Optimality Conditions and Scalarization of Approximate Quasi Weak Efficient Solutions for Vector Equilibrium Problem

Abstract

This paper is devoted to the investigation of optimality conditions for approximate quasi weak efficient solutions for a class of vector equilibrium problem (VEP). First, a necessary optimality condition for approximate quasi weak efficient solutions to VEP is established by utilizing the separation theorem with respect to the quasirelative interior of convex sets and the properties of the Clarke subdifferential. Second, the concept of approximate pseudoconvex function is introduced and its existence is verified by a concrete example. Under the assumption of introduced convexity, a sufficient optimality condition for VEP in sense of approximate quasi weak efficiency is also presented. Finally, by using Tammer’s function and the directed distance function, the scalarization theorems of the approximate quasi weak efficient solutions of the VEP are proposed.

1. Introduction

Vector equilibrium, which is closely related to complementarity problems, variational inequalities, and fixed point theory, is one of the momentous contents in the field of applied mathematics. The characteristics and optimality conditions of various solutions are the key study of vector equilibrium problems. For instance, the optimality conditions for efficient solutions to vector equilibrium problem were presented in [1]; the literatures [2, 3] derived the optimality conditions of weakly efficient solutions; some optimality conclusions related to several properly efficient solutions were established in [4–7]. In practical applications, the majority of solutions obtained by numerical algorithms are approximate solutions. Undoubtedly, it is of great theoretical and practical significance to study the approximate solutions of vector equilibrium problem. In recent years, the concept of approximate weak efficient solutions for vector equilibrium problem was introduced and its properties were discussed in [8, 9]. Das and Nahak [10] presented the concept of approximate quasi weak efficient solutions to vector equilibrium problem and examined its optimality conditions by generalized derivatives. One of the main purposes of this paper is to establish the necessary optimality condition for approximate quasi weak efficient solutions to vector equilibrium problem via the quasirelative interior-type separation theorem of convex sets. It is worth mentioning that our method is different from that of Das and Nahak [10].

Convexity and its generalization play a critical role in optimization and vector equilibrium theory, especially in establishing the sufficient optimality conditions. For instance, Gong [11, 12] derived the sufficient optimality condition to approximate efficient solutions for vector equilibrium problem under the cone convexity; under the assumptions of arcwise connected functions, the sufficient optimality conditions with regard to properly efficient solutions to vector equilibrium problem are presented in the literature [13]; based on the assumption of generalized cone subconvexlikeness, the literature [14] proposed the properties of globally efficient solutions to vector equilibrium problem. In this paper, we will introduce notion of approximate quasi-pseudoconvex function in terms of subdifferential, and under its assumption, we establish the sufficient optimality condition of approximate quasi weak efficient solutions to vector equilibrium problem, which is another aim of this paper.

Scalarization is to transform a vector problem into a numerical (scalar) problem which is equivalent to primal vector problem under mild conditions. There is no doubt that scalarization is one of the core topics in the study of vector equilibrium problem. In present paper, we will utilize Tammer’s nonlinear scalar function and the directed distance function to deal with the scalarization theorems for the approximate quasi weak efficient solutions to vector equilibrium problem.

In the view of the above discussion, the paper will examine the optimality conditions and scalarization theorems in sense of approximate quasi weak efficient solutions to vector equilibrium problem. The article is arranged as follows: in section 2, some symbols, concepts, and lemmas will be presented, which will be used in the subsequent sections; Section 3 is devoted to establish the optimality conditions for approximate quasi weak efficient solutions to the discussed vector equilibrium problem; in section 4, the scalarization theorems will be proven.

2. Preliminaries

Throughout the paper, we set

Let and be real Banach spaces with topological dual spaces and , respectively, and stands for the open ball of radius around . For all and , the value of linear functional at be denoted by . Let be a pointed closed convex cone in , then the dual cone of be defined as (see [15])

Without other specifications, we always suppose that is a pointed closed convex cone in . We will use the following properties of .

