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

Allahviranloo, T.; Balooch Shahryari, M. R.; Sedaghatfar, O.; Shahriari, M. R.; Saadati, R.; Noeiaghdam, S.; Fernandez-Gamiz, U.

doi:https://doi.org/10.1155/2024/8335864

Advances in Fuzzy Systems

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 8335864 | https://doi.org/10.1155/2024/8335864

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

T. Allahviranloo,¹M. R. Balooch Shahryari,²O. Sedaghatfar,³M. R. Shahriari,^1,4R. Saadati,⁵S. Noeiaghdam,^6,7and U. Fernandez-Gamiz⁸

Academic Editor: Ramin Ranjbarzadeh

Received18 Aug 2022

Revised24 Aug 2023

Accepted18 Mar 2024

Published10 Apr 2024

Abstract

In this paper, we are thus motivated to define and introduce the extended fuzzy-valued convex functions that can take the singleton fuzzy values and at some points. Such functions can be characterized using the notions of effective domain and epigraph. In this way, we study important concepts such as fuzzy indicator function and fuzzy infimal convolution for extended fuzzy-valued functions. Finally, we introduce the concept of directional generalized derivative for extended above functions and its properties. Eventually, we give a practical example that will illustrate well the directional -derivative for the extended fuzzy-valued convex function.

1. Introduction

Since Zadeh [1] began to study the essential concepts and definitions 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 [4–8] brought up the concepts of Hukuhara differentiability (H-differentiability in short) and integrability for fuzzy mappings. The 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 [10–13]. 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 definitions 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 field. A generalization of the Hukuhara difference (H-difference in short), called the generalized Hukuhara difference (-difference in short), was proposed by Stefanini in 2010 [18] because the H-difference exists between two fuzzy numbers only under very restricted positions. Compared to the H-difference, the -difference exists in more cases but does not always exist. To solve this difficulty, Bede and Stefanini [19] introduced the generalized difference (-difference in short), which always exists. It should be noted that this difference in some cases does not maintain the convexity condition of fuzzy numbers, therefore may not be a fuzzy number. So, this difficulty is resolved by considering the convex hull of the resulting set by Gomes and Barros [20]. Based on these two differences, the generalized Hukuhara differentiability (-differentiability in short), level-wise generalized Hukuhara differentiability (-differentiability in short), and generalized differentiability (-differentiability 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 fuzzy-valued 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 defined over the whole space (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 number-valued functions that produce extended fuzzy-valued functions. An example is a fuzzy-valued function of the form.where is an infinite index set and can take the fuzzy value even if the functions are fuzzy number-valued. Furthermore, we will encounter functions that are fuzzy-valued convex over convex subset and cannot be extended to functions that are fuzzy number-valued and convex over the entire space (e.g., the fuzzy number-valued function defined by ).

In such situations, it may be convenient, instead of restricting the domain of to the subset where takes fuzzy numbers values, to extend the domain to all of , but allow to take fuzzy values . This process of extension enables us to treat fuzzy number-valued convex functions with different domains as fuzzy-valued convex functions with extended fuzzy values in and defined throughout . A difficulty in defining extended fuzzy-valued convex functions that can take both fuzzy values and is that the term arising in earlier papers for the fuzzy-valued convex case may involve the forbidden fuzzy sum (this, of course, may happen only if is fuzzy improper but fuzzy improper function may arise on occasion in proofs or other analyses). So, the notions of effective domain and epigraph provide an effective way of dealing with this difficulty. Furthermore, we present some of the essential concepts such as the fuzzy indicator function, the epigraph, the fuzzy infimal convolution, the directional generalized derivatives, and their properties for extended fuzzy values.

This paper is divided into seven sections; in Section 2, several definitions besides the results about fuzzy numbers and the -difference and -differentiability are expressed at first. Moreover, in Section 3, we introduced the specific case of fuzzy Jensen’s inequality for fuzzy-valued convex functions, and in Section 4, the -differentiability for extended fuzzy-valued convex function is considered. Then, the concepts of fuzzy indicator function and the epigraph are discussed, and some outcomes are gained in Section 5. Furthermore, the fuzzy infimal 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.

