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Globally proper efficiency of setvalued optimization and vector variational inequality involving the generalized contingent epiderivative
Journal of Inequalities and Applications volumeÂ 2018, ArticleÂ number:Â 299 (2018)
Abstract
In this paper, firstly, a new property of the cone subpreinvex setvalued map involving the generalized contingent epiderivative is obtained. As an application of this property, a sufficient optimality condition for constrained setvalued optimization problem in the sense of globally proper efficiency is derived. Finally, we establish the relations between the globally proper efficiency of the setvalued optimization problem and the globally proper efficiency of the vector variational inequality.
1 Introduction
It is well known that convexity plays a crucial role in setvalued optimization. To generalize convexity of setvalued maps, some scholars introduced different kinds of generalized convex setvalued maps. Bhatia and Mehra [1] introduced the cone preinvex setvalued function which is a generalization of the cone convex setvalued map and derived a Lagrangian type duality for a fractional programming problem involving the cone preinvex setvalued map. Jia [2] defined the cone subpreinvex setvalued map and discussed the optimality condition and duality of setvalued optimization problems. Qiu [3] proposed the generalized cone preinvex setvalued map and used the conedirected contingent derivative given by [4] to obtain necessary and sufficient optimality conditions of setvalued optimization problems in the sense of weak efficiency and strong efficiency, respectively. Zhou et al. [5], in real normed spaces, introduced a new concept of generalized cone convex setvalued map and established optimality conditions of setvalued optimization problem in the sense of Henig proper efficiency by applying the contingent epiderivative proposed by [6] and the generalized cone convexity of setvalued maps. Other generalized convexity of setvalued maps and their applications can be found in [7â€“10] and references therein.
On the other hand, Giannessi [11] initially introduced the notion of vector variational inequality in the finite dimensional Euclidean space. Since then, different types of extensions and generalizations of variational inequalities have been proposed in different settings, see [6, 12â€“27] and references therein. In [6], optimality conditions are given by a certain kind of vector variational inequality in real normed spaces. We notice that such vector variational inequality is characterized by the contingent epiderivative of a setvalued map. Later, by virtue of the contingent epiderivative of setvalued maps, Liu and Gong [13] investigated the relations between the proper efficiency of setvalued optimization problem (for short, (SVOP)) and the proper efficiency of vector variational inequality based on the convexity assumption. By considering the generalized cone preinvexity and the contingent epiderivative of setvalued maps, Yu [20] disclosed the relations between Henig global efficiency of (SVOP) and Henig global efficiency of a kind of vector variational inequality. Yu and Kong [21] discussed the relations between a weakly approximate minimizer of a cone subinvex setvalued optimization problem and a weakly approximate solution of a kind of vector variational inequality characterized by the contingent epiderivative. We remark that the contingent epiderivative and the cone convexity (or the generalized cone convexity) of setvalued maps are used to deal with the relations between proper efficiency of (SVOP) and proper efficiency of vector variational inequality. However, it is worth noting that Chen and Jahn [12] pointed out that the existence of the contingent epiderivative of a setvalued map in a general setting is still an open question. To overcome this difficulty, Chen and Jahn [12] introduced a generalized contingent epiderivative of setvalued maps and derived the existence of the generalized contingent epiderivative under some standard assumptions. Therefore, it is interesting to study the optimality conditions of (SVOP) in the sense of globally proper efficiency via the generalized contingent epiderivative.
Motivated by the works [2, 5, 6, 13, 20, 21, 28], in this paper, we first propose a property of the cone subpreinvex setvalued map involving the generalized contingent epiderivative of setvalued maps which improves and generalizes the corresponding results in [5, 12]. A example is presented to illustrate this property. Second, we introduce a new type of generalized vector variational inequality problem (shortly, (GVVIP)) by virtue of the generalized contingent epiderivative of setvalued maps, propose the notion of globally proper efficiency of (GVVIP) and obtain the optimality conditions of the globally proper efficiency of (GVVIP).
This paper is organized as follows. In Sect.Â 2, we recall some definitions including the generalized contingent epiderivative of setvalued maps, the globally proper efficient point of a set and some generalized cone convexity of setvalued maps. In Sect.Â 3, a new property of the cone subpreinvex setvalued map is obtained and a sufficient optimality condition of constrained setvalued optimization problem in the sense of globally proper efficiency is derived. In Sect.Â 4, the relations between the globally proper efficiency of (SVOP) and the globally proper efficiency of (GVVIP) are disclosed.
