find𝑥₀∈𝐾suchthat⟨𝐹₁𝑥₀,𝑥−𝑥₀⟩+𝑔₁(𝑥)−𝑔₁𝑥₀≥0,forany𝑥∈𝐾, (5) and the variational inequality (VI)(VI)Find𝑥₀∈𝐾suchthat⟨𝐹₁𝑥₀,𝑥−𝑥₀⟩≥0,forany𝑥∈𝐾, (6) 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: (GVI)_𝜉Find𝑥₀∈𝐾suchthat_𝑚𝑖∑₌₁𝜉_𝑖𝐹_𝑖𝑥₀,𝑥−𝑥₀+∑^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖(𝑥)−∑^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖𝑥₀()≥0forall𝑥∈𝐾,VI_𝜉Find𝑥₀∈𝐾suchthat_𝑚𝑖∑₌₁𝜉_𝑖𝐹_𝑖𝑥₀,𝑥−𝑥₀≥0forall𝑥∈𝐾. (7) 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)If 𝑔_𝑖 is an affine function for every 𝑖, then, 𝑠𝑜𝑙(_{𝜉∈int𝑅𝑚+}𝐺𝑉𝐼)_𝜉⊂𝑠𝑜𝑙(𝐺𝑉𝑉𝐼)=_{𝜉∈𝑅𝑚+⧵{0}}𝑠𝑜𝑙(𝐺𝑉𝐼)_𝜉. (8)(ii)If 𝐹_𝑖 and 𝑔_𝑖 are continuous functions for every 𝑖, then 𝑠𝑜𝑙(𝐺𝑉𝑉𝐼) is a closed set.

Proof. (i) We first prove the inclusion. In fact, take any 𝑥₀∈sol(GVI)_𝜉, where 𝜉∈int𝑅^𝑚₊. Then, for any 𝑥∈𝐾, ^𝑚_𝑖=1𝜉_𝑖𝐹_𝑖𝑥₀,𝑥−𝑥₀+^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖𝑥₀=^𝑚_𝑖=1𝜉_𝑖𝐹_𝑖𝑥₀,𝑥−𝑥₀+𝑔_𝑖(𝑥)−𝑔_𝑖𝑥₀=𝜉,𝐹𝑥₀,𝑥−𝑥₀+𝑔(𝑥)−𝑔𝑥₀≥0. (9) Thus, there cannot exist 𝑥∈𝐾 such that ⟨𝐹𝑥₀,𝑥−𝑥₀⟩+𝑔(𝑥)−𝑔𝑥₀∈−int𝑅^𝑚₊, (10) which means that 𝑥₀∈sol(GVVI).
Now, we prove the equality in (8). If 𝑥₀∈sol(GVVI), then, ⟨𝐹𝑥₀,𝑥−𝑥₀⟩+𝑔(𝑥)−𝑔𝑥₀∩∶𝑥∈𝐾−int𝑅^𝑚₊=∅. (11) Moreover, since each 𝑔_𝑖 is an affine function ⟨𝐹𝑥₀,𝑥−𝑥₀⟩+𝑔(𝑥)−𝑔𝑥₀∶𝑥∈𝐾 (12) is a convex set. Thus, by using the separation theorem (see [16, Theorem 11.3]), there exists 𝜉∈𝑅^𝑚⧵{0} such that inf_𝑥∈𝐾𝜉^𝑇⟨𝐹𝑥₀,𝑥−𝑥₀⟩+𝑔(𝑥)−𝑔𝑥₀≥sup_{𝑣∈−int𝑅𝑚+}𝜉^𝑇𝑣. (13) This means that 𝜉∈𝑅^𝑚⧵{0}, and for any 𝑥∈𝐾, 𝜉^𝑇⟨𝐹𝑥₀,𝑥−𝑥₀⟩+𝑔(𝑥)−𝑔𝑥₀≥0. (14) 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 Δ∶=𝑅^𝑚⧵−int𝑅^𝑚₊. (15) Then, Δ is a closed set. For any 𝑥∈𝐾, let 𝐾(𝑥)∶=𝑦∈𝐾∶⟨𝐹𝑦,𝑥−𝑦⟩+𝑔(𝑥)−𝑔𝑦∈Δ. (16) Since 𝐹 and 𝑔 are continuous, 𝐾(𝑥) is a closed set. Moreover, since sol(GVVI)=_𝑥∈𝐾𝐾(𝑥), (17) we get that sol(GVVI) is a closed set. The proof is complete.