2. Preliminaries

In this section, the basic definitions and concepts which will be used throughout the paper will be presented and introduced. Also, we use to denote the fuzzy numbers set, that is normal, quasiconcave, upper semicontinuous, and compactly supported fuzzy sets that are defined on the real line. Suppose that is a fuzzy number; for , the -cuts of are described by , and for by is illustrated. Moreover, is explained, so the -cut is a closed interval for all . If , and , the addition and scalar multiplication are described as having the -cuts of and , relatively. By , a trapezoidal fuzzy number defined so that , and has -cuts for ; if , we have the triangular fuzzy number. The support of fuzzy numbers is specified as follows:

The standard Hukuhara difference (H-difference ) is defined by , being + the standard fuzzy addition; if exists, its -cuts are . It is outstanding that (here 0 signifies the singleton ) for any fuzzy number but .whenever is uniquely determined by (19). It is called the -derivative of at .

Definition 1. The family of all closed and bounded intervals in is demonstrated by , i.e.,

Definition 2. (see [9]). A singleton fuzzy number like can be defined for each as follows: can be embedded in .

Definition 3. Let us consider the singleton fuzzy values and such that , if and , if , also, , if and , if .

Remark 4. Throughout this paper, we use the extended fuzzy numbers, i.e., by adjoining the singleton fuzzy elements and .

Definition 5. (see [23]). Suppose that , the partial order relations between two fuzzy numbers, i.e.,If and . And and .

Proposition 6 (see [19]). is a fuzzy number which is entirely distinguished by the pair of as functions , denoting by the endpoints of the -cuts, fulfilling the below situations:(1)As a function of , is a bounded monotonic increasing left-continuous function for all (0, 1] and right-continuous at ;(2)As a function of , is a bounded monotonic decreasing left-continuous function for all (0, 1] and right-continuous at ;(3) for , which supplies , .The addendum outcome is well known [24]:

Proposition 7 (see [19]). Suppose that an arbitrary real interval collection that satisfied the below situations:(1) for every (0, 1];(2)if then ;(3)For any increasing sequence (0, 1] is given, such that , then .Therefore, there exists a unique LU-fuzzy quantity (L for lower, U for upper) with , (0, 1] and .

Lemma 8 (see [19]). Suppose that , and . Then, if(1) uniformly w.r.t. ,(2)The collections of satisfy the situations in Proposition 6 or equivalently satisfy the situations in Proposition 7, therefore , with

Definition 9. (see [19]). For each , the -difference is determined by the formIn terms of -cuts,and conditions for the entity of are as the formIt is obvious that the conditions (9) and (10) are both satisfied if and only if is a crisp number.

Definition 10. (see [19]). Suppose that and with , then the function with -derivative at is described by the formIf fulfilling (9) exists, it is said to be is generalized Hukuhara differentiable (-differentiable in short) at .

Definition 11 (see [19]). Suppose that and in terms of -cuts and are differentiable at . Then,(1)if is [(9) ]-differentiable at (2)if is [(10) ]-differentiable at

Definition 12. (see [19]). The -difference of in terms of -cuts is as follows:such that the -difference of intervals is denoted by .

Proposition 13 (see [19]). The -difference (4) in terms of -cuts as the form

Remark 14. (see [19]). Assume that as well as .

Proposition 15 (see [19]). The -difference of every is denoted by .

Proposition 16 (see [19]). Suppose that , then(1), if the right side exists; particularly ;(2);(3);(4); also, .

Definition 17. Whenever , then(1);(2);(3).

Definition 18. (see [19]). For every (0, 1], suppose that is a fuzzy number. We can describe the Hausdorff distance on as the formwhere, for an interval , is the norm on ,Note that the metric is well defined because of the interval -difference, evermore exists. Therefore, with the Hausdorff distance becomes a complete metric space. This definition is equivalent to the usual definitions for metric fuzzy numbers spaces, e.g., [2, 25, 26].

Proposition 19 (see [19]). For all where .

Remark 20. (see [19]). Note that since whenever the right expression exists, we also consummate , whenever exists.

Definition 21. (see [19]). Suppose that and with , then the level-wise -derivative (-derivative in short) of a function at is described as the interval-valued -derivatives set if they exist,If , for all , it is said to be is -differentiable at , and the intervals’ collection is the -derivative of at and indicated by .

