 Research
 Open access
 Published:
Coderivatives of gap function for Minty vector variational inequality
Journal of Inequalities and Applications volumeÂ 2015, ArticleÂ number:Â 285 (2015)
Abstract
The purpose of this paper is to investigate coderivatives of the gap function involving the Minty vector variational inequality. First, we discuss the regular coderivative, the normal coderivative, and the mixed coderivative of a class of setvalued maps. Then, by using the relationships between the coderivatives of a setvalued map and its efficient points setvalued map, we obtain the coderivatives of the gap function for the Minty vector variational inequality.
1 Introduction
The vector variational inequality (for short, VVI) and the Minty vector variational inequality (for short, MVVI) have been of great interest in the academic and professional communities ever since the pathbreaking paper [1] in the early 1980s. Enormous results on the existence (see [2, 3]) and stabilities (see [4, 5]) have been obtained. There are some applications to be found in vector traffic equilibrium problems (see [6, 7]).
It is well known that the concept of gap functions is very important for the study of (VVI) and (MVVI). From the vector optimization point of view, Chen et al. [8] defined the gap function for the (VVI) problem as a setvalued map. Under some suitable coerciveness conditions, Li et al. [9] discussed the differential and sensitivity properties of the setvalued gap functions defined in [8] for (VVI). They also obtained an explicit expression of the contingent derivative for a class of setvalued maps, and some optimality conditions for (VVI) and weak (VVI) by virtue of the gap functions. Later, by the definition of the gap function for Minty vector variational inequalities, some differential and sensitivity results for Minty vector variational inequalities were also obtained in [10]. Highorder optimality conditions and differential and sensitivity properties for gap functions of weak (VVI) were also considered (see [11]).
The generalized derivatives mentioned above for setvalued maps are generated by tangent cones to their graphs in primal spaces. Another derivativelike construction for setvalued maps has been introduced by Mordukhovich [12], which is called coderivatives and is generated by normal cones to their graphs in dual spaces. There are numerous applications of coderivatives and the corresponding subdifferential to derive necessary conditions and existence properties in various vector optimization problems, such as [13â€“15]. Coderivatives have also been applied to sensitivity analysis of scalar (singleobjective) optimization problems. We refer the readers to [16â€“19] for just a few of them.
Recently, Li and Xue [20] discussed the differential and sensitivity properties of the setvalued gap functions defined in [8] for (VVI) via coderivatives. First, they established an explicit expression for computing the normal coderivative and mixed coderivative of a class of setvalued map. Then, through discussing the relations between a setvalued map and its efficient points setvalued map, they investigated sensitivity properties of the gap function for VVI. They also obtained some optimality conditions for (VVI).
Motivated by the work reported in [10, 20], in this paper, we make an effort to investigate the coderivatives of Minty vector variational inequality problem in general Banach spaces. First, we establish an explicit expression for computing the regular coderivatives, normal coderivative, and mixed coderivative of a class of setvalued maps. Then, using the relations between coderivatives of a setvalued map and its efficient points setvalued map, we obtain the coderivatives of the gap function for the Minty vector variational inequality. We also give some examples to illustrate the results.
The rest of the paper is organized as follows. In Section 2, we recall the basic definitions and notations from the vector variational inequality, setvalued analysis, and variational analysis. In Section 3, we establish the coderivative of a class of setvalued map. Under some mild conditions, we first give the including relations of the coderivatives of setvalued maps. Then we obtain the explicit expressions under some stronger conditions. In Section 4, we give the coderivatives of the gap function for (MVVI).
Throughout the paper we use the standard notation, with special symbols introduced where they are defined. Unless otherwise stated, all spaces considered are Banach spaces, whose norms are always denoted by \(\\cdot\\). For any space X, we consider its dual space \(X^{*}\) equipped with the weak* topology \(w^{*}\), where \(\langle\cdot,\cdot\rangle \) means the canonical pairing. The closed ball with center x and radius Î· is denoted by \(B_{\eta}(x)\). The symbol \(A^{*}\) denotes the adjoint operator of a linear continuous operator A. If \(F:X\rightrightarrows Y\) is a setvalued map, we denote by \(\operatorname{dom} F=\{x\in X\mid F(x)\neq\emptyset\}\) and \(\operatorname{gph}F=\{(x,y)\in X\times Y\mid y\in F(x)\}\), the domain and graph of F, respectively. The notation \(\overset{\ast}{\rightarrow}\) stands for weak^{âˆ—} convergence in a dual space, while \(x_{n}\overset{S}{\rightarrow}x\) means that the sequence \(x_{n}\) is contained in the subset S and converges to x. For the setvalued map \(F:X\rightrightarrows X^{*}\) the expression
signifies the sequential PainlevÃ©Kuratowski upper (outer) limit with respect to the norm topology in X and the weak* topology in \(X^{*}\). The origins of all real normed spaces are denoted by 0.
