DOI: 10.24193/subbmath.2021.3.14
Convexity-preserving properties of set-valued ratios of affine functions
Alexandru Orzan and Nicolae Popovici
Abstract. The aim of this paper is to introduce some classes of set-valued func- tions that preserve the convexity of sets by direct and inverse images. In particu- lar, we show that the so-called set-valued ratios of affine functions represent such a class. To this aim, we characterize them in terms of vector-valued selections that are ratios of affine functions in the classical sense of Rothblum.
Mathematics Subject Classification (2010):54C60, 26B25.
Keywords:Set-valued affine function, single-valued selection, ratio of affine func- tions, generalized convexity.
1. Introduction
Various classes of fractional type real-valued or vector-valued functions have been introduced in the literature, being nowadays well recognized for their important applications in scalar and vector optimization (see, e.g., Avrielet al.[2], Cambini and Martein [4], G¨opfertet al.[6], Stancu-Minasian [13], and the references therein).
According to Rothblum [11], a vector-valued function f : D → Rm, defined on a nonempty convex setD ⊆Rn, is said to be a ratio of affine functions if there exist a vector-valued affine function g :Rn →Rm and a real-valued affine function h:Rn→R, such that
D⊆ {x∈Rn|h(x)>0}
and
f(x) = g(x)
h(x), ∀x∈D. (1.1)
These functions are known to preserve the convexity of sets by direct and inverse images. Moreover, they transform the line segments in line segments (possibly de- generated into singletons). This concept along with the above mentioned convexity- preserving properties can be naturally extended for vector-valued functions acting between general real linear spaces.
Recently, Orzan [10] investigated a class of set-valued ratios of affine functions, defined similarly to (1.1), by replacing g with a set-valued function that is affine in the sense of Tan [14]. As shown in [10], these functions preserve the convexity of sets by direct images. A natural question is whether they preserve the convexity of sets by inverse images too. The aim of this paper is not only to give a positive answer to this question, but also to identify some broader classes of set-valued functions that preserve the convexity of sets by direct and inverse images.
We start by recalling some basic definitions of set-valued and convex analysis in Section 2. The concept of set-valued affine function, as defined by Tan [14], is investigated in Section 3. In particular, we show that the inverse of such a function is affine as well (in contrast to other concepts of affine set-valued functions, cf. Kuroiwa et al. [7, Ex. 2]). Section 4 is devoted to the ratios of affine functions. In Section 4.1 we briefly present how the classical results of Rothblum [11] can be extended from finite-dimensional Euclidean spaces to general linear spaces. Subsection 4.2 is devoted to our main results. First we introduce the class of set-valued ratios of affine functions, by refining the definition proposed in [10]. Theorem 4.8 gives an explicit representation of these functions, which plays a key role in establishing the convexity- preserving properties of the set-valued ratios of affine functions. Moreover, Theorems 4.10 and 4.11 show that these properties are still valid in some broader classes of set-valued functions. We conclude the paper by rising an interesting open question in Section 5.
2. Preliminaries
Throughout this paper we assume that X and Y are two real linear spaces. As usual in set-valued analysis (see, e.g., Aubin and Frankowska [1]), for any set-valued functionF:X→ P(Y) we denote by
domF ={x∈X |F(x)6=∅}
the domain of F. We say thatF is proper if domF 6=∅.The (direct) image of a set A⊆X byF is defined as
F(A) = [
x∈A
F(x).
There are different manners to define the inverse image of a setB⊆Y by a set-valued mapF, two of them being currently used in set-valued analysis [1], namely:
F−1(B) ={x∈X |F(x)∩B6=∅}, (2.1)
F+1(B) ={x∈X |F(x)⊆B}. (2.2)
The setF−1(B) is called the inverse image ofB byF andF+1(B) is called the core ofB byF (also known as the lower inverse image and the upper inverse image of B byF, respectively). They are related by
F+1(B) =X\F−1(Y \B). (2.3)
Remark 2.1. LetF :X → P(Y) be a set-valued function. Given a setB⊆Y we have
F−1(B)⊆domF, (2.4)
F+1(B) ={x∈domF|F(x)⊆B} ∪(X\domF). (2.5) Notice that, instead of (2.2), Berge [3] defined a slightly different concept of up- per inverse image of B, namely {x ∈ domF | F(x) ⊆ B}, which actually means F+1(B)∩domF.
