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Prederivatives of gamma paraconvex setvalued maps and Pareto optimality conditions for set optimization problems
Journal of Inequalities and Applications volume 2017, Article number: 243 (2017)
Abstract
Prederivatives play an important role in the research of set optimization problems. First, we establish several existence theorems of prederivatives for γparaconvex setvalued mappings in Banach spaces with \(\gamma>0\). Then, in terms of prederivatives, we establish both necessary and sufficient conditions for the existence of Pareto minimal solution of set optimization problems.
1 Introduction
Let X and Y be Banach spaces. We say that \(G: X\rightrightarrows Y\) is a setvalued mapping if \(G(x)\) is a subset of Y for all \(x\in X\). Setvalued problems occur in many situations, such as control problems, feasibility problems, optimality problems, equilibrium problems and variational inequality problems. A powerful tool dealing with setvalued problems is setvalued analysis. We refer the reader to the references [1–4] for more knowledge about setvalued analysis and its applications.
In a pioneering work [5], Ioffe introduced a notion of prederivative which can be viewed as an extension of Clarke generalized gradient. It is well known that the prederivative is an effective tool in dealing with nondifferentiable mapping of nonsmooth analysis. In contrast with the derivative, the prederivative may not be unique. However, in terms of prederivatives, one can establish an inverse function theorem and implicit theorem and solve nondifferential inclusion problems [6]. In the later publication of Pang [7, 8], and Gaydu, Geoffroy and JeanAlexis [9], some notions of prederivatives were posed and further studied. In 2016, Gaydu, Geoffroy and Marcelin [10] studied the existence of some kinds of prederivatives of convex setvalued mappings and established necessary and sufficient optimality conditions for the weak minimizers and the strong minimizers of set optimization problems. γparaconvex setvalued mappings are an extension of convex setvalued mappings, and were studied by some researchers [11, 12]. Moreover, in set optimization problems, Pareto minimizers are more suitable than weak minimizers and strong minimizers in practice [13, 14]. Now, two natural questions are posed. Can we establish some existence results of some kinds of prederivatives for γparaconvex setvalued mappings? Can we give optimality conditions for the Pareto minimizers of set optimization problems by prederivatives?
In this paper, we firstly establish several existence theorems of prederivatives for γparaconvex setvalued mappings and coneγparaconvex setvalued mappings. Then we establish necessary and sufficient optimality conditions for the Pareto minimizers of set optimization problems in terms of prederivatives.
2 Preliminaries
Throughout this paper, unless stated otherwise, we always assume that X and Y are real Banach spaces and \(G:X\rightrightarrows Y\) is a setvalued mapping. The domain of G is defined by
The graph of G is defined by
We say that G is a closed setvalued mapping if \(\operatorname {Gr}(G)\) is a closed subset of \(X\times Y\). We say that G has convex values if \(G(x)\) is a convex subset of Y for any \(x \in X\). Let Ω be a subset of X; we use \(\operatorname {cl}(\Omega)\) to denote the closure of Ω, \(\operatorname {int}(\Omega)\) to denote the interior of Ω. We use \(B_{X}\) and \(B_{Y}\) to denote the closed unit ball of X and Y, respectively. Let \(\bar{x}\in X\). We use \(N(\bar{x})\) to denote all open neighborhoods of x̄. Let \(C\subseteq Y\) be a nonempty set. We say that C is a cone, if \(\lambda c\in C\) for any \(c\in C\) and \(\lambda\geq0\). We say that C is pointed if \(C\cap(C)=\{0\}\). Define \(G+C:X\rightrightarrows Y\) as
The following definition is needed in the sequel.
Definition 2.1
[1]
Let \(\Phi :X\rightrightarrows Y\) be a setvalued mapping. We say that Φ is positively homogeneous if \(0\in\Phi(0)\) and \(\Phi(\lambda x)=\lambda\Phi(x)\), \(\forall x \in X\), \(\forall \lambda>0\).