2 Basic definitions and preliminaries
Throughout this paper, let \(L(X,Y)\) be the set of all linear continuous operators from X to Y. For any \(A\in L(X,Y)\), we introduce norm
Since X, Y are Banach spaces, \(L(X,Y)\) is also a Banach space with the norm.
Definition 2.1
Let \(F:X\rightarrow L(X,Y)\) be a vectorvalued function. F is said to be FrÃ©chet differentiable at \(x_{0}\) if and only if there exists a linear continuous operator \(\Phi:X\rightarrow L(X,Y)\), such that
Obviously, Î¦ is unique. We denote derivative Î¦ of F at \(x_{0}\) by \(\nabla F(x_{0})\). If, for any \(x\in K\), F is FrÃ©chet differentiable at x, F is said to be FrÃ©chet differentiable on K. Therefore, \(\nabla F(\cdot):X\rightarrow L(X,Y)\) is a vectorvalued function.
In the following of this section, we introduce the basic concepts and constructions of variational analysis and generalized differentiation needed for formulations and justifications of the main results of the paper. Most of the concepts and properties can be found in [21].
Definition 2.2
Let \(\Omega\subset X\) be a nonempty subset of a Banach space.

(i)
Given \(\bar{x}\in\Omega\) and \(\varepsilon\geq0\). The set of Îµnormals to Î© at \(\bar{x}\in\Omega\) is defined by
$$ \hat{N}_{\varepsilon}(\bar{x},\Omega)= \biggl\{ x^{*}\in X^{*}\Bigm \limsup_{x\overset{\Omega}{\rightarrow}\bar{x}}\frac{\langle x^{*},x\bar{x}\rangle}{\x\bar{x}\}\leq \varepsilon \biggr\} . $$(1)When \(\varepsilon=0\), the set (1) is a cone that is called the regular normal cone (or the prenormal cone) to Î© at xÌ„ and is denoted by \(\hat{N}(\bar{x},\Omega)\). We put \(\hat{N}_{\varepsilon}(\bar{x},\Omega)=\emptyset\) for all \(\varepsilon\geq0\) if \(\bar{x}\notin\Omega\).

(ii)
The Mordukhovich normal cone (or basic normal cone) to \(\Omega\subset X\) at xÌ„ is defined through the PainlevÃ©Kuratowski upper (outer) limit as
$$ N(\bar{x},\Omega)=\mathop{\operatorname{Limsup}}\limits _{x_{k}\rightarrow \bar{x},\varepsilon_{k}\rightarrow0_{+}} \hat{N}_{\varepsilon_{k}}(x_{k},\Omega). $$(2)
Definition 2.3
Consider a setvalued map \(\Phi: X\rightrightarrows Y\) between Banach spaces.

(i)
The Îµcoderivative \(\hat{D}^{*}_{\varepsilon}\Phi(\bar{x},\bar{y})\) at \((\bar{x},\bar{y})\) is defined through the Îµnormal set (1) to the graph as
$$ \hat{D}^{*}_{\varepsilon}\Phi(\bar{x},\bar{y}) \bigl(y^{*} \bigr)= \bigl\{ x^{*}\in X^{*}\mid \bigl(x^{*},y^{*} \bigr)\in\hat{N}_{\varepsilon} \bigl(( \bar{x}, \bar{y}),\operatorname{gph}\Phi \bigr) \bigr\} . $$(3)When \(\varepsilon=0\), the positive homogeneous setvalued map of \(y^{*}\) in (3) is called the regular coderivative of Î¦ at \((\bar{x},\bar{y})\) and denoted by \(\hat{D}^{*}\Phi(\bar{x},\bar{y})(\cdot)\).

(ii)
The normal (Mordukhovich) coderivative of Î¦ at \((\bar{x},\bar{y})\) is
$$ D^{*}_{N}\Phi(\bar{x},\bar{y}) \bigl(y^{*} \bigr)= \bigl\{ x^{*} \in X^{*}\mid \bigl(x^{*},y^{*} \bigr)\in N \bigl((\bar{x},\bar{y}), \operatorname{gph}\Phi \bigr) \bigr\} , $$(4)that is, \(D^{*}_{N}\Phi(\bar{x},\bar{y})(y^{*})\) is the collection of all \(x^{*}\) for which there are sequences \(\varepsilon_{k}\rightarrow0_{+}\), \((x_{k},y_{k})\rightarrow (\bar{x},\bar{y})\), \((x_{k}^{*},y_{k}^{*})\overset{*}{\rightarrow}(x^{*},y^{*})\) with \((x_{k},y_{k})\in\operatorname{gph}\Phi\) and \(x_{k}^{*}\in \hat{D}^{*}_{\varepsilon_{k}}\Phi(x_{k},y_{k})(y^{*}_{k})\).