Remark 2.2. LetF :X → P(Y) be a set-valued function. Then, for all sets A⊆X andB⊆Y, the following equivalence holds:
A⊆F+1(B) ⇐⇒ F(A)⊆B.
Remark 2.3. According to [1], the inverse ofF is the set-valued functionF−1:Y → P(X) defined for anyy∈Y by
F−1(y) :={x∈X |y∈F(x)}.
Notice that domF−1=F(X) andF−1(Y) = domF. Also, for every setB⊆Y, we have
F−1(B) = [
y∈B
F−1(y) ={x∈X |F(x)∩B6=∅}. (2.6) It is important to notice that, according to (2.1) and (2.6), one can use without any confusion the same notation, F−1(B), for both the lower inverse image of B by F and the direct image ofB byF−1.
Remark 2.4. Every vector-valued function f : D → Y defined on a nonempty set D⊆X can be identified with a set-valued functionF :X→ P(Y),
F(x) =
( {f(x)} if x∈D
∅ if x∈X\D.
Obviously, domF =D and, for allA⊆X andB ⊆Y, we have F(A) =f(A∩D) :={f(x)|x∈A∩D}, F−1(B) =f−1(B) :={x∈D|f(x)∈B},
F+1(B) =f−1(B)∪(X\D), i.e., F+1(B)∩D=f−1(B).
In particular, the second relation shows that one can define the set-valued function f−1:Y → P(X) asf−1(y) :=F−1(y) for ally∈Y.
The aim of this paper is to study some classes of set-valued functions that preserve the convexity of sets by direct and inverse images. Therefore we will adopt the following conventional notations in a real linear spaceV (in particular,X orY).
GivenS, S0 ⊆V,λ∈Randv0∈V, we set:
S+S0:={v+v0|(v, v0)∈S×S0}, v0+S :={v0}+S, λS:={λv|v∈S}, and S
λ := 1
λS wheneverλ6= 0.
Recall that a set S ⊆V is called convex if (1−t)S+tS ⊆S for all t ∈[0,1]. The convex hull of a set S, i.e., the smallest convex subset of V that contains S, will be denoted by convS. For convenience, when the convex hull applies only to the first term of a sum of two sets we will simply write convS+S0instead of (convS) +S0. A setS⊆X is called affine if (1−t)S+tS⊆S for allt∈R. IfS is nonempty, thenS is affine if and only if there exists a (unique) linear subspaceLofV such thatS =v+L for allv∈S.
3. Affine set-valued functions
Recall that a vector-valued function, a:E →Y, defined on a nonempty affine setE⊆X, is said to be affine if for anyx1, x2∈E andt∈Rwe have
a((1−t)x1+tx2) = (1−t)a(x1) +ta(x2).
This concept has been generalized for set-valued affine functions in different ways (see, e.g. Deutsch and Singer [5], Nikodem and Popa [9], Tan [14], and the references therein). The following definition, inspired from [14], is suitable for the purposes of our paper.
Definition 3.1. A set-valued functionG:X → P(Y) is said to be affine if
G((1−t)x1+tx2) = (1−t)G(x1) +tG(x2) (3.1) for allx1, x2∈domGandt∈R.
Remark 3.2. It can be easily seen that ifG:X → P(Y) is a set-valued affine function, then domGis affine, since for any x1, x2 ∈ domG and t ∈ [0,1], the equality (3.1) implies G((1−t)x1+tx2) 6= ∅, i.e., (1−t)x1+tx2 ∈ domG. Moreover, for every x∈X, the set G(x) is affine. Indeed, lettingx1=x2=xin (3.1) we get
G(x) =G((1−t)x+tx) = (1−t)G(x) +tG(x) for allt∈R, henceG(x) is affine.
Example 3.3. Letg:E→Y be a vector-valued affine function, defined on a nonempty affine setE⊆X. In view of Remark 2.4, we can identifygwith the set-valued function G:X → P(Y), given by
G(x) =
( {g(x)} if x∈E
∅ if x∈X\E.
It is easy to check thatGis affine.