Taking 𝑔=0 in Lemma 1, we can easily get the following result.

Corollary 2 (see [9]). The following properties hold.(i)𝑠𝑜𝑙(_{𝜉∈int𝑅𝑚+}𝑉𝐼)_𝜉⊂𝑠𝑜𝑙(𝑉𝑉𝐼)=_{𝜉∈𝑅𝑚+⧵{0}}𝑠𝑜𝑙(𝑉𝐼)_𝜉. (18)(ii)If 𝐹_𝑖 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 function 𝜓∶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 function 𝜑∶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 Lemma 1 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 𝜓(𝑥)=inf_{𝜉∈𝑆𝑚}sup_𝑦∈𝐾^𝑚_𝑖=1𝜉_𝑖𝐹_𝑖(𝑥),𝑥−𝑦−^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖𝑦+^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖(𝑥,𝜑(𝑥))=inf_{𝜉∈𝑆𝑚}sup_𝑦∈𝐾^𝑚_𝑖=1𝜉_𝑖𝐹_𝑖(𝑥),𝑥−𝑦, (19)(20) respectively. The symbol 𝑆^𝑚 in the above expression denotes the unit simplex in 𝑅^𝑚₊; that is, it is given as 𝑆^𝑚=𝑥∈ℝ^𝑚₊∶^𝑚_𝑖=1𝑥_𝑖=1. (21) The use of 𝑆^𝑚 in the above expression is to stress the fact that the vector 𝜉≠0, and we just express the normalized version. Further, use of 𝑆^𝑚 has an advantage since if additionally 𝐾 is compact and each 𝑔_𝑖 is convex for any 𝑖=1,2,…,𝑚, then, the functions 𝜓 and 𝜑 are finite.

Theorem 5. If 𝑔_𝑖 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 that 𝜓(𝑥)≥0 for all 𝑥∈𝐾.
(ii) If there exists 𝑥∈𝐾 such that 𝜓(𝑥)=0, set 𝜃(𝑥,𝜉)=sup_𝑦∈𝐾^𝑚_𝑖=1𝜉_𝑖𝐹_𝑖(𝑥),𝑥−𝑦−^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖𝑦. (22) Then, 𝜓(𝑥)=inf_{𝜉∈𝑆𝑚}𝜃(𝑥,𝜉)+^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖(𝑥). (23) 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^∗∈𝑆^𝑚 such that 𝜃𝑥,𝜉^∗+^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖(𝑥)=𝜓(𝑥)=0, (24) which follows that for all 𝑦∈𝐾, ^𝑚_𝑖=1𝜉_𝑖𝐹_𝑖(𝑥),𝑥−𝑦+^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖𝑦≤0. (25) Then, ^𝑚_𝑖=1𝜉_𝑖𝐹_𝑖(𝑥),𝑦−𝑥+^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖𝑦−^𝑚_𝑖=1𝜉_𝑖𝑔_𝑖(𝑥)≥0. (26) This means that 𝑥 solves (GVI)_𝜉. Thus, using Lemma 1, we get that 𝑥 is a solution of (GVVI).
Conversely, let 𝑥∈sol(GVVI). By Lemma 1, there exists 𝜉^′∈𝑆^𝑚 such that 𝑥∈sol(GVI)_𝜉′. Then, for all 𝑦∈𝐾, ^𝑚_𝑖=1𝜉^′_𝑖𝐹_𝑖(𝑥),𝑦−𝑥+^𝑚_𝑖=1𝜉^′_𝑖𝑔_𝑖𝑦−^𝑚_𝑖=1𝜉^′_𝑖𝑔_𝑖(𝑥)≥0; (27) that is, ^𝑚_𝑖=1𝜉^′_𝑖𝐹_𝑖(𝑥),𝑥−𝑦+^𝑚_𝑖=1𝜉^′_𝑖𝑔_𝑖(𝑥)−^𝑚_𝑖=1𝜉^′_𝑖𝑔_𝑖𝑦≤0. (28) Then, 𝜃𝑥,𝜉^′+^𝑚_𝑖=1𝜉^′_𝑖𝑔_𝑖(𝑥)≤0. (29) So, 𝜓(𝑥)≤0. Moreover, as 𝜓(𝑥)≥0 for all 𝑥∈𝐾, then, 𝜓(𝑥)=0 and the proof is complete.

By Theorem 5, it is easy to see that the following result holds.

Corollary 6. The function 𝜑 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.

Acknowledgments

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).