(iii)
The mixed coderivative \(D^{*}_{M}\Phi(\bar{x},\bar{y})\) of a setvalued map \(\Phi:X\rightrightarrows Y\) at \((\bar{x},\bar{y})\) is the setvalued map \(D^{*}_{M}\Phi(\bar{x},\bar{y}):Y^{*}\rightrightarrows X^{*}\) defined by
$$ D^{*}_{M}\Phi(\bar{x},\bar{y}) \bigl(y^{*} \bigr)= \mathop{ \operatorname{Limsup}}\limits _{(x_{k},y_{k},y_{k}^{*})\rightarrow (\bar{x},\bar{y},y^{*}),\varepsilon_{k}\rightarrow 0_{+}}\hat{D}_{\varepsilon_{k}}^{*} \Phi(x_{k},y_{k}) \bigl(y_{k}^{*} \bigr), $$(5)i.e., \(x^{*}\in D^{*}_{M}\Phi(\bar{x},\bar{y})(y^{*})\) if and only if there are sequences \(\varepsilon_{k}\rightarrow0_{+}\), \((x_{k},y_{k},y^{*}_{k})\rightarrow(\bar{x},\bar{y},y^{*})\), \(x_{k}^{*}\overset{*}{\rightarrow}x^{*}\) with \((x_{k},y_{k})\in\operatorname{gph}\Phi\), and \(x_{k}^{*}\in\hat{D}^{*}_{\varepsilon_{k}}\Phi(x_{k},y_{k})(y^{*}_{k})\).
It follows from the definitions that \(D^{*}_{M}\Phi(\bar{x},\bar{y})(y^{*})\subset D^{*}_{N}\Phi(\bar{x},\bar{y})(y^{*})\) when the equality obviously holds if Y is finitedimensional. We say that Î© is regular at \(\bar{x}\in\Omega\) if \(N(\bar{x},\Omega)=\hat{N}(\bar{x},\Omega)\) and Î¦ is Nregular (resp. Mregular) at \((\bar{x},\bar{y})\) if and only if \(D_{N}^{*}\Phi(\bar{x},\bar{y})=\hat{D}^{*}\Phi(\bar{x},\bar{y})\) (resp. \(D_{M}^{*}\Phi(\bar{x},\bar{y})=\hat{D}^{*}\Phi(\bar{x},\bar{y})\)) (see [24]). The following proposition gives a sufficient condition for the regularity of Î¦ and special representations of the coderivatives.
Proposition 2.1
[21]
Let \(\Phi:X\rightarrow Y\) be FrÃ©chet differentiable at xÌ„. Then
Moreover, if Î¦ is strictly differentiable at xÌ„, i.e., Î¦ is singlevalued around xÌ„ and
then Î¦ is Nregular at xÌ„ and one has
We also need some Lipschitzian notions in the following study.
Definition 2.4
[25]
Let \(f:X\rightarrow Y\) be a singlevalued map and \(\bar{x}\in\operatorname {dom}f\). f is said to be local upper Lipschitzian at xÌ„ if there are numbers \(\eta>0\) and \(L>0\) such that
We say that a setvalued map \(F:X\rightrightarrows Y\) admits a local upper Lipschitzian selection at \((\bar{x},\bar{y})\in\operatorname{gph}F\) if there is a singlevalued map \(f:\operatorname{dom}F\rightarrow Y\) which is local upper Lipschitzian at xÌ„ satisfying \(f(\bar{x})=\bar{y}\) and \(f(x)\in F(x)\) for all \(x\in\operatorname{dom}F\) in a neighborhood of xÌ„.
Definition 2.5
[26]
We say that the domination property holds for the multifunction \(F:X\rightrightarrows Y\) around xÌ„ if there exists a neighborhood U of xÌ„ such that
or
3 Coderivatives of a setvalued map
In subsequent sections, let K be a closed subset of X, \(F:X\rightarrow L(X,Y)\) be a continuous vectorvalued map, and
We will discuss the coderivative properties of G. First, recall that a setvalued map \(H:X\rightrightarrows Y\) is said to be inner semicontinuous at \((\hat{x},\hat{y})\in\operatorname{gph}H\) if for every sequence \(x_{k}\rightarrow\hat{x}\) with \(x_{k}\in\operatorname{dom}H\) there is a sequence \(y_{k}\in H(x_{k})\) converging to Å· as \(k\rightarrow\infty\). H is inner semicompact at xÌ‚ if for every sequence \(x_{k}\rightarrow\hat{x}\) with \(x_{k}\in\operatorname{dom}H\) there is a sequence \(y_{k}\in H(x_{k})\) that contains a convergent subsequence as \(k\rightarrow\infty\).