Definition 2.2
[12]
Let \(C\subseteq Y\) be a convex cone, \(\gamma>0\) and \(\eta>0\). We say that G is a Cγparaconvex setvalued mapping with modulus η, if
for all \(x , u\in X\), \(\theta\in[0, 1]\). We say that G is a γparaconvex setvalued mapping if \(C=\{0\}\).
Remark 2.1
In the special case of \(\eta={0}\) and \(C=\{0\}\), Cγparaconvex setvalued mappings reduce to convex setvalued mappings.
Definition 2.3
[15]
We say that G is Lipschitz continuous at x̄ if there exist \(l>0\) and \(U \in N(\bar{x})\) such that
If the above equation holds on \(U=\Omega\), then we say that G is Lipschitz continuous on Ω.
Definition 2.4
[10]
Let \(\Phi: X\rightrightarrows Y\) be a positively homogeneous setvalued mapping, \(\bar{x}\in X\) and \(\bar{y}\in G(\bar{x})\).

(i)
Φ is called an outer prederivative of G at x̄, if for any \(\delta>0\) there exists \(U \in N(\bar{x})\) such that
$$ G(u)\subseteq G(\bar{x})+\Phi(u\bar{x})+\delta \Vert u\bar {x}\Vert B_{Y}, \quad \forall u\in U . $$ 
(ii)
Φ is called a strict prederivative of G at x̄, if for any \(\delta>0\) there exists \(U \in N(\bar{x})\) such that
$$ G(u)\subseteq G\bigl(u'\bigr)+\Phi\bigl(uu'\bigr)+ \delta\bigl\Vert uu'\bigr\Vert B_{Y},\quad \forall u, u'\in U. $$ 
(iii)
Φ is called a pseudo strict prederivative of G at \((\bar{x}, \bar{y})\), if for any \(\delta>0\) there exist \(U\in N(\bar {x})\) and \(V \in N(\bar{y})\) such that
$$ G(u)\cap V\subseteq G\bigl(u'\bigr)+\Phi\bigl(uu' \bigr)+\delta\bigl\Vert uu'\bigr\Vert B_{Y}, \quad \forall u, u'\in U. $$
3 Prederivatives of gamma paraconvex setvalued mappings
In this section, we establish the existence results of pseudo strict prederivatives for γparaconvex setvalued mappings and strict prederivatives for Cγparaconvex setvalued mappings, respectively.
Lemma 3.1
[11]
Let \(G:X\rightrightarrows Y\) be a closed setvalued mapping, \(y_{0}\in G(X)\), \(x_{0}\in X\), \(\eta>0\), \(\delta>0\), \(\gamma>0\), \(G^{1}\) be a γparaconvex setvalued mapping with modulus r, and \(y_{0}+\eta B_{Y}\subseteq G(x_{0}+\delta B_{X})\). Let \(\eta_{1}>0\), \(\eta_{2}>0\) with \(\eta_{1}+\eta_{2}=\eta\). Then, for each \(y\in y_{0}+\eta_{1}B_{Y}\),
Theorem 3.1
Let \(\eta>0\), \(\delta>0\), \(r>0\), \(\gamma>0\), \(G:X\rightrightarrows Y\) be a closed γparaconvex setvalued mapping with modulus r, \((\bar{x}, \bar{y})\in \operatorname {Gr}(G)\) and \(\bar{x}+\eta B_{X}\subseteq G^{1}(\bar{y}+\delta B_{Y})\). Then G has a pseudo strict prederivative Φ at \((\bar{x}, \bar{y})\) with \(\Phi(\cdot )=L\Vert \cdot \Vert B_{Y}\), where \(L=(\delta+r(\frac{\eta }{2})^{\gamma}+\frac{\eta}{2})\frac{2}{\eta}>0\).
