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Arcwise connected conequasiconvex setvalued mappings and Pareto reducibility in vector optimization
Journal of Inequalities and Applicationsvolume 2015, Article number: 317 (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.
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].
Preliminaries
Let X be a real linear space and Y be a real Banach spaces, $D\subset Y$ be a closed convex cone. Considering the partially order induced by D, defined as follows:
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]
A subset $S\subset X$ is said to be an arcwise connected set, if for every $x_{1}\in S$, $x_{2}\in S$, there exists a continuous vectorvalued function $H_{x_{1},x_{2}}: [0,1]\rightarrow S$, called an arc, such that
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
Let
Clearly, S is not convex. However, S is an arcwise connected set with respect to the arc $H_{x,z}$, defined by
Let $\varphi: S\subset X\rightarrow\mathbb{R}$ be a realvalued function. Recall that φ is said to be arcwise connected quasiconvex, if for all $x_{1},x_{2}\in X$ and $t\in [0,1]$ there exists an arc $H_{x_{1},x_{2}}: [0,1]\rightarrow S$ such that $H_{x_{1},x_{2}}(0)=x_{1}$, $H_{x_{1},x_{2}}(1)=x_{2}$, and
which means that for every $\lambda\in\mathbb{R}$ the following level set is arcwise connected:
In the literature [5, 10], the notion of arcwise connected quasiconvexity has been extended to the vectorvalued functions and setvalued mappings, respectively. For any setvalued map $F: S\rightarrow2^{Y}$ and every set $A\subset Y$, we denote by $\operatorname{dom}(F)=\{x\in S: F(x)\neq\emptyset\}$ and $F^{1}(A):=\{x\in S: F(x) \cap A\neq\emptyset\}$ efficient domain of F and the inverse image of A by F, respectively. Throughout this paper, we always assume that $\operatorname{dom}F=S$ for the setvalued mapping $F: S\subset X\rightarrow2^{Y}$. A function $f: S\subset X\rightarrow Y$ is called a selection of F if $f(x)\in F(x)$ for all $x\in S$.
Definition 2.3
[10]
Let $F: S\subset X\rightarrow2^{Y}$ be a setvalued mapping. It is said that F is arcwise connected Dquasiconvex if the generalized level set
is an arcwise connected set for every point $y\in Y$. Actually, F is arcwise connected Dquasiconvex if for all $x_{1},x_{2}\in X$ and $t\in[0,1]$ there exists an arc $H_{x_{1},x_{2}}: [0,1]\rightarrow S$ such that $H_{x_{1},x_{2}}(0)=x_{1}$, $H_{x_{1},x_{2}}(1)=x_{2}$ and
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
Let $X=\mathbb{R}$, Y be the space of all real sequences y in $\mathbb{R}$
in which $\lim_{n\rightarrow\infty}\xi_{n}=0$. Let D be the set of all nonnegative sequences in Y. Define
Then Y is a Banach space. For $S=\mathbb{R}_{+}$, the setvalued mapping $F: S\rightarrow2^{Y}$ defined by
is arcwise connected Dquasiconvex with respect to the arc
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
In fact, for any $y\in Y$, suppose that the generalized level set of F with respect to the ordering cone $D_{1}$
is arcwise connected. Then, for any $x_{1},x_{2}\in\{x\in S: y\in F(x)+D_{1}\}$, there exists an arc $H_{x_{1},x_{2}} [0,1]\rightarrow S$ such that $H_{x_{1},x_{2}}(0)=x_{1}$, $H_{x_{1},x_{2}}(1)=x_{2}$, and $H_{x_{1},x_{2}}(t)\in \operatorname{Lev}_{F}^{D_{1}}(y)$ for all $t\in\,]0,1[$. Therefore, we get $y\in F (H_{x_{1},x_{2}}(t) )+D_{1}\subset F (H_{x_{1},x_{2}}(t) )+D_{2}$ for all $t\in[0,1]$. This shows that
is arcwise connected, as desired. □
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
Taking $x_{1},x_{2}\in S$ arbitrary, there exist $y_{1}\in F(x_{1})$ and $y_{2}\in F(x_{2})$. Since D generates the space Y, there exists $y\in Y$ such that $y_{1}\leqslant_{D} y$ and $y_{2}\leqslant_{D} y$. This means
Then we see from the arcwise connected Dquasiconvexity of F that there exists an arc $H_{x_{1},x_{2}}: [0,1]\rightarrow S$ such that $H_{x_{1},x_{2}}(0)=x_{1}$, $H_{x_{1},x_{2}}(2)=x_{2}$, and $H_{x_{1},x_{2}}(t)\in \operatorname{Lev}_{F}(y)\subset S$ for all $t\in\,]0,1[$. Hence, S is arcwise connected. □
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
In fact, for $x_{1},x_{2}\in X$ there exists an arc $H_{x_{1},x_{2}}: [0,1]\rightarrow S$ such that $H_{x_{1},x_{2}}(0)=x_{1}$, $H_{x_{1},x_{2}}(1)=x_{2}$, and
for any $t\in\,]0,1[$. Hence, it follows that
□
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
In fact, for $x_{1},x_{2}\in X$ and $t\in[0,1]$ there exists an arc $H_{x_{1},x_{2}}: [0,1]\rightarrow S$ such that $H_{x_{1},x_{2}}(0)=x_{1}$, $H_{x_{1},x_{2}}(2)=x_{2}$, and
□
Remark 2.9
In fact, the assumption $F(x)\subset f(x)+D$ for all $x\in S$ had been used in [10] by La Torre to derive the similar characterization for arcwise connected coneconvex setvalued maps; see Theorem 4 and Theorem 5 in [10]. Let us give a simple example, in which the assumption $F(x)\subset f(x)+D$ is satisfied. Let $X=Y=\mathbb{R}$, $D=\mathbb{R}_{+}$, and $F: X\rightarrow2^{Y}$ be defined by
It is obviously that $f(x)=x^{2}$ is a selection of F and $F(x)\subset f(x)+D$.
