Coderivatives of gap function for Minty vector variational inequality
- Xiaowei Xue^{1}Email author and
- Yu Zhang^{2}
https://doi.org/10.1186/s13660-015-0810-5
© Xue and Zhang 2015
Received: 13 January 2015
Accepted: 5 June 2015
Published: 17 September 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 set-valued maps. Then, by using the relationships between the coderivatives of a set-valued map and its efficient points set-valued map, we obtain the coderivatives of the gap function for the Minty vector variational inequality.
Keywords
MSC
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 path-breaking 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 set-valued map. Under some suitable coerciveness conditions, Li et al. [9] discussed the differential and sensitivity properties of the set-valued gap functions defined in [8] for (VVI). They also obtained an explicit expression of the contingent derivative for a class of set-valued 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]. High-order optimality conditions and differential and sensitivity properties for gap functions of weak (VVI) were also considered (see [11]).
The generalized derivatives mentioned above for set-valued maps are generated by tangent cones to their graphs in primal spaces. Another derivative-like construction for set-valued 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 [13–15]. Coderivatives have also been applied to sensitivity analysis of scalar (single-objective) optimization problems. We refer the readers to [16–19] for just a few of them.
Recently, Li and Xue [20] discussed the differential and sensitivity properties of the set-valued 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 set-valued map. Then, through discussing the relations between a set-valued map and its efficient points set-valued 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 set-valued maps. Then, using the relations between coderivatives of a set-valued map and its efficient points set-valued 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, set-valued analysis, and variational analysis. In Section 3, we establish the coderivative of a class of set-valued map. Under some mild conditions, we first give the including relations of the coderivatives of set-valued maps. Then we obtain the explicit expressions under some stronger conditions. In Section 4, we give the coderivatives of the gap function for (MVVI).
2 Basic definitions and preliminaries
Definition 2.1
Obviously, Φ is unique. We denote derivative Φ of F at \(x_{0}\) by \(\nabla F(x_{0})\). If, for any \(x\in K\), F is Fréchet differentiable at x, F is said to be Fréchet differentiable on K. Therefore, \(\nabla F(\cdot):X\rightarrow L(X,Y)\) is a vector-valued 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
- (i)Given \(\bar{x}\in\Omega\) and \(\varepsilon\geq0\). The set of ε-normals to Ω at \(\bar{x}\in\Omega\) is defined byWhen \(\varepsilon=0\), the set (1) is a cone that is called the regular normal cone (or the prenormal cone) to Ω at x̄ and is denoted by \(\hat{N}(\bar{x},\Omega)\). We put \(\hat{N}_{\varepsilon}(\bar{x},\Omega)=\emptyset\) for all \(\varepsilon\geq0\) if \(\bar{x}\notin\Omega\).$$ \hat{N}_{\varepsilon}(\bar{x},\Omega)= \biggl\{ x^{*}\in X^{*}\Bigm| \limsup_{x\overset{\Omega}{\rightarrow}\bar{x}}\frac{\langle x^{*},x-\bar{x}\rangle}{\|x-\bar{x}\|}\leq \varepsilon \biggr\} . $$(1)
- (ii)The Mordukhovich normal cone (or basic normal cone) to \(\Omega\subset X\) at x̄ is defined through the Painlevé-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
- (i)The ε-coderivative \(\hat{D}^{*}_{\varepsilon}\Phi(\bar{x},\bar{y})\) at \((\bar{x},\bar{y})\) is defined through the ε-normal set (1) to the graph asWhen \(\varepsilon=0\), the positive homogeneous set-valued map of \(y^{*}\) in (3) is called the regular coderivative of Φ at \((\bar{x},\bar{y})\) and denoted by \(\hat{D}^{*}\Phi(\bar{x},\bar{y})(\cdot)\).$$ \hat{D}^{*}_{\varepsilon}\Phi(\bar{x},\bar{y}) \bigl(y^{*} \bigr)= \bigl\{ x^{*}\in X^{*}\mid \bigl(x^{*},-y^{*} \bigr)\in\hat{N}_{\varepsilon} \bigl(( \bar{x}, \bar{y}),\operatorname{gph}\Phi \bigr) \bigr\} . $$(3)