Proof
Clearly, \(G^{1}\) is a closed setvalued mapping since G is a closed setvalued mapping. Since G is a γparaconvex setmapping and \(\bar{x}+\eta B_{X}\subseteq G^{1}(\bar{y}+\delta B_{Y})\), it follows from Lemma 3.1 that, for each \(x\in\bar{x}+\frac{\eta}{2}B_{X}\),
Then, for any \(y\in\bar{y}+\frac{\eta}{2} B_{Y}\),
Let \(L:=(\delta+r(\frac{\eta}{2})^{\gamma}+\frac{\eta}{2})\frac {2}{\eta}\). We have
This implies that
Then, for any \(\tilde{\delta}>0\),
Therefore \(\Phi(\cdot):=L\Vert \cdot \Vert B_{Y}\) is a pseudo strict prederivative of G at \((\bar{x}, \bar{y})\). □
Remark 3.1
Theorem 3.1 extends [10, Theorem 3.3] from convex setvalued mappings to γparaconvex setvalued mappings. Recall [16] that G is said to be open at \((\bar{x}, \bar{y})\in \operatorname {Gr}(G)\) if \(G(U)\) is a neighborhood of ȳ for every neighborhood U of x̄. The assumption \(\bar{x}+\eta B_{X}\subseteq G^{1}(\bar{y}+\delta B_{Y})\) means that \(G^{1}(\bar{y}+\delta B_{Y})\) is a neighborhood of x̄, which is very close to the openness property of \(G^{1}\) at \((\bar{x}, \bar{y})\). However, the coefficients δ and η are fixed in our assumption.
Lemma 3.2
[17, Lemma 1]
Let A, B and D are subsets of X. If B is a closed convex set, D is a bounded set and \(A+D\subseteq B+D\), then \(A\subseteq B\).
Theorem 3.2
Let \(C\subseteq Y\) be a nonempty closed convex cone, \(\eta>0\), \(\gamma\geq1\), \(r>0\), \(\alpha>0\), \(G:X\rightrightarrows Y\) be a Cγparaconvex setvalued mapping with modulus r, \(\bar{x}+\alpha B_{X}\subseteq \operatorname {Dom}(G)\). Assume that
where \(A(x)\) is a subset of Y and \(E(x)\) is a convex set with \(A(x)\subseteq\eta B_{Y}\) and \(0\in E(x)\subseteq C\). Let \(\Phi:X\rightrightarrows Y\) be defined by
Then the following conclusions hold:

(i)
\(G+C\) is Lipschitz on \(\bar{x}+\frac{\alpha}{4}B_{X}\) with modulus \(\frac{16\eta}{\alpha}+r(\frac{3\alpha}{4})^{\gamma1}\);

(ii)
Φ is a strict prederivative of \(G+C\) at each \(x\in\bar {x}+\frac{\alpha}{8}B_{X}\);

(iii)
\(\Phi+C\) is a strict prederivative of G at each \(x\in\bar {x}+\frac{\alpha}{8}B_{X}\).
Proof
(i) Take \(x_{1}, x_{2}\in\bar{x}+\frac{\alpha}{4}B_{X}\) with \(x_{1}\neq x_{2}\). Let \(\lambda:=\frac{\alpha}{4}\) and define
Then
Let \(\theta:=\frac{\Vert x_{2}x_{1}\Vert }{\Vert x_{2}x_{1}\Vert +\lambda}\). Then \(\theta\in(0,1)\) and \(x_{1}=(1\theta)x_{2}+\theta z\). Since \(zx_{2}=\frac{x_{1}x_{2}}{\theta}\) and \(\gamma\geq1\), we have
Since G is a Cγparaconvex setvalued mapping with modulus r, it is easy to verify that \(G+C\) is a γparaconvex setvalued mapping with modulus r. Taking into account inequality (3.2), we have
Due to the convexity of \((G+C)(x)\) for each \(x\in X\), we have
Adding \(\theta(G+C)(x_{2})\) on both sides of equation (3.3), and using (3.4), we get
Clearly, \(C+\theta C=C+C=2C=C\) since C is a convex cone. Therefore, the above equation can be rewritten as
