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# Approximate Benson efficient solutions for set-valued equilibrium problems

*Journal of Inequalities and Applications*
**volume 2020**, Article number: 87 (2020)

## Abstract

In locally convex Hausdorff topological vector spaces, the approximate Benson efficient solution is proposed for set-valued equilibrium problems and its relationship to the Benson efficient solution is discussed. Under the assumption of generalized convexity, by using a separation theorem for convex sets, Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued equilibrium problems are established, respectively.

## Introduction

The vector equilibrium problem is a broad problem in many practical fields. It covers many typical mathematical problems, for instance, vector optimization, variational inequality, vector Nash equilibrium, vector complementarity, and so on. It is widely used in investment decision-making, quantitative economy, optimal control, and engineering technology. Because of the universality and unity of the problems involved and the profundity of solving them, vector equilibrium has become a hot issue in the field of nonlinear analysis and operational research [1–6]. In Banach spaces, Feng et al. [1] established Kuhn–Tucker-like conditions for weakly efficient solutions of vector equilibrium problems with constraints by using the Gerstewitz’s functional, and obtained sufficient conditions of weakly efficient solutions under the assumption of generalized invexity. You et al. [2] established Lagrangian-type sufficient optimality conditions for general constrained vector optimization problems by applying Gerstewitz’s function, and, under suitable restriction qualifications, by virtue of Clarke subdifferentials, they obtained Karush–Kuhn–Tucker necessary conditions. Luu et al. [3] derived necessary conditions for efficient solutions to vector equilibrium problems with equality and inequality constraints. Under the assumption of cone-convexity, Gong [4] obtained necessary and sufficient optimality conditions for several efficient solutions to constrained vector equilibrium problems. By using asymptotic analysis, Iusem et al. [5] studied vector equilibrium problems and noncoercive pseudomonotone equilibrium problems.

In recent years, approximate solutions of the set-valued optimization problem have attracted people’s attention [6–8]. In real ordered linear spaces, Zhou et al. [6, 7] studied several kinds of approximate properly efficient solutions of set-valued optimization problems, including *ϵ*-weakly, *ϵ*-global, *ϵ*-Benson, *ϵ*-super properly efficient solutions, and derived the relationship between *ϵ*-Benson properly efficient solutions and *ϵ*-global properly efficient solutions. Dhingra et al. [8] established existence and scalarization using a generalized Gerstewitz’s function for approximate solutions.

On the other hand, convexity is vital for studying the vector equilibrium problem. Sach [10] proposed a new type of convexity named ic-cone-convexness in 2005. Yang et al. [9] introduced another type of convexity called near cone-subconvexlikeness in 2001, which is a generalization of cone-subconvexlikeness and cone-convexness. Xu et al. [11] certified that near cone-subconvexlikeness is also an extension of ic-coneconvexness in 2011. So far, near cone-subconvexlikeness is regarded as the most universal convexity property.

The above discussions motivate the aim of this paper—discussing the relationship between approximate Benson efficient solutions and Benson efficient solutions, and establishing Lagrange-type and Kuhn–Tucker-type optimality conditions for approximate Benson efficient solutions.

## Preliminaries

Throughout this paper, let *X* be a real topological vector space; let *Y* and *Z* be real locally convex Hausdorff topological vector spaces, respectively, let \(S\subset Y\) and \(K\subset Z\) be pointed closed convex cones with nonempty interiors. Let \(X_{0}\) be a nonempty subset of *X*, and \(\varUpsilon:X_{0}\times X_{0}\rightarrow 2^{Y}\) and \(G: X_{0}\rightarrow 2^{Z}\) be maps. Furthermore, \(0_{Y}\) denotes the zero element in *Y*; \(Y^{*}\) and \(Z^{*}\) denote the topological dual space of *Y* and *Z*, respectively; \(S^{*}\) and \(S^{*i}\) denote the positive dual cone and strictly positive dual cone of *S*, respectively, that is,

### Definition 2.1

([12])

The map \(F: X_{0}\rightarrow 2^{Y}\) is called generalized *S*-subconvexlike on \(X_{0}\) if and only if there exists \(\theta \in \operatorname{int} S\) such that, for all \(x_{1},x_{2}\in X_{0}\), \(\lambda \in [0,1]\), and \(\alpha >0\), there exist \(x_{3}\in X_{0}\) and \(\rho >0\) such that

### Definition 2.2

([9])

The map \(F:X_{0}\rightarrow 2^{Y}\) is called nearly *S*-subconvexlike on \(X_{0}\) iff \(\mathrm{clcone}(F(X_{0})+S)\) is convex.

### Lemma 2.1

([13])

*Let**C**and**D**be two cones in**Y*, \(C\cap D = \{0_{Y}\}\). *If**D**is closed and**C**has a compact base*, *then there exists a pointed convex cone**M**such that*\(C\backslash \{0_{Y}\}\subset \operatorname{int} M\)*and*\(M\cap D = \{0_{Y}\}\).

### Lemma 2.2

([14])

*If*\(f\in S^{*}\backslash \{0_{Y^{*}}\}\), \(s\in \operatorname{int} S\), *then*\(f(s)>0\).