In the rest of this paper, let \(D^{*}\) stand either for the normal coderivative (4) or for the mixed coderivative (5). Since the proof methods of normal coderivative and mixed coderivative are similar, we only show the case of normal coderivative in the following.
Theorem 3.1
Let \(\hat{x}\in\operatorname{dom}G\), \(\hat{y} \in G(\hat{x})\), and

(i)
For any \(y^{*}\in Y^{*}\),
$$ \hat{D}^{*}G(\hat{x},\hat{y}) \bigl(y^{*} \bigr)\subset\bigcap _{\hat{z}\in M(\hat {x},\hat{y})}F(\hat{z})^{*}y^{*}+\hat{N}(\hat{x},\operatorname{dom}G). $$(8) 
(ii)
If M is inner semicompact at \((\hat{x},\hat{y})\), then, for any \(y^{*}\in Y^{*}\),
$$ D^{*}G(\hat{x},\hat{y}) \bigl(y^{*} \bigr)\subset\bigcup _{\hat{z}\in M(\hat{x},\hat {y})}F(\hat{z})^{*}y^{*}+N(\hat{x},\operatorname{dom}G). $$(9) 
(iii)
Given \(\hat{z}\in M(\hat{x},\hat{y})\), if M is inner semicontinuous at \((\hat{x},\hat{y},\hat{z})\), then, for any \(y^{*}\in Y^{*}\),
$$ D^{*}G(\hat{x},\hat{y}) \bigl(y^{*} \bigr)\subset F(\hat{z})^{*}y^{*}+N( \hat{x},\operatorname{dom}G). $$(10)
Proof
(i) For any \(x^{*}\in\hat{D}^{*}G(\hat {x},\hat{y})(y^{*})\), by the definitions of regular coderivative and regular normal cone,
Then, for any \(z_{k}\in K\) satisfying \(z_{k}\rightarrow\hat{z}\), we have
Especially, let \(z_{k}=\hat{z}\), we have
Thus,
This ensures that \(x^{*}F(\hat{z})^{*}y^{*}\in\hat{N}(\hat{x},\operatorname {dom}G)\). So we have
(ii) For any \(x^{*}\in D^{*}G(\hat{x},\hat{y})(y^{*})\), by the definitions of coderivatives and normal cone, there are sequences of vector \((x_{k},y_{k})\rightarrow(\hat{x},\hat{y})\), \((x_{k}^{*},y_{k}^{*})\overset {*}{\rightarrow}(x^{*},y^{*})\) (\(y_{k}^{*}\rightarrow y^{*}\) for the case of a mixed coderivative) and \(\varepsilon_{k}\downarrow0\) such that \(y_{k}\in G(x_{k})\) and
Since M is inner semicompact at \((\hat{x},\hat{y})\), for the above \((x_{k},y_{k})\), there exists a sequence of \(z_{k}\in M(x_{k},y_{k})\) that contains a subsequence converging to some áº‘. Since K is closed, we have \(\hat{z}\in K\). Let \(k\rightarrow\infty\) in \(y_{k}=F(z_{k})(x_{k}z_{k})\), we have \(\hat{y}=F(\hat{z})(\hat{x}\hat{z})\), which implies \(\hat{z}\in M(\hat{x},\hat{y})\). Then, for any \(z_{k_{i}}\in K\) satisfying \(z_{k_{i}}\rightarrow z_{k}\), we have
Especially, let \(z_{k_{i}}=z_{k}\), we have
Thus,
where \(\varepsilon_{k}^{\prime}=(1+\F(z_{k})\)\varepsilon_{k}\rightarrow0 \) as \(k\rightarrow\infty\). This ensures that \(x^{*}F(\hat{z})^{*}y^{*}\in N(\hat{x},\operatorname{dom}G)\). So we have
(iii) It can be proved similarly to the case (ii), since for any sequence \((x_{k} , y_{k} )\rightarrow (\hat{x},\hat{y})\), by the inner semicontinuous assumption of M, there exists a sequence \(z_{k} \in M(x_{k} , y_{k})\) converging to áº‘. This complete the proof.â€ƒâ–¡
Now we turn to the converse inclusion.