2 Preliminaries
Throughout the paper, \(\mathbb{R}_{+}^{m}\) represents the nonnegative orthant of \(\mathbb{R}^{m}\) and \(\mathbb{R}_{+}:=\mathbb{R}_{+}^{1}\), where \(\mathbb{R}^{m}\) represents the mdimensional Euclidean space. Let X be a linear space, Y and Z be two real normed spaces, respectively. For a set \(K\subset Y\), intK denotes the interior of K. Let 0 denote the zero element for every space.
Let \(C\subset Y\) and \(D\subset Z\) be two closed pointed convex cones with \(\operatorname{int}C\neq\emptyset\) and \(\operatorname{int}D\neq \emptyset\). The cones C and D induce partial ordering of Y and Z, respectively. We denote by \(Y^{*}\) and \(Z^{*}\) the topological dual spaces of Y and Z, respectively. The topological dual cone \(C^{+}\) of C is defined as
and the quasiinterior \(C^{+i}\) of C is defined as
where \(\mu(y)\) denotes the value of the linear continuous functional Î¼ at y. The meaning of \(D^{+}\) is similar.
Let Î“ be a nonempty subset in X and \(F:\varGamma\rightrightarrows 2^{Y}\) be a setvalued map. Let
The graph and epigraph of F are, respectively, defined as
and
Now, we recall some basic definitions which will be used in the sequel.
Definition 2.1
([29])
Let Î© be a nonempty subset in X, \(\bar{z}\in \operatorname{cl}\varOmega\) and \(h\in X\). The contingent cone of Î© at zÌ„ is \(T(\varOmega,\bar{z}):=\{h\in X\exists t_{n}\downarrow 0,\exists h_{n}\rightarrow h,\mbox{ such that }\bar{z}+t_{n}h_{n}\in\varOmega ,\forall n\in\mathbb{N}\}\), or equivalently, \(T(\varOmega,\bar{z}):=\{h\in X\exists\lambda_{n}\rightarrow+\infty,\exists z_{n}\in\varOmega,\mbox{ such that }\lim_{n\rightarrow\infty}z_{n}=\bar{z}\mbox{ and }\lim_{n\rightarrow \infty}\lambda_{n}(z_{n}\bar{z})=h\}\).
Definition 2.2
Let S be a nonempty subset in Y.

(i)
A point \(\bar{y}\in S\) is called a minimal point of S iff \((\bar{y}C)\cap S=\{\bar{y}\}\). The set of all minimal points of S with respect to C is denoted by MinS.

(ii)
A point \(\bar{y}\in S\) is called a globally proper efficient point of S iff there exists a pointed convex cone \(C'\subset Y\) with \(C\backslash\{0\}\subset\operatorname{int}C'\) such that \((S\bar{y})\cap(C'\backslash\{0\})=\emptyset\). The set of all globally proper efficient points of S with respect to C is denoted by \(\operatorname{GPE}(S,C)\).
Remark 2.1
Notice that Yu and Liu [32] presented an equivalent characterization of the globally proper efficient point: For a nonempty subset S in Y, a point \(\bar{y}\in S\) is called a globally proper efficient point of S iff there exists a pointed convex cone \(C'\subset Y\) with \(C\backslash\{0\}\subset\operatorname{int}C'\) such that \((S\bar{y})\cap(\operatorname{int}C')=\emptyset\).
Definition 2.3
([30])

(a)
The cone C is called Daniell iff any decreasing sequence in Y having a lower bound converges to its infimum.

(b)
A subset A of Y is said to be minorized iff there exists \(y\in Y\) such that \(A\subset\{y\}+C\).

(c)
The domination property holds for a subset A of Y iff \(A\subset\operatorname{Min}A+C\).
Definition 2.4
([12])
Let Î“ be a nonempty subset in X, \(F:\varGamma \rightrightarrows2^{Y}\) be a setvalued map and \((\bar{x},\bar{y})\in \operatorname{gr}(F)\). A setvalued map \(D_{g}F(\bar{x},\bar{y}): \varGamma \rightrightarrows2^{Y}\), defined by
is called the generalized contingent epiderivative of F at \((\bar {x},\bar{y})\).
Definition 2.5
([33])
A subset \(\varGamma\subset X\) is called an invex set with respect to \(\eta:\varGamma\times\varGamma\rightarrow X\) iff for each \(x,y\in \varGamma\) and \(\lambda\in[0,1]\), \(y+\lambda\eta(x,y)\in\varGamma\).
