Arcwise connected conequasiconvex setvalued mappings and Pareto reducibility in vector optimization
 GuoLin Yu^{1}Email author
https://doi.org/10.1186/s1366001508386
© Yu 2015
Received: 10 June 2015
Accepted: 23 September 2015
Published: 6 October 2015
Abstract
The aim of this note is twofold: first, to show that arcwise connected conequasiconvex setvalued mappings can be characterized in terms of classical arcwise connected quasiconvexity of certain realvalued functions, defined by Gerstewitz’s scalarization function; second, by making use of the recent result concerning the Pareto reducibility in multicriteria arcwise connected conequasiconvex optimization problems to establish similar setvalued optimization problems under appropriate assumptions.
Keywords
MSC
1 Introduction
It is well known that convexity and its various generalizations play a dominant role in optimization. In order to relax the convexity assumption, many kinds of generalized convexity have been introduced by many authors. Among the different notions of generalized convexity, quasiconvexity and arcwise connected convexity have found many important applications; see for instance [1–13]. In convex analysis, the new generalized convexity can be derived by combining two or more existing types of generalized convexity. A good example is the arcwise connected quasiconvexity, which was presented by mixing arcwise connected convexity together with quasiconvexity. In fact, the realvalued arcwise connected quasiconvex functions had already appeared in early works, such as [1]. In [11], La Torre and Popovici extended this notion to vectorvalued functions taking values in real partially ordered vector spaces, and applied it to study the contractibility of efficient sets and Pareto reducibility in multicriteria optimization. The notion of Pareto reducibility, introduced by Popovici in [14], is to represent the weakly efficient solution set as the union of the sets of efficient solutions of all subproblems obtained from the original one by selecting certain criteria. Popovici in [9] extended the Pareto reducibility in multicriteria optimization problems to explicitly quasiconvex setvalued optimization; for more details related to the Pareto reducibility, refer to [8, 9, 11, 14, 15]. La Torre in [10] introduced the arcwise connected conequasiconvex setvalued mapping and investigated the optimality conditions for a setvalued optimization involving this type of data. A natural question arises: is the arcwise connected conequasiconvex setvalued optimization problem also Pareto reducible? One aim of this paper is to show, with the help of results obtained for multicriteria optimization in [11], that the answer is positive.
Another interesting topic in vector optimization is to characterize the generalized coneconvexity of the vectorvalued (or setvalued) objective functions in terms of usual generalized convexity of certain realvalued functions, by means of some appropriate scalarization functionals. For instance, it was presented in [2, 16] that coneconvex and conequasiconvex functions can be characterized by means of the extreme directions of a polar cone and Gerstewitz’s scalarization functions; similar characterizations of weakly coneconvex and weakly conequasiconvex functions were given in [12]; scalar characterizations of coneconvex functions in variable domination structures were proposed in [17]. For characterizations of coneconvexity and conequasiconvexity for setvalued maps, we refer to [3, 9] and the references therein. The second aim of this note is to show that the arcwise connected conequasiconvex setvalued mappings can also be characterized by means of Gerstewitz’s scalarization functions.
We begin in Section 2 by recalling some definitions and preliminary results concerning arcwise connected conequasiconvex setvalued mappings. In addition, several properties for arcwise connected conequasiconvex setvalued mappings, which will be used in the sequel, are discussed. Section 3 is devoted to the characterizations of arcwise connected conequasiconvex setvalued mappings by means of Gerstewitz’s scalarization function. Finally, in Section 4, by restricting our attention on setvalued optimization with the value of objective function in a finite dimensional Euclidean space, we get the sufficient condition for the Pareto reducibility with the help of the results derived for arcwise connected conequasiconvex multicriteria optimization in [11].
2 Preliminaries
From now on, we always assume that S is a nonempty subset of X. Let us recall the notions of arcwise connected convexity for a set and arcwise connected conequasiconvexity for a setvalued mapping.
Definition 2.1
[1]
In this paper, the empty set ∅ is assumed to be an arcwise connected set. On the other hand, obviously, if a set is convex, then it is arcwise connected set. In general, the opposite is not true.
