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

Authors: Yameng Zhang; Guolin Yu; Wenyan Han
Journal: Complexity
 (2020)

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## 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.


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## Body

## 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 Clarke 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 section2, 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(1)ℝ+n=x1,x2,…,xn:xi≥0,i=1,2,…,n.LetX and Y be real Banach spaces with topological dual spaces X∗ and Y∗, respectively, and Bx¯,r stands for the open ball of radius r>0 around x¯∈X. For all x∈X and x∗∈X∗, the value of linear functional x∗ at x be denoted by x∗,x. Let Q be a pointed closed convex cone in Y, then the dual cone of Q be defined as (see [15])(2)Q∗=y∗∈Y∗:y∗,y≥0,∀y∈Q.Without other specifications, we always suppose thatQ is a pointed closed convex cone in Y. We will use the following properties of Q.Lemma 1 (see [16]).
Ify∗∈Q∗/0 and y∈intQ, then y∗,y>0, where int represents the interior of a set.LetK be a nonempty subset of X, and the Clarke contingent cone (see [1]) to set K at point x¯∈K is defined as(3)Tx¯;K=y∈X:∃tn⟶0,yn⟶y,s.tx¯+tnyn∈K.TheClarke normal cone (see [1]) associated with Tx¯;K is denoted by(4)Nx¯;K=ξ∈X∗:ξ,y≤0,∀y∈Tx¯;K.Especially whenK be a convex set, the Clarke contingent cone to set K at x¯ is given by (see [15])(5)Tx¯;K=cly∈X:y=βx−x¯,x∈K,β>0.TheClarke normal cone to set K at x¯ is(6)Nx¯;K=ξ∈X∗:ξ,x−x¯≤0,∀x∈K,where cl stands for the closure of a set.LetF:X⟶Y be a mapping. F is said to be locally Lipschitz at x¯∈X, if there exist constant L>0 and r>0 such that(7)Fx1−Fx2≤Lx1−x2,∀x1,x2∈Bx¯,r.If for anyx∈X, F is locally Lipschitz at x, then F is called locally Lipschitz mapping. In particular, for a real-valued locally Lipschitz function f:X⟶ℝ (ℝ denotes real number), the Clarke generalized directional derivative of f at x¯∈X in the direction d∈X is given by (see [15])(8)f°x¯;d=limy⟶x¯supλ⟶0+fy+λd−fyλ,∂fx¯=ξ∈X∗:f°x¯;d≥ξ,d,∀d∈X,which is defined as the Clarke subdifferential of f at x¯.We present below some significant properties of locallyLipschitz function that we shall use in the sequel.Lemma 2 (see [15,17]).
Let functionf:K⊂X⟶ℝ is locally Lipschitz at x¯∈K, if x¯ is the minimum value point of f on K, then(9)0∈∂fx¯+Nx¯;K.Lemma 3 (see [15]).
Letfi:X⟶ℝ,i=1,…,m, be locally Lipschitz at x¯∈X, then function φ⋅:=maxfi⋅:i=1,…,m is also locally Lipschitz at x¯, and(10)∂φx¯⊂∪∑i=1mλi∂fix¯:λi≥0,i=1,2,…m,∑i=1mλi=1,λifix¯−φx¯=0,∂f1+⋯+fmx¯⊂∂f1x¯+⋯+∂fmx¯.LetK⊂X be a nonempty subset, and F:K×K⟶Y be a mapping. Consider the following vector equilibrium problem (VEP):(11)VEPfindx¯∈K,such thatFx¯,x∉−Q\0,∀x∈K.Givenx¯∈K, Fx¯:K⟶Y be vector-valued mapping of one variable, which is defined by(12)Fx¯x≔Fx¯,x,∀x∈K.Throughout this paper, it is always assumed thatFx¯x¯=0 and(13)Fx¯K=Fx¯,K=∪x∈KFx¯,x.Definition 1 (see [10]).
LetK⊂X be a nonempty subset, ε≥0, e∈intQ. x¯∈K is called an εe-quasi weak efficient solution to VEP, if(14)Fx¯,x+εx−x¯e∉−intQ,∀x∈K.
The notion ofεe-quasi weak efficient solution is illustrated by the following example.Example 1.
LetX=Y=ℝ, K=ℝ+, Q=ℝ+2, and x¯∈K. Consider the following questions:(15)Fx¯,x=−x−x¯,x−x¯−x−x¯,∀x∈K.
Takingε=1 and e=1,1, then(16)Fx¯,x+εx−x¯e=−x−x¯,x−x¯−x−x¯+x−x¯,x−x¯=0,x−x¯.
Takingx¯=0, for all x∈K, we obtain(17)Fx¯,x+εx−x¯e=0,x∉−intQ.
Hence, 0 is anεe-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]).
LetK⊂X is a convex subset; the quasirelative interior of K denoted by qriK is defined as(18)qriK=x∈K:clconeK−xis a linear subset ofX,where cl and cone stand for closure and cone hull.Lemma 4 (see [19]).
LetM and N be nonempty convex subsets of Y, qriM≠∅and qriN≠∅, and clconeqriM−qriN is not a linear subset of Y, then there exists λ∈Y∗0 such that(19)λ,m≤λ,n,∀m∈M,∀n∈N.