Remark 2.2
Obviously, a convex set is an invex set by taking \(\eta(x,y)=xy\) in DefinitionÂ 2.5. In general, the converse is not true (see Example 2.1 in [34]).
Definition 2.6
([1])
Let Î“ be a nonempty invex subset in X with respect to Î·. The setvalued map \(F:\varGamma\rightrightarrows2^{Y}\) is called Cpreinvex on Î“ with respect to Î· iff
Remark 2.3
It is clear that the Cconvexity of the setvalued map F is the Cpreinvexity of the setvalued map F. However, the converse is not necessarily true (see Example 2.1 in [1]).
Now, we will give the concept of cone subpreinvex setvalued maps, which will be needed in the sequel.
Definition 2.7
([2])
Let Î“ be a nonempty invex subset in X with respect to Î·. The setvalued map \(F:\varGamma\rightrightarrows2^{Y}\) is called Csubpreinvex on Î“ with respect to Î· iff, \(\exists \theta\in\operatorname{int}C\) such that, \(\forall x,y\in \varGamma\), \(\forall\lambda\in[0,1]\), \(\forall\varepsilon>0\),
Remark 2.4
Clearly, the Cpreinvex setvalued map on Î“ is a Csubpreinvex setvalued map on Î“. However, ExampleÂ 2.1 shows that the converse is not necessarily true. Therefore, the Csubpreinvex setvalued map is a proper generalization of the Cpreinvex setvalued map.
Example 2.1
Let \(X=Y=\mathbb{R}\), \(C=\mathbb{R}_{+}\) and \(\varGamma=\{x\in\mathbb {R}x\in[2,2]\backslash\{0\}\}\). The setvalued map \(F:\varGamma \rightrightarrows2^{Y}\) and the map \(\eta:\varGamma\times\varGamma\rightarrow X\) are, respectively, defined as follows:
and
It is easy to check that Î“ is an invex set with respect to Î· and there exists \(\theta=1\in\operatorname{int}\mathbb{R}_{+}\), such that \(\forall x,y\in\varGamma\), \(\forall\lambda\in[0,1]\), \(\forall\varepsilon>0\),
Therefore, F is Csubpreinvex on Î“ with respect to Î·. On the other hand, if we take \(\hat{x}=2\), \(\hat{y}=1\) and \(\hat {\lambda}=\frac{1}{3}\), then \(0\in[0,\frac{5}{6}[=\hat{\lambda}F(\hat {x})+(1\hat{\lambda})F(\hat{y})\). However, \(0\notin\,]0,\frac {1}{2}[\,{}+C=F(\hat{y}+\hat{\lambda}\eta(\hat{x},\hat{y}))+C\). Thus, F is not Cpreinvex on Î“ with respect to Î·.
Remark 2.5
Zhou et al. [5] introduced the concept of generalized Cconvex setvalued functions (see Defintion 2.4 in [5]) and pointed out that the generalized Cconvex setvalued map is a proper generalization of the Cconvex setvalued map (see RemarkÂ 2.1 in [5]). Clearly, if \(\eta(x,y)=xy\), that is, the invex set Î“ is a convex set, then DefinitionÂ 2.7 reduces to DefinitionÂ 2.4 in [5]. However, if the invex set Î“ is not a convex set in X, then the Csubpreinvexity of setvalued maps cannot imply the generalized Cconvexity of setvalued maps. Therefore, the setvalued map F in ExampleÂ 2.1 is not the generalized Cconvex on Î“.
A chain of inclusion relations can be established now:
3 Optimality condition
In this section, firstly, we will use the generalized contingent epiderivative of setvalued maps to present a property of the Csubpreinvex setvalued map. Secondly, as an application of this property, we will give a sufficient optimality condition of constrained setvalued optimization problem in the sense of globally proper efficiency.
