Optimality for set-valued optimization in the sense of vector and set criteria
- Xiangyu Kong^{1},
- GuoLin Yu^{1}Email author and
- Wei Liu^{1}
https://doi.org/10.1186/s13660-017-1319-x
© The Author(s) 2017
Received: 13 October 2016
Accepted: 10 February 2017
Published: 17 February 2017
Abstract
The vector criterion and set criterion are two defining approaches of solutions for the set-valued optimization problems. In this paper, the optimality conditions of both criteria of solutions are established for the set-valued optimization problems. By using Studniarski derivatives, the necessary and sufficient optimality conditions are derived in the sense of vector and set optimization.
Keywords
MSC
1 Introduction
Another criterion, the set criterion, also called set optimization, was proposed by Kuroiwa [5, 6] in 1999 for the first time. Set optimization consists of looking for image sets \(F(\bar{x})\), with \(\bar{x}\in S\), which satisfy some set-relations between the rest of the image sets \(F(x)\) with \(x\in S\). In [3], p. 378, Jahn states that the set relation approach opens a new and exciting field of research. Although the set criterion seems to be more natural and interesting than the traditional one, the study of set optimization is very limited, such as papers on existence conditions, see [7–12]; on duality theory, see [13–16]; on optimality conditions, see [17–19]; on scalarization [20–23]; on well-posedness properties, see [24, 25]; on Ekeland variational principles, see [26, 27].
The investigation of optimality conditions, especially as regards the vector criterion, has received tremendous attention in the research of optimization problems and has been studied extensively. As everyone knows most of the problems under consideration are nonsmooth, which leads to introducing many kinds of generalized derivatives by the authors. A meaningful concept is the Studniarski derivative [28], which has some properties similar to the contingent derivative [1] and the Hadamard derivative [29]. Recently, much attention has been paid to optimality conditions and related topics for vector optimization by using Studniarski derivatives; see [30, 31].
Inspired by the above observations, in this paper, the optimality conditions are established for problem (P) in both the vector criterion and set criterion by using Studniarski derivatives. The rest of the paper is organized as follows: In Section 2, some well-known definitions and results used in sequel are recalled. In Section 3 and Section 4, the necessary and sufficient optimality conditions are given in the sense of the vector criterion and set criterion, respectively.
2 Preliminaries
Definition 2.1
see [1]
For a set-valued mapping \(F: X\rightarrow2^{Y}\), the upper and lower Studniarski derivatives of the map F is defined as follows:
Definition 2.2
see [28]
- (i)The upper Studniarski derivatives of F at \((\bar{x},\bar {y})\) is defined by$$ \overline{D} F(\bar{x},\bar{y}) (x):=\limsup_{(t,x')\rightarrow (0^{+},x)} \frac{F(\bar{x}+tx')-\bar{y}}{t}. $$
- (ii)The lower Studniarski derivative of F at \((\bar{x},\bar {y})\) is defined by$$ \underline{D} F(\bar{x},\bar{y}) (x):=\liminf_{(t,x')\rightarrow (0^{+},x)} \frac{F(\bar{x}+tx')-\bar{y}}{t}. $$
Remark 2.3
- (a)It is easily to see from Definition 2.2 that$$\begin{aligned} \overline{D} F(\bar{x},\bar{y}) (x)={}& \bigl\{ y\in Y: \exists (t_{n}) \rightarrow0^{+}, \exists(x_{n},y_{n})\rightarrow(x,y)\\ &\mbox{s.t. } \forall n, \bar{y}+t_{n} y_{n}\in F( \bar{x}+t_{n} x_{n}) \bigr\} , \\ \underline{D} F(\bar{x},\bar{y}) (x)={}& \bigl\{ y\in Y: \forall (t_{n}) \rightarrow0^{+}, \forall(x_{n})\rightarrow x, \exists (y_{n}) \rightarrow y\\ &\mbox{s.t. } \forall n, \bar{y}+t_{n} y_{n}\in F( \bar {x}+t_{n} x_{n}) \bigr\} . \end{aligned}$$
- (b)
The upper Studniarski derivative is an exactly contingent derivative [1] or an upper Hadamard derivative [29], and the lower Studniarski derivative is the lower Hadamard derivative introduced by Penot [29].
Lemma 2.4 is an useful property associated with C-pseudoconvexity of a set-valued mapping, which will be used in the next two sections.
