Approximate weakly efficient solutions of set-valued vector equilibrium problems

In this paper, we introduce a new kind of approximate weakly efficient solutions to the set-valued vector equilibrium problems with constraints in locally convex Hausdorff topological vector spaces; then we discuss a relationship between the weakly efficient solutions and approximate weakly efficient solutions. Under the assumption of near cone-subconvexlikeness, by using the separation theorem for convex sets we establish Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued vector equilibrium problems, respectively.


Introduction
Vector optimization problems, vector variational inequality problems, vector complementarity problems, and vector saddle point problems are particular cases of vector equilibrium problems. As an extensive mathematical model, the vector equilibrium problem is a hot topic in the fields of operations research and nonlinear analysis (see [1][2][3][4][5][6][7][8]). Gong [2][3][4] obtained optimality conditions for vector equilibrium problems with constraints under the assumption of cone-convexity, and by using a nonlinear scalarization function and Ioffe subdifferentiability he derived optimality conditions for weakly efficient solutions, Henig solutions, super efficient solutions, and globally efficient solutions to nonconvex vector equilibrium problems. Long et al. [5] obtained optimality conditions for Henig efficient solutions to vector equilibrium problems with functional constrains under the assumption of near cone-subconvexlikeness. Luu et al. [7,8] established sufficient and necessary conditions for efficient solutions to vector equilibrium problems with equality and inequality constraints and obtained the Fritz John and Karush-Kuhn-Tucker necessary optimality conditions for locally efficient solutions to vector equilibrium problems with constraints and sufficient conditions under assumptions of appropriate convexities.
It is well known that models describe only simplified versions of real problems and numerical algorithms generate only approximate solutions. Hence it is interesting and meaningful to have a theoretical analysis of the notion of an approximate solution. For example, Loridan [9,10] introduced the concept of -solutions in general vector optimization problems.
As far as we know, there are few papers dealing with approximate weakly efficient solutions to the set-valued vector equilibrium problems. Li et al. [11] introduced a new kind of approximate solution set of a vector approximate equilibrium problem; it is uncertain if tends to zero, whether or not the approximate solution set equals to the original solution set? It is a natural question how to define approximate weakly efficient solutions to the set-valued vector equilibrium problems and under what condition the set of approximate weakly efficient solutions equals to the set of weakly efficient solutions? This has great theoretical significance and applicable value in the research of optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems.
On the other hand, convexity plays an important role in the study of vector equilibrium problems. In 2001, Yang et al. [12] introduced a new convexity, named near conesubconvexlikeness, and proved that it is a generalization of cone-convexness and conesubconvexlikeness. In 2005, Sach (see [13]) introduced another new convexity called iccone-convexness, Xu et al. [14] proved that near cone-subconvexlikeness is also a generalization of ic-coneconvexness. Up to now, near cone-subconvexlikeness is considered to be the most generalized convexity.
Motivated by works in [3,12,15], in this paper, we introduce a new kind of approximate weakly efficient solutions to the set-valued vector equilibrium problems and reveal the relationship between weakly efficient solutions and approximate weakly efficient solutions.
We establish Kuhn-Tucker type and Lagrange-type optimality conditions for set-valued vector equilibrium problems under the assumption of the near cone-subconvexlikeness.
The organization of the paper is as follows. Some preliminary facts are given in Sect. 2 for our later use. Section 3 is devoted to the relationship between weakly efficient solutions and approximate weakly efficient solutions. In Sect. 4, we establish Kuhn-Tucker-type sufficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems. In Sect. 5, we establish Lagrange-type sufficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems. At the end of the paper, we draw some conclusions.