Since \(A(x)\subseteq\eta B_{Y}\) for any \(x\in\bar{x}+\alpha B_{X}\), we obtain
Next, we show that \(z\in\bar{x}+\alpha B_{X}\). Since \(x_{1}=(1\theta )x_{2}+\theta z\), we have
Therefore \(z\in\bar{x}+\alpha B_{X}\). As \(x_{2}, z\in\bar{x}+\alpha B_{X}\), combined with (3.6), we have
By the assumption (3.1), \(G(z)=A(z)+E(z)\), \(G(x_{2})=A(x_{2})+E(x_{2})\), it follows from (3.5) that
Since \(0\in E(x_{2})\subseteq C\) and \(0\in E(z)\subseteq C\), we have
Equation (3.7) yields
Since \(\operatorname {cl}(G(x_{1})+2\eta\theta B_{Y}+C+r(\frac{3\alpha}{4})^{\gamma 1}\Vert x_{1}x_{2}\Vert B_{Y})\) is a closed convex set and \(A(z)\) is a bounded set, it follows from Lemma 3.2 and (3.8) that
where the last inequality holds since \(\lambda=\frac{\alpha}{4}\). Therefore, \(G+C\) is Lipschitz with modulus \(\frac{16\eta}{\alpha }+r(\frac{3\alpha}{4})^{\gamma1}\) on \(\bar{x}+\frac{\alpha}{4}B_{X}\) since \(x_{1}\) and \(x_{2}\) are two arbitrary elements of \(\bar{x}+\frac{\alpha}{4}B_{X}\).
(ii) Let \(\Phi: X\rightrightarrows Y\) be defined by
Clearly, Φ is a positively homogeneous mapping with bounded closed values. By (3.9), we get
for any \(\delta>0\). This implies that Φ is a strict prederivative of \(G+C\) at each \(x\in\bar{x}+\frac{\alpha}{8}B_{X}\).
(iii) Since C is a cone, it follows from (3.10) that \(0\in (\Phi+C)(0)\), and for any \(t>0\) and \(x\in X\),
and hence \(\Phi+C\) is positively homogeneous. Let \(\tilde{x}\in\bar{x}+\frac{\alpha}{8}B_{X}\). Then there exists \(\tilde{r}>0\) such that \(\tilde{x}+\tilde{r}B_{X}\subseteq\bar{x}+\frac{\alpha}{4}B_{X}\). Since \(0\in C\), it follows from (3.11) that for any \(\delta>0\),
Therefore, \(\Phi+C\) is a strict prederivative of G at x̃. □
Remark 3.2
In [10, Theorem 3.8], Gaydu, Geoffroy and Marcelin proved the following result. Let Y be a finite dimensional Banach space, \(C\subseteq Y\) be a nonempty closed convex cone, \(G:X\rightrightarrows Y\) be a Cconvex setvalued mapping, \(\bar {x}\in \operatorname {int}(\operatorname {dom}(G))\). Assume that there exist \(\alpha>0\) and \(\eta>0\) such that \(G(x)+\operatorname {cl}(C)\) is a closed set and \(G(x)\subseteq\eta B_{Y}\) for all \(x\in\bar{x}+\alpha B_{X}\). Then there exists \(U\in N(\bar{x})\) such that

(i)
\(G+C\) is Lipschitz on U;

(ii)
there exists a positively homogeneous mapping \(\Phi: X\rightrightarrows Y\) with bounded closed values such that Φ is a strict prederivative of \(G+C\) at each \(x\in U\);

(iii)
\(\Phi+C\) is a strict prederivative of G at each \(x\in U\).
In contrast with [10, Theorem 3.8], Theorem 3.2 has some improvements. Firstly, we extend Y from finite dimensional spaces to general Banach spaces. Secondly, we extend G from Cconvex setvalued mappings to Cγparaconvex setvalued mappings. Thirdly, we do not need the boundedness of \(G(x)\).
In the following, we give an example to illustrate Theorem 3.2.