Theorem 3.2
Let \(\hat{x}\in\operatorname{dom}G\), \(\hat{y} \in G(\hat{x})\), and \(\hat {z}\in M(\hat{x},\hat{y})\). If F is local upper Lipschitzian at áº‘, and M admits a local upper Lipschitzian selection at \((\hat {x},\hat{y},\hat{z})\), then, for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat {z})^{*}y^{*}\in\hat{D}^{*}F(\hat{z})(y^{*}(\hat{x}\hat{z})^{*})\),
Moreover, if domG is regular at xÌ‚, then, for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat{z})^{*}y^{*}\in\hat{D}^{*}F(\hat{z})(y^{*}(\hat {x}\hat{z})^{*})\),
Proof
Let \(F(\hat{z})^{*}y^{*}\in\hat{D}^{*}F(\hat {z})(y^{*}(\hat{x}\hat{z})^{*})\) and \(x^{*}\in\hat{N}(\hat{x},\operatorname {dom}G)\). Then we have
and
This means that, for any \(\varepsilon>0\), one can find \(\eta_{1}>0\), \(\eta _{2}>0\) such that
and
for all \(z\in B_{\eta_{1}}(\hat{z})\), \(x\in B_{\eta_{2}}(\hat{x})\).
Since M admits a local upper Lipschitzian selection at \((\hat{x},\hat {y},\hat{z})\), for any \((x,y)\overset{\operatorname{gph}G}{\longrightarrow}(\hat{x},\hat{y})\), there are a constant \(t>0\) and \(z\in M(x,y)\) such that \(\z\hat{z}\\leq t(\x\hat{x}\+\y\hat{y}\)\). Furthermore, the locally upper Lipschitzian assumption of F ensures that \(\ F(z)F(\hat{z})\_{L}\leq L\z\hat{z}\\) for some \(L>0\) whenever \(z\rightarrow\hat{z}\). Thus, for any \((x,y)\overset{\operatorname {gph}G}{\longrightarrow}(\hat{x},\hat{y})\), we have the inequalities
Since \(\varepsilon> 0\) is chosen arbitrarily, we have
By the definition of regular coderivative, \(x^{*}+F(\hat{z})^{*}y^{*}\in\hat {D}^{*}G(\hat{x},\hat{y})(y^{*})\), and then
Moreover, if K is regular at xÌ‚, then \(N(\hat{x},\operatorname {dom}G)=\hat{N}(\hat{x},\operatorname{dom}G)\) and the residual part is obvious.â€ƒâ–¡
Remark 3.1
In fact, since the local upper Lipschitzian selection of M implies the inner semicontinuity and inner semicompactness of M, the converse include relations in Theorem 3.2 can be written as equalities.
Corollary 3.1
Let \(\hat{z}\in M(\hat{x},\hat{y})\). If F is FrÃ©chet differentiable at áº‘, and M admits a local upper Lipschitzian selection at \((\hat{x},\hat{y},\hat{z})\), then, for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat{z})^{*}y^{*}= \nabla F(\hat{z})^{*}(y^{*}(\hat{x}\hat {z})^{*})\), we have
Moreover, if domG is regular at xÌ‚, then G is Nregular at \((\hat{x},\hat{y})\) and for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat{z})^{*}y^{*}= \nabla F(\hat{z})^{*}(y^{*}(\hat{x}\hat{z})^{*})\), we have
Proof
Since F is FrÃ©chet differentiable at áº‘, then F is locally upper Lipschitzian at xÌ‚, and for any \(y^{*}\in Y^{*}\), \(\hat{D}^{*}F(\hat{z})(y^{*}(\hat{x}\hat{z})^{*})=\nabla F(\hat {z})^{*}(y^{*}(\hat{x}\hat{z})^{*})\). The first equality relation immediately follows from Theorems 3.1, 3.2.
Now assume that domG is regular at xÌ‚, we prove the second part of the corollary. Obviously, the upper Lipschitz selection property of M implies that M is inner semicontinuous at \((\hat {x},\hat{y},\hat{z})\). Thus, for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat {z})^{*}y^{*}= \nabla F(\hat{z})^{*}(y^{*}(\hat{x}\hat{z})^{*})\), we have
The proof is completed.â€ƒâ–¡
We give an example to illustrate Theorems 3.1 and 3.2.
Example 3.1
Let \(X=Y=\mathbb{R}\), \(K=[1,1]\), \(S=\mathbb{R}_{+}\), and \(F(x)=x^{2}\). Let \(\hat{x}=1\), \(\hat{y}=0\). Then we have
By direct computing, we have \(M(\hat{x},\hat{y})=\{1,0\}\), \(\hat {N}(\hat{x},\operatorname{dom}G)=N(\hat{x},\operatorname{dom}G)=\{0\}\),
and
Then, for any \(y^{*}\in\mathbb{R}\), we have
and
However, for any \(\hat{z}\in M(\hat{x},\hat{y})\), the inclusion
does not hold. This is so because M is not inner semicontinuous at any \((\hat{x},\hat{y},\hat{z})\). For example, let \((\hat{x},\hat{y},\hat {z})=(1,0,1)\), we can find a sequence \((x_{k},y_{k})=(1\frac{1}{k},0)\) which does not have a sequence \(z_{k}\in M(x_{k},y_{k})\) converging to áº‘ as \(k\rightarrow\infty\).