Example 2.2
Definition 2.3
[10]
If the setvalued mapping F degenerates to a vectorvalued function, then the definition of arcwise connected Dquasiconvexity coincides with the definition of ‘arcwise Dquasiconvexity’ introduced by La Torre and Popovici in [11]. In order to unify the terminology, we still use ‘arcwise connected conequasiconvexity’ in the case of singlevalued functions. Let us see an example of arcwise connected conequasiconvex setvalued map.
Example 2.4
Next, some basic properties concerning arcwise connected conequasiconvex setvalued maps can easily be obtained and some of them will be used in the sequel.
Proposition 2.5
Let \(F: S\subset X\rightarrow2^{Y}\) be a setvalued map and \(D_{1}\), \(D_{2}\) be two convex cones of Y, \(D_{1}\subset D_{2}\). If F is arcwise connected \(D_{1}\)quasiconvex, then F is also arcwise connected \(D_{2}\)quasiconvex.
Proof
Proposition 2.6
If \(F: S\subset X\rightarrow2^{Y}\) is arcwise connected Dquasiconvex and the ordering cone D generates the space Y, i.e. \(DD=Y\), then S is arcwise connected.
Proof
Proposition 2.7
Let \(F: S\subset X\rightarrow2^{Y}\) be a setvalued mapping. Suppose that f is a selection of F and \(F(x)\subset f(x)+D\) for all \(x\in S\). If f is arcwise connected Dquasiconvex, then F is arcwise connected Dquasiconvex.
Proof
Proposition 2.8
Let \(F: S\subset X\rightarrow2^{Y}\) be a setvalued mapping. Suppose that f is a selection of F and \(F(x)\subset f(x)+D\) for all \(x\in S\). If F is arcwise connected Dquasiconvex, then f is arcwise connected Dquasiconvex.
Proof
Remark 2.9
Corollary 2.10 shows that an acwise connected conequasiconvex setvalued mapping \(F S\subset X\rightarrow2^{Y}\) is characterized by a selection f satisfying \(F(x)\subset f(x)+D\) for all \(x\in S\), which can be obtained directly from Proposition 2.7 and Proposition 2.8.
Corollary 2.10
Let \(F S\subset X\rightarrow2^{Y}\) be a setvalued mapping. Suppose that f is a selection of F satisfying \(F(x)\subset f(x)+D\) for all \(x\in S\). Then F is arcwise connected Dquasiconvex if and only if f is arcwise connected Dquasiconvex.
3 Scalarization by means of Gerstewitz’s function
Lemma 3.1
[18]
For fixed \(e\in\operatorname{int}D\), any \(v\in Y\) and \(r\in\mathbb{R}\), we have \(h_{e,v}(y)\leq r \Leftrightarrow y\in re+vD\).
Proposition 3.2
Let \(F: S\subset X\rightarrow2^{Y}\) be arcwise connected Dquasiconvex setvalued mapping. Then for any \(v\in Y\), \(h_{e,v}\circ F\) is arcwise connected \(\mathbb {R}_{+}\)quasiconvex, where \(h_{e,v}\circ F(x):=h_{e,v} (F(x) )=\bigcup_{y\in F(x)}h_{e,v}(y)\), for all \(x\in S\).
Proof
Proposition 3.3
Let \(F: S\subset X\rightarrow2^{Y}\) be a setvalued mapping. If for any \(v\in Y\), \(h_{e,v}\circ F\) is arcwise connected \(\mathbb{R}_{+}\)quasiconvex, then F is arcwise connected Dquasiconvex.
Proof
According to Proposition 3.2 and Proposition 3.3, we conclude this section by presenting Corollary 3.4, which is a characterization of arcwise conequasiconvex setvalued maps in terms of scalar arcwise connected quasiconvexity.
Corollary 3.4
 (I)
f is arcwise connected Dquasiconvex.
 (II)
For every point \(v\in Y\) the composite mapping \(h_{e,v}\circ F\) is arcwise connected \(\mathbb{R}_{+}\)quasiconvex.