- (ii)The normal (Mordukhovich) coderivative of Φ at \((\bar{x},\bar{y})\) isthat 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 \(x_{k}^{*}\in \hat{D}^{*}_{\varepsilon_{k}}\Phi(x_{k},y_{k})(y^{*}_{k})\).$$ D^{*}_{N}\Phi(\bar{x},\bar{y}) \bigl(y^{*} \bigr)= \bigl\{ x^{*} \in X^{*}\mid \bigl(x^{*},-y^{*} \bigr)\in N \bigl((\bar{x},\bar{y}), \operatorname{gph}\Phi \bigr) \bigr\} , $$(4)
- (iii)The mixed coderivative \(D^{*}_{M}\Phi(\bar{x},\bar{y})\) of a set-valued map \(\Phi:X\rightrightarrows Y\) at \((\bar{x},\bar{y})\) is the set-valued map \(D^{*}_{M}\Phi(\bar{x},\bar{y}):Y^{*}\rightrightarrows X^{*}\) defined byi.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 \(x_{k}^{*}\in\hat{D}^{*}_{\varepsilon_{k}}\Phi(x_{k},y_{k})(y^{*}_{k})\).$$ D^{*}_{M}\Phi(\bar{x},\bar{y}) \bigl(y^{*} \bigr)= \mathop{ \operatorname{Limsup}}\limits _{(x_{k},y_{k},y_{k}^{*})\rightarrow (\bar{x},\bar{y},y^{*}),\varepsilon_{k}\rightarrow 0_{+}}\hat{D}_{\varepsilon_{k}}^{*} \Phi(x_{k},y_{k}) \bigl(y_{k}^{*} \bigr), $$(5)
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 finite-dimensional. We say that Ω is regular at \(\bar{x}\in\Omega\) if \(N(\bar{x},\Omega)=\hat{N}(\bar{x},\Omega)\) and Φ is N-regular (resp. M-regular) 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 Φ and special representations of the coderivatives.
Proposition 2.1
[21]
We also need some Lipschitzian notions in the following study.
Definition 2.4
[25]
We say that a set-valued map \(F:X\rightrightarrows Y\) admits a local upper Lipschitzian selection at \((\bar{x},\bar{y})\in\operatorname{gph}F\) if there is a single-valued map \(f:\operatorname{dom}F\rightarrow Y\) which is local upper Lipschitzian at x̄ satisfying \(f(\bar{x})=\bar{y}\) and \(f(x)\in F(x)\) for all \(x\in\operatorname{dom}F\) in a neighborhood of x̄.
Definition 2.5
[26]
3 Coderivatives of a set-valued map
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
- (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
(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 ẑ. This complete the proof. □
Now we turn to the converse inclusion.
Theorem 3.2
Proof
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
Proof
Since F is Fréchet differentiable at ẑ, then F is locally upper Lipschitzian at x̂, 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.
We give an example to illustrate Theorems 3.1 and 3.2.
Example 3.1
4 Coderivatives of gap functions
Definition 4.1
- (a)
\(0\in N(\hat{x})\) if and only if x̂ solves (MVVI);
- (b)
\(N(x)\cap(-S\setminus\{0\})=\emptyset\), \(\forall x\in K\).
In this section, we discuss the coderivative \(D^{*}N\).
Theorem 4.1
- (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)
Proof
(i) First, we show that G is compact at any \(x\in K\), which ensures that G is locally compact around x̂. 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 \(y_{i}=F(z_{i})(x-z_{i})\). 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 \(y_{i}=F(z_{i})(x-z_{i})\rightarrow F(z)(x-z):=y\in G(x)\). Therefore, G is compact at any \(x\in K\) and then \(G(x)\) is a compact set for any \(x\in K\).
Example 4.1
Theorem 4.2
- (i)If F is local upper Lipschitzian relative to K at ẑ, 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 x̂, then, for any \(y^{*}\in Y^{*}\) satisfying \(F(\hat{z})^{*}y^{*}\in \hat{D}^{*}F(\hat{z})(y^{*}(\hat{x}-\hat{z})^{*})\) and \(\sup_{s\in S\setminus \{0\}} \frac{\langle y^{*},s\rangle}{\|s\|}=:v<0\),$$F(\hat{z})^{*}y^{*}+ N(\hat{x},K)\subset D^{*}N(\hat{x},\hat{y}) \bigl(y^{*} \bigr). $$
Similarly to Corollary 3.1 we have the following result.
Corollary 4.1
- (i)If F is Fréchet differentiable at ẑ, 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 x̂, then N is M-regular 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 N-regular 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 Fréchet differentiable at ẑ, 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.
(iii) It can be proved similar to case (ii). This completes the proof. □
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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