Let \(\varOmega \subset X_{0}\). Consider the following constrained set-valued equilibrium problem (for short, *ϒ*-SEPC): find \(\hat{x}\in \varOmega \) such that

where \(H\cup \{0\}\) is a convex cone in *Y*.

### Definition 2.3

A vector \(\hat{x}\in \varOmega \) is called a Benson efficient solution to (*ϒ*-SEPC) if

The set of all Benson efficient solutions to (*ϒ*-SEPC) is denoted by \(X_{\mathrm{Ben}}(\varUpsilon, \varOmega )\).

### Definition 2.4

Let \(\varepsilon \in S\). A vector \(\hat{x}\in \varOmega \) is said to be an *ε*-Benson efficient solution to (*ϒ*-SEPC) if

The set of all *ε*-Benson efficient solutions to (*ϒ*-SEPC) is denoted by *ε*-\(X_{ \mathrm{Ben}}(\varUpsilon,\varOmega )\).

In what follows, we discuss the relationship between Benson and *ε*-Benson efficient solution sets to constrained set-valued equilibrium problems.

### Proposition 2.1

*For any*\(\varepsilon \in S\), *we have*\(X_{\mathrm{Ben}}(\varUpsilon,\varOmega )\subset \varepsilon \text{-}X_{ \mathrm{Ben}}(\varUpsilon,\varOmega )\).

### Proof

Let \(x\in X_{\mathrm{Ben}}(\varUpsilon,\varOmega )\), then

Since \(\varepsilon \in S\) and *S* is a cone, we have \(\varepsilon +S\subset S+S\subset S \). Then

Hence, \(0_{Y}\in \mathrm{clcone}(\varUpsilon (x,\varOmega )+\varepsilon +S) \subset \mathrm{clcone}(\varUpsilon (x,\varOmega )+S) \). Together with (2.1), this yields \(\mathrm{clcone}(\varUpsilon (x,\varOmega )+\varepsilon +S)\cap (-S)=\{0_{Y} \} \). Then, \(x\in \varepsilon \text{-}X_{\mathrm{Ben}}(\varUpsilon,\varOmega )\). Hence we obtain \(X_{\mathrm{Ben}}(\varUpsilon,\varOmega )\subset \varepsilon \text{-}X_{ \mathrm{Ben}}(\varUpsilon,\varOmega ) \). □

### Remark 2.1

From Proposition 2.1, it follows that

However, the reversed inclusion is not necessarily true. The following example illustrates the case.

### Example 2.1

Let \(X=Y=\mathbb{R}^{2}, X_{0}=\mathbb{R}^{2}, S=\{(x_{1},x_{2})\in \mathbb{R}^{2}:x_{1}\geq 0, x_{2}\geq 0\}, \varOmega =\{(x_{1},x_{2}) \in \mathbb{R}^{2}:x_{2}\geq x_{1}^{2}\}\). Consider the set-valued map \(\varUpsilon:X_{0}\times X_{0}\rightarrow 2^{Y}\) defined by \(\varUpsilon (x,u)=u-x \). If \(\hat{x}=(0,0)\), then \(\varUpsilon (\hat{x},\varOmega )=\varOmega \). Then for any \(\varepsilon \in S\backslash \{0_{Y}\}\), one has

however,

Hence,

### Proposition 2.2

*For any*\(\varepsilon _{1},\varepsilon _{2}\in S\), *if*\(\varepsilon _{2}-\varepsilon _{1}\in S\), *then*

### Proof

Similar to the proofs of Proposition 2.1 and [16], Proposition 3.2. □

In the above, \(\varOmega \subset X_{0}\) is the normal constraint set in the constrained set-valued equilibrium problem. In the following, we give a specific constraint set, that is, \(E=\{x\in X_{0}:G(x)\cap (-K)\neq \emptyset \}\).

Let \(F:X_{0}\rightarrow 2^{Y}\). Consider the following set-valued optimization problem:

### Definition 2.5

([15])

A vector \(\hat{x}\in E\) is called a Benson efficient solution to (SOP) if there exists \(\hat{y}\in F(\hat{x})\) such that

In this case, \((\hat{x},\hat{y})\) is called a Benson efficient pair to (SOP).

## Kuhn–Tucker-type optimality conditions

In this part, under the assumption of near cone-subconvexlikeness, we present Kuhn–Tucker-type optimality conditions for *ε*-Benson efficient solutions to constrained set-valued equilibrium problems.

If \(\emptyset \neq Q\subset Y\), \(\emptyset \neq W\subset Y\), \(\psi \in Y^{*}\), then

### Definition 3.1

([16])

Let \(\hat{x}\in X_{0}\) and define an ordered pair map \(\varphi:X_{0}\rightarrow 2^{Y\times Z}\) as follows:

### Remark 3.1

([16])

Note that *φ* is nearly \(S\times K\)-subconvexlike on \(X_{0}\) iff \(\mathrm{clcone}(\varphi (X_{0})+S\times K)\) is convex, where \(\varphi (X_{0})=\bigcup_{x\in X_{0}}\varphi (x)=\bigcup_{x\in X_{0}}(\varUpsilon (\hat{x},x)+\varepsilon,G(x))\).