Theorem 3.1
Let C be a closed pointed convex cone being Daniell, Î“ be a nonempty invex subset in X with respect to Î· and the setvalued map \(F:\varGamma\rightrightarrows2^{Y}\) be Csubpreinvex on Î“ with respect to Î·. Let \((\bar{x},\bar{y})\in\operatorname{gr}(F)\). For any \(x\in\varGamma\), write \(\varPhi(\eta(x,\bar{x})):=\{y\in Y(\eta(x,\bar {x}),y)\in T(\operatorname{epi}(F),(\bar{x},\bar{y}))\}\). If \(\varPhi(\eta(x,\bar {x}))\) is minorized and fulfills the domination property for any \(x\in \varGamma\), then
Proof
Take any \(x\in\varGamma\) and \(y\in F(x)\). Let \(\{t_{n}\}\) be a sequence in \(\mathbb{R}\) such that \(t_{n}\in(0,1)\) with \(\lim_{n\rightarrow\infty}t_{n}=0\). Since F is a Csubpreinvex setvalued map on Î“ with respect to Î·, there exists \(\theta \in\operatorname{int}(C)\), for \(x,\bar{x}\in\varGamma\), \(y\in F(x)\) and \(\bar {y}\in F(\bar{x})\),
Now, we define two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) as follows:
By (3.1) and (3.2), we have \((x_{n},y_{n})\in\operatorname{epi}(F)\) and \(\lim_{n\rightarrow\infty}(x_{n},y_{n})=(\bar{x},\bar {y})\). It follows from the definitions of \(\{x_{n}\}\) and \(\{y_{n}\}\) that
Therefore,
i.e.,
Since \(\varPhi(\eta(x,\bar{x}))\) satisfies the domination property, we obtain
Thus,
â€ƒâ–¡
Remark 3.1
In Theorem 3.1, the conditions that C is Daniell and \(\varPhi(\eta(x,\bar{x}))\) is minorized guarantee the existence of \(D_{g}F(\bar{x},\bar{y})(\eta(x,\bar{x}))\).
Remark 3.2
Notice that Theorem 3.1 generalizes and improves Lemma 1 in [12] and Theorem 3.1 in [5] in the following aspects: (i) The convex set and the Cconvexity of Lemma 1 in [12] are extended to the invex set and the Csubpreinvexity of Theorem 3.1, respectively. (ii) The convex set, the Cconvexity and the contingent epiderivative of Theorem 3.1 in [5] are replaced by the invex set, the Csubpreinvexity and the generalized contingent epiderivative of Theorem 3.1, respectively.
The following example is used to illustrate TheoremÂ 3.1.
Example 3.1
Let \(X=Y=\mathbb{R}\), \(C=\mathbb{R}_{+}\) and \(\varGamma=\{x\in\mathbb {R}x\in[2,2]\backslash\{0\}\}\). The setvalued map \(F:\varGamma \rightrightarrows2^{Y}\) and the map \(\eta:\varGamma\times\varGamma\rightarrow X\) are, respectively, defined as follows:
and
It is easy to check that Î“ is an invex set with respect to Î· and F is Csubpreinvex on Î“ with respect to Î·. Therefore, Lemma 1 in [12] and Theorem 3.1 in [5] cannot be applied. On the other hand, if we take \((\bar{x},\bar{y})=(\frac {1}{3},0)\), then \(T(\operatorname{epi}(F),(\bar{x},\bar{y}))=\mathbb{R}\times \mathbb{R}_{+}\). Moreover, for any \(x\in\varGamma\), we have \(\varPhi(\eta (x,\bar{x}))=\mathbb{R}_{+}\). Clearly, \(\varPhi(\eta(x,\bar{x}))\) is minorized and fulfills the domination property. It follows from Definition 2.4 that \(D_{g}F(\bar{x},\bar{y})(\eta(x,\bar {x}))=0\). Thus,
Now we consider the following constrained setvalued optimization problem:
where \(F:\varGamma\rightrightarrows2^{Y}\) and \(G:\varGamma\rightrightarrows 2^{Z}\) are two setvalued maps with nonempty value. We always assume that the feasible set E is nonempty.
Definition 3.1
([31])
\(\bar{x}\in E\) is called a globally proper efficient solution of (SVOP) iff there exists \(\bar{y}\in F(\bar{x})\) such that \(\bar{y}\in\operatorname{GPE}(F(E),C)\). The pair \((\bar{x},\bar{y})\) is called a globally proper efficient element of (SVOP).
Definition 3.2
([35])
Let \(\bar{x}\in E\) and \(\bar{y}\in F(\bar{x})\). \((\bar {x},\bar{y})\) is called a positive properly efficient element of (SVOP) iff there exists \(\mu\in C^{+i}\) such that \(\mu(F(x)\bar{y})\geq0\) for all \(x\in E\).
Lemma 3.1
([35])
A positive properly efficient element of (SVOP) must be a globally proper efficient element of (SVOP).
By applying Lemma 3.1, Gong et al. [28] obtained a sufficient condition involving multiplier functionals for a globally proper efficient solution of (SVOP) (see Theorem 3.4 in [28]). Next, we will use LemmaÂ 3.1 to establish a sufficient optimality condition characterized by the generalized contingent epiderivative of setvalued maps in the sense of globally proper efficiency.
