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Prederivatives of gamma paraconvex set-valued maps and Pareto optimality conditions for set optimization problems
- Hui Huang^{1}Email authorView ORCID ID profile and
- Jixian Ning^{1, 2}
https://doi.org/10.1186/s13660-017-1519-4
© The Author(s) 2017
Received: 2 August 2017
Accepted: 18 September 2017
Published: 2 October 2017
Abstract
Prederivatives play an important role in the research of set optimization problems. First, we establish several existence theorems of prederivatives for γ-paraconvex set-valued 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.
Keywords
MSC
1 Introduction
Let X and Y be Banach spaces. We say that \(G: X\rightrightarrows Y\) is a set-valued mapping if \(G(x)\) is a subset of Y for all \(x\in X\). Set-valued problems occur in many situations, such as control problems, feasibility problems, optimality problems, equilibrium problems and variational inequality problems. A powerful tool dealing with set-valued problems is set-valued analysis. We refer the reader to the references [1–4] for more knowledge about set-valued 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 Jean-Alexis [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 set-valued mappings and established necessary and sufficient optimality conditions for the weak minimizers and the strong minimizers of set optimization problems. γ-paraconvex set-valued mappings are an extension of convex set-valued 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 γ-paraconvex set-valued 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 γ-paraconvex set-valued mappings and cone-γ-paraconvex set-valued mappings. Then we establish necessary and sufficient optimality conditions for the Pareto minimizers of set optimization problems in terms of prederivatives.
2 Preliminaries
The following definition is needed in the sequel.
Definition 2.1
[1]
Let \(\Phi :X\rightrightarrows Y\) be a set-valued 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]
Remark 2.1
In the special case of \(\eta={0}\) and \(C=\{0\}\), C-γ-paraconvex set-valued mappings reduce to convex set-valued mappings.
Definition 2.3
[15]
Definition 2.4
[10]
- (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$$ G(u)\subseteq G\bigl(u'\bigr)+\Phi\bigl(u-u'\bigr)+ \delta\bigl\Vert u-u'\bigr\Vert B_{Y},\quad \forall u, u'\in U. $$
- (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$$ G(u)\cap V\subseteq G\bigl(u'\bigr)+\Phi\bigl(u-u' \bigr)+\delta\bigl\Vert u-u'\bigr\Vert B_{Y}, \quad \forall u, u'\in U. $$
3 Prederivatives of gamma paraconvex set-valued mappings
In this section, we establish the existence results of pseudo strict prederivatives for γ-paraconvex set-valued mappings and strict prederivatives for C-γ-paraconvex set-valued mappings, respectively.
Lemma 3.1
[11]
Theorem 3.1
Let \(\eta>0\), \(\delta>0\), \(r>0\), \(\gamma>0\), \(G:X\rightrightarrows Y\) be a closed γ-paraconvex set-valued 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
Remark 3.1
Theorem 3.1 extends [10, Theorem 3.3] from convex set-valued mappings to γ-paraconvex set-valued 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
- (i)
\(G+C\) is Lipschitz on \(\bar{x}+\frac{\alpha}{4}B_{X}\) with modulus \(\frac{16\eta}{\alpha}+r(\frac{3\alpha}{4})^{\gamma-1}\);
- (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
Remark 3.2
- (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 C-convex set-valued mappings to C-γ-paraconvex set-valued 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
Corollary 3.1
- (i)
\(G+C\) is Lipschitz on \(\bar{x}+\frac{\alpha}{4}B_{X}\) with modulus \(\frac{16\eta}{\alpha}+r(\frac{3\alpha}{4})^{\gamma-1}\);
- (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 \(x\in\bar {x}+\frac{\alpha}{8}B_{X}\).
4 Pareto minimizer and prederivative
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
Proof
Definition 4.2
[18]
The following theorem provides a sufficient optimality condition for a Pareto minimizer of the optimization problem (SP).
Theorem 4.2
Proof
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
5 Conclusion
In this paper, we establish two existence theorems of prederivatives for γ-paraconvex set-valued 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.
Declarations
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.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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.
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