On the other hand, since M does not admit a local upper Lipschitzian selection at \((\hat{x},\hat{y},\hat{z})=(1,0,1)\), the converse inclusions
and
do not hold for any \(y^{*}\in \mathbb{R}\) though F is continuous differential at áº‘ and domG is regular at xÌ‚.
4 Coderivatives of gap functions
Let X, Y be Banach spaces and \(S\subset Y\) be a closed convex and point cone. Given a nonempty closed set \(K\subset X\) and a map \(F:K\rightarrow L(X,Y)\), where \(L(X,Y)\) is a set of all linear continuous operators from X to Y, the Minty vector variational inequality is to find \(x^{*}\in K\) such that
Definition 4.1
Let S be a closed convex and pointed cone in Y with nonempty interior. A setvalued map \(N:X\rightrightarrows Y\) is said to be a gap function of (MVVI) if and only if

(a)
\(0\in N(\hat{x})\) if and only if xÌ‚ solves (MVVI);

(b)
\(N(x)\cap(S\setminus\{0\})=\emptyset\), \(\forall x\in K\).
Consider setvalued map \(N:X\rightrightarrows Y\) defined by
where the symbol \(\operatorname{Max}_{S}\) denotes the collection of efficient points. That is, for a set \(A\in Y\),
By Theorem 4.2 in [10], N is a gap function of (MVVI). So, (MVVI) is equivalent to the following setvalued optimization problem:
In this section, we discuss the coderivative \(D^{*}N\).
Theorem 4.1
Let K be a compact set, \(\hat{x}\in K\), \(\hat{y} \in N(\hat{x})\), and

(i)
For any \(y^{*}\in Y^{*}\) satisfying \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle}{\s\}=:v<0\),
$$ \hat{D}^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr)\subset\bigcap _{\hat{z}\in M(\hat {x},\hat{y})}F(\hat{z})^{*}y^{*}+\hat{N}(\hat{x},K). $$(11) 
(ii)
If M is inner semicompact at \((\hat{x},\hat{y})\), then, for any \(y^{*}\in Y^{*}\) satisfying \(\sup_{s\in S\setminus\{0\}} \frac {\langle y^{*},s\rangle}{\s\}=:v<0\),
$$ D_{M}^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr)\subset\bigcup _{\hat{z}\in M(\hat{x},\hat {y})}F(\hat{z})^{*}y^{*}+N(\hat{x},K). $$(12) 
(iii)
Given \(\hat{z}\in M(\hat{x},\hat{y})\), if M is inner semicontinuous at \((\hat{x},\hat{y},\hat{z})\), then, for any \(y^{*}\in Y^{*}\) satisfying \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle }{\s\}=:v<0\),
$$ D_{M}^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr)\subset F( \hat{z})^{*}y^{*}+N(\hat{x},K). $$(13)
In addition to the assumption of case (ii) and (iii), respectively, if K has a compact base, then the including relations in case (ii) and (iii) hold for normal coderivatives.
Proof
(i) First, we show that G is compact at any \(x\in K\), which ensures that G is locally compact around xÌ‚. Give a sequence \(\{ (x_{i},y_{i})\}\subset\operatorname{gph} G\) satisfying \(x_{i}\rightarrow x\). By the construction of G, there exist \(z_{i}\in K\) such that \(y_{i}=F(z_{i})(xz_{i})\). Since K is a compact set, we assume without loss of generality that \(z_{i}\rightarrow z\in K\). The continuity of F implies that \(y_{i}=F(z_{i})(xz_{i})\rightarrow F(z)(xz):=y\in G(x)\). Therefore, G is compact at any \(x\in K\) and then \(G(x)\) is a compact set for any \(x\in K\).
Since G is compact at any \(x\in K\), \(G(x)\) is a compact set for any \(x\in K\). So the domination property for âˆ’G holds around xÌ‚. That is, \(G(x)\subset\operatorname{Min}_{S} (G(x))+S\) for any \(x\in K\). Therefore, we have \(G(x)+S=\operatorname{Min}_{S} (G(x))+S\), which implies \(G(x)+S=N(x)+S\). Thus, for any \(y^{*}\in Y^{*}\), by the definitions of regular coderivatives, we have
Obviously âˆ’G is also compact at any \(x\in K\). According to [20], Proposition 4.3, this implies that âˆ’N is locally order semicontinuous around \((\hat{x},\hat{y})\) and then âˆ’G is order semicontinuous at \((\hat{x},\hat{y})\). Employing [20], Proposition 4.2, we have, for any \(y^{*}\in Y^{*}\) satisfying \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle}{\s\}=:v<0\),
Note that \(\operatorname{dom}G=K\), the inclusions follow from Theorem 3.1.
