The Investigation of Some Essential Concepts of Extended Fuzzy-Valued Convex Functions and Their Applications

function and fuzzy


Introduction
Since Zadeh [1] began to study the essential concepts and defnitions of fuzzy theory, many studies have concentrated on the theoretical and practical aspects of fuzzy numbers.In this way, fuzzy numbers have been extensively researched by many researchers.For instance, Diamond and Kloeden [2], Puri and Ralescu [3], and many other researchers [4][5][6][7][8] brought up the concepts of Hukuhara diferentiability (Hdiferentiability in short) and integrability for fuzzy mappings.Te fuzzy convex analysis is one of the fundamental concepts in fuzzy optimal control and fuzzy optimization.Nanda and Kar [9] proposed the concept of convexity for fuzzy mapping in 1992.Accordingly, various studies on convexity for fuzzy mappings and application in fuzzy optimization have been introduced [10][11][12][13].Yan and Xu [12] have explored the concepts of convexity and quasiconvexity of fuzzy-valued functions.Syau and Lee [14] have studied the concepts of quasiconvex and pseudoconvex multivariable fuzzy functions.Convexity and Lipschitz continuity of fuzzy-valued functions have been discussed by Furukawa [15].Accordingly, some defnitions for various types of convexity or generalized convexity of fuzzy mapping have been proposed, and their properties have been perused [10,16].Noor [17] has expressed the concept and properties of fuzzy preinvex functions in the R feld.A generalization of the Hukuhara diference (H-diference in short), called the generalized Hukuhara diference (gH-diference in short), was proposed by Stefanini in 2010 [18] because the Hdiference exists between two fuzzy numbers only under very restricted positions.Compared to the H-diference, the gH-diference exists in more cases but does not always exist.To solve this difculty, Bede and Stefanini [19] introduced the generalized diference (g-diference in short), which always exists.It should be noted that this diference in some cases does not maintain the convexity condition of fuzzy numbers, therefore may not be a fuzzy number.So, this difculty is resolved by considering the convex hull of the resulting set by Gomes and Barros [20].Based on these two diferences, the generalized Hukuhara diferentiability (gH-diferentiability in short), level-wise generalized Hukuhara diferentiability (L gH -diferentiability in short), and generalized diferentiability (g-diferentiability in short) have been introduced [19].For more recent interesting results related to Jensen's and related inequalities, we recommended [21,22].
In this paper, we consider a generalization for a fuzzyvalued convex function whose range can be the extended fuzzy values.Also, we investigate some essential concepts of extended fuzzy-valued convex functions.We are thus motivated to introduce the extended fuzzy-valued convex functions that can take the singleton fuzzy values −  ∞ and +  ∞ at some points.
Hereupon, the theoretical aspect of extended fuzzy number-valued functions is described, and our aim is not to consider the real applications.It is clear that this research has many applications in dynamic systems of biomedical science, such as problems with cancer, problems with drug release, and so on.
In the following, we describe a comparative study between the convex functions with fuzzy values and the extended fuzzy-valued functions.In general, we prefer to work with fuzzy convex functions containing fuzzy numbers defned over the whole space R n (and not only over a convex subset).However, in some situations, arising mainly in the context of fuzzy optimization and fuzzy conjugation or fuzzy duality, we will encounter operations with fuzzy numbervalued functions that produce extended fuzzy-valued functions.An example is a fuzzy-valued function of the form.
where Λ is an infnite index set and can take the fuzzy value +  ∞ even if the functions f i are fuzzy number-valued.Furthermore, we will encounter functions f that are fuzzyvalued convex over convex subset and cannot be extended to functions that are fuzzy number-valued and convex over the entire space R n (e.g., the fuzzy number-valued function In such situations, it may be convenient, instead of restricting the domain of f to the subset where f takes fuzzy numbers values, to extend the domain to all of R n , but allow f to take fuzzy values +  ∞.Tis process of extension enables us to treat fuzzy number-valued convex functions with diferent domains as fuzzy-valued convex functions with extended fuzzy values in +  ∞ and defned throughout R n .A difculty in defning extended fuzzy-valued convex functions φ that can take both fuzzy values −  ∞ and +  ∞ is that the term θ ⊙ φ(x) ⊕ (1 − θ) ⊙ φ(y) arising in earlier papers for the fuzzy-valued convex case may involve the forbidden fuzzy sum −  ∞ ⊕ +  ∞ (this, of course, may happen only if φ is fuzzy improper but fuzzy improper function may arise on occasion in proofs or other analyses).So, the notions of efective domain and epigraph provide an efective way of dealing with this difculty.Furthermore, we present some of the essential concepts such as the fuzzy indicator function, the epigraph, the fuzzy infmal convolution, the directional generalized derivatives, and their properties for extended fuzzy values.
Tis paper is divided into seven sections; in Section 2, several defnitions besides the results about fuzzy numbers and the g-diference and g-diferentiability are expressed at frst.Moreover, in Section 3, we introduced the specifc case of fuzzy Jensen's inequality for fuzzy-valued convex functions, and in Section 4, the g-diferentiability for extended fuzzy-valued convex function is considered.Ten, the concepts of fuzzy indicator function and the epigraph are discussed, and some outcomes are gained in Section 5. Furthermore, the fuzzy infmal convolution is considered in Section 6.At the end of this paper, in Section 7, the directional generalized derivatives with their properties for extended fuzzy-valued convex function are presented, and eventually, the above concepts are presented with several examples.