Definition 22. Suppose that is said to be -continuous at , if for every with , then we have

Definition 23. (see [19]). Let be a point of and with . Then, is said to be -differentiable at such that

Theorem 24 (see [19]). Suppose that , the collection of interval is uniformly -differentiable at . Then, has a -derivative at and

Theorem 25 (see [19]). Suppose that with . If the real-valued functions and are both differentiable w.r.t. , uniformly w.r.t. , then has a -derivative at ,

3. The Fuzzy-Valued Convex Function in Sense of Jensen’s Inequality

Therein-after, all of these below inequalities are now called the fuzzy-valued convex function Jensen’s inequality. So, we shall designate by a (closed, open, or half-open, finite or infinite) interval in . Also, we denoted the interior of by .

Definition 26. (see [23]). Suppose that , then is said to be a fuzzy-valued convex function ifThe basic fuzzy inequality equation (22) is sometimes called fuzzy Jensen’s inequality.
Closely related to fuzzy convexity is the following concept.

Definition 27. Suppose that is a midpoint fuzzy-valued convex function ifNote that if is a fuzzy-valued convex function, then is the midpoint fuzzy-valued convex function.

Theorem 28 (see [23]). Suppose that in terms of , then is a fuzzy-valued convex function if and only if for any fixed , the convex functions and are both real-valued of .

Definition 29. Suppose that is a fuzzy-valued convex function. Let and with then, we haveexists on . If and satisfying (24) and (25) exist, then is said to be right and left -differentiable at on .

4. The Extended Fuzzy-Valued Convex Functions and -Differentiability

In the previous section, we consider the fuzzy-valued convex function in sense of Jensen’s inequality with fuzzy values in . Now, in this part, we shall consider more general fuzzy-valued functions, with fuzzy values in . In other words, we want to define the fuzzy-valued convex functions whose range of them be the extended fuzzy numbers in . Throughout our paper, we consider for convenience extended fuzzy-valued functions, which take fuzzy values in . The usual conventions of the fuzzy arithmetic are that if , if , if , but also the following less obvious one is as follows:

Also, we discussed the -differentiability and the basic facts of the -differentiability for the extended fuzzy-valued convex functions that can be easily visualized. The expression is undefined.

Definition 30. An extended fuzzy-valued function is called convex, if for all and such that , we have ,

Lemma 31. Suppose that , then is the fuzzy-valued convex function in Definition 26 if and only if is the fuzzy-valued convex function in Definition 30.

Proof. Suppose that is convex fuzzy-valued function, and let , , . Then,Conversely, suppose that Definition 30 holds. Then, for any , let and , then by the hypothesis, we haveas , and soHence, is a fuzzy-valued convex function. It comes upon that Definition 30 is a generalization of Definition 26.

Definition 32. The extended fuzzy-valued convex function effective domain of , denoted by , is the set of .

Lemma 33. The extended fuzzy-valued convex function effective domain of is a convex set.

Proof. Suppose that is a fuzzy-valued convex function, let , we haveAnd soHence, dom is a convex set.

Definition 34. The extended fuzzy-valued function is called fuzzy proper, if , and .

Definition 35. The extended fuzzy-valued function is called fuzzy improper, if is not fuzzy proper, i.e., or there exists such that .

The below theorem is the class of fuzzy-valued improper convex functions that is easy to describe.

Theorem 36. Suppose that be the fuzzy-valued improper convex function. Then, whenever .

Proof. The statement is trivially true if , i.e., for all , then therefore . Henceforth, on . Let , there exists such that , then . Let , be arbitrary. There exists and so that . By Definition 30, for each therefore and each Since . Letting , we see that . The proof is complete.

By the following lemma, it is often convenient to extend a fuzzy-valued convex function to all of by defining its fuzzy value to be outside its domain.

Lemma 37. (The fuzzy-valued convex extension) Suppose that is a fuzzy number-valued convex function, where is a convex set. We define its extended fuzzy-valued of , as follows:

Then, the extension is a fuzzy-valued convex function that defines on all and takes the fuzzy values in .

Proof. Let , be arbitrary.