Proposition 3.4. For any set-valued function G:X → P(Y) the following assertions are equivalent:
1◦ Gis affine.
2◦ For allx1, x2∈domG andt∈Rwe have
(1−t)G(x1) +tG(x2)⊆G((1−t)x1+tx2).
Proof. The implication 1◦=⇒2◦is obvious, while 2◦=⇒1◦ can be seen as a partic- ular instance of a known result by Nikodem and Popa [9, Prop. 2.11], applied to the restriction ofGto E:= domG, which is an affine set in view of Remark 3.2.
Remark 3.5. In the paper by Tan [14] the set-valued affine functions are defined on a nonempty affine set E ⊆ X taking values in P0(Y) := P(Y)\ {∅}. Actually, if G : X → P(Y) is a proper set-valued affine function, then the set E := domG is nonempty and affine, therefore the restriction ofGto E, i.e.,G|domG :E→ P0(Y), is affine in the sense of Tan.
Theorem 3.6. Let G:X → P(Y)be a set-valued affine function. Then the inverse of G, i.e., the set-valued functionG−1:Y → P(X), is affine.
Proof. According to Proposition 3.4, we just need to show that for any y1, y2 ∈ domG−1=G(X) andt∈Rthe following inclusion holds:
(1−t)G−1(y1) +tG−1(y2)⊆G−1((1−t)y1+ty2). (3.2) To this aim, let x∈(1−t)G−1(y1) +tG−1(y2). Then there exist x1 ∈G−1(y1) and x2∈G−1(y1) such thatx= (1−t)x1+tx2. In view of (2.4), we havex1, x2∈domG.
Taking into account that functionGis affine, we deduce that
(1−t)y1+ty2∈(1−t)G(x1) +tG(x2) =G((1−t)x1+tx2) =G(x), which entailsx∈G−1((1−t)y1+ty2). Thus (3.2) holds.
Corollary 3.7. Ifg:E →Y is a vector-valued affine function, defined on a nonempty affine set E⊆X, then the set-valued function g−1:Y → P(X)is affine.
Proof. Follows by Theorem 3.6, in view of Remark 2.4.
The following two results are based on Tan [14, Props. 4 and 5].
Proposition 3.8. If G:X → P(Y) is a proper set-valued affine function, then there is a unique linear subspaceM ⊆Y such that
G(x) =y+M (3.3)
for allx∈domGandy∈G(x).
Proposition 3.9. If G : X → P(Y) is a proper set-valued affine function, then G possesses an affine selection, i.e., there exists a vector-valued affine function g: domG→Y such that, for all x∈domG,
g(x)∈G(x).
Corollary 3.10. If G: X → P(Y) is a proper set-valued affine function, then there exist a vector-valued affine function g : domG→ Y and a linear subspace M ⊆ Y such that, for allx∈domG,
G(x) =g(x) +M.
Proof. It is a straightforward consequence of Propositions 3.8 and 3.9.
Remark 3.11. If Y = R, then there are only two types of proper set-valued affine functions G : R → P(R), namely: (i) G(x) = {g(x)} for all x ∈ domG, where g: domG→Ris an affine function, or (ii)G(x) =Rfor allx∈domG. In particular, when X = R the domain domG is either a singleton or the entire R, in view of Remark 3.2.
4. Ratios of affine functions
4.1. Vector-valued ratios of affine functions
We begin this section by extending the notion of vector-valued ratios of affine functions, originally introduced by Rothblum [11] within finite-dimensional Euclidean spaces, to the framework of general real linear spaces.
Definition 4.1. A vector-valued functionf :D →Y, defined on a nonempty convex setD⊆X, is said to be a ratio of affine functions if there exist a vector-valued affine functiong:X→Y and a real-valued affine functionh:X →R, such that
D⊆ {x∈X |h(x)>0}
and
f(x) = g(x)
h(x), ∀x∈D. (4.1)
Remark 4.2. SinceD is assumed to be nonempty in Definition 4.1, it is understood that the set{x∈X|h(x)>0}is nonempty.
The following propositions extend to the framework of general real linear spaces some results obtained within Rn by Rothblum (see [11, Props. 1, 2 and 3] along with subsequent remarks). Their proofs are omitted, since they follow the main lines in [11].