Preliminaries
In this section, the basic defnitions and concepts which will be used throughout the paper will be presented and introduced.Also, we use R F to denote the fuzzy numbers set, that is normal, quasiconcave, upper semicontinuous, and compactly supported fuzzy sets that are defned on the real line.Suppose that X ∈ R F is a fuzzy number; for r ∈ [0, 1], the r-cuts of X are described by and θ ∈ R, the addition and scalar multiplication are described as having the we have the triangular fuzzy number.Te support of fuzzy numbers X is specifed as follows: Te standard Hukuhara diference (H-diference Advances in Fuzzy Systems X⊖ H X � 0 (here 0 signifes the singleton 0 { }) for any fuzzy number X but X − X ≠ 0.whenever φ g ′ (x 0 ) ∈ R F is uniquely determined by (19).It is called the g-derivative of φ at x 0 .Defnition 1. Te family of all closed and bounded intervals in R is demonstrated by K C , i.e., Defnition 2 (see [9]).A singleton fuzzy number like  a can be defned for each a ∈ R as follows: R can be embedded in R F .

Defnition 3. Let us consider the singleton fuzzy values
by adjoining the singleton fuzzy elements +  ∞ and −  ∞.
Defnition 5 (see [23]).Suppose that X, Y ∈ R F , the partial order relations between two fuzzy numbers, i.e., (5) Proposition 6 (see [19]).X is a fuzzy number which is entirely distinguished by the pair of X � (X − , X + ) as functions X − , X + : [0, 1] ⟶ R, denoting by the endpoints of the r-cuts, fulflling the below situations: (1) As a function of r, X − : r ⟶ X − r ∈ R is a bounded monotonic increasing left-continuous function for all r ∈ (0, 1] and right-continuous at r � 0; (2) As a function of r, X + : r ⟶ X + r ∈ R is a bounded monotonic decreasing left-continuous function for all r ∈ (0, 1] and right-continuous at r � 0; . Te addendum outcome is well known [24]: Proposition 7 (see [19]).Suppose that an arbitrary real interval collection C r | r∈ (0, 1]   that satisfed the below situations: (3) For any increasing sequence r n ∈ (0, 1] is given, such that lim n⟶∞ r n � r > 0, then C r � ⋃ ∞ n�1 C r n .Terefore, there exists a unique LU-fuzzy quantity X (L for lower, U for upper) with [X] r � C r , ∀r∈ (0, 1] and Lemma 8 (see [19]).Suppose that φ : R ⟶ R F , and x 0 ∈ R. Ten, if (1) (2) Te collections of C − r , C + r satisfy the situations in Proposition 6 or equivalently C r satisfy the situations in Proposition 7,therefore Defnition 9 (see [19]).For each X, Y ∈ R F , the gH-difference is determined by the form In terms of r-cuts, and conditions for the entity of It is obvious that the conditions ( 9) and ( 10) are both satisfed if and only if Z is a crisp number.