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.
Scalarization by means of Gerstewitz’s function
In this section, we assume that the closed convex cone $D\subset Y$ is solid, i.e. $\operatorname{int}D\neq\emptyset$. For a fixed $e\in \operatorname{int}D$, $v\in Y$ and any $y\in Y$ the set $\{t\in\mathbb{R}: y\in te+vD\}$ is nonempty, closed and bounded from below (see [5]). The wellknown Gerstewitz’s function $h_{e,v}: Y\rightarrow\mathbb {R}$ is defined by
We need its following salient property.
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
For any $v\in Y$, we have to check, for any $\lambda_{0}\in\mathbb{R}$, whether the set
is arcwise connected set. Without loss of generality, we suppose that $\operatorname{Lev}_{(h_{e,v}\circ F)} (\lambda_{0} )\neq\emptyset$, let $x_{1}$, $x_{2}$ be any two points of the level set $\operatorname{Lev}_{(h_{e,v}\circ F)} (\lambda_{0} )$ and $t\in[0,1]$. Then there exist $y_{1}\in F(x_{1})$ and $y_{2}\in F(x_{2})$ such that $\lambda_{1}=h_{e,v}(y_{1}) \leq \lambda_{0}$ and $\lambda_{2}=h_{e,v}(y_{2}) \leq\lambda_{0}$. Noticing that D is a closed convex cone, then we get
Since $\lambda_{1}=h_{e,v}(y_{1})\leq\lambda_{0}$ and $\lambda _{2}=h_{e,v}(y_{2})\leq\lambda_{0}$, it follows from Lemma 3.1 that
which means
By the arcwise connected Dquasiconvexity of F, there exist an arc $H_{x_{1},x_{2}}: [0,1]\rightarrow S$ and $y_{t}\in F (H_{x_{1},x_{2}}(t) )$ such that
that is,
By Lemma 3.1 again, we get
Therefore, $H_{x_{1},x_{2}}(t)\in\operatorname{Lev}_{(h_{e,v}\circ F)} (\lambda _{0} )$, which shows that $\operatorname{Lev}_{(h_{e,v}\circ F)} (\lambda _{0} )$ is an arcwise connected set. □
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
We proceed by contradiction. Suppose that F is not arcwise connected Dquasiconvex, i.e. there exist $x_{1},x_{2}\in X$, $y_{1}\in F(x_{1})$, $y_{2}\in F(x_{2})$, and $y\in Y$ with $y_{1}\in yD$ and $y_{2}\in yD$, and for any arc $H_{x_{1},x_{2}}: [0,1]\rightarrow S$ and $t\in[0,1]$, there exists $y_{t}\in F (H_{x_{1},x_{2}}(t) )$ such that
Taking $v=y$ and noticing that $y_{1}\in yD$ and $y_{2}\in yD$, we get from Lemma 3.1
On the other hand, one finds from $y_{t}\in F (H_{x_{1},x_{2}}(t) )$ with $y_{t}\notin yD$ and Lemma 3.1 that
which shows that $h_{e,y}\circ F$ is not arcwise connected $\mathbb {R}_{+}$quasiconvex. This is a contradiction. □
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
Let $F: S\subset X\rightarrow2^{Y}$ be a setvalued map and $\operatorname{int}D\neq\emptyset$. For every $e\in\operatorname {int}D$ the following assertions are equivalent:

(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].
Setvalued optimization problems
In this section, we will restrict our attention to the particular case where $Y=\mathbb{R}^{n}$ is the ndimensional Euclidean space with $n\geq2$, partially ordered by the standard ordering cone $D=\mathbb{R}^{n}_{+}$. For any subset A of $\mathbb{R}^{n}$ we denote by
the sets of efficient points and weakly efficient points of A, respectively. Let $F: S\subset X\rightarrow2^{\mathbb{R}^{n}}$, considering the setvalued optimization problem
The efficient solutions and the weakly efficient solutions of problem (SVOP) are defined as the following sets, respectively:
Let $I_{n}:=\{1,2,\ldots,n\}$ be the set of indices, for every selection of indices, $\emptyset\neq I\subset I_{n}$, we consider the polyhedral cone:
For any subset A of $\mathbb{R}^{n}$, the set of efficient points of A with respect to $D_{I}$ is defined by
Then the set $\operatorname{Eff}_{I}(SF):=F^{1} (\operatorname{Min}_{I}F(S) )$ represents the set of efficient solutions of the following setvalued optimization problem associated to (SVOP):
Definition 4.1
[9]
It is said that problem (SVOP) is Pareto reducible if its weakly efficient solutions can be represented as the union of the efficient solutions of all associated problems of type (SVOP)_{ I }, i.e.
In the literature [11], for every $i\in I_{n}$, the convex cone $K_{i}$ in $\mathbb{R}^{n}$, defined by
was introduced. Let us recall some basic definitions in vector optimization. A set $\Omega\subset\mathbb{R}^{n}$ is called:
 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.
Proof
It is a straightforward consequence of Lemma 4.4 and Proposition 4.5. □
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.
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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.
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MSC
 90C29
 90C46
 26B25
Keywords
 arcwise connected conequasiconvexity
 setvalued optimization
 Gerstewitz’s scalarization function
 Pareto reducibility