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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.
1 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.
2 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])
LetCandDbe two cones inY, \(C\cap D = \{0_{Y}\}\). IfDis closed andChas a compact base, then there exists a pointed convex coneMsuch 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).
3 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\(\hat{x}\in E\), Shas a compact base, φis nearly\(S\times K\)-subconvexlike on\(X_{0}\), and that there exists\(x'\in X_{0}\)such that\(G(x')\cap (-\operatorname{int}K)\neq \emptyset \). Ifx̂is anε-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 intP 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\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), Shas 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\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). Ifx̂is 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
$$ \gamma ^{*}(y)+\gamma ^{*}(\varepsilon )+\omega ^{*}(z)\geq 0,\quad \forall x\in E, y\in \varUpsilon (\hat{x},x), z\in G(x). $$
Thenx̂is anε-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
$$ \gamma ^{*}(y)+\omega ^{*}(z)\geq 0,\quad \forall x\in E, y\in \varUpsilon (\hat{x},x), z\in G(x). $$
Thenx̂is 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\(\hat{x}\in E\), Shas a compact base, φis nearly\(S\times K\)-subconvexlike on\(X_{0}\), and that there exists\(x'\in X_{0}\)such that\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). Thenx̂is anε-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\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), Shas 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\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). Thenx̂is 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 \(\inf \omega ^{*}(G(\hat{x}))=0\).
4 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\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), Shas a compact base, φis nearly\(S\times K\)-subconvexlike on\(X_{0}\), and that there exists\(x'\in X_{0}\)such that\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). Ifx̂is anε-Benson efficient solution to (ϒ-SEPC), then there exists\(\hat{T}\in L_{+}(Z,Y)\)such thatx̂is anε-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\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), Shas 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\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). Ifx̂is a Benson efficient solution to (ϒ-SEPC), then there exists\(\hat{T}\in L_{+}(Z,Y)\)such thatx̂is 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 thatx̂is anε-Benson efficient solution to (Ψ-USEP), where\(\varPsi:X_{0}\times X_{0}\rightarrow 2^{Y}\)is defined as
$$ \varPsi (y,x)=\varUpsilon (y,x)+\hat{T} \bigl(G(x) \bigr). $$
Thenx̂is anε-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 thatx̂is a Benson efficient solution to (Ψ-USEP), where\(\varPsi:X_{0}\times X_{0}\rightarrow 2^{Y}\)is defined as
$$ \varPsi (y,x)=\varUpsilon (y,x)+\hat{T} \bigl(G(x) \bigr). $$
Thenx̂is 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\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), Shas a compact base, φis nearly\(S\times K\)-subconvexlike on\(X_{0}\), and that there exists\(x'\in X_{0}\)such that\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). Thenx̂is anε-Benson efficient solution to (ϒ-SEPC) if and only if there exists\(\hat{T}\in L_{+}(Z,Y)\)such thatx̂is anε-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\(\hat{x}\in E\), \(0\in \varUpsilon (\hat{x},\hat{x})\), Shas 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\(G(x')\cap (-\operatorname{int} K)\neq \emptyset \). Thenx̂is a Benson efficient solution to (ϒ-SEPC) if and only if there exists\(\hat{T}\in L_{+}(Z,Y)\)such thatx̂is 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\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\), Shas 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 anε-Benson efficient pair to (SOP), then there exists\(\hat{T}\in L_{+}(Z,Y)\)such that\((\hat{x},\hat{y})\)is anε-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 anε-Benson efficient pair to\((\mathrm{USOP})_{\hat{\mathrm{T}}}\).
Thenx̂is anε-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\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\), Shas 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 anε-Benson efficient pair to (SOP) if and only if there exists\(\hat{T}\in L_{+}(Z,Y)\)such that\((\hat{x},\hat{y})\)is anε-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\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\), Shas 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\(\hat{x}\in E\), \(\hat{y}\in F(\hat{x})\), Shas 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].
5 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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The authors gratefully acknowledge the anonymous referees and editors for their constructive comments and advice on the earlier version for this paper.
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Yihong Xu (1969-), Professor, Doctor, major field of interest is in the area of set-valued optimization.
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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).
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YX conceived and designed the study. SH wrote the paper. ZN reviewed and edited the manuscript. All authors read and approved the manuscript.
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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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DOI: https://doi.org/10.1186/s13660-020-02352-6