Lemma 2.4
Proof
Example 2.5
3 Optimality conditions with vector criterion
Let \(S\subset X\) be a nonempty set, \(F: S\rightarrow2^{Y}\) be a set-valued mapping and \((\bar{x},\bar{y})\in\operatorname{graph}(F)\).
From now on, for convention we always assume that the upper and lower Studniarski derivatives of the map F at \((\bar{x},\bar{y})\) exist and \(\operatorname{dom}\overline{D}F(\bar{x},\bar{y})= \operatorname {dom}\underline{D}F(\bar{x},\bar{y})=S\).
Definition 3.1
Theorems 3.2 and 3.3 are necessary optimality conditions for the weak minimizer of problem (VP).
Theorem 3.2
Proof
Theorem 3.3
Proof
Now, we present a sufficient optimality condition for problem (VP) under the assumption of pesudoconvexity of F.
Theorem 3.4
Proof
4 Optimality conditions with set criterion
This section works with the optimality conditions for problem (P) in the sense of the set criterion. Firstly, let us recall the concepts of set relations.
Definition 4.1
see [5]
Let A and B be two nonempty sets of Y. We write \(A\leqslant^{\mathrm{l}} B\), if for all \(b\in B\) there exists \(a\in A\) such that \(a\leqslant b\). Here, ‘\(\leqslant^{\mathrm{l}}\)’ is called a lower relation.
Remark 4.2
The above defined lower relation is equivalent to \(B\subset A+C\) and it is the generalization of the order induced by a pointed convex cone C in Y, in the following sense: \(a\leqslant b\) if \(b\in a+C\).
Definition 4.3
see [14]
Let A and B be two nonempty sets of Y. We write \(A< ^{\mathrm{lw}} B\), if \(B\subset A+\operatorname {int}C\). Here, ‘\(< ^{\mathrm{lw}}\)’ is named a lower weak relation.
Definition 4.4
see [14]
- (1)
\(A\in\mathcal{A}\) is named a lower minimal of \(\mathcal {A}\), if for each \(B\in\mathcal{A}\) such that \(B\leqslant^{\mathrm{l}} A\), it satisfies \(A\leqslant^{\mathrm{l}} B\);
- (2)
\(A\in\mathcal{A}\) is called a lower weak minimal of \(\mathcal{A}\), if for each \(B\in\mathcal{A}\) such that \(B< ^{\mathrm{lw}} A\), it satisfies \(A< ^{\mathrm{lw}} B\).
Example 4.5
Definition 4.6
see [19]
Let \(F(\bar{x})\) be a lower weak minimal of (SP). \(F(\bar{x})\) is named strict lower weak minimal of (SP), if there exists a neighborhood \(U(\bar{x})\) of x̄ such that \(F(x)\nless^{\mathrm{lw}} F(\bar{x})\), for all \(x\in U(\bar{x})\cap S\). Then x̄ is called a strict lower weak minimum of (SP).
Lemma 4.7
see [19]
- (1)
\(F(x)\in[F(\bar{x})]^{\mathrm{lw}}\).
- (2)
There exists \(\bar{y}\in F(\bar{x})\) such that \((F(x)-\bar{y} )\cap(-\operatorname{int}C)=\emptyset\).
Definition 4.8
see [19]
(Domination property). It is called that a subset \(A\subset Y\) has the D-weak minimal property, if for all \(y\in A\) there exists a weakly minimal a of A such that \(a-y\in(-\operatorname{int} C)\cup\{0\}\).
Lemma 4.9
Proof
Theorem 4.10
Proof
Theorem 4.11
Proof
Based upon Theorem 4.10, we see that equation (4.2) holds. The rest of the proof can be followed by similar arguments to that of Theorem 3.3. □
Theorem 4.12
Proof
Remark 4.13
By comparing the results derived in Section 3, it can be found that the optimality for the set criterion and vector criterion, with the suitable conditions, possesses the same forms in terms of Studniarshi derivatives.
5 Conclusions
We have studied the optimality conditions for both the vector criterion and set criterion for a set-valued optimization problem in this note. We have presented two sufficient optimality conditions and a necessary condition for a weak minimizer in terms of Studniarski derivatives. In the set optimization criterion, utilizing the known results, we have proved two necessary optimality conditions for a strict lower weak minimum and a lower weak minimum. A sufficient optimality condition has been proved under the assumption of pseudoconvexity.
Declarations
Acknowledgements
This research was supported by Natural Science Foundation of China under Grant No. 11361001; Natural Science Foundation of Ningxia under Grant No. NZ14101. The authors are grateful to the anonymous referees who have contributed to improve the quality of 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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