## 3. Optimality Conditions

In this section, first, we propose a necessary optimality condition forεe-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 K⊂X be a nonempty convex set.Theorem 1.
In VEP, letx¯∈K, ε≥0, and e∈intQ. Assume that x¯ be an εe-quasi weak efficient solution of VEP and Fx¯:X⟶Y is locally Lipschitz mapping at x¯. In addition, qriFx¯K≠∅ and clconeqricoFx¯K+qriQ is not a linear subspace of Y. Then, there exist λ∈Q∗0 such that(20)0∈∂λ∘Fx¯x¯+Nx¯;K+λ,eεB,where co⋅ stands for the convex hull, B:=B0,1, and λ°Fx¯⋅:=λ,Fx¯⋅.Proof 1.
Since(21)qriQ=−qri−Q,and clconeqricoFx¯K+qriQ is not a linear subspace of Y, then clconeqricoFx¯K−qri−Q is not a linear subspace of Y. Moreover,(22)qriFx¯K≠∅.
Thus,(23)qricoFx¯K≠∅.
Noticing thatqriQ≠∅, it follows from Lemma 4 that there exists λ∈Y∗\0 such that(24)λ,q≤λ,x,∀q∈−Q,∀x∈coFx¯K,which means(25)λ,q≤λ,Fx¯x,∀q∈−Q,∀x∈K.
Takingx=x¯ in the above formula, we obtain(26)λ,q≤0,∀q∈−Q.
Hence,λ∈Q∗\0. Since(27)Fx¯,x+εx−x¯e∉−intQ,∀x∈K,and Fx¯,x¯=0, it leads to(28)Fx¯,x−Fx¯,x¯+εx−x¯e∉−intQ,∀x∈K.
It follows fromλ∈Q∗\0 and equation (28) that(29)λ,Fx¯,x−Fx¯,x¯+εx−x¯e≥0,∀x∈K,that is(30)λ,Fx¯x−λ,Fx¯x¯+λ,eεx−x¯≥0,∀x∈K.
On the other hand, letf:=λ∘Fx¯. Since Fx¯ is locally Lipschitz at x¯, it is obvious that f is a locally Lipschitz function at x¯. We set(31)φx=fx−fx¯+λ,eεx−x¯,∀x∈K.
It follows from equation (29) that(32)φx≥0=φx¯,∀x∈K,which shows that x¯ is the minimum point of φx on K. Taking account of Lemma 2, we arrive at(33)0∈∂φx¯+Nx¯;K.
Sincef is a locally Lipschitz function at x¯, by Lemma 3, we have(34)∂φx¯⊂∂f+λ,eε⋅−x¯x¯,⊂∂λ∘Fx¯x¯+λ,eεB.
Together with equation (33), we obtain(35)0∈∂λ∘Fx¯x¯+Nx¯;K+λ,eεB.
Next, we introduce the concept of approximate quasi-pseudoconvex function, and under the assumption of this generalized convexity, a sufficient optimality condition forεe-quasi weak efficient solutions to VEP is derived.Definition 3.
Letε≥0 and the function f:X⟶ℝ be locally Lipschitz at x¯∈X. f is said to be ε-quasi-pseudoconvex at x¯, if there exists ξ∈∂fx¯ such that for each x∈X satisfying(36)ξ,x−x¯+εx−x¯≥0⟹fx−fx¯+εx−x¯≥0.Example 2.
LetX=ℝ, then f:ℝ⟶ℝ is defined by(37)fx=23x2+x,ifx<0,lnx+1,ifx≥0.
Takingε=1 and x¯=0, by a simple computation, we derive ∂fx¯=1. For any x∈ℝ, 1=ξ∈∂f0, if(38)ξ,x−x¯+εx−x¯=1⋅x+1⋅x≥0,then(39)fx−fx¯+εx−x¯=fx+1⋅x=23x2≥0,x<0,lnx+1+x≥0,x≥0.
Thus,f is a 1-quasi-pseudoconvex at 0.Theorem 2.
