- Research
- Open Access
Approximate weakly efficient solutions of set-valued vector equilibrium problems
- Jian Chen^{1},
- Yihong Xu^{1}Email author and
- Ke Zhang^{1}
https://doi.org/10.1186/s13660-018-1773-0
© The Author(s) 2018
- Received: 5 May 2018
- Accepted: 22 June 2018
- Published: 20 July 2018
Abstract
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.
Keywords
- Set-valued vector equilibrium problem
- Approximate weakly efficient solution
- Near cone-subconvexlikeness
- Optimality condition
MSC
- 90C33
- 90C46
- 90C59
1 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 [1–8]). Gong [2–4] 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 ϵ-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 ϵ 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 cone-subconvexlikeness, and proved that it is a generalization of cone-convexness and cone-subconvexlikeness. In 2005, Sach (see [13]) introduced another new convexity called ic-cone-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.
2 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 \(C\subset Y\) and \(D\subset Z\) be pointed closed convex cones with \(\operatorname{int}C\neq \emptyset \) and \(\operatorname{int}D\neq \emptyset \). The dual cones \(C^{*}\) of C and \(D^{*}\) of D are defined as \(C^{*}=\{\phi \in Y^{*}:\phi (c)\geq 0, \forall c\in C\}\) and \(D^{*}=\{\psi \in Z^{*}:\psi (d)\geq 0,\forall d\in D\}\), respectively. Let \(X_{0}\) be a nonempty convex subset in X, and let \(G:X_{0}\rightarrow 2^{Z}\) and \(\Phi:X_{0}\times X_{0}\rightarrow 2^{Y}\) be mappings.
We denote by \(L(Z,Y)\) the set of all continuous linear operators from Z to Y. A subset \(L^{+}(Z,Y)\) of \(L(Z,Y)\) is defined as \(L^{+}(Z,Y)=\{T\in L(Z,Y):T(D)\subset C\}\).
Definition 2.1
We assume that the feasible set \(A\subset X_{0}\) of (SOP) is nonempty.
Definition 2.2
A feasible solution x̄ of (SOP) is said to be a weakly efficient solution of (SOP) if there exists \(\bar{y}\in F(\bar{x})\) such that \((F(A)-\bar{y})\cap (-\operatorname{int}C)=\emptyset\). In this case, \((\bar{x},\bar{y})\) is said to be a weakly efficient pair to (SOP).
Definition 2.3
Let \(\epsilon \in C\). A feasible solution x̄ of (SOP) is said to be an ϵ-weakly efficient solution of (SOP) if there exists \(\bar{y}\in F(\bar{x})\) such that \((F(A)-\bar{y}+\epsilon)\cap ( -\operatorname{int}C)=\emptyset\). In this case, \((\bar{x},\bar{y})\) is said to be an ϵ-weakly efficient pair to (SOP).
Definition 2.4
A vector \(\bar{x}\in X_{0}\) is said to be a weakly efficient solution of \((\mathrm{USOP})_{\bar{T}}\) if there exists \(\bar{y} \in F(\bar{x})\) such that \((L(X_{0},\bar{T})-\bar{y})\cap ( -\operatorname{int}C)=\emptyset\), where \(L(X_{0},\bar{T})=\bigcup_{x\in X_{0}}L(x,\bar{T})\). In this case, \((\bar{x},\bar{y})\) is said to be a weakly efficient pair to \((\mathrm{USOP})_{\bar{T}}\).
Definition 2.5
Let \(\epsilon \in C\). A vector \(\bar{x}\in X_{0}\) is said to be an ϵ-weakly efficient solution of \((\mathrm{USOP})_{\bar{T}}\) if \(\exists \bar{y}\in F(\bar{x})\) such that \((L(X_{0},\bar{T})-\bar{y}+ \epsilon)\cap (-\operatorname{int}C)=\emptyset\), where \(L(X_{0},\bar{T})= \bigcup_{x\in X_{0}}L(x,\bar{T})\). In this case, \((\bar{x}, \bar{y})\) is said to be an ϵ-weakly efficient pair to \((\mathrm{USOP})_{\bar{T}}\).
Several definitions of generalized convexities have been introduced in the literature.