(ii) Using [20], Proposition 4.1, we get, for any \(y^{*}\in Y^{*}\) satisfying \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle}{\ s\}=:v<0\),
and
Thus, case (ii) and (iii) can be proved similar to case (i).
Moreover, if for K we have the compact case, then according to [27], Proposition 3.5, for any \(y^{*}\in Y^{*}\) satisfying \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle}{\s\}=:v<0\), we have
and
Thus, the including relations in case (ii) and (iii) hold for normal coderivatives. This completes the proof.â€ƒâ–¡
Example 4.1
Let X, Y, K, F, \((\hat{x},\hat{y})\) as in Example 3.1. Obviously
By direct computing, we have \(\hat{N}(\hat{x},K)=N(\hat{x},K)=\mathbb {R}_{}\) and
Then, for any \(y^{*}\in Y^{*}\) satisfying \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle}{\s\}=:v<0\), i.e., \(y^{*}<0\),
and
Theorem 4.2
Let K be a compact set, \(\hat{x}\in K\), \(\hat{y}\in N(\hat{x})\), and \(\hat{z}\in M(\hat{x},\hat{y})\).

(i)
If F is local upper Lipschitzian relative to K at áº‘, and M admits a local upper Lipschitzian selection at \((\hat {x},\hat{y},\hat{z})\), then, for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat {z})^{*}y^{*}\in\hat{D}^{*}F(\hat{z})(y^{*}(\hat{x}\hat{z})^{*})\) and \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle}{\s\}=:v<0\),
$$F(\hat{z})^{*}y^{*}+ \hat{N}(\hat{x},K)\subset\hat{D}^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr). $$ 
(ii)
In addition to the conditions in (i), if K is regular at xÌ‚, then, for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat{z})^{*}y^{*}\in \hat{D}^{*}F(\hat{z})(y^{*}(\hat{x}\hat{z})^{*})\) and \(\sup_{s\in S\setminus \{0\}} \frac{\langle y^{*},s\rangle}{\s\}=:v<0\),
$$F(\hat{z})^{*}y^{*}+ N(\hat{x},K)\subset D^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr). $$
Proof
It immediately follows from Theorem 3.2 and the proof of Theorem 4.1.â€ƒâ–¡
Similarly to Corollary 3.1 we have the following result.
Corollary 4.1
Let K be a compact set, \(\hat{x}\in K\), \(\hat{y}\in N(\hat{x})\), and \(\hat{z}\in M(\hat{x},\hat{y})\).

(i)
If F is FrÃ©chet differentiable at áº‘, and M admits a local upper Lipschitzian selection at \((\hat{x},\hat{y},\hat {z})\), then, for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat{z})^{*}y^{*}= \nabla F(\hat{z})^{*}(y^{*}(\hat{x}\hat{z})^{*})\) and \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle}{\s\}=:v<0\), we have
$$\hat{D}^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr)=F(\hat{z})^{*}y^{*}+ \hat{N}( \hat{x},K). $$ 
(ii)
In addition to the conditions in case (i), if K is regular at xÌ‚, then N is Mregular at \((\hat{x},\hat{y})\), and for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat{z})^{*}y^{*}= \nabla F(\hat {z})^{*}(y^{*}(\hat{x}\hat{z})^{*})\) and \(\sup_{s\in S\setminus\{0\}} \frac {\langle y^{*},s\rangle}{\s\}=:v<0\), we have
$$ D_{M}^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr)=F(\hat{z})^{*}y^{*}+ N( \hat{x},K). $$ 
(iii)
In addition to the conditions in case (ii), if K has a compact base, then N is Nregular at \((\hat{x},\hat{y})\), and for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat{z})^{*}y^{*}= \nabla F(\hat{z})^{*}(y^{*}(\hat {x}\hat{z})^{*})\) and \(\sup_{s\in S\setminus\{0\}} \frac{\langle y^{*},s\rangle}{\s\}=:v<0\), we have
$$ D_{N}^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr)=F(\hat{z})^{*}y^{*}+ N( \hat{x},K). $$
Proof
(i) Since F is FrÃ©chet differentiable at áº‘, we get \(\hat{D}^{*}F(\hat{z})(y^{*}(\hat {x}\hat{z})^{*})=\nabla F(\hat{z})^{*}(y^{*}(\hat{x}\hat{z})^{*})\). The result immediately follows from Theorem 4.2.