In VEP, letε≥0, e∈intQ, x¯∈K, and Fx¯:K⟶Y be locally Lipschitz at x¯. Suppose that there exists λ∈Q∗\0 such that(40)0∈∂λ∘Fx¯x¯+Nx¯;K+λ,eεB.
Ifλ∘Fx¯:K⟶ℝ is λ,eε-quasi-pseudoconvex at x¯, then x¯ is εe-quasi weak efficient solutions of VEP.Proof 2.
It follows from (40) that there exist ξ∈∂λ∘Fx¯x¯, σ∈Nx¯;K, and b∈B such that(41)ξ+σ+λ,eεb=0,which implies for each x∈K,(42)ξ+σ+λ,eεb,x−x¯=0,which is equivalent to(43)ξ,x−x¯+σ,x−x¯+λ,eεb,x−x¯=0,∀x∈K.
SinceK is a convex set, according to the definition of contingent cone to set K at x¯,(44)Tx¯;K=cly∈X:y=βx−x¯,x∈K,β>0.
Therefore,(45)σ,x−x¯≤0,∀x∈K.
Combining (43) and (44), it is not difficult to find(46)ξ,x−x¯+λ,eεb,x−x¯≥0,∀x∈K.
Becauseb∈B, we obtain b=1. Hence,(47)b,x−x¯≤x−x¯,∀x∈K.
Together with equation (46), it leads to(48)ξ,x−x¯+λ,eεx−x¯≥0,∀x∈K.
Sinceλ∘Fx¯ is λ,eε-quasi-pseudoconvex at x¯, by Definition 3, we obtain(49)λ∘Fx¯x−λ∘Fx¯x¯+λ,eεx−x¯≥0,∀x∈K.
In view ofFx¯x¯=0, we arrive at(50)λ∘Fx¯x+λ,eεx−x¯≥0,∀x∈K.
Suppose thatx¯ is not εe-quasi weak efficient solutions of VEP, then there exists x^∈K such that(51)Fx¯,x^+εx^−x¯e∈−intQ.
Sinceλ∈Q∗\0, it yields from Lemma 1 that(52)λ,Fx¯,x^+εx^−x¯e<0,which means(53)λ,Fx¯,x^+λ,εx^−x¯e<0.
That is,(54)λ∘Fx¯x^+λ,eεx^−x¯<0,which contradicts (50). Hence, x¯ is εe-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]).
LetQ⊂Y is a pointed closed convex cone and e∈intQ≠∅ is a fixed element, then Tammer’s function ΨeQ:Y⟶ℝ (ℝ represents the set of real number) is defined by(55)ΨeQy=inft∈R:y∈te−Q,y∈Y.
Then,ΨeQ is continuous sublinear functional and(56)y∈Y:ΨeQy<0=−intQ.Definition 4.
LetK be a nonempty subset of X, ε≥0, and f:X⟶ℝ is a real-valued function. Define optimization problem P as follows:(57)Pinffx,s.t.x∈K.
x¯ is called a ε-quasi-optimality solution of P if(58)fx−fx¯+εx−x¯≥0,∀x∈K.
Letx¯∈K and e∈intQ. Based on VEP and Tammer’s function ΨeQ, consider the following scalarization problem PΨeQ:(59)PΨeQinfΨeQFx¯x,s.t.x∈K.Theorem 3.
Letε≥0 and e∈intQ. If x¯∈K is εe-quasi weak efficient solutions of VEP, then x¯ is ε-quasi-optimality solutions of scalarization problem PΨeQ.Proof 3.
Sincex¯∈K is εe-quasi weak efficient solutions of VEP, then(60)Fx¯,x+εx−x¯e∉−intQ,∀x∈K.
Considering Tammer’s nonlinear scalarization functionΨeQ,(61)ΨeQy=inft∈ℝ:y∈te−Q,y∈Y.
According to Lemma5 and combining (60) and (61) yield that(62)ΨeQFx¯,x+εx−x¯e≥0,∀x∈K.
SinceΨeQ is continuous sublinear functional, it holds that(63)ΨeQFx¯,x+εx−x¯≥0,∀x∈K.