### Theorem 3.1

*Assume that*\(\hat{x}\in E\), *S**has a compact base*, *φ**is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int}K)\neq \emptyset \). *If**x̂**is an**ε*-*Benson efficient solution to* (*ϒ*-SEPC), *then there exist*\(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\)*such that*

### Proof

Since *x̂* is an *ε*-Benson efficient solution to (*ϒ*-SEPC), we have

Since \(\mathrm{clcone}(\varUpsilon (\hat{x},E)+\varepsilon +S)\) is a closed cone, and *S* has a compact base, from Lemma 2.1 we deduce that there exists a pointed convex cone *P* such that \(-S\backslash \{0_{Y}\}\subset -\operatorname{int} P\) and

From \(0_{Y}\notin -\operatorname{int} P\), we have \(\operatorname{int} P\subset P\backslash \{0_{Y}\}\), hence

Next, we prove

where \(\varphi (X_{0})=\bigcup_{x\in X_{0}}\varphi (x)=\bigcup_{x\in X_{0}}(\varUpsilon (\hat{x},x)+\varepsilon,G(x))\).

If not, then there exists \((\tilde{y},\tilde{z})\in Y\times Z\) such that

Thus, \(\tilde{y}\in -\operatorname{int} P\), \(\tilde{z}\in -\operatorname{int} K\), and there exist \(\lambda _{n}>0\), \(x_{n}\in X_{0}\), \((y_{n},z_{n})\in (\varUpsilon (\hat{x},x_{n}),G(x_{n}))\) and \((s_{n},k_{n})\in S\times K\) such that

and

Since \(-\operatorname{int} K\) is open and *K* is a cone, we have \(z_{n}+k_{n}\in -K\), and from \(k_{n}\in K\), we get \(z_{n}\in -K\). Thus, \(z_{n}\in G(x_{n})\cap (-K)\), therefore, \(x_{n}\in E\), combining with \(\tilde{y}\in -\operatorname{int} P\), we obtain \(\tilde{y}\in \mathrm{clcone}(\varUpsilon (\hat{x},E)+\varepsilon +S) \cap (-\operatorname{int} P) \), which contradicts (3.1). Hence \(\mathrm{clcone}(\varphi (X_{0})+S\times K)\cap (-(\operatorname{int} P \times \operatorname{int} K))=\emptyset \). From Remark 3.1, we know \(\mathrm{clcone}(\varphi (X_{0})+S\times K)\) is convex. By the separation theorem for convex sets, there exists \((\gamma ^{*},\omega ^{*})\in Y^{*}\times Z^{*}\backslash \{(0_{Y^{*}},0_{Z^{*}}) \}\) such that

Since \(\mathrm{clcone}(\varphi (X_{0})+S\times K)\) is a cone and on which \((\gamma ^{*},\omega ^{*})\) has a lower bound, we conclude

It follows from \((0_{Y},0_{Z})\in S\times K\) and (3.3) that \((\gamma ^{*},\omega ^{*})(\varphi (X_{0}))\geq 0\). In other words,

In view of \((0_{Y},0_{Z})\in \mathrm{clcone}(\varphi (X_{0})+S\times K)\) and (3.2), one has

From (3.3), we conclude that for any \(x\in X_{0}\), \(y\in \varUpsilon (\hat{x},x)\), \(z\in G(x)\), \(\beta _{1}, \beta _{2}\geq 0\), \(s\in S\) and \(k\in K\), one has

(i) Firstly, we prove that \(\omega ^{*}\in K^{*}\), i.e., \(\omega ^{*}(k)\geq 0, \forall k\in K\).

If not, then there exists \(k_{0}\in K\) such that \(\omega ^{*}(k_{0})<0\). When \(\beta _{2}\) is large enough, there exist \(x_{2}\in X_{0}\), \(y_{2}\in \varUpsilon (\hat{x},x_{2})\), \(z_{2}\in G(x_{2})\), \(\beta _{1}'\geq 0\), and \(s_{2}\in S\) such that

which contradicts (3.6). Hence we obtain \(\omega ^{*}(k)\geq 0, \forall k\in K \).

(ii) Next, we prove that \(\gamma ^{*}\neq 0_{Y^{*}}\).

If not, then \(\gamma ^{*}=0_{Y^{*}}\). Since \((\gamma ^{*},\omega ^{*})\neq (0_{Y^{*}},0_{Z^{*}})\), we have \(\omega ^{*}\neq 0_{Z^{*}}\), and then \(\omega ^{*}\in K^{*}\backslash \{0_{Z^{*}}\}\). From \(\gamma ^{*}=0_{Y^{*}}\) and (3.4), we get

On the other hand, from \(G(x')\cap (-\operatorname{int} K)\neq \emptyset \), there exists \(z'\in G(x')\) such that \(z'\in -\operatorname{int} K\), hence combining with Lemma 2.2, we have \(\omega ^{*}(z')<0\), which contradicts (3.7). Hence we get \(\gamma ^{*}\neq 0_{Y^{*}} \).

(iii) Finally, we prove that \(\gamma ^{*}\in S^{*i}\).