Lemma 1 (see [16]). If and , then , where represents the interior of a set.

Let be a nonempty subset of , and the Clarke contingent cone (see [1]) to set at point is defined as

The normal cone (see [1]) associated with is denoted by

Especially when be a convex set, the contingent cone to set at is given by (see [15])

The normal cone to set at iswhere stands for the closure of a set.

Let be a mapping. is said to be locally at , if there exist constant and such that

If for any , is locally at , then is called locally mapping. In particular, for a real-valued locally function ( denotes real number), the generalized directional derivative of at in the direction is given by (see [15])which is defined as the subdifferential of at .

We present below some significant properties of locally function that we shall use in the sequel.

Lemma 2 (see [15,17]). Let function is locally at , if is the minimum value point of on , then

Lemma 3 (see [15]). Let , be locally at , then function is also locally at , and

Let be a nonempty subset, and be a mapping. Consider the following vector equilibrium problem (VEP):

Given , be vector-valued mapping of one variable, which is defined by

Throughout this paper, it is always assumed that and

Definition 1 (see [10]). Let be a nonempty subset, , . is called an -quasi weak efficient solution to VEP, ifThe notion of -quasi weak efficient solution is illustrated by the following example.

Example 1. Let , , , and . Consider the following questions:Taking and , thenTaking , for all , we obtainHence, 0 is an -quasi weak efficient solution of VEP.
It is well known that, for a nonempty convex set, its interior may be empty, but its quasirelative interior is always nonempty (see [18]). In this paper, we will prove the optimality condition of VEP by the separation theorem with respect to the quasirelative interior of convex sets (see [19]).

Definition 2 (see [18]). Let is a convex subset; the quasirelative interior of denoted by is defined aswhere cl and cone stand for closure and cone hull.

Lemma 4 (see [19]). Let and be nonempty convex subsets of , , and is not a linear subset of , then there exists such that

3. Optimality Conditions

In this section, first, we propose a necessary optimality condition for -quasi weak efficient solutions to VEP by using separation theorem in terms of quasirelative interiors of a convex set. Second, the concept of approximate quasi-pseudoconvex function is introduced and a sufficient optimality conditions is established under the introduced generalized convexity. Throughout this section, let be a nonempty convex set.

Theorem 1. In VEP, let , , and . Assume that be an -quasi weak efficient solution of VEP and is locally Lipschitz mapping at . In addition, and is not a linear subspace of . Then, there exist such thatwhere stands for the convex hull, , and .

Proof 1. Sinceand is not a linear subspace of , then is not a linear subspace of . Moreover,Thus,Noticing that , it follows from Lemma 4 that there exists such thatwhich meansTaking in the above formula, we obtainHence, . Sinceand , it leads toIt follows from and equation (28) thatthat isOn the other hand, let . Since is locally at , it is obvious that is a locally function at . We setIt follows from equation (29) thatwhich shows that is the minimum point of on . Taking account of Lemma 2, we arrive atSince is a locally function at , by Lemma 3, we haveTogether with equation (33), we obtainNext, we introduce the concept of approximate quasi-pseudoconvex function, and under the assumption of this generalized convexity, a sufficient optimality condition for -quasi weak efficient solutions to VEP is derived.

Definition 3. Let and the function be locally at . is said to be -quasi-pseudoconvex at , if there exists such that for each satisfying

Example 2. Let , then is defined byTaking and , by a simple computation, we derive . For any , , ifthenThus, is a 1-quasi-pseudoconvex at 0.

Theorem 2. In VEP, let , , , and be locally at . Suppose that there exists such thatIf is -quasi-pseudoconvex at , then is -quasi weak efficient solutions of VEP.