The Fuzzy-Valued Convex Function in Sense of Jensen's Inequality
Terein-after, all of these below inequalities are now called the fuzzy-valued convex function Jensen's inequality.So, we shall designate by I a (closed, open, or half-open, fnite or infnite) interval in R. Also, we denoted the interior of I by int(I).
Defnition 26 (see [23]).Suppose that φ : then φ is said to be a fuzzy-valued convex function if Te basic fuzzy inequality equation ( 22) is sometimes called fuzzy Jensen's inequality.
Closely related to fuzzy convexity is the following concept.
Defnition 27.Suppose that φ : Note that if φ is a fuzzy-valued convex function, then φ is the midpoint fuzzy-valued convex function.
Theorem 28 (see [23]).Suppose that φ : then φ is a fuzzy-valued convex function if and only if for any fxed r ∈ [0, 1], the convex functions φ − r (x) and φ + r (x) are both real-valued of x.

The Extended Fuzzy-Valued Convex Functions and g-Differentiability
In the previous section, we consider the fuzzy-valued convex function in sense of Jensen's inequality with fuzzy values in R F .Now, in this part, we shall consider more general fuzzy-valued functions, with fuzzy values in In other words, we want to defne the fuzzy-valued convex functions whose range of them be the extended fuzzy Troughout our paper, we consider for convenience extended fuzzy-valued functions, which take fuzzy values in x < 0, but also the following less obvious one is as follows: Also, we discussed the g-diferentiability and the basic facts of the g-diferentiability for the extended fuzzy-valued convex functions that can be easily visualized.Te expression −  ∞ ⊕ +  ∞ is undefned.
Defnition 30.An extended fuzzy-valued function Lemma 31.Suppose that φ : R ⟶ R F , then φ is the fuzzyvalued convex function in Defnition 26 if and only if φ is the fuzzy-valued convex function in Defnition 30.

□ Defnition 32. Te extended fuzzy-valued convex function efective domain of
Lemma 33.Te extended fuzzy-valued convex function effective domain of φ : And so Hence, dom (φ) is a convex set.
Te below theorem is the class of fuzzy-valued improper convex functions that is easy to describe.

□
By the following lemma, it is often convenient to extend a fuzzy-valued convex function to all of R by defning its fuzzy value to be +  ∞   outside its domain.
Lemma 37. (Te fuzzy-valued convex extension) Suppose that φ : I ⊆ R ⟶ R F is a fuzzy number-valued convex function, where I is a convex set.We defne its extended fuzzyvalued of φ : R ⟶ R F ∪ +  ∞  , as follows: Ten, the extension φ is a fuzzy-valued convex function that defnes on all R and takes the fuzzy values in R F ∪ +  ∞  .
Note that, by replacing the domain of a proper fuzzyvalued convex function with efective domain, we can convert it into a fuzzy-valued function.
In the below theorem, the sufcient conditions of left and right L gH -diferentiability for right and left g-diferentia- proper fuzzy-valued convex functions in terms of r-cut are stated.