Preliminaries
Let X be a real topological vector space, and let Y and Z be real locally convex Hausdorff topological vector spaces with topological dual spaces Y * and Z * , respectively. Let C ⊂ Y and D ⊂ Z be pointed closed convex cones with int C = ∅ and int D = ∅. The dual cones C * of C and D * of D are defined as C * = {φ ∈ Y * : φ(c) ≥ 0, ∀c ∈ C} and D * = {ψ ∈ Z * : ψ(d) ≥ 0, ∀d ∈ D}, respectively. Let X 0 be a nonempty convex subset in X, and let G : X 0 → 2 Z and : X 0 × X 0 → 2 Y be mappings.
We denote by L(Z, Y ) the set of all continuous linear operators from Z to Y . A subset We denote the feasible set by Consider the set-valued vector equilibrium problem with constraints (for short, - is called a weakly efficient solution to the -SVEPC. The set of all weakly efficient solutions to the -SVEPC is denoted by X W min ( , A).
Let F : X 0 → 2 Y be a set-valued map. We consider the following set-valued optimization problem: We assume that the feasible set A ⊂ X 0 of (SOP) is nonempty. Definition 2.2 A feasible solutionx of (SOP) is said to be a weakly efficient solution of (SOP) if there existsȳ ∈ F(x) such that (F(A) -ȳ) ∩ (-int C) = ∅. In this case, (x,ȳ) is said to be a weakly efficient pair to (SOP).

Definition 2.3
Let ∈ C. A feasible solutionx of (SOP) is said to be an -weakly efficient solution of (SOP) if there existsȳ ∈ F(x) such that (F(A) -ȳ + ) ∩ (-int C) = ∅. In this case, (x,ȳ) is said to be an -weakly efficient pair to (SOP).
LetT ∈ L + (Z, Y ). Consider the following unconstrained set-valued optimization problem induced by (SOP): In this case, (x,ȳ) is said to be a weakly efficient pair to (USOP)T . T). In this case, (x,ȳ) is said to be an -weakly efficient pair to (USOP)T .
Several definitions of generalized convexities have been introduced in the literature.

Approximate weakly efficient solutions
Firstly, we introduce approximate weakly efficient solutions to the set-valued vector equilibrium problems with constraints.
is called an -weakly efficient solution to the -SVEPC. The set of all -weakly efficient solutions to the -SVEPC is denoted by -X W min ( , A).
Let ϒ : X 0 × X 0 → 2 Y be a mapping. Consider the following unconstrained set-valued vector equilibrium problem (for short, ϒ-USVEP): find x ∈ X 0 such that is called an -weakly efficient solution to the ϒ-USVEP. The set of all -weakly efficient solutions to the ϒ-USVEP is denoted by -X W min (ϒ, X 0 ).

Proposition 3.1
For any ∈ C, we have Thus there existsz ∈ (x,ȳ) such that Hence and thus x / ∈ X W min ( , A). Then we obtain Next, we show that in the proposition the relationship may be strict when ∈ C \ {0}.

Proposition 3.2 For any
Thus there exists z 1 ∈ (x, y 1 ) such that Since C is a convex cone, from 2 -1 ∈ C and (3.2) we have that Hence (x, y 1 ) and thus x / ∈ 1 -X W min ( , A), so we obtain In what follows, we discuss the relationship between the approximate weakly efficient solutions and weakly efficient solutions to the set-valued vector equilibrium problems with constraints.

Proposition 3.3 We have:
Proof Firstly, we prove that From Proposition 3.1 we can see that, for any ∈ C \ {0}, we have and hence we can find z 0 ∈ (x, y 0 ) such that Then there exists a neighborhood U 0 of 0 in Y such that Since z 0 ∈ (x, y 0 ), we have Hencex / ∈ 0 -X W min ( , A), and thereforē From this we obtain ∈C\{0} -X W min ( , A) = X W min ( , A).

Kuhn-Tucker-type optimality conditions
In this section, under the assumption of near C-subconvexlikeness, we establish Kuhn-Tucker-type sufficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems, which generalize the relevant results given by Gong [2] and Yang [17].