From (3.5) we derive \(\gamma ^{*}(\operatorname{int} P)\geq \omega ^{*}(-\operatorname{int} K)\). Since int*P* is a cone on which \(\gamma ^{*}\) has a lower bound, we conclude that \(\gamma ^{*}(\operatorname{int} P)\geq 0\). Since *P* is a convex cone, we have \(P\subset \mathrm{cl}P=\mathrm{cl}(\operatorname{int} P)\). Then, for any \(p\in P\), there exists a net \(\{p_{\alpha }\}\subset \operatorname{int} P\) such that \(p=\lim p_{\alpha }\). Thus \(\gamma ^{*}(p)=\gamma ^{*}(\lim p_{\alpha })=\lim \gamma ^{*}( p_{\alpha })\geq 0\), which implies \(\gamma ^{*}(P)\geq 0\), therefore \(\gamma ^{*}\in P^{*} \). It follows from (ii) that \(\gamma ^{*}\in P^{*}\backslash \{0_{Y^{*}}\} \). From Lemma 2.2, we have \(\gamma ^{*}(\operatorname{int} P)>0 \). By \(S\backslash \{0_{Y}\}\subset \operatorname{int} P\), we get \(\gamma ^{*}(S\backslash \{0_{Y}\})>0 \). Thus, \(\gamma ^{*}\in S^{*i}\). □

### Corollary 3.1

*Assume that*\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), *S**has a compact base*, \((\varUpsilon (\hat{x},\cdot ),G(\cdot ))\)*is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *If**x̂**is a Benson efficient solution to* (*ϒ*-SEPC), *then there exist*\(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\)*such that*\(\min \omega ^{*}(G(\hat{x}))=0\)*and*

### Proof

In Theorem 3.1, letting \(\varepsilon =0\), we see that there exist \(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\) such that

According to \(\hat{x}\in E\), we obtain \(G(\hat{x})\cap (-K)\neq \emptyset \). Thus there exists \(\hat{z}\in G(\hat{x})\) such that \(\hat{z}\in -K\), and since \(\omega ^{*}\in K^{*}\) we know

In equation (3.8), letting \(x=\hat{x}\), from \(0\in \varUpsilon (\hat{x},\hat{x})\) we get

Since \(\hat{z}\in G(\hat{x})\), we know \(\omega ^{*}(\hat{z})\geq 0\), which, together with (3.9), implies \(\omega ^{*}(\hat{z})=0\). Then

It follows from (3.10) that \(\min \omega ^{*}(G(\hat{x}))=0 \). It follows from \(0\in \varUpsilon (\hat{x},\hat{x})\) and (3.11) that \(0\in \gamma ^{*}(\varUpsilon (\hat{x},\hat{x}))+\omega ^{*}(G(\hat{x})) \). From (3.8), we derive

□

### Theorem 3.2

*Suppose that*

- (i)
\(\hat{x}\in E\);

- (ii)
*there exist*\(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\)*such that*$$ \gamma ^{*}(y)+\gamma ^{*}(\varepsilon )+\omega ^{*}(z)\geq 0,\quad \forall x\in E, y\in \varUpsilon (\hat{x},x), z\in G(x). $$

*Then**x̂**is an**ε*-*Benson efficient solution to* (*ϒ*-SEPC).

### Proof

Let \(s\in \mathrm{clcone}(\varUpsilon (\hat{x},E)+\varepsilon +S)\cap (-S)\), then there exist \(\lambda _{n}>0\), \(y_{n}\in \varUpsilon (\hat{x},E)\), and \(s_{n}\in S\) such that

Thus,

It follows from (ii) that

By \(x\in E\), there exists \(z_{x}\in G(x)\) such that \(z_{x}\in -K\), combining with \(\omega ^{*}\in K^{*}\), we know \(\omega ^{*}(z_{x})\leq 0\), hence \(\omega ^{*}(G(x))\cap (-\infty,0]\neq \emptyset \). Then, from (3.13) we have

Thus \(\gamma ^{*}(y_{n})+\gamma ^{*}(\varepsilon )\geq 0\), and since \(\gamma ^{*}\in S^{*i}\), we derive \(\gamma ^{*}(s_{n})\geq 0\). Hence, by (3.12) we obtain \(\gamma ^{*}(s)\geq 0\). On the other hand, from \(s\in -S\) we know \(\gamma ^{*}(s)\leq 0\). Thus \(\gamma ^{*}(s)=0\), combining with \(\gamma ^{*}\in S^{*i}\), we know \(s=0_{Y} \). Then,

Thus, *x̂* is an *ε*-Benson efficient solution to (*ϒ*-SEPC). □

### Corollary 3.2

*Suppose that*

- (i)
\(\hat{x}\in E\);

- (ii)
*there exist*\(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\)*such that*$$ \gamma ^{*}(y)+\omega ^{*}(z)\geq 0,\quad \forall x\in E, y\in \varUpsilon (\hat{x},x), z\in G(x). $$

*Then**x̂**is a Benson efficient solution to* (*ϒ*-SEPC).