Proposition 4.3. Given a vector-valued function f :D →Y defined on a nonempty convex set D⊆X, the following assertions are equivalent:
1◦ convf(S)⊆f(convS) for every setS ⊆D.
2◦ f(A) is convex for every convex set A ⊆D, i.e., f preserves the convexity of sets by direct images.
Proposition 4.4. Given a vector-valued function f :D →Y defined on a nonempty convex set D⊆X, the following assertions are equivalent:
1◦ f(convS)⊆convf(S) for every setS ⊆D.
2◦ f−1(B) is convex for every convex set B ⊆ Y, i.e., function f preserves the convexity of sets by inverse images.
Proposition 4.5. LetD⊆Xbe a nonempty convex set. Iff :D→Y is a vector-valued ratio of affine functions, then
convf(S) =f(convS) for every set S⊆D.
Therefore f preserves the convexity of sets by direct and inverse images.
4.2. Set-valued ratios of affine functions
In this section we introduce a class of set-valued ratios of affine functions, by slightly modifying the one proposed by us in [10, Def. 3.3].
Definition 4.6. LetF :X → P(Y) be a set-valued function, whose domain domF =:
D ⊆ X is a nonempty convex set. We say that F is a ratio of affine functions if there exist a proper set-valued affine functionG:X→ P(Y) and a real-valued affine functionh:X→R, such that
D⊆ {x∈X|h(x)>0} ∩domG and
F(x) =
G(x)
h(x) if x∈D
∅ if x∈X\D.
(4.2)
Remark 4.7. As we have pointed out in Remark 4.2, since D is nonempty, the set {x∈X |h(x) >0} is nonempty as well. In particular, we deduce that any proper set-valued affine function F :X → P(Y) is a ratio of affine functions of type (4.2).
Indeed, in this case,D:= domF is a nonempty affine hence convex set,G(x) =F(x) andh(x) = 1 for allx∈X.
Theorem 4.8. Let F :X → P(Y) be a set-valued ratio of affine functions defined by (4.2). Then there exist a vector-valued ratio of affine functions f : D → Y and a linear subspaceM ⊆Y, such that
F(x) =
f(x) +M if x∈D
∅ if x∈X\D.
(4.3) Proof. In view of Corollary 3.10, we can find a vector-valued affine functiong:X → Y and a linear subspace M ⊆ Y, such that G(x) = g(x) +M for any x ∈ D.
Consequently, we get F(x) =G(x)
h(x) = g(x) +M
h(x) = g(x) h(x)+ M
h(x) = g(x)
h(x)+M, ∀x∈D.
Thus, we can define a vector-valued ratio of affine functions,f :D→Y, by f(x) := g(x)
h(x), ∀x∈D,
which satisfies (4.3).
The following result is a set-valued counterpart of Proposition 4.5 and recovers a similar result obtained in [10, Th. 3.1].
Corollary 4.9. Let F :X → P(Y)be a set-valued ratio of affine functions defined by (4.2). Then, for any setS⊆D we have
convF(S) =F(convS). (4.4)
Proof. According to Theorem 4.8, there exist a vector-valued ratio of affine functions f :D →Y and a linear subspace M ⊆Y such that F has the form (4.3). Consider an arbitrary setS⊆D. On the one hand, we have F(S) =f(S) +M, hence
convF(S) = conv (f(S) +M). (4.5)
On the other hand, we haveF(convS) =f(convS) +M, which in view of Proposition 4.5 means
F(convS) = convf(S) +M. (4.6)
Taking into account thatM is convex, it is a simple exercise to check that
convf(S) +M = conv (f(S) +M). (4.7)
Thus, (4.4) follows by (4.5), (4.6) and (4.7).
In what follows we will show that, similarly to vector-valued ratios of affine functions, the set-valued ratios of affine functions preserve the convexity of sets by direct and inverse images. To this aim we establish two preliminary results for more general classes of set-valued functions.
Theorem 4.10. Consider a set-valued function F : X → P(Y) of type (4.3), where D ⊆X andM ⊆Y are nonempty convex sets, whilef :D →Y is a vector-valued function that preserves the convexity of sets by direct images. Then, for every convex set A ⊆X, the set F(A) is convex, i.e., F preserves the convexity of sets by direct images.