Definition 2.6
Definition 2.7
([16])
Definition 2.8
([17])
Definition 2.9
([12])
The map \(F:X_{0}\rightarrow 2^{Y}\) is called nearly C-subconvexlike on \(X_{0}\) iff clcone\((F(X_{0})+C)\) is convex.
3 Approximate weakly efficient solutions
Firstly, we introduce approximate weakly efficient solutions to the set-valued vector equilibrium problems with constraints.
Definition 3.1
Definition 3.2
Proposition 3.1
Proof
Next, we show that in the proposition the relationship may be strict when \(\epsilon \in C\setminus {\{0\}}\).
Example 3.1
Let \(X=R^{1}\), \(A=[0,2]\), \(Y=R^{2}\), \(C=R_{+}^{2}\), and \(\epsilon =(x_{0},y_{0})\in C\setminus {\{0\}}\). Let \(\Phi:A\times A \longrightarrow 2^{Y}\) be defined by \(\Phi (x,y)=\{(p,q):q\geq p^{2}-x \}\cap ([-y,y]\times [0,+\infty))\), \(\forall x,y\in A\). It is obvious that \(X_{W\min }(\Phi,A)=\{0\}\); however, \(\epsilon \text{-}X_{W \min }(\Phi,A)=[0,\delta ]\), where \(\delta =\min \{\max \{x_{0}^{2},y _{0}\},2\}\).
Proposition 3.2
Proof
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
Proof
4 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
Let \(\bar{x}\in X_{0}\), and let \(\varphi:X_{0}\rightarrow 2^{Y\times Z}\) be an ordered pair mapping defined as \(\varphi (x)=(\Phi (\bar{x},x)+ \epsilon,G(x))\), \(\forall x\in X_{0}\).
By definition, φ is nearly \(C\times D\)-subconvexlike on \(X_{0}\) if and only if \(\operatorname{cl}(\operatorname{cone}(\varphi (X_{0})+C\times D))\) is convex, where \(\varphi (X_{0})=\bigcup_{x\in X_{0}}\varphi (x)= \bigcup_{x\in X_{0}}(\Phi (\bar{x},x)+\epsilon,G(x))\).
Lemma 4.1
([15])
If \(y^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\), \(c_{0}\in \operatorname{int}C\), then \(y^{*}(c_{0})>0\).
Theorem 4.1
Proof
Corollary 4.1
Proof
Theorem 4.2
- (i)
\(\bar{x}\in A\) and
- (ii)there exist \(s^{*}\in C^{*}\setminus \{0_{Y ^{*}}\}\) and \(k^{*}\in D^{*}\) such that$$ s^{*}(y)+s^{*}(\epsilon)+k^{*}(z)\geq 0, \quad \forall x\in X_{0},y\in \Phi (\bar{x},x),z\in G(x). $$
Proof
Corollary 4.2
- (i)
\(\bar{x}\in A\) and
- (ii)there exist \(s^{*}\in C^{*}\setminus \{0_{Y ^{*}}\}\) and \(k^{*}\in D^{*}\) such that$$ s^{*}(y)+k^{*}(z)\geq 0, \quad \forall x\in X_{0},y\in \Phi (\bar{x},x),z \in G(x). $$
Proof
Letting \(\epsilon =0\) in Theorem 4.2, we get the conclusion. □
Corollary 4.3
Remark 4.1
Corollary 4.4
Proof
Remark 4.2
From Remarks 3.1 and 3.3 in [18] we can see that if \((F-\bar{y},G)\) is generalized \(C\times D\)-subconvexlike on \(X_{0}\), then \((F-\bar{y},G)\) is nearly \(C\times D\)-subconvexlike on \(X_{0}\). Thus, Corollary 4.4 generalizes Theorem 4.2 in [17].
From Theorems 4.1 and 4.2 we obtain the following result.
Corollary 4.5
5 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].
Theorem 5.1
Proof
From the proof of Theorem 4.1 we see that there exist \(s^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\) and \(k^{*}\in D^{*}\) satisfying (4.7).