### Proof

In Theorem 3.2, letting \(\varepsilon = 0\), we derive that the conclusion is true. □

### Remark 3.2

Let \(\varUpsilon (y,x)=F(x)-\hat{y}\). Since \(\hat{y}\in F(\hat{x})\) and \(\hat{x}\in E\), \(\varUpsilon (\cdot,\cdot )\) depends only on the second variable. Then Theorem 2.3 in [15] is a special case of Corollary 3.2.

### Corollary 3.3

*Suppose that*\(\hat{x}\in E\), *S**has a compact base*, *φ**is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *Then**x̂**is an**ε*-*Benson efficient solution to* (*ϒ*-SEPC) *if and only if there exist*\(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\)*such that*

### Proof

It follows from Theorems 3.1 and 3.2 that the conclusion is true. □

### Corollary 3.4

*Suppose that*\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), *S**has a compact base*, \((\varUpsilon (\hat{x},\cdot ),G(\cdot ))\)*is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *Then**x̂**is a Benson efficient solution to* (*ϒ*-SEPC) *if and only if there exist*\(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\)*such that*\(\min \omega ^{*}(G(\hat{x}))=0\)*and*

### Proof

By Corollaries 3.1 and 3.2, we can easily see that the conclusions are true. □

### Corollary 3.5

*Assume that*

- (i)
\(\hat{x}\in E\);

- (ii)
*there exist*\(\hat{y}\in F(\hat{x})\), \(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\)*such that*$$ \gamma ^{*}(\hat{y})=\min \bigl\{ \gamma ^{*}(y)+\omega ^{*}(z): x\in E, y \in F(x), z\in G(x) \bigr\} . $$

*Then*\((\hat{x},\hat{y})\)*is a Benson efficient pair to* (SOP).

### Proof

From (ii), we have

Letting \(\varUpsilon (y,x)=F(x)-\hat{y}\), then \(\varUpsilon (\hat{x},x)=F(x)-\hat{y}\). From \(\hat{y}\in F(\hat{x})\) we get \(0\in F(\hat{x})-\hat{y}\). Then, \(0\in \varUpsilon (\hat{x},\hat{x})\) and \(F(x)-\hat{y}=\varUpsilon (\hat{x},x)\). Thus

It follows from Corollary 3.2 that *x̂* is a Benson efficient solution to (*ϒ*-SEPC). Hence, \((\hat{x},\hat{y})\) is a Benson efficient pair to (SOP). □

### Remark 3.3

Comparing with Theorem 2.5 in [15], this corollary does not require \(\inf \omega ^{*}(G(\hat{x}))=0\).

## Lagrange-type optimality conditions

In this section, we establish Lagrange-type optimality conditions for *ε*-Benson efficient solutions to unconstrained set-valued equilibrium problems.

Let \(L(Z,Y)\) be the space of continuous linear operators from *Z* to *Y*, and let

Let \(\varTheta:X_{0}\times X_{0}\rightarrow 2^{Y}\). We consider unconstrained set-valued equilibrium problem (in brief, *Θ*-USEP): find \(\hat{x}\in X_{0}\) such that

where \(H\cup \{0\}\) is a convex cone on *Y*.

Next, we introduce Benson and *ε*-Benson efficient solutions to unconstrained set-valued equilibrium problems.

### Definition 4.1

A vector \(\hat{x}\in X_{0}\) is said to be a Benson efficient solution to (*Θ*-USEP) if

### Definition 4.2

Let \(\varepsilon \in S\). A vector \(\hat{x}\in X_{0}\) is called an *ε*-Benson efficient solution to (*Θ*-USEP) if

Let \(F:X_{0}\rightarrow 2^{Y}\), \(\hat{T}\in L_{+}(Z,Y)\). We consider the following unconstrained set-valued optimization problem:

where \(\zeta (x,\hat{T})=F(x)+\hat{T}(G(x))\).

### Definition 4.3

([18])

A vector \(\hat{x}\in X_{0}\) is called a Benson efficient solution to \((\mathrm{USOP})_{\hat{\mathrm{T}}}\) if there exists \(\hat{y}\in F(\hat{x})\) such that

where \(\zeta (X_{0},\hat{T})=\bigcup_{x\in X_{0}}\zeta (x,\hat{T})= \bigcup_{x\in X_{0}}(F(x)+\hat{T}(G(x)))\). In this case, \((\hat{x},\hat{y})\) is called a Benson efficient pair to \((\mathrm{USOP})_{\hat{\mathrm{T}}}\).

### Definition 4.4

Let \(\varepsilon \in S\). A vector \(\hat{x}\in X_{0}\) is called an *ε*-Benson efficient solution to \((\mathrm{USOP})_{\hat{\mathrm{T}}}\) if there exists \(\hat{y}\in F(\hat{x})\) such that

where \(\zeta (X_{0},\hat{T})=\bigcup_{x\in X_{0}}\zeta (x,\hat{T})= \bigcup_{x\in X_{0}}(F(x)+\hat{T}(G(x)))\). In this case, \((\hat{x},\hat{y})\) is called an *ε*-Benson efficient pair to \((\mathrm{USOP})_{\hat{\mathrm{T}}}\).