Theorem 3.2
Let C and D be two closed pointed convex cones being Daniell, and let Î“ be a nonempty invex subset in X with respect to Î·. Let \(\bar{x}\in E\), \(\bar{y}\in F(\bar{x})\) and \(\bar{z}\in G(\bar {x})\cap(D)\). For any \(x\in\varGamma\), write \(\varPsi(\eta(x,\bar{x})):=\{ (y,z)\in Y\times Z(\eta(x,\bar{x}),y,z)\in T(\operatorname{epi}(F,G),(\bar {x},\bar{y},\bar{z}))\}\). The set \(\varPsi(\eta(x,\bar{x}))\) is minorized and fulfills the domination property for any \(x\in\varGamma\). Suppose that the following conditions are satisfied:

(i)
The setvalued map \((F,G)\) is \(C\times D\)subpreinvex on Î“ with respect to Î·;

(ii)
There exists \((\mu,\nu)\in C^{+i}\times D^{+}\) such that
$$\begin{aligned} (\mu,\nu) \bigl(D_{g}(F,G) (\bar{x},\bar{y},\bar{z}) \bigl(\eta(x,\bar{x})\bigr)\bigr)\geq 0,\quad \forall x\in\varGamma \end{aligned}$$(3.3)and
$$\begin{aligned} \nu(\bar{z})=0. \end{aligned}$$(3.4)Then \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP).
Proof
From Theorem 3.1, we have
It follows from (3.5) that for any \(x\in\varGamma\) and \((y,z)\in (F,G)(x)\), there exist \(c\in C\) and \(d\in D\) such that
i.e.,
Since \((\mu,\nu)\in C^{+i}\times D^{+}\), we have \(\mu(c)+\nu(d)\geq0\), which together with (3.4) and (3.7) yields
Now, we prove \((\bar{x},\bar{y})\) is a positive properly efficient element of (SVOP). Otherwise, for \(\mu\in C^{+i}\), there exist \(\hat{x}\in E\) and \(\hat{y}\in F(\hat{x})\) such that
Since \(\hat{x}\in E\), there exists \(\hat{z}\in G(\hat{x})\) such that \(\hat{z}\inD\). Therefore,
which contradicts (3.8). Hence, \((\bar{x},\bar{y})\) is a positive properly efficient element of (SVOP). By Lemma 3.1, \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP).â€ƒâ–¡
Remark 3.3
Compared with Theorem 4.1 in [36], Theorem 3.2 has some weaker conditions and a stronger conclusion. Therefore, Theorem 3.2 is a proper generalization of Theorem 4.1 in [36].
4 Generalized vector variational inequality
In this section, we will introduce a generalized vector variational inequality problem (GVVIP) and disclose the close relations between the globally proper efficiency of (SVOP) and the globally proper efficiency of (GVVIP).
Let \(\bar{x}\in E\) and \(\bar{y}\in F(\bar{x})\). We always assume that \(\eta(x,\bar{x})\) belongs to the domain of \(D_{g}F(\bar{x},\bar{y})\) for any \(x\in E\). The generalized vector variational inequality problem is to find \(\bar{x}\in E\) and \(\bar{y}\in F(\bar{x})\) such that
Definition 4.1
A pair \((\bar{x},\bar{y})\) is called a globally proper efficient element of (GVVIP) iff there exists a pointed convex cone \(C'\subset Y\) with \(C\backslash\{0\}\subset\operatorname{int}C'\) such that
Remark 4.1
When \(\eta(x,\bar{x})=x\bar{x}\) and the generalized contingent epiderivative of the setvalued map becomes the contingent epiderivative of the setvalued map, Definition 4.1 reduces to Definition 18 in [13] and Definition 2.10 in [20].
We will use the standard assumptions: Let C be a closed pointed convex cone being Daniell and \(\operatorname{int}C\neq\emptyset\), and suppose that \(\varPhi(\eta(x,\bar{x}))\) given by TheoremÂ 3.1 is minorized and fulfills the domination property for any \(x\in\varGamma\).
Theorem 4.1
Let the standard assumptions hold. If \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP), then \((\bar{x},\bar{y})\) is a globally proper efficient element of (GVVIP).