Definition 4.1
Letx ∈ X 0 , and let ϕ : X 0 → 2 Y ×Z be an ordered pair mapping defined as  (4.1) Next, we prove that Suppose to the contrary that there existλ ≥ 0 andx ∈ X 0 such that Thus, From 0 / ∈ -int D we haveλ > 0. Since D is a convex cone, combining with (4.4), we have It is clear that Since C is a convex cone, we have which contradicts (4.1), and thus we obtain (4.2).
Since -int C and -int D are open sets, combining with (4.2), we have Since ϕ is nearly C × D-subconvexlike on X 0 , by Definition 4.1 we can see that cl(cone(ϕ(X 0 ) + C × D)) is convex. By the separation theorem for convex sets, there exists Since cl(cone(ϕ(X 0 ) + C × D)) is a cone, and there is a lower bound of (s * , k * ) on cl(cone(ϕ(X 0 ) + C × D)), we have It is clear that From (4.6) we get (4.8) Next, we prove that Suppose to the contrary that there exists c 0 ∈ C such that s * (c 0 ) < 0.
Then we need to prove that Suppose to the contrary that (4.7) we can see that On the other hand, there exists x 0 ∈ X 0 such that G(x 0 ) ∩ (-int D) = ∅, and thus there exists p ∈ G(x 0 ) ∩ (-int D), so that by Lemma 4.1 we obtain k * (p) < 0, which contradicts (4.9). Hence The proof is complete.
Ifx is a weakly efficient solution to the -SVEPC, then there exist s * ∈ C * \ {0 Y * } and k * ∈ D * such that min k * (G(x)) = 0 and Proof In the proof of Theorem 4.1, letting = 0, we see that Thus there exists q ∈ G(x) such that q ∈ -D, and since k * ∈ D * , we have Letting x =x in (4.11), by 0 ∈ (x,x) we have From q ∈ G(x) we have k * (q) ≥ 0. Combining with (4.12), we obtain k * (q) = 0.

Theorem 4.2 Assume that (i)x ∈ A and
(ii) there exist s * ∈ C * \ {0 Y * } and k * ∈ D * such that Thenx is an -weakly efficient solution to the -SVEPC.

Corollary 4.2 Assume that (i)x ∈ A and
(ii) there exist s * ∈ C * \ {0 Y * } and k * ∈ D * such that Thenx is a weakly efficient solution to the -SVEPC.
Proof Letting = 0 in Theorem 4.2, we get the conclusion.

Lagrange-type optimality conditions
In this section, we present Lagrange-type sufficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems, which generalize the relevant results given by Rong [15]. C)), andx is an -weakly efficient solution to the -USVEP, where : Proof From the proof of Theorem 4.1 we see that there exist s * ∈ C * \ {0 Y * } and k * ∈ D * satisfying (4.7). Since s * ∈ C * \ {0 Y * }, we can find c 0 ∈ int C such that s * (c 0 ) = 1. Define the operator Letting x =x in (4.7), since 0 ∈ (x,x), we have Noticing that z ∈ -D, we obtain Next, we prove Finally, we prove thatx is an -weakly efficient solution to the -USVEP.
In fact, by the definition ofT and (4.7) we obtain Since s * (-int C) < 0, we have Consequently, It is evident thatx is an -weakly efficient solution to the -USVEP.

Theorem 5.2 Assume that
Thenx is an -weakly efficient solution to the -SVEPC.
Proof Sincex is an -weakly efficient solution to the -USVEP, we have Since C is a convex cone, we obtain On the other hand, for any It follows from (5.6) that Since 0 ∈ C, it is evident that Hencex is an -weakly efficient solution to the -SVEPC.
From Theorems 5.1 and 5.2 we obtain the following result.
Since (x,ȳ) is an -weakly efficient pair to the (SOP), we see thatx is an -weakly efficient solution to the -SVEPC.
Since (x,ȳ) is an -weakly efficient pair to the (USOP)T , we see thatx is an -weakly efficient solution to the -USVEP. Combining this with Theorem 5.2, we conclude that x is an -weakly efficient solution to the -SVEPC; it is clear that (x,ȳ) is an -weakly efficient pair to the (SOP).
Remark 5.2 Comparing with Theorem 3.2 in [15], this corollary is not required for the convexity of (F, G).

Conclusions
In this paper, we discuss some relationships between approximate weakly efficient solutions and weakly efficient solutions of set-valued vector equilibrium problems. We conclude that ∈C\{0} -X W min ( , A) = X W min ( , A), and hence it is really "approximate". The optimality conditions for set-valued vector equilibrium problems are established, and the results we obtained generalize those of Gong [2], Yang [17], and Rong [15]. As an extensive mathematical model, further research on approximate weakly efficient solutions of set-valued vector equilibrium problems seems to be of interest and value.