### Theorem 4.1

*Assume that*\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), *S**has a compact base*, *φ**is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *If**x̂**is an**ε*-*Benson efficient solution to* (*ϒ*-SEPC), *then there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that**x̂**is an**ε*-*Benson efficient solution to* (*Ψ*-USEP) *and*

*where*\(\varPsi:X_{0}\times X_{0}\rightarrow 2^{Y}\)*is defined as*\(\varPsi (y,x)=\varUpsilon (y,x)+\hat{T}(G(x))\).

### Proof

It follows from Theorem 3.1 that there exist \(\gamma ^{*}\in S^{*i}\), \(\omega ^{*}\in K^{*}\) such that

From \(\gamma ^{*}\in S^{*i}\), we obtain that there exists \(s_{0}\in S\backslash \{0_{Y}\}\subset S\) such that \(\gamma ^{*}(s_{0})=1\). Define the operator \(\hat{T}:Z\rightarrow Y\) as follows:

Thus, \(\hat{T}(K)=\omega ^{*}(K)s_{0}\subset S\), which implies \(\hat{T}\in L_{+}(Z,Y) \). In equation (4.1), letting \(x=\hat{x}\), from \(0\in \varUpsilon (\hat{x},\hat{x})\) we obtain

From \(z\in -K\), we know \(-\hat{T}(z)=-\omega ^{*}(z)s_{0}\in S \). Then,

In the following, we prove \(-\hat{T}(G(\hat{x})\cap (-K))\cap (\varepsilon +(S\backslash \{0_{Y} \}))=\emptyset \).

If not, then there exists \(\hat{z}\in G(\hat{x})\cap (-K)\) such that

Hence, \(-\hat{T}(\hat{z})-\varepsilon \in S\backslash \{0_{Y}\}\). It follows from (4.2) and the definition of *T̂* that

On the other hand, from \(\gamma ^{*}\in S^{*i}\) we have \(\gamma ^{*}(S\backslash \{0_{Y}\})>0 \). Then, \(-\hat{T}(\hat{z})-\varepsilon \notin S\backslash \{0_{Y}\} \), which contradicts (4.4). Hence, \(-\hat{T}(G(\hat{x})\cap (-K))\cap (\varepsilon +(S\backslash \{0_{Y} \}))=\emptyset \), and, combining with (4.3), we have \(-\hat{T}(G(\hat{x})\cap (-K))\subset S\backslash (\varepsilon +(S \backslash \{0_{Y}\})) \).

Ultimately, we prove \(\mathrm{clcone}(\varPsi (\hat{x},X_{0})+\varepsilon +S)\cap (-S)=\{0_{Y} \}\).

Let \(s\in \mathrm{clcone}(\varPsi (\hat{x},X_{0})+\varepsilon +S)\cap (-S)\), then there exist \(\lambda _{n}>0\), \(y_{n}\in \varPsi (\hat{x},X_{0})\), and \(s_{n}\in S\) such that

Thus,

From \(\gamma ^{*}\in S^{*i}\) we get \(\gamma ^{*}(s_{n})\geq 0\). By the definition of *T̂*, \(\gamma ^{*}(s_{0})=1\) and (4.1), we get that for any \(x\in X_{0}\),

Hence, by (4.5) we obtain \(\gamma ^{*}(s)\geq 0\). On the other hand, from \(s\in -S\) we know \(\gamma ^{*}(s)\leq 0\). Thus, \(\gamma ^{*}(s)=0\), together with \(\gamma ^{*}\in S^{*i}\), this yields \(s=0_{Y} \). Then,

Thus, *x̂* is an *ε*-Benson efficient solution to (*Ψ*-USEP). □

### Corollary 4.1

*Assume that*\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), *S**has a compact base*, \((\varUpsilon (\hat{x},\cdot ),G(\cdot ))\)*is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *If**x̂**is a Benson efficient solution to* (*ϒ*-SEPC), *then there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that**x̂**is a Benson efficient solution to* (*Ψ*-USEP) *and*

*where*\(\varPsi:X_{0}\times X_{0}\rightarrow 2^{Y}\)*is defined as*\(\varPsi (y,x)=\varUpsilon (y,x)+\hat{T}(G(x))\).

### Proof

In Theorem 4.1, letting \(\varepsilon =0\), we get that the conclusions hold. □

### Theorem 4.2

*Assume that*

- (i)
\(\hat{x}\in E\);

- (ii)
*there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that**x̂**is an**ε*-*Benson efficient solution to*(*Ψ*-USEP),*where*\(\varPsi:X_{0}\times X_{0}\rightarrow 2^{Y}\)*is defined as*$$ \varPsi (y,x)=\varUpsilon (y,x)+\hat{T} \bigl(G(x) \bigr). $$

*Then**x̂**is an**ε*-*Benson efficient solution to* (*ϒ*-SEPC).