SinceFx¯,x¯=0, then(64)ΨeQFx¯,x−ΨeQFx¯,x¯+εx−x¯≥0,∀x∈K.
Therefore,x¯ is ε-quasi-optimality solutions of scalarization problem PΨeQ.Theorem 4.
Letε≥0, e∈intQ, and x¯∈K. Suppose function φ:Y⟶ℝ satisfying the following:(i)
φ is monotone with respect to the pointed closed convex cone Q, that is, if y1−y2∈Q, then φy1≥φy2(ii)
φ0=0, and φ is positively homogeneous functional that means φay=aφy, a>0(iii)
φ−εe<−εLet scalarization problemPφ be defined by(65)PφinfφFx¯x,s.t.x∈K.Ifx¯ is ε-quasi-optimality solutions of scalarization problem Pφ, then x¯ is εe-quasi weak efficient solutions of VEP.Proof 4.
Ifx¯ is not εe-quasi weak efficient solutions of VEP, there would exist x^∈K such that(66)Fx¯,x^+εx^−x¯e∈−intQ,which is equivalent to(67)−εx^−x¯e−Fx¯,x^∈intQ.
Then, we have(68)−εx^−x¯e−Fx¯,x^∈Q.
Sinceφ is monotone with respect to Q,(69)φFx¯,x^≤φ−εx^−x¯e.
Noticing thatx¯ is ε-quasi-optimality solutions of problem Pφ, we obtain(70)φFx¯,x^−φFx¯,x¯+εx^−x¯≥0.
BecauseFx¯,x¯=0 and φ0=0, it holds that(71)φFx¯,x^+εx^−x¯≥0.
Combining (69) and (71), we obtain(72)φ−εx^−x¯e≥−εx^−x¯.
Sinceφ is positively homogeneous functional,(73)φ−εe≥−ε,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]).
LetA⊆Y is a nonempty subset, then the directed distance function ΔA:Y⟶ℝ be defined as(74)ΔAy=dAy−dY\Ay,y∈Y,where(75)dAy=infx∈Ay−x.Lemma 6 (see [21]).
LetA⊆Y is a nonempty subset, then the following properties hold:(i)
ΔA is real-valued Lipschitz function of rank 1.(ii)
IfA is closed, then A=y:ΔAy≤0; if A is a cone, then △A is positively homogeneous; if A is pointed closed convex cone, then ΔA is sublinear.(iii)
IfA is closed convex cone, then ΔA is nonincreasing with respect to A, that is, if y1,y2∈Y, then(76)y1−y2∈A⟹ΔAy1≤ΔAy2;y1−y2∈intA⟹ΔAy1<ΔAy2.Lete∈intQ and x¯∈K. Based on VEP and the directed distance function Δ−Q, consider the following scalarization problem PΔ−Q:(77)PΔ−QinfΔ−QFx¯x,s.t.x∈K.Theorem 5.
Letε>0, e∈intQ with e=1, and x¯∈K. If x¯ is εe-quasi weak efficient solutions to VEP, then for any x∈K, x¯ is an ε-quasi-optimality solution of scalarization problem PΔ−Q.Proof 5.
By the given conditions, we have(78)Fx¯,x+εx−x¯e∉−intQ,∀x∈K.
SinceQ is a pointed closed convex cone, by Lemma 6, it yields that(79)Δ−QFx¯,x+εx−x¯−Qe≥Δ−QFx¯,x+x−x¯e≥0,∀x∈K.
Noticing thatΔ−Qe=e=1, Fx¯,x¯=0, and Δ−Q0=0, we arrive at(80)Δ−QFx¯,x−Δ−QFx¯,x+εx−x¯≥0,∀x∈K.
Hence,x¯ is an ε-quasi-optimality solution of scalarization problem PΔ−Q.