Theorem 5.2
- (i)
\(\bar{x}\in A\) and
- (ii)there exists \(\bar{T}\in L^{+}(Z,Y)\) such that x̄ is an ϵ-weakly efficient solution to the Ψ-USVEP, where \(\Psi:X_{0}\times X_{0}\rightarrow 2^{Y}\) is defined by$$ \Psi (y,x)=\Phi (y,x)+\bar{T}\bigl(G(x)\bigr). $$
Proof
From Theorems 5.1 and 5.2 we obtain the following result.
Corollary 5.1
Corollary 5.2
Suppose that \((F-\bar{y},G)\) is nearly \(C\times D\)-subconvexlike on \(X_{0}\) and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap ( -\operatorname{int}D)\neq \emptyset \). If \((\bar{x},\bar{y})\) is an ϵ-weakly efficient pair to the (SOP), then there exists \(\bar{T}\in L^{+}(Z,Y)\) such that \(-\bar{T}(G(\bar{x})\cap (-D)) \subset ((\operatorname{int}C\cap \{0\})\setminus (\epsilon +\operatorname{int}C))\), and \((\bar{x},\bar{y})\) is an ϵ-weakly efficient pair to the \((\mathrm{USOP})_{\bar{T}}\).
Proof
Letting \(\Phi (y,x)=F(x)-\bar{y}\), since \(\bar{y}\in F(\bar{x})\), it is evident that \(0\in \Phi (\bar{x},\bar{x})\) and \((F(x)-\bar{y},G(x))=( \Phi (\bar{x},x),G(x))\).
Since \((\bar{x},\bar{y})\) is an ϵ-weakly efficient pair to the (SOP), we see that x̄ is an ϵ-weakly efficient solution to the Φ-SVEPC.
Remark 5.1
If \((F,G)\) is \(C\times D\)-subconvexlike, then \((F-\bar{y},G)\) is nearly \(C\times D\)-subconvexlike. Thus Corollary 5.2 generalizes Theorem 3.1 in [15].
Corollary 5.3
- (i)
\(\bar{x}\in A, \bar{y}\in F(\bar{x})\), and
- (ii)
there exists \(\bar{T}\in L^{+}(Z,Y)\) such that \((\bar{x},\bar{y})\) is an ϵ-weakly efficient pair to the \((\mathrm{USOP}) _{\bar{T}}\).
Proof
Letting \(\Phi (y,x)=F(x)-\bar{y}\), since \(\bar{y}\in F(\bar{x})\), it is evident that \(0\in \Phi (\bar{x},\bar{x})\) and \((F(x)-\bar{y},G(x))=( \Phi (\bar{x},x),G(x))\).
Since \((\bar{x},\bar{y})\) is an ϵ-weakly efficient pair to the \((\mathrm{USOP})_{\bar{T}}\), we see that x̄ is an ϵ-weakly efficient solution to the Ψ-USVEP. Combining this with Theorem 5.2, we conclude that x̄ is an ϵ-weakly efficient solution to the Φ-SVEPC; it is clear that \((\bar{x}, \bar{y})\) is an ϵ-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 \((F, G)\).
Corollary 5.4
Suppose that \(\bar{x}\in A\), \(\bar{y}\in F(\bar{x})\), \((F-\bar{y},G)\) is nearly \(C\times D\)-subconvexlike on \(X_{0}\), and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap (-\operatorname{int}D)\neq \emptyset \). Then \((\bar{x},\bar{y})\) is an ϵ-weakly efficient pair to the (SOP) if and only if there exists \(\bar{T}\in L^{+}(Z,Y)\) such that \((\bar{x},\bar{y})\) is an ϵ-weakly efficient pair to the \((\mathrm{USOP}) _{\bar{T}}\).
6 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 \(\bigcap_{\epsilon \in C\setminus {\{0\}}}\epsilon \text{-}X_{W \min }(\Phi,A)=X_{W\min }(\Phi,A)\), 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.
Declarations
Authors’ information
Yihong Xu (1969-), Professor, Doctor, the major field of interest is in the area of set-valued optimization. Jian Chen, email: 1169898604@qq.com. Ke Zhang, email: 2768482283@qq.com.
Funding
This research was supported by the National Natural Science Foundation of China Grant (11461044), the Natural Science Foundation of Jiangxi Province (20151BAB 201027), and the Educational Commission of Jiangxi Province (GJJ12010).
Authors’ contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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