### Proof

Since *x̂* is an *ε*-Benson efficient solution to (*Ψ*-USEP), we gain

For any \(x\in E\), we know \(G(x)\cap (-K)\neq \emptyset \). Thus, there exists \(z_{x}\in G(x)\) such that \(z_{x}\in -K\). Since \(\hat{T}\in L_{+}(Z,Y)\), we get \(-\hat{T}(z_{x})\in S\), hence \(S-\hat{T}(z_{x})\in S+S\subset S\), and thus \(S\subset \hat{T}(z_{x})+S\). It follows from \(z_{x}\in G(x)\) that \(S\subset \hat{T}(G(x))+S\). Hence

Thus

Together with (4.6), this yields

Thus *x̂* is an *ε*-Benson efficient solution to (*ϒ*-SEPC). □

### Corollary 4.2

*Assume that*

- (i)
\(\hat{x}\in E\);

- (ii)
*there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that**x̂**is a Benson efficient solution to*(*Ψ*-USEP),*where*\(\varPsi:X_{0}\times X_{0}\rightarrow 2^{Y}\)*is defined as*$$ \varPsi (y,x)=\varUpsilon (y,x)+\hat{T} \bigl(G(x) \bigr). $$

*Then**x̂**is a Benson efficient solution to* (*ϒ*-SEPC).

### Proof

In Theorem 4.2, letting \(\varepsilon =0\), we gain that the conclusion is true. □

### Remark 4.1

Let \(\varUpsilon (y,x)=F(x)-\hat{y}\). Since \(\hat{y}\in F(\hat{x})\) and \(\hat{x}\in E\), \(\varUpsilon (\cdot,\cdot )\) depends only on the second variable. Then Theorem 5.2 in [18] is a special case of Corollary 4.2.

### Corollary 4.3

*Suppose that*\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), *S**has a compact base*, *φ**is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *Then**x̂**is an**ε*-*Benson efficient solution to* (*ϒ*-SEPC) *if and only if there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that**x̂**is an**ε*-*Benson efficient solution to* (*Ψ*-USEP), *where*\(\varPsi:X_{0}\times X_{0}\rightarrow 2^{Y}\)*is defined as*

### Proof

This proof follows immediately from Theorems 4.1 and 4.2. □

### Corollary 4.4

*Suppose that*\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), *S**has a compact base*, \((\varUpsilon (\hat{x},\cdot ),G(\cdot ))\)*is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *Then**x̂**is a Benson efficient solution to* (*ϒ*-SEPC) *if and only if there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that**x̂**is a Benson efficient solution to* (*Ψ*-USEP), *where*\(\varPsi:X_{0}\times X_{0}\rightarrow 2^{Y}\)*is defined as*

### Proof

It follows from Corollaries 4.1 and 4.2 that the conclusion is true. □

### Corollary 4.5

*Suppose that*\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\), *S**has a compact base*, \((F-\hat{y},G)\)*is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *If*\((\hat{x},\hat{y})\)*is an**ε*-*Benson efficient pair to* (SOP), *then there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that*\((\hat{x},\hat{y})\)*is an**ε*-*Benson efficient pair to*\((\mathrm{USOP})_{\hat{\mathrm{T}}}\)*and*

### Proof

Since \((\hat{x},\hat{y})\) is an *ε*-Benson efficient pair to (SOP), we have

Letting \(\varUpsilon (y,x)=F(x)-\hat{y}\), then \(\varUpsilon (\hat{x},x)=F(x)-\hat{y}\). From \(\hat{y}\in F(\hat{x})\) we have \(0\in F(\hat{x})-\hat{y}\). Then, \(0\in \varUpsilon (\hat{x},\hat{x})\) and \(F(E)-\hat{y}=\varUpsilon (\hat{x},E)\). Thus

Therefore, *x̂* is an *ε*-Benson efficient solution to (*ϒ*-SEPC). By Theorem 4.1, we can see that there exists \(\hat{T}\in L_{+}(Z,Y)\) such that *x̂* is an *ε*-Benson efficient solution to (*Ψ*-SEPC) and \(-\hat{T}(G(\hat{x})\cap (-K))\subset S\backslash (\varepsilon +(S \backslash \{0_{Y}\})) \). Then

Consequently,

From \(\varUpsilon (\hat{x},x)=F(x)-\hat{y}\), we get

that is, \(\mathrm{clcone}(\zeta (X_{0},\hat{T})-\hat{y}+\varepsilon +S)\cap (-S)= \{0_{Y}\} \). Hence, \((\hat{x},\hat{y})\) is an *ε*-Benson efficient pair to \((\mathrm{USOP})_{\hat{\mathrm{T}}}\). □

### Corollary 4.6

*Assume that*

- (i)
\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\);

- (ii)
*there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that*\((\hat{x},\hat{y})\)*is an**ε*-*Benson efficient pair to*\((\mathrm{USOP})_{\hat{\mathrm{T}}}\).

*Then**x̂**is an**ε*-*Benson efficient pair to* (SOP).