Example 3.1
Let \(X=Y=\mathbb{R}\), \(C=\mathbb {R}_{+}\), \(G: \mathbb{R}\rightrightarrows\mathbb{R}\) be defined by
It follows from [12, Example 3.1] that G is a C1paraconvex setvalued mapping with modulus 1, but not a Cconvex setvalued mapping. Take \(\bar{x}=0\), \(A(x)=\vert \vert x\vert 1\vert \) and \(E(x)=\mathbb{R}_{+}\). Then
with \(A(x)\subseteq[1,1]\) for all \(x\in\bar{x}+B_{X}\). All conditions of Theorem 3.2 are justified. By Theorem 3.2, \(G+C\) is Lipschitz on \(\bar{x}+\frac{1}{4}B_{X}\) with modulus 17, and \(\Phi(\cdot)=17\vert \cdot \vert B_{Y}\) satisfies (ii) and (iii) of Theorem 3.2 on \(\bar{x}+\frac{1}{8}B_{X}\).
Corollary 3.1
Let \(C\subseteq Y\) be a nonempty closed convex cone, \(\eta>0\), \(\gamma\geq1\), \(r>0\), \(\alpha>0\), \(G:X\rightrightarrows Y\) be a Cγparaconvex setvalued mapping with modulus r and \(\bar{x}+\alpha B_{X}\subseteq \operatorname {Dom}(G)\). Assume that
Let \(\Phi:X\rightrightarrows Y\) be defined by
Then the following conclusions hold:

(i)
\(G+C\) is Lipschitz on \(\bar{x}+\frac{\alpha}{4}B_{X}\) with modulus \(\frac{16\eta}{\alpha}+r(\frac{3\alpha}{4})^{\gamma1}\);

(ii)
Φ is a strict prederivative of \(G+C\) at each \(x\in\bar {x}+\frac{\alpha}{8}B_{X}\);

(iii)
\(\Phi+C\) is a strict prederivative of G at each \(x\in\bar {x}+\frac{\alpha}{8}B_{X}\).
Proof
Take \(A(x)=G(x)\), \(E(x)=\{0\}\) for all \(x\in X\) in Theorem 3.2. Then the conclusions follow from Theorem 3.2 directly. □
4 Pareto minimizer and prederivative
In this section, we always assume that C is a pointed closed convex cone of Y. Consider the following set optimization problem:
where Ω is a nonempty closed subset of X with \(\operatorname {Dom}(G)\cap \Omega\neq\emptyset\).
Definition 4.1
[13]
We say that \((\bar{u}, \bar {v})\in \operatorname {Gr}(G)\) is a Pareto minimizer of the optimization problem (SP), if \(\bar{u}\in\Omega\) and \((\bar{v}C)\cap G(\Omega)=\{\bar {v}\}\).
First, we establish a necessary condition for Pareto minimizers of the optimization problem (SP).
Theorem 4.1
Let \(\bar{u}\in \operatorname {int}(\Omega)\), \((\bar{u}, \bar{v})\in \operatorname {Gr}(G)\). Suppose that Φ is a pseudo strict prederivative of G at \((\bar {u},\bar{v})\) and \((\bar{u}, \bar{v})\) is a Pareto minimizer of the optimization problem (SP). Then, for any \(\delta>0\) and \(u\in\Omega\),
Proof
Let \((\bar{u}, \bar{v})\) be a Pareto minimizer of the optimization problem (SP). Suppose that the conclusion is not true. Then there exist \(\delta_{0}>0\) and \(u_{0}\in\Omega\) such that
Since Φ is a pseudo strict prederivative of G at \((\bar{u},\bar {v})\), there exist \(\eta_{1}>0\) and \(\eta_{2}>0\) such that
Choose \(\theta>0\) such that \(\bar{u}+\theta(u_{0}\bar{u})\in(\bar{u}+\eta_{1}B_{X})\cap\Omega\). By the above equation, we have
Since \(\bar{v}\in G(\bar{u})\cap(\bar{v}+\eta_{2}B_{Y}) \), the above equation implies that there exists \(\hat{v}\in G(\bar{u}+\theta(u_{0}\bar{u}))\) such that
Since Φ is positively homogeneous, we get
Combined with (4.2), we have \(\bar{v}\hat{v}\in C\setminus\{ 0\}\). Noting that \(\hat{v}\in G(\bar{u}+\theta(u_{0}\bar{u}))\subseteq G(\Omega)\), we have
This contradicts the assumption that \((\bar{u}, \bar{v})\) is a Pareto minimizer of the optimization problem (SP). Therefore (4.1) holds. □
Definition 4.2
[18]
Let \(G:X\rightrightarrows Y\) be a setvalued mapping, \(\bar{u}\in \operatorname {Dom}(G)\). We say that G is Cstarshaped at ū, if for any \(u\in X\), \(\theta\in[0, 1]\),
The following theorem provides a sufficient optimality condition for a Pareto minimizer of the optimization problem (SP).
