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

*is 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.


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. e 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][5][6][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. ere 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. e 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.

Preliminaries
roughout the paper, we set Let X and Y be real Banach spaces with topological dual spaces X * and Y * , respectively, and B(x, 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]) Q * � y * ∈ Y * : 〈y * , y〉 ≥ 0, ∀ y ∈ Q . (2) Without other specifications, we always suppose that Q is a pointed closed convex cone in Y. We will use the following properties of Q.
Let K be a nonempty subset of X, and the Clarke contingent cone (see [1]) to set K at point x ∈ K is defined as T(x; K) � y ∈ X: ∃ t n ⟶ 0, y n ⟶ y, s.t x + t n y n ∈ K .
(3) e Clarke normal cone (see [1]) associated with T(x; K) is denoted by N(x; K) � ξ ∈ X * : 〈ξ, y〉 ≤ 0, ∀ y ∈ T(x; K) . (4) Especially when K be a convex set, the Clarke contingent cone to set K at x is given by (see [15]) where cl stands for the closure of a set. Let F: 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 If for any x ∈ X, F is locally Lipschitz at x, then F is called locally Lipschitz mapping. In particular, for a realvalued 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]) which is defined as the Clarke subdifferential of f at x. We present below some significant properties of locally Lipschitz function that we shall use in the sequel.
Let K ⊂ X be a nonempty subset, and F: K × K ⟶ Y be a mapping. Consider the following vector equilibrium problem (VEP): 2 Complexity Given x ∈ K, F x : K ⟶ Y be vector-valued mapping of one variable, which is defined by roughout this paper, it is always assumed that F x (x) � 0 and Definition 1 (see [10]). Let K ⊂ X be a nonempty subset, e notion of εe-quasi weak efficient solution is illustrated by the following example.
, and x ∈ K. Consider the following questions: Taking ε � 1 and e � (1, 1), then Hence, 0 is an εe-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 K ⊂ X is a convex subset; the quasirelative interior of K denoted by qriK is defined as where cl and cone stand for closure and cone hull.

Optimality Conditions
In this section, first, we propose a necessary optimality condition for εe-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. roughout this section, let K ⊂ X be a nonempty convex set.
and clcone [qri(coF x (K)) + qriQ] is not a linear subspace of Y, then clcone [qri(coF x (K)) − qri(− Q)] is not a linear subspace of Y. Moreover, us, Noticing that qriQ ≠ ∅, it follows from Lemma 4 that which means Taking x � x in the above formula, we obtain and F(x, x) � 0, it leads to It follows from λ ∈ Q * \ 0 { } and equation (28) that On the other hand, let f : � λ ∘ F x . Since F x is locally Lipschitz at x, it is obvious that f is a locally Lipschitz function at x. We set It follows from equation (29) that which shows that x is the minimum point of φ(x) on K.
Taking account of Lemma 2, we arrive at 0 ∈ zφ(x) + N(x; K). (33) Since f is a locally Lipschitz function at x, by Lemma 3, we have Together with equation (33), we obtain Next, we introduce the concept of approximate quasipseudoconvex function, and under the assumption of this generalized convexity, a sufficient optimality condition for εe-quasi weak efficient solutions to VEP is derived.
Definition 3. Let ε ≥ 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 ∈ zf(x) such that for each x ∈ X satisfying Taking ε � 1 and x � 0, by a simple computation, we us, f is a 1-quasi-pseudoconvex at 0.
which implies for each x ∈ K, which is equivalent to (43) Since K is a convex set, according to the definition of contingent cone to set K at x, (45) Combining (43) and (44), it is not difficult to find Because b ∈ B, we obtain ‖b‖ � 1. Hence, Together with equation (46), it leads to Since λ ∘ F x is 〈λ, e〉ε-quasi-pseudoconvex at x, by Definition 3, we obtain In view of F x (x) � 0, we arrive at Suppose that x is not εe-quasi weak efficient solutions of VEP, then there exists x ∈ K such that Since λ ∈ Q * \ 0 { }, it yields from Lemma 1 that which means at is, which contradicts (50). Hence, x is εe-quasi weak efficient solutions of VEP.

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.

Scalarization via Tammer's Function
Lemma 5 (see [20]). Let Q ⊂ Y is a pointed closed convex cone and e ∈ intQ ≠ ∅ is a fixed element, then Tammer's function Ψ Q e : Y ⟶ R (R represents the set of real number) is defined by en, Ψ Q e is continuous sublinear functional and Definition 4. Let K be a nonempty subset of X, ε ≥ 0, and f: X ⟶ R is a real-valued function. Define optimization problem (P) as follows: x is called a ε-quasi-optimality solution of (P) if Let x ∈ K and e ∈ intQ. Based on VEP and Tammer's function Ψ Q e , consider the following scalarization problem Theorem 3. Let ε ≥ 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 (P Ψ Q e ).
Proof 3. Since x ∈ K is εe-quasi weak efficient solutions of VEP, then Considering Tammer's nonlinear scalarization function According to Lemma 5 and combining (60) and (61) yield that Since Ψ Q e is continuous sublinear functional, it holds that Since F(x, x) � 0, then erefore, x is ε-quasi-optimality solutions of scalarization problem (P Ψ Q e ).
which is equivalent to Since φ is monotone with respect to Q, Noticing that x is ε-quasi-optimality solutions of problem (P φ ), we obtain Because F(x, x) � 0 and φ(0) � 0, it holds that Combining (69) and (71), we obtain Since φ is positively homogeneous functional, which contradicts to condition (iii).

Scalarization via the Directed Distance Function.
Let us introduce the concept of directed distance function.
Definition 5 (see [21]). Let A ⊆ Y is a nonempty subset, then the directed distance function Δ A : Y ⟶ R be defined as where Complexity Lemma 6 (see [21]). Let A ⊆ Y is a nonempty subset, then the following properties hold: Let e ∈ intQ and x ∈ K. Based on VEP and the directed distance function Δ − Q , consider the following scalarization problem (P Δ − Q ): Theorem 5. Let ε > 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 (P Δ − Q ).

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.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.