Case 1. If , then by (11), we haveConsiderNote that since is the fuzzy-valued convex function on and by (34), we haveby (35),Then,

Case 2. If , henceforth by (11),Also, , then by (34),Therefore.by (40),Hence,

Case 3. If and , afterward by (11),Since and , then , by (34), we haveTherefore, we haveby (45),So,Thus, in all three cases, the definition of a fuzzy-valued convex function was established for ; hence, it is a fuzzy-valued convex function on .

Note that, by replacing the domain of a proper fuzzy-valued convex function with effective domain, we can convert it into a fuzzy-valued function.

In the below theorem, the sufficient conditions of left and right -differentiability for right and left -differentiability for proper fuzzy-valued convex functions in terms of -cut are stated.

Theorem 41. Suppose that is a proper fuzzy-valued convex function, and is right and left uniformly -differentiable at . Then, has the right and left -derivative throughout , provided the fuzzy values and are permitted.

Proof. The proof is the same as Theorem 5.2 in [23] and Theorem 35 in [19] since the -quotient is nondecreasing and bounded from below on ; therefore, there exists a subsequence , in which the members of subsequence are and as converges to equals to , i.e., there exists a subsequence such thatSimilarly, since the -quotient is nonincreasing and bounded from above on , therefore there exists a subsequence , in which the members of subsequence are and as converges to equals to , i.e., there exists a subsequence such thatThus, the left -derivative for the case where exists. Hence, by the above concepts, exists whenever (a, b], and exists whenever (a, b]. But for any , we have , so the quotienthence, , for any , we have the quotientand so .

5. 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.

Definition 42. Suppose that is a set. Define the fuzzy indicator function of as follows:is given by

Theorem 43. Let . Then, is a convex set if and only if is a fuzzy-valued convex function.

Proof. Suppose that is a convex set. Let , . If , since is a convex set, then . Therefore,If , then , . Since is a convex set, then , thereforeThus,Hence, is the fuzzy-valued convex function.
Conversely, suppose that is a fuzzy-valued convex function. Let , , then , . ConsiderthenTherefore, is a convex set. The proof is complete.

Definition 44. Let . The epigraph of ( in short) is a subset by

Definition 45. Let . The fuzzy strictly epigraph of ( in short) is a subset by

Definition 46. An extended fuzzy-valued function is said to be convex, if for all , , and such that , ,

Theorem 47. Let , the following conditions are equivalent:(1) is a fuzzy-valued convex function.(2) is a convex set.(3) is a convex set.

Proof. : Let , , , then , . Bring upwe have is convex set if and only ifwhenever , .
Applying Definition 26, the equivalence of (9) and (11) is as follows.: Supposing that is a convex set. Let , , . Because , we can select so that . We have . Since is a convex set, henceforthand soIt follows thatWe conclude that is a convex set.
Vice versa, suppose that is a convex set. Let , , then , , therefore for each thusSince is a convex set, thenHence,It follows thatthenand we consummate that is a convex set.

Definition 48. Suppose that is a convex set. We define a fuzzy-valued function byi.e., is the greatest fuzzy-valued convex function on which the epigraph contains .

Theorem 49. Suppose that is a convex set, and letThen is a fuzzy-valued convex function.

Proof. We show that is a convex set. Let , then , . By definition of infimum, there exists so that , and there exists so that , . Since is a convex set, thenby Definition 30, thenthereforeso is a convex set, hence by Theorem 47, is a fuzzy-valued convex function.

6. The Fuzzy Infimal Convolution

Now, in the following, we introduce the fuzzy infimal convolution as a subset for extended fuzzy-valued convex functions and that denote by .

Definition 50. Let be the fuzzy-valued functions. Define the fuzzy infimal convolution and as follows:byNote that if and are fuzzy-valued convex functions, then , in are convex sets, therefore in is a convex set, hence by Theorem 49, is a fuzzy-valued convex function. The terminology is motivated by the case where and are fuzzy-valued functions, . Then, can also be defined asHence.which is analogous to the formula for fuzzy integral convolutionand is exact at , if , i.e., there exists so that is exact if it is exact at every point of its domain, in which case it is denoted by .

Proposition 51. Let , , and be the extended fuzzy-valued functions from to . Then, the following cases hold:(1);(2).

Proof. For the proof of (1), we can see thatHence, . Also, for the proof of (10), we haveOn the other hand, we haveHence, .