(ii) Since K is regular at xÌ‚, we have \(\hat {N}(\hat{x},K)=N(\hat{x},K)\). Then
Thus, we have
which implies N is Mregular at \((\hat{x},\hat{y})\).
(iii) It can be proved similar to case (ii). This completes the proof.â€ƒâ–¡
References
Giannessi, F: Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, RW, Giannessi, F, Lions, JL (eds.) Variational Inequalities and Complementarity Problems, pp.Â 151186. Wiley, Chichester (1980)
Chen, GY: Existence of solutions for a vector variational inequality: an extension of HartmanStampacchia theorem. J.Â Optim. Theory Appl. 74, 445456 (1992)
Lee, GM, Kim, DS, Lee, BS: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 34, 745765 (1998)
Cheng, YH, Zhu, DL: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543550 (2005)
Li, SJ, Chen, GY, Teo, KL: On the stability of generalized vector quasivariational inequality problems. J. Optim. Theory Appl. 113, 283295 (2002)
Daniele, P, Maugeri, A: Vector variational inequalities and modeling of a continuum traffic equilibrium problem. In: Giannessi, F (ed.) Vector Variational Inequality and Vector Equilibria: Mathematical Theories, pp.Â 97109. Kluwer Academic, Boston (2000)
Goh, CJ, Yang, XQ: Vector equilibrium problem and vector optimization. Eur. J. Oper. Res. 116, 615628 (1999)
Chen, GY, Goh, CJ, Yang, XQ: On gap functions for vector variational inequalities. In: Giannessi, F (ed.) Vector Variational Inequality and Vector Equilibria: Mathematical Theories, pp.Â 5570. Kluwer Academic, Boston (2000)
Li, SJ, Yan, H, Chen, GY: Differential and sensitivity properties of gap functions for vector variational inequalities. Math. Methods Oper. Res. 57, 377391 (2003)
Meng, KW, Li, SJ: Differential and sensitivity properties of gap functions for Minty vector variational inequalities. J.Â Math. Anal. Appl. 337, 386398 (2008)
Li, MH, Li, SJ: Secondorder differential and sensitivity properties of weak vector variational inequalities. J. Optim. Theory Appl. 144, 7687 (2010)
Mordukhovich, BS: Metric approximations and necessary optimality conditions for general classes of extremal problems. Sov. Math. Dokl. 22, 526530 (1980)
Bao, TQ, Mordukhovich, BS: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program., Ser. A 122, 301347 (2010)
Dutta, J, Tammer, C: Lagrangian conditions for vector optimization in Banach spaces. Math. Methods Oper. Res. 64, 521541 (2006)
Ye, JJ, Zhu, QJ: Multiobjective optimization problem with variational inequality constraints. Math. Program., Ser. A 96, 139160 (2003)
Levy, AB, Mordukhovich, BS: Coderivatives in parametric optimization. Math. Program., Ser. A 99, 311327 (2004)
Lucet, Y, Ye, JJ: Sensitivity analysis of the value function for optimization problems with variational inequality constraints. SIAM J. Control Optim. 40, 699723 (2001); Erratum SIAM J. Control Optim. 41, 13151319 (2002)
Outrata, JV: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24, 627644 (1999)
Zhang, R: Weakly upper Lipschitzian multifunctions and applications to parametric optimization. Math. Program., Ser.Â A 102, 153166 (2005)
Li, SJ, Xue, XW: Sensitivity analysis of gap functions for vector variational inequality via coderivatives. Optimization 63(7), 10751098 (2014)
Mordukhovich, BS: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)
Kruger, AY, Mordukhovich, BS: Extremal points and the Euler equation in nonsmooth optimization. Dokl. Akad. Nauk SSSR 24, 684687 (1980)
Mordukhovich, BS: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960969 (1976)
Mordukhovich, BS: Coderivatives of setvalued mappings: calculus and applications. Nonlinear Anal. 30, 30593070 (1997)
Robinson, SM: Generalized equations and their solutions, I: basic theory. Math. Program. Stud. 10, 128141 (1979)
Tanino, T: Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56, 479499 (1988)
Chuong, TD: Normal subdifferentials of efficient point multifunctions in parametric vector optimization. Optim. Lett. 7, 10871117 (2013)
Acknowledgements
The first author was supported by the Scientific Research Fund for Advanced Talents of Nanyang Normal University. The second author was supported by the National Natural Science Foundation of Yunnan Province (No. 2014FD023). The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authorsâ€™ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Xue, X., Zhang, Y. Coderivatives of gap function for Minty vector variational inequality. J Inequal Appl 2015, 285 (2015). https://doi.org/10.1186/s1366001508105
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366001508105