Proof. LetA⊆X be a convex set. Since domF =D, we have
F(A) =F(A∩D)∪F(A\D) =F(A∩D) =f(A∩D) +M.
By hypothesis,f(A∩D) is convex as being the image of the convex set A∩D byf. SinceM is convex too, we conclude thatF(A) is convex.
Theorem 4.11. Consider a set-valued function F : X → P(Y) of type (4.3), where D ⊆X andM ⊆Y are nonempty convex sets, whilef :D →Y is a vector-valued function that preserves the convexity of sets by inverse images. Then, for every convex set B ⊆Y, the sets F−1(B)and F+1(B)∩domF are convex, i.e., F preserves the convexity of sets by lower inverse images as well as by upper inverse images in the sense of Berge.
Proof. First notice that domF =D. For every convex setB⊆Y we have F−1(B) = {x∈X |F(x)∩B6=∅}
= {x∈D|(f(x) +M)∩B6=∅}
= [
m∈M
{x∈D|f(x) +m∈B}
= [
m∈M
f−1(B−m)
= f−1(B−M).
Since B −M is convex (by convexity of B and M), its inverse image by f, i.e., f−1(B−M), is convex as well. Consequently, the set F−1(B) is convex.
In order to prove thatF+1(B)∩domF, i.e.,F+1(B)∩D, is convex, let x1, x2∈F+1(B)∩D.
We have to show that
conv{x1, x2} ⊆F+1(B)∩D.
SinceDis convex set, we just have to check that conv{x1, x2} ⊆F+1(B). In view of Remark 2.2, this actually means that F(conv{x1, x2})⊆B, which by (4.3) reduces to
f(conv{x1, x2}) +M ⊆B. (4.8) Observe that, by applying Proposition 4.4 forS:={x1, x2}, we have
f(conv{x1, x2}) +M ⊆ convf({x1, x2}) +M
= conv{f(x1), f(x2)}+M.
Taking into account thatM is convex, we can deduce that
conv{f(x1), f(x2)}+M = conv (f(x1) +M)∪(f(x2) +M)
= conv F(x1)∪F(x2) .
Finally, recalling that x1, x2 ∈F+1(B), we have F(x1)∪F(x2)⊆B, which by con- vexity ofB yields
conv F(x1)∪F(x2)
⊆B.
Hence, the desired inclusion (4.8) holds.
As a direct consequence of Theorems 4.8, 4.10 and 4.11 we obtain the following result.
Corollary 4.12. If F :X → P(Y)is a set-valued ratio of affine functions defined by (4.2), then the following assertions hold:
1◦ F preserves the convexity of sets by direct images.
2◦ F preserves the convexity of sets by lower inverse images.
3◦ F preserves the convexity of sets by upper inverse images in the sense of Berge.
Remark 4.13. Corollary 4.12 (1◦) extends a result obtained in [10, Cor. 3.1].
Remark 4.14. Assertions 2◦ and 3◦ of Corollary 4.12, can be interpreted in terms of generalized convexity. They actually show that the restriction F|domF : D → P0(Y) of any set-valued ratio of affine functions defined by (4.2) is quasiconvex and quasiconcave in the sense of Nikodem [8, Th. 7.2]. Assertion 3◦ also means thatF is (u1)-type{0Y}-quasiconvex in the sense of Seto, Kuroiwa and Popovici [12, Def. 3.3], where 0Y stands for the origin ofY. Actually, the more general set-valued functions involved in Theorem 4.11 also satisfy these generalized convexity properties as well.
Remark 4.15. In contrast to Corollary 4.12 (3◦), if F : X → P(Y) is a set-valued (even single-valued) ratio of affine functions defined by (4.2), the upper inverse image F+1(B) of a convex setB ⊆Y in the sense of Aubin-Frankowska is not necessarily convex, as the following example shows.
Example 4.16. LetX =Y =Rand D= [0,1]. LetF :R→ P(R) be the set-valued function defined as
F(x) =
( {x} if x∈[0,1]
∅ if x /∈[0,1].
In view of Remarks 3.11 and 4.7, F is a set-valued ratio of affine functions of type (4.2), whereG(x) ={x}andh(x) = 1 for allx∈R. Consider the convex setB={0}.