Theorem 4.2
Let \(\bar{u}\in\Omega\), Φ be an outer prederivative of G at ū, \(\bar{v}\in G(\bar{u})\), \(G(\bar{u})\subseteq\bar{v}+C\) and G be Cstarshaped at ū. If there exist \(\delta>0\) and \(\eta>0\) such that
for all \(u\in\bar{u}+\eta B_{X}\), then \((\bar{u}, \bar{v})\) is a Pareto minimizer of the optimization problem (SP).
Proof
Let \(u\in\Omega\). Since Φ is an outer prederivative of G at ū, for the given δ in the assumption, there exists \(\bar{\eta}\in(0, \eta)\) such that
Choose \(\theta\in(0, 1)\) such that \((1\theta)\bar{u}+\theta u\in \bar{u}+\bar{\eta}B_{X}\). Since G is Cstarshaped at ū, we have
Combined (4.4) with (4.5), we get
As \(G(\bar{u})\subseteq\bar{v}+C\), we get
that is,
Since \((1\theta)\bar{u}+\theta u\in\bar{u}+\bar{\eta} B_{X}\), with (4.3), we have
It follows from (4.6) that
Therefore, \((\bar{u}, \bar{v})\) is a Pareto minimizer of the optimization problem (SP) since u is an arbitrary element of Ω. □
Remark 4.1
In [10], Gaydu, Geoffroy and Marcelin established necessary condition and sufficient conditions for the weak minimizers and the strong minimizers of the optimization problem (SP). It is well known that each strong minimizer is a Pareto minimizer and each Pareto minimizer is a weak minimizer, but the converses are not true.
In the following, we give an example to illustrate Theorem 4.2.
Example 4.1
In (SP), let \(X=\mathbb{R}\), \(Y=\mathbb {R}^{2}\), \(C=\mathbb{R}^{2}_{+}\), \(\Omega=(\infty, 0]\), \(G: X\rightrightarrows Y\) be defined by
Take \(\bar{u}=0\), \(\bar{v}=(0,0)\in G(\bar{u})\) and \(\delta=\frac {1}{2}\). Then \(G(\bar{u})\subseteq\bar{v}+\mathbb{R}^{2}_{+}\). It is easy to verify that G is Cstarshaped at ū, \(\Phi =(1,1)\) is an outer prederivative of G at ū and
All conditions of Theorem 4.2 are verified. Therefore, \((\bar {u}, \bar{v})\) is a Pareto minimizer of the optimization problem (SP).
5 Conclusion
In this paper, we establish two existence theorems of prederivatives for γparaconvex setvalued mappings, and give optimality conditions for the Pareto minimizers of set optimization problems. These results improve the corresponding one obtained in [10]. Moreover, the coefficients in Theorems 3.1 and 3.2 can be calculated. Theorems 3.1 and 3.2 give sufficient conditions for the existence of Φ of Theorem 4.1.
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Acknowledgements
The research was supported by the National Natural Science Foundation of China under Grant No. 11461080. The authors are greatly indebted to the reviewers and the editor for their valuable comments.
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Huang, H., Ning, J. Prederivatives of gamma paraconvex setvalued maps and Pareto optimality conditions for set optimization problems. J Inequal Appl 2017, 243 (2017). https://doi.org/10.1186/s1366001715194
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DOI: https://doi.org/10.1186/s1366001715194