Example 1. Consider to be a nonempty convex set and . The fuzzy distance of from is defined byWe show that is a fuzzy-valued convex function. Since is a convex set, then by Theorem 43, is a fuzzy-valued convex function. Let . Therefore, is a fuzzy-valued convex function. By Definition 50, hence is a fuzzy-valued convex function. ConsiderSo, . According to Definition 50, is a fuzzy-valued convex function.

Example 2. Consider is an extended fuzzy-valued function and let . We show that , whereis the translation of the extended fuzzy-valued function by .
Let be arbitrary. ThenHence, .

7. The Directional -Derivative for Extended Fuzzy-Valued Convex Functions

Now, we introduce the directional -derivative for extended fuzzy-valued convex functions and their properties are discussed.

Definition 52. Suppose that is an extended fuzzy-valued function, and . The directional -derivative of at in the direction is as follows:If satisfying (93) exist. Note that if it exists ( and being allowed as limits).

Theorem 53. Suppose that is a proper fuzzy-valued convex function and . Then,(1) exist, .(2) is fuzzy positively homogeneous and fuzzy-valued convex.

Proof. For proof of (1), let be arbitrary. Define bySince by the hypothesis, is the proper fuzzy-valued convex function, then is a proper fuzzy-valued convex function, therefore by Theorem 41, exists where it is the right -derivative of the proper fuzzy-valued convex function at , as follows:Hence exists, for all . For proof of (10), let . Considerget , when Hence, .
The convexity of this function is as follows:
Let . ConsiderLet , thenHence, is an extended fuzzy-valued convex function.

Proposition 54. Let be a proper fuzzy-valued convex function and be so that . Assume that and of the proper real-valued convex functions are directional differentiable at in the direction of , uniformly w.r.t. . Then, has a directional -derivative at in the direction of as follows:

Proof. According to Proposition 13, we getSince the proper real-valued convex functions and are directional differentiable at in the direction of , we haveAlso, let us consider that if the functions and are left continuous w.r.t. (0, 1] and right continuous at 0. From the definition of the directional derivative, for any , there exists a sequence such that the quotientsas functions of are left continuous at (0, 1] and right continuous at 0. Also, for any , there exists a sequence such that , then the functionsandsatisfying the above properties. Thus, it follows thatandas functions of are left continuous at (0, 1] and right continuous at 0. It is obvious to see that the function is increasing function w.r.t. and the function is decreasing function w.r.t. , by Proposition 6, they define a fuzzy number. Consequently, the -cuts define a fuzzy number, by Lemma 8, the directional -derivative with extended fuzzy-valued exists at in the direction .

Below, we give a practical example that will illustrate well the directional -derivative for the extended fuzzy-valued convex function.

Example 3. Let be an extended fuzzy-valued convex function defined byits -cuts, , are defined byFor all , the functions and are extended real-valued differentiable at each point , then and . Now, for all , the two functions and are not differentiable at . However, for all , the functions and are real-valued directional differentiable at in any direction and satisfy the conditions in Proposition 6, indeed for any direction and we haveit follows thatHence, is directionally -differentiable at in any direction .

8. Conclusion

The concepts of -difference and -differentiability were introduced for fuzzy-valued functions in 2013 by Bede and Stefanini [19] which is the generalization concept of -difference and -differentiability. Here, we defined 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, infimal convolution, and directional -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 -subgradient and -subdifferential for the extended fuzzy-valued convex function. In this way, in the next studies and research, we propose the concepts of -subgradient and -subdifferential, which play an important role in extended fuzzy-valued optimization.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

TA is the supervisor of this study and was a major contributor to methodology, investigation, and validation. RS and MRBS worked on resources, investigation, and formal analysis of this study. OS, UFG, MRS, and SN worked on the software, writing reviews, and editing and validating the results. All authors have main contributions in writing the original draft preparation and in also writing a review and editing the paper. All authors read and approved the final manuscript.

Acknowledgments

The work of U.F.-G. was supported by the Government of the Basque Country for the ELKARTEK21/10 KK-2021/00014 and ELKARTEK22/85 research programs, respectively. The research was supported by the Ministry of Science and Higher Education of Russian Federation (Project No. FZZS-2024-0003).

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