### Proof

It follows from (ii) that \(\mathrm{clcone}(\zeta (X_{0},\hat{T})-\hat{y}+\varepsilon +S)\cap (-S)= \{0_{Y}\} \), that is,

Letting \(\varUpsilon (y,x)=F(x)-\hat{y}\), then \(\varUpsilon (\hat{x},x)=F(x)-\hat{y}\). From \(\hat{y}\in F(\hat{x})\) we have \(0\in F(\hat{x})-\hat{y}\). Then, \(0\in \varUpsilon (\hat{x},\hat{x})\) and \(F(x)-\hat{y}=\varUpsilon (\hat{x},x)\). Thus

Thus, \(\mathrm{clcone}(\varPsi (\hat{x},X_{0})+\varepsilon +S)\cap (-S)=\{0_{Y} \} \), Hence, *x̂* is an *ε*-Benson efficient solution to (*Ψ*-USEP). By Theorem 4.2, we can see that *x̂* is an *ε*-Benson efficient solution to (*ϒ*-SEPC), that is, \(\mathrm{clcone}(\varUpsilon (\hat{x},E)+\varepsilon +S)\cap (-S)=\{0_{Y} \} \). From \(\varUpsilon (\hat{x},E)=F(E)-\hat{y}\), we get \(\mathrm{clcone}(F(E)-\hat{y}+\varepsilon +S)\cap (-S)=\{0_{Y}\} \). Thus, \((\hat{x},\hat{y})\) is an *ε*-Benson efficient pair to (SOP). □

### Corollary 4.7

*Suppose that*\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\), *S**has a compact base*, \((F-\hat{y},G)\)*is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *Then*\((\hat{x},\hat{y})\)*is an**ε*-*Benson efficient pair to* (SOP) *if and only if there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that*\((\hat{x},\hat{y})\)*is an**ε*-*Benson efficient pair to*\((\mathrm{USOP})_{\hat{\mathrm{T}}}\).

### Proof

It follows from Corollaries 4.5 and 4.6 that the conclusion holds. □

### Corollary 4.8

*Suppose that*\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\), *S**has a compact base*, \((F-\hat{y},G)\)*is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *If*\((\hat{x},\hat{y})\)*is a Benson efficient pair to* (SOP), *then there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that*\((\hat{x},\hat{y})\)*is a Benson efficient pair to*\((\mathrm{USOP})_{\hat{\mathrm{T}}}\)*and*

### Proof

In Corollary 4.5, letting \(\varepsilon =0\), we get that the conclusions are true. □

### Remark 4.2

Corollary 4.8 generalizes Theorem 5.1 of [18] at the following points:

- (i)
If \((F,G)\) is \(S\times K\)-subconvexlike on \(X_{0}\), then \((F-\hat{y},G)\) is nearly \(S\times K\)-subconvexlike on \(X_{0}\);

- (ii)
Comparing with Theorem 5.1 in [18], this corollary does not require the convexity of

*F*.

### Corollary 4.9

*Suppose that*\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\), *S**has a compact base*, \((F-\hat{y},G)\)*is nearly*\(S\times K\)-*subconvexlike on*\(X_{0}\), *and that there exists*\(x'\in X_{0}\)*such that*\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). *Then*\((\hat{x},\hat{y})\)*is a Benson efficient pair to* (SOP) *if and only if there exists*\(\hat{T}\in L_{+}(Z,Y)\)*such that*\((\hat{x},\hat{y})\)*is a Benson efficient pair to*\((\mathrm{USOP})_{\hat{\mathrm{T}}}\)*and*

### Proof

It follows from Corollary 4.8 and Remark 4.1 that the conclusions are true. □

### Remark 4.3

Corollary 4.9 is different from Theorem 5.1 of [19] at the following points:

- (i)
The vector-valued function is extended to a set-valued function;

- (ii)
According to Remarks 3.1 and 3.3 in [17], we know that if \((F-\hat{y},G)\) is generalized \(S\times K\)-subconvexlike on \(X_{0}\), then \((F-\hat{y},G)\) is nearly \(S\times K\)-subconvexlike on \(X_{0}\). Hence, Corollary 4.9 generalizes Theorem 5.1 in [19].

## Conclusions

In this paper, we investigated the relationship between Benson and *ε*-Benson efficient solutions, and established Kuhn–Tucker-type and Lagrange-type optimality conditions to set-valued equilibrium problems. The results we obtained generalize those of Liu [15], Li [18], and Chen [19], respectively. As a mathematical topic, further research on *ε*-Benson efficient solutions seems to be of value and interest.

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### Acknowledgements

The authors gratefully acknowledge the anonymous referees and editors for their constructive comments and advice on the earlier version for this paper.

### Availability of data and materials

Not applicable.

### Authors’ information

Yihong Xu (1969-), Professor, Doctor, major field of interest is in the area of set-valued optimization.

## Funding

This research was supported by the National Natural Science Foundation of China Grant (11961047), the Natural Science Foundation of Jiangxi Province (20192BAB201010) and the Scientific Research Training Program of Nanchang University (55001942).

## Author information

### Affiliations

### Contributions

YX conceived and designed the study. SH wrote the paper. ZN reviewed and edited the manuscript. All authors read and approved the manuscript.

### Corresponding author

Correspondence to Yihong Xu.

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### Competing interests

The authors declare that they have no competing interests.

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### Cite this article

Hu, S., Xu, Y. & Niu, Z. Approximate Benson efficient solutions for set-valued equilibrium problems.
*J Inequal Appl* **2020, **87 (2020). https://doi.org/10.1186/s13660-020-02352-6

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### MSC

- 90C33
- 90C46
- 90C59

### Keywords

- Approximate Benson efficient solution
- Near cone-subconvexlikeness
- Optimality condition
- Set-valued equilibrium problem