a proper fuzzy-valued convex function, and φ is right and left uniformly L gH -diferentiable at x. Ten, φ has the right and left g-derivative throughout do m(φ), provided the fuzzy values + 􏽥
∞ and −  ∞ are permitted.
Proof.Te proof is the same as Teorem 5.2 in [23] and Teorem 35 in [19] since the g-quotient φ(x) ⊖ g φ(a)/x − a is nondecreasing and bounded from below on do m(φ) � [a, b]; therefore, there exists a subsequence x n > a, in which the members of subsequence are φ(x n ) ⊖ g φ(a)/x n − a and as φ(x n ) ⊖ g φ(a)/x n − a converges to inf x>a φ(x) ⊖ g φ(a)/x − a equals to φ +g ′ (a), i.e., there exists a subsequence x n > a such that Similarly, since the g-quotient φ(x) ⊖ g φ(a)/x − a is nonincreasing and bounded from above on dom(φ) � [a, b], therefore there exists a subsequence x n < a, in which the members of subsequence are φ(x n ) ⊖ g φ(a)/x n − a and as n ⟶ ∞ converges to sup x<a φ(x) ⊖ g φ(a)/x − a equals to φ −g ′ (a), i.e., there exists a subsequence x n < a such that Tus, the left g-derivative φ −g ′ for the case where do m(φ) � [a, b] exists.Hence, by the above concepts, φ +g ′ (x) exists whenever x ∈ (a, b], and φ −g ′ (x) exists whenever x∈ (a, b].But for any x < a, we have φ(x) � +  ∞, so the quotient hence, φ −g ′ (a) � −  ∞, for any x > b, we have the quotient and so φ +g ′ (b) � +  ∞. □

The Fuzzy Concepts of Indicator Function and Epigraph
Now, we introduce the fuzzy indicator function and the epigraph for the extended fuzzy-valued convex function by the forms.
Defnition 42.Suppose that C ⊆ R n is a set.Defne the fuzzy indicator function of C as follows: is given by Tus, (58) Hence,  I C is the fuzzy-valued convex function.
Conversely, suppose that  I C is a fuzzy-valued convex function.Let x, y ∈ C, 0 < θ < 1, then  I C (x) �  0,  I C (y) �  0. Consider then Defnition 46.An extended fuzzy-valued function , the following conditions are equivalent: (1) φ is a fuzzy-valued convex function.( 2 we have epi s (φ) is convex set if and only if whenever φ(x) ≺  α, φ(y) ≺  β, 0 < θ < 1. Applying Defnition 26, the equivalence of ( 9) and ( 11) is as follows.( 2 and so It follows that We conclude that epi s (φ) is a convex set.Vice versa, suppose that epi Since epi s (φ) is a convex set, then Advances in Fuzzy Systems It follows that then and we consummate that epi(φ) is a convex set.
We defne a fuzzy-valued function φ : i.e., φ(x) is the greatest fuzzy-valued convex function on R n which the epigraph contains A.
Ten φ is a fuzzy-valued convex function.

The Fuzzy Infimal Convolution
Now, in the following, we introduce the fuzzy infmal convolution as a subset R n × R F for extended fuzzy-valued convex functions φ and g that denote by φ□g.
Defnition 50.Let φ, g : be the fuzzy-valued functions.Defne the fuzzy infmal convolution φ and g as follows: Note that if φ and g are fuzzy-valued convex functions, then epi(φ), epi(g) in R n × R F are convex sets, therefore epi(φ) ⊕ epi(g) in R n × R F is a convex set, hence by Teorem 49, φ□g is a fuzzy-valued convex function.Te terminology is motivated by the case where φ and g are fuzzy-valued functions, φ, g : . Ten, φ□g can also be defned as Advances in Fuzzy Systems Hence.
which is analogous to the formula for fuzzy integral convolution and φ□g is exact at x ∈ R n , if (φ□g)(x) � min y ∈R n φ(y) ⊕ g(x − y), i.e., there exists y ∈ R n so that φ□g is exact if it is exact at every point of its domain, in which case it is denoted by φ ⊡ g .
Proof.For the proof of (1), we can see that Hence, φ□g � g□φ.Also, for the proof of (10), we have On the other hand, we have Hence, (φ□g)□h � φ□(g□h).

