1. IntroductionVector 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 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. PreliminariesThroughout the paper, we set(1)ℝ+n=x1,x2,…,xn:xi≥0, i=1,2,…,n.
Let X 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 that Q is a pointed closed convex cone in Y. We will use the following properties of Q.
Lemma 1 (see [16]).If y∗∈Q∗/0 and y∈intQ, then y∗,y>0, where int represents the interior of a set.
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(3)Tx¯;K=y∈X: ∃ tn⟶0, yn⟶y, s.t x¯+tnyn∈K.
The Clarke normal cone (see [1]) associated with Tx¯;K is denoted by(4)Nx¯;K=ξ∈X∗: ξ,y≤0, ∀ y∈Tx¯;K.
Especially when K 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.
The Clarke 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.
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(7)Fx1−Fx2≤Lx1−x2, ∀ x1,x2∈Bx¯,r.
If for any x∈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+λ d−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 locally Lipschitz function that we shall use in the sequel.
Lemma 2 (see [15,17]).Let function f: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]).Let fi: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¯.
Let K⊂X be a nonempty subset, and F:K×K⟶Y be a mapping. Consider the following vector equilibrium problem (VEP):(11)VEP find x¯∈K,such that Fx¯,x∉−Q\0, ∀ x∈K.
Given x¯∈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 that Fx¯x¯=0 and(13)Fx¯K=Fx¯,K=∪x∈KFx¯,x.
Definition 1 (see [10]).Let K⊂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 of εe-quasi weak efficient solution is illustrated by the following example.
Example 1.Let X=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.
Taking ε=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¯.
Taking x¯=0, for all x∈K, we obtain(17)Fx¯,x+εx−x¯e=0, x∉−intQ.
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(18)qriK=x∈K: clconeK−x is a linear subset of X,where cl and cone stand for closure and cone hull.
Lemma 4 (see [19]).Let M and N be nonempty convex subsets of Y, qriM≠∅ and qriN≠∅, and clcone qriM−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 ConditionsIn 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. Throughout this section, let K⊂X be a nonempty convex set.
Theorem 1.In VEP, let x¯∈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 clcone qricoFx¯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 clcone qricoFx¯K+qriQ is not a linear subspace of Y, then clcone qricoFx¯K−qri−Q is not a linear subspace of Y. Moreover,(22)qriFx¯K≠∅.
Thus,(23)qricoFx¯K≠∅.
Noticing that qriQ≠∅, 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.
Taking x=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 from λ∈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, let f:=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.
Since f 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 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∈∂fx¯ such that for each x∈X satisfying(36)ξ,x−x¯+εx−x¯≥0⟹fx−fx¯+εx−x¯≥0.
Example 2.Let X=ℝ, then f: ℝ⟶ℝ is defined by(37)fx=23x2+x,if x<0,lnx+1,if x≥0.
Taking ε=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, let ε≥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.
If λ∘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.
Since K 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.
Because b∈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.
Since λ∘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 of Fx¯x¯=0, we arrive at(50)λ∘Fx¯x+λ,eεx−x¯≥0, ∀ x∈K.
Suppose that x¯ is not εe-quasi weak efficient solutions of VEP, then there exists x^∈K such that(51)Fx¯,x^+εx^−x¯e∈−intQ.
Since λ∈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.