Proof 2. It follows from (40) that there exist , , and such thatwhich implies for each ,which is equivalent toSince is a convex set, according to the definition of contingent cone to set at ,Therefore,Combining (43) and (44), it is not difficult to findBecause , we obtain . Hence,Together with equation (46), it leads toSince is -quasi-pseudoconvex at , by Definition 3, we obtainIn view of , we arrive atSuppose that is not -quasi weak efficient solutions of VEP, then there exists such thatSince , it yields from Lemma 1 thatwhich meansThat is,which contradicts (50). Hence, is -quasi weak efficient solutions of VEP.

4. Scalarization

In this section, the scalarization theorems for approximate quasi weak efficient solutions to VEP are established by using Tammer’s function and the directed distance function, respectively.

4.1. Scalarization via Tammer’s Function

Lemma 5 (see [20]). Let is a pointed closed convex cone and is a fixed element, then Tammer’s function ( represents the set of real number) is defined byThen, is continuous sublinear functional and

Definition 4. Let be a nonempty subset of , , and is a real-valued function. Define optimization problem as follows: is called a -quasi-optimality solution of ifLet and . Based on VEP and Tammer’s function , consider the following scalarization problem :

Theorem 3. Let and . If is -quasi weak efficient solutions of VEP, then is -quasi-optimality solutions of scalarization problem .

Proof 3. Since is -quasi weak efficient solutions of VEP, thenConsidering Tammer’s nonlinear scalarization function ,According to Lemma 5 and combining (60) and (61) yield thatSince is continuous sublinear functional, it holds thatSince , thenTherefore, is -quasi-optimality solutions of scalarization problem .

Theorem 4. Let , , and . Suppose function satisfying the following:(i) is monotone with respect to the pointed closed convex cone , that is, if , then (ii), and is positively homogeneous functional that means , (iii)

Let scalarization problem be defined by

If is -quasi-optimality solutions of scalarization problem , then is -quasi weak efficient solutions of VEP.

Proof 4. If is not -quasi weak efficient solutions of VEP, there would exist such thatwhich is equivalent toThen, we haveSince is monotone with respect to ,Noticing that is -quasi-optimality solutions of problem , we obtainBecause and , it holds thatCombining (69) and (71), we obtainSince is positively homogeneous functional,which contradicts to condition (iii).

4.2. Scalarization via the Directed Distance Function

Let us introduce the concept of directed distance function.

Definition 5 (see [21]). Let is a nonempty subset, then the directed distance function be defined aswhere

Lemma 6 (see [21]). Let is a nonempty subset, then the following properties hold:(i) is real-valued function of rank 1.(ii)If is closed, then ; if is a cone, then is positively homogeneous; if is pointed closed convex cone, then is sublinear.(iii)If is closed convex cone, then is nonincreasing with respect to , that is, if , then

Let and . Based on VEP and the directed distance function , consider the following scalarization problem :

Theorem 5. Let , with , and . If is -quasi weak efficient solutions to VEP, then for any , is an -quasi-optimality solution of scalarization problem .

Proof 5. By the given conditions, we haveSince is a pointed closed convex cone, by Lemma 6, it yields thatNoticing that , , and , we arrive atHence, is an -quasi-optimality solution of scalarization problem .

5. Conclusions

Making use of the quasirelative interior-type separation theorem of convex set, we have examined the optimality condition of the approximate quasi weak efficient solutions of VEP. In addition, the scalarization theorems of approximate quasi weak efficient solutions to VEP are also established via using Tammer’s function and directed distance function, respectively, and scalarization theorems realize the purpose that solving the approximate quasi weak efficient solutions of vector equilibrium problem is equivalent to solving the approximate quasi-optimality solution of a specific scalar optimization problem.

Abbreviations

VEP:	Vector equilibrium problem.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final version of the manuscript.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grant no. 11861002, Natural Science Foundation of Ningxia under Grant no. NZ17112, The Key Project of North Minzu University under Grant no. ZDZX201804, and Nonlinear Analysis and Financial Optimization Research Center of North Minzu University.