□
Example 1.Consider C ⊆ R n to be a nonempty convex set and x 0 ∈ R n .Te fuzzy distance of x 0 from C is defned by We show that d(x 0 , C) is a fuzzy-valued convex function.Since C is a convex set, then by Teorem 43,  I C is a fuzzy-valued convex function.Let φ(x) � 〈−1, 0, 1〉 ⊙ ‖x‖, ∀x ∈ R n .Terefore, φ is a fuzzy-valued convex function.By Defnition 50, hence φ□  I C is a fuzzy-valued convex function.Consider is an extended fuzzy-valued function and let y ∈ R n .We show that  I y { } ⊡ φ � ι y φ, where is the translation of the extended fuzzy-valued function φ by y ∈ R n .Let x ∈ R n be arbitrary.Ten Hence,  I y { } ⊡ φ � ι y φ.

The Directional g-Derivative for Extended Fuzzy-Valued Convex Functions
Now, we introduce the directional g-derivative for extended fuzzy-valued convex functions and their properties are discussed.
Defnition 52.Suppose that φ is an extended fuzzy-valued function, φ : and x 0 , x ∈ R n .Te directional g-derivative of φ at x 0 in the direction x is as follows: satisfying (93) exist.Note that if it exists (+  ∞ and −  ∞ being allowed as limits).
Proof.For proof of (1), let x ∈ R n be arbitrary.Defne Since by the hypothesis, φ is the proper fuzzy-valued convex function, then g : R ⟶ R is a proper fuzzy-valued convex function, therefore by Teorem 41, g +g ′ (0) exists where it is the right g-derivative of the proper fuzzy-valued convex function at t � 0, as follows: Hence φ g ′ (x 0 , x) exists, for all x ∈ R n .For proof of (10), let θ > 0, x ∈ R n .Consider Advances in Fuzzy Systems Hence, φ g ′ (x 0 , x) is an extended fuzzy-valued convex function.

□
be a proper fuzzy-valued convex function and x 0 ∈ do m(φ) be so that Assume that φ − r (x) and φ + r (x) of the proper real-valued convex functions are directional diferentiable at x 0 in the direction of x, uniformly w.r.t.r ∈ [0, 1].Ten, φ has a directional g-derivative at x 0 in the direction of x as follows: Proof.According to Proposition 13, we get Since the proper real-valued convex functions φ − r (x) and φ + r (x) are directional diferentiable at x 0 in the direction of x, we have lim Also, let us consider that if the functions φ − r (x) and φ + r (x) are left continuous w.r.t.r ∈ (0, 1] and right continuous at 0. From the defnition of the directional derivative, for any n ≥ 1, there exists a sequence t n > 0 such that t n ⟶ 0 + the quotients as functions of r∈ [0, 1] are left continuous at r ∈ (0, 1] and right continuous at 0. Also, for any n ≥ 1, there exists a sequence t n > 0 such that t n ⟶ 0 + , then the functions by Proposition 6, they defne a fuzzy number.Consequently, the r-cuts φ g ′ (x 0 , x) r defne a fuzzy number, by Lemma 8, the directional g-derivative with extended fuzzy-valued φ g ′ (x 0 , x) exists at x 0 in the direction x.

Conclusion
Te concepts of g-diference and g-diferentiability were introduced for fuzzy-valued functions in 2013 by Bede and Stefanini [19] which is the generalization concept of gH-diference and gH-diferentiability.Here, we defned the fuzzy-valued convex functions whose range is extended fuzzy numbers, and some of their properties were expressed.Moreover, several important fuzzy concepts such as indicator function, epigraph, infmal convolution, and directional g-derivative with their properties for the extended fuzzy-valued convex functions have been stated and discussed.It is worth pursuing follow-up research by considering g-subgradient and g-subdiferential for the extended fuzzy-valued convex function.In this way, in the next studies and research, we propose the concepts of g-subgradient and g-subdiferential, which play an important role in extended fuzzy-valued optimization.

Remark 4 .
Troughout this paper, we use the extended fuzzy numbers