Proof
Suppose that \((\bar{x},\bar{y})\) is not a globally proper efficient element of (GVVIP), then for any pointed convex cone \(\hat{C}\subset Y\) with \(C\backslash\{0\}\subset\operatorname{int}\hat{C}\), there exist \(x'\in E\) and \(y'\in D_{g}F(\bar{x},\bar{y})(\eta(x',\bar{x}))\) such that
It follows from Definition 2.4 that \((\eta(x',\bar{x}),y')\in T(\operatorname{epi}(F),(\bar{x},\bar{y}))\). Therefore, there exist a sequence \(\{(x_{n},y_{n})\}_{n\in\mathbb{N}}\) in \(\operatorname{epi}(F)\) and a sequence \(\{t_{n}\}_{n\in\mathbb{N}}\) of positive real numbers such that \((\bar{x},\bar{y})=\lim_{n\rightarrow \infty}(x_{n},y_{n})\) and
According to (4.2), we get
By (4.1) and (4.3), there exists \(n_{0}\in\mathbb{N}\) such that
which implies that
Since \(\{(x_{n},y_{n})\}_{n\in\mathbb{N}}\subset\operatorname{epi}(F)\), there exist \(y_{n}^{*}\in F(x_{n})\) and \(c_{n}\in C\) such that \(y_{n}=y_{n}^{*}+c_{n}\), which together with (4.4) yields
Therefore, \(y_{n}^{*}\bar{y}\in(F(E)\bar{y})\cap(\operatorname {int}\hat{C})\), which contradicts the fact that \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP). Hence, \((\bar{x},\bar {y})\) is a globally proper efficient element of (GVVIP).â€ƒâ–¡
Remark 4.2
It is worth noting that the contingent epiderivative of Theorem 8 in [13] is replaced with the generalized contingent epiderivative. Therefore, Theorem 4.1 improves Theorem 8 in [13].
Theorem 4.2
Let the standard assumptions hold. Let Î“ be a nonempty invex subset in X with respect to Î· and the setvalued map \(F:\varGamma \rightrightarrows2^{Y}\) be Csubpreinvex on Î“ with respect to Î·. If \((\bar{x},\bar{y})\) is a globally proper efficient element of (GVVIP), then \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP).
Proof
Since \((\bar{x},\bar{y})\) is a globally proper efficient element of (GVVIP), there exists a pointed convex cone \(C'\subset Y\) with \(C\backslash\{0\}\subset\operatorname{int}C'\) such that
Suppose that \((\bar{x},\bar{y})\) is not a globally proper efficient element of (SVOP), then for the pointed convex cone \(C'\subset Y\) with \(C\backslash\{0\}\subset\operatorname{int}C'\), there exist \(\hat{x}\in E\) and \(\hat{y}\in F(\hat{x})\) such that
It follows from Theorem 3.1 that there exist \(\hat{a}\in D_{g}F(\bar{x},\bar{y})(\eta(\hat{x},\bar{x}))\) and \(\hat{c}\in C\) such that
which contradicts (4.5). Hence, \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP).â€ƒâ–¡
Remark 4.3
Theorem 4.2 generalizes and improves Theorem 7 in [13] in the following three aspects: (i) The convex set becomes an invex set. (ii) The contingent epiderivative is generalized to the generalized contingent epiderivative. (iii) The Cconvexity of F is extended to Csubpreinvexity of F.
5 Conclusions
In this paper, based on the generalized contingent epiderivative of setvalued maps, we obtained a new property of the cone subpreinvex setvalued map. By applying this property, we derived a sufficient optimality condition in the sense of globally proper efficiency in the constrained setvalued optimization problem. We also introduced a new kind of generalized variational inequality problem. Moreover, the relations between the globally proper efficiency of the setvalued optimization problem and the globally proper efficiency of the generalized variational inequality problem are disclosed. These results are new and are extensions of the corresponding ones in setvalued optimization.
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Zhiang Zhou (1972), Professor, Doctor, the major field of interest is in the area of setvalued optimization.
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This research was supported by the National Nature Science Foundation of China (11431004, 11861002, 11471291), the Key Project of Chongqing Frontier and Applied Foundation Research (cstc2017jcyjBX0055, cstc2015jcyjBX0113) and the Graduate Innovation Foundation of Chongqing University of Technology (ycx2018256).
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Chen, W., Zhou, Z. Globally proper efficiency of setvalued optimization and vector variational inequality involving the generalized contingent epiderivative. J Inequal Appl 2018, 299 (2018). https://doi.org/10.1186/s1366001818918
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DOI: https://doi.org/10.1186/s1366001818918