AlthoughF+1(B)∩domF ={0}is convex, the setF+1(B) =R\]0,1] is not convex.
Remark 4.17. LetF :X → P(Y) be a proper set-valued affine function, whose domain is the affine setD := domF. Notice thatF is a set-valued ratio of affine functions, in view of Remark 4.7. Then, for every convex setB ⊆Y, the sets
F−1(B) and F+1(B)∩D are convex, according to Corollary 4.12 (2◦and 3◦).
On the other hand, by Theorem 3.6, the inverse ofF, i.e., the set-valued function F−1 :Y → P(X), is also affine with domF−1 =F(X) =F(D). Thus, by applying the above arguments to F−1 in the role of F, we deduce that for any convex set A⊆X the sets
(F−1)−1(A) and (F−1)+1(A)∩F(D)
are convex. Of course, the convexity of (F−1)−1(A) is simply recovered by Corollary 4.12 (1◦) applied toF−1in the role ofF, since (F−1)−1(A) =F(A). However, the con- vexity of (F−1)+1(A)∩F(D) is not a simple consequence of the convexity-preserving properties ofF. Indeed, by applying (2.3) forF−1 in the role ofF, we get
(F−1)+1(A) =Y \F(X\A) =Y \F(D\A), hence
(F−1)+1(A)∩F(D) = (Y \F(D\A))∩F(D)
= F(D)\F(D\A).
Notice that F(D)\F(D\A) ⊆ F(D∩A) and F(D∩A) is convex, according to Corollary 4.12 (1◦), sinceD∩Ais convex. However,F(D)\F(D\A)6=F(D∩A) in general, as for instance whenF :R→ P(R) is the constant ratio of affine functions defined byF(x) ={1}for allx∈RandA={0} ⊆D=R.
5. Conclusions
Theorem 4.8 gives a useful characterization of the set-valued ratios of affine functions, as reflected in its Corollaries 4.9 and 4.9. Actually, the special structure of these functions, given by Theorem 4.8, suggested us to consider in Theorems 4.10 and 4.11 some broader classes of set-valued functions that enjoy the convexity-preserving properties of the set-valued ratios of affine functions pointed out in Corollary 4.9.
An interesting question arises, namely whether it would be possible to extend these classes by replacing the convex set M in Theorems 4.10 and 4.11 (which can be seen as a constant affine set-valued function) by appropriate set-valued functions.
In general, this is not true even if M is replaced by a single-valued function that
preserves the convexity of sets by direct and inverse images. For instance, consider D = [0,1] ⊆ X = R, Y = R2 and let F : R → P(R2) be the set-valued function defined as
F(x) =
( f(x) +M(x) if x∈[0,1]
∅ if x /∈[0,1],
wheref : [0,1]→R2is a vector-valued ratio of affine functions given by f(x) = g(x)
h(x) = (1,1)
x+ 1 for all x∈[0,1]
andM :R→ P(R2) is a single-valued affine function, given by M(x) ={(x,−x)} for all x∈R.
Consider the convex setsA= [0,1] andB = convF({0,1}). It is easy to check that B is a line segment with end points (1,1) and (3/2,−1/2), while F(A) is an arc of hyperbola joining these points. On the other hand, we have F−1(B) = {0,1} and F+1(B) =R\]0,1[. Obviously, the setsF(A),F−1(B) andF+1(B) are not convex. It is worth to mention that similar examples, whereF preserves the convexity of sets by direct and inverse images, can be given by considering ratios af affine functionsf = g
h andM = G
h with the same denominator. However, such a configuration always leads to the trivial case whereF itself is a ratio of affine functions satisfying the properties in demand by virtue of Corollary 4.9.
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Alexandru Orzan Babe¸s-Bolyai University,
Faculty of Mathematics and Computer Sciences, 1, Kog˘alniceanu Street,
400084 Cluj-Napoca, Romania e-mail:[email protected] Nicolae Popovici
Babe¸s-Bolyai University,
Faculty of Mathematics and Computer Sciences, 1, Kog˘alniceanu Street,
400084 Cluj-Napoca, Romania e-mail:[email protected]