## 4.1. Scalarization via Tammer’s Function

Lemma 5 (see [20]).
LetQ⊂Y is a pointed closed convex cone and e∈intQ≠∅ is a fixed element, then Tammer’s function ΨeQ:Y⟶ℝ (ℝ represents the set of real number) is defined by(55)ΨeQy=inft∈R:y∈te−Q,y∈Y.
Then,ΨeQ is continuous sublinear functional and(56)y∈Y:ΨeQy<0=−intQ.Definition 4.
LetK be a nonempty subset of X, ε≥0, and f:X⟶ℝ is a real-valued function. Define optimization problem P as follows:(57)Pinffx,s.t.x∈K.
x¯ is called a ε-quasi-optimality solution of P if(58)fx−fx¯+εx−x¯≥0,∀x∈K.
Letx¯∈K and e∈intQ. Based on VEP and Tammer’s function ΨeQ, consider the following scalarization problem PΨeQ:(59)PΨeQinfΨeQFx¯x,s.t.x∈K.Theorem 3.
Letε≥0 and e∈intQ. If x¯∈K is εe-quasi weak efficient solutions of VEP, then x¯ is ε-quasi-optimality solutions of scalarization problem PΨeQ.Proof 3.
Sincex¯∈K is εe-quasi weak efficient solutions of VEP, then(60)Fx¯,x+εx−x¯e∉−intQ,∀x∈K.
Considering Tammer’s nonlinear scalarization functionΨeQ,(61)ΨeQy=inft∈ℝ:y∈te−Q,y∈Y.
According to Lemma5 and combining (60) and (61) yield that(62)ΨeQFx¯,x+εx−x¯e≥0,∀x∈K.
SinceΨeQ is continuous sublinear functional, it holds that(63)ΨeQFx¯,x+εx−x¯≥0,∀x∈K.
SinceFx¯,x¯=0, then(64)ΨeQFx¯,x−ΨeQFx¯,x¯+εx−x¯≥0,∀x∈K.
Therefore,x¯ is ε-quasi-optimality solutions of scalarization problem PΨeQ.Theorem 4.
Letε≥0, e∈intQ, and x¯∈K. Suppose function φ:Y⟶ℝ satisfying the following:(i)
φ is monotone with respect to the pointed closed convex cone Q, that is, if y1−y2∈Q, then φy1≥φy2(ii)
φ0=0, and φ is positively homogeneous functional that means φay=aφy, a>0(iii)
φ−εe<−εLet scalarization problemPφ be defined by(65)PφinfφFx¯x,s.t.x∈K.Ifx¯ is ε-quasi-optimality solutions of scalarization problem Pφ, then x¯ is εe-quasi weak efficient solutions of VEP.Proof 4.
Ifx¯ is not εe-quasi weak efficient solutions of VEP, there would exist x^∈K such that(66)Fx¯,x^+εx^−x¯e∈−intQ,which is equivalent to(67)−εx^−x¯e−Fx¯,x^∈intQ.
Then, we have(68)−εx^−x¯e−Fx¯,x^∈Q.
Sinceφ is monotone with respect to Q,(69)φFx¯,x^≤φ−εx^−x¯e.
Noticing thatx¯ is ε-quasi-optimality solutions of problem Pφ, we obtain(70)φFx¯,x^−φFx¯,x¯+εx^−x¯≥0.
BecauseFx¯,x¯=0 and φ0=0, it holds that(71)φFx¯,x^+εx^−x¯≥0.
Combining (69) and (71), we obtain(72)φ−εx^−x¯e≥−εx^−x¯.
Sinceφ is positively homogeneous functional,(73)φ−εe≥−ε,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]).
LetA⊆Y is a nonempty subset, then the directed distance function ΔA:Y⟶ℝ be defined as(74)ΔAy=dAy−dY\Ay,y∈Y,where(75)dAy=infx∈Ay−x.Lemma 6 (see [21]).
LetA⊆Y is a nonempty subset, then the following properties hold:(i)
ΔA is real-valued Lipschitz function of rank 1.(ii)
IfA is closed, then A=y:ΔAy≤0; if A is a cone, then △A is positively homogeneous; if A is pointed closed convex cone, then ΔA is sublinear.(iii)
IfA is closed convex cone, then ΔA is nonincreasing with respect to A, that is, if y1,y2∈Y, then(76)y1−y2∈A⟹ΔAy1≤ΔAy2;y1−y2∈intA⟹ΔAy1<ΔAy2.Lete∈intQ and x¯∈K. Based on VEP and the directed distance function Δ−Q, consider the following scalarization problem PΔ−Q:(77)PΔ−QinfΔ−QFx¯x,s.t.x∈K.Theorem 5.
Letε>0, e∈intQ with e=1, and x¯∈K. If x¯ is εe-quasi weak efficient solutions to VEP, then for any x∈K, x¯ is an ε-quasi-optimality solution of scalarization problem PΔ−Q.Proof 5.
By the given conditions, we have(78)Fx¯,x+εx−x¯e∉−intQ,∀x∈K.
SinceQ is a pointed closed convex cone, by Lemma 6, it yields that(79)Δ−QFx¯,x+εx−x¯−Qe≥Δ−QFx¯,x+x−x¯e≥0,∀x∈K.
Noticing thatΔ−Qe=e=1, Fx¯,x¯=0, and Δ−Q0=0, we arrive at(80)Δ−QFx¯,x−Δ−QFx¯,x+εx−x¯≥0,∀x∈K.
Hence,x¯ is an ε-quasi-optimality solution of scalarization problem PΔ−Q.


## 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.



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