Remark 3.5
The Gerstewitz scalarizing function plays a very important role in vector optimization with setvalued mappings. Recently, it has been extended to functions mapping from the family of nonempty subsets of Y to \(\mathbb{R}\), and whether the extended Gerstewitz functions can characterize the generalized convex setvalued mappings is also an interesting question; for more details related to the extended Gerstewitz functions, we refer to [19–22].
4 Setvalued optimization problems
Definition 4.1
[9]
 1^{∘} :

upward, if \(\Omega+\mathbb{R}_{+}^{n}=\Omega\);
 2^{∘} :

Kradiant, where K is a cone of \(\mathbb{R}^{n} \), if$$\operatorname{ray}(y_{1},y_{2}):=y_{1}+ \mathbb{R}_{+}(y_{2}y_{1})\subset\Omega \quad \mbox{for all } y_{1},y_{2}\in\Omega, y_{1} \leqslant_{K} y_{2}. $$
Remark 4.2
Let K be a cone of \(\mathbb{R}^{n}\), \(\Omega _{1}\), \(\Omega_{2}\) be two subsets of \(\mathbb{R}^{n}\), and \(\Omega_{1}\subset \Omega_{2}\). From the definition of Kradiant, it is easy to see that if \(\Omega_{1}\) is Kradiant then \(\Omega_{2}\) is Kradiant.
Lemma 4.3
[11]
Let \(f: S\rightarrow\mathbb{R}^{n}\) be a function. If f is continuous and arcwise connected \(K_{i}\)quasiconvex for every \(i\in I_{n}\), then \(\operatorname{WMin} (f(S)+\mathbb{R}_{+}^{n} )\) is \(\mathbb{R}_{+}^{n}\)radiant.
Lemma 4.4
[9]
In problem (SVOP), if \(\operatorname{WMin} (F(S)+\mathbb{R}_{+}^{n} )\) is \(\mathbb{R}_{+}^{n}\)radiant, then the problem (SVOP) is Pareto reducible.
Proposition 4.5
Let \(F: S\subset X\rightarrow 2^{\mathbb{R}^{n}}\) be a setvalued mapping. Suppose that f is a continuous selection of F and \(F(x)\subset f(x)+\mathbb{R}_{+}^{n}\). If F is arcwise connected \(K_{i}\)quasiconvex for every \(i\in I_{n}\), then \(\operatorname{WMin} (F(S)+\mathbb{R}_{+}^{n} )\) is \(\mathbb {R}_{+}^{n}\)radiant.
Proof
It follows from Proposition 2.8 that the continuous function f is arcwise connected \(K_{i}\)quasiconvex for every \(i\in I_{n}\). Then we see from Lemma 4.3 that \(\operatorname{WMin} (f(S)+\mathbb{R}_{+}^{n} )\) is \(\mathbb{R}_{+}^{n}\)radiant. Finally, we see from Remark 4.2 that \(\operatorname{WMin} (F(S)+\mathbb{R}_{+}^{n})\) is \(\mathbb{R}_{+}^{n}\)radiant. □
Proposition 4.6
Let \(F: S\subset X\rightarrow 2^{\mathbb{R}^{n}}\) be a setvalued mapping. Suppose that f is a continuous selection of F and \(F(x)\subset f(x)+\mathbb{R}_{+}^{n}\). If F is arcwise connected \(K_{i}\)quasiconvex for every \(i\in I_{n}\), then the problem (SVOP) is Pareto reducible.
5 Conclusions
In this note, some properties of the arcwise connected conequasiconvex setvalued mapping have been carried out. We point out that an arcwise connected conequasiconvex setvalued mapping is characterized by a selection satisfying suitable conditions. On the other hand, we show that the arcwise connected conequasiconvex setvalued mappings can also be characterized by means of Gerstewitz scalarization functions. Finally, in the setting of finite dimensional Euclidean space, the sufficient condition for the Pareto reducibility for arcwise connected conequasiconvex multicriteria optimization is presented.
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 author would like to extend sincere gratitude to Prof. Dai Yuhong (Academy of Mathematics and System Science, Chinese Academy of Sciences, China) for his freehanded assistance. The author is 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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