The Karush-Kuhn-Tucker Optimality Conditions for the Fuzzy Optimization Problems in the Quotient Space of Fuzzy Numbers

Yu, Nanxiang; Qiu, Dong

doi:https://doi.org/10.1155/2017/1242841

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Fuzzy Calculus Theory and Its Applications

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Volume 2017 | Article ID 1242841 | https://doi.org/10.1155/2017/1242841

The Karush-Kuhn-Tucker Optimality Conditions for the Fuzzy Optimization Problems in the Quotient Space of Fuzzy Numbers

Nanxiang Yu¹and Dong Qiu¹

Academic Editor: Omar Abu Arqub

Received13 Apr 2017

Revised04 Jun 2017

Accepted11 Jul 2017

Published07 Aug 2017

Abstract

We propose the solution concepts for the fuzzy optimization problems in the quotient space of fuzzy numbers. The Karush-Kuhn-Tucker (KKT) optimality conditions are elicited naturally by introducing the Lagrange function multipliers. The effectiveness is illustrated by examples.

1. Introduction

The fuzzy set theory was introduced initially in 1965 by Zadeh [1]. After that, to use this concept in topology and analysis many authors have expansively developed the theory of fuzzy sets and application. The fuzziness occurring in the optimization problems is categorized as the fuzzy optimization problems. Bellman and Zadeh [2] inspired the development of fuzzy optimization by providing the aggregation operators, which combined the fuzzy goals and fuzzy decision space. After this motivation and inspiration, there come out a lot of works dealing with the fuzzy optimization problems.

Zimmermann and Rödder initially applied fuzzy sets theory to the linear programing problems and linear multiobjective programing problems by using the aspiration level approach [3–6]. Durea and Tammer [7] derived the Lagrange multiplier rules for fuzzy optimization problems using the concept of abstract subdifferential. Bazine et al. [8] developed some fuzzy optimality conditions for fractional multiobjective optimization problems. In 2013, the solution approach for the lower level fuzzy optimization problem and the fuzzy bilevel optimization problem was investigated by Budnitzki [9]. Panigrahi et al. [10] extended and generalized these concepts to fuzzy mappings of several variables using the approach due to Buckley and Feuring [11] for fuzzy differentiation and derived the KKT conditions for the constrained fuzzy minimization problems. Wu [12, 13] presented the KKT conditions for the optimization problems with convex constraints and fuzzy-valued objective functions on the class of all fuzzy numbers by considering the concepts of Hausdorff metric and Hukuhara difference. Chalco-Cano et al. [14] discussed the KKT optimality conditions for a class of fuzzy optimization problems using strongly generalized differentiable fuzzy-valued functions, which is a concept of differentiability for fuzzy mappings more general than the Hukuhara differentiability.

These above results of fuzzy optimization are based on well-known and widely used algebraic structures of fuzzy numbers and the differentiability of fuzzy mappings was based on the concept of Hukuhara difference. However these operations can have some disadvantages for both theory and practical application. In [15], Qiu et al. intuitively showed a method of finding the inverse operation in the quotient space of fuzzy numbers based on the Mareš equivalence relation [16, 17], which have the desired group properties for the addition operation [18–20] midpoint function. As an application of the main results, it is shown that if we identify every fuzzy number with the corresponding equivalence class, there would be more differentiable fuzzy functions than what is found in the literature. In [21] Qiu et al. further investigated the differentiability properties of such functions in the quotient space of fuzzy numbers. In this paper, the KKT optimality conditions for the constrained fuzzy optimization problems in the quotient space of fuzzy numbers are derived.

2. Preliminaries

We start this section by recalling some pertinent concepts and key lemmas from the function of bounded variation, fuzzy numbers, and fuzzy number equivalence classes which will be used later.

Definition 1 (see [22]). Let be a function. is said to be of bounded variation if there exists a such that for every partition on . The set of all functions of bounded variation on is denoted by

Definition 2 (see [22]). Let be a function of bounded variation. The total variation of on , denoted by , is defined by where represents all partitions of

Lemma 3 (see [22]). Let , and then we have the following:(1) and for any contents (2) and

Lemma 4 (see [22]). Every monotonic function is of bounded variation and

Any mapping will be called a fuzzy set on Its -level set of is for each Specifically, for , the set is defined by , where denotes the closure of a crisp set A fuzzy set is said to be a fuzzy number if it is normal, fuzzy convex, and upper semicontinuous and the set is compact.

Let be the set of all fuzzy numbers on Then for an it is well known that the -level set is a nonempty bounded closed interval in for all , where denotes the left-hand end point of and denotes the right one. For any and , owing to Zadeh’s extension principle [23], the addition and scalar multiplication can be, respectively, defined for any by We say that a fuzzy number is symmetric if [16]. We denote the set of all symmetric fuzzy numbers by .

Definition 5 (see [15]). Let , and we define a function by assigning the midpoint of each -level set to for all ; that is,Then the function will be called the midpoint function of the fuzzy number

Lemma 6 (see [15]). For any , the midpoint function is continuous from the right at and continuous from the left on . Furthermore, it is a function of bounded variation on

Definition 7 (see [24]). Let , and we say that is equivalent to , if there exist two symmetric fuzzy numbers such that and then we denote this by

It is easy to verify that the equivalence relation defined above is reflexive, symmetric, and transitive [16]. Let denote the fuzzy number equivalence class containing the element and denote the set of all fuzzy number equivalence classes by

Definition 8 (see [17]). Let and let be a fuzzy number such that for some , and if for some and , then Then the fuzzy number will be called the Mareš core of the fuzzy number

Definition 9 (see [21]). Let , and we define the midpoint function by for all , where is the Mareš core of

Definition 10 (see [21]). Let , and we define the sum of this two fuzzy number equivalence classes as a fuzzy equivalence class , which satisfies the condition for all and we denote this by

Remark 11. The addition operation defined by Definition 10 is a group operation over the set of fuzzy number equivalence classes up to the equivalence relation in Definition 7. For the details of the discussion, please see [25, 26].

Definition 12 (see [15]). Let , and we say that is the product of and if their midpoint functions satisfy for all and we denote this by

Definition 13 (see [21]). For any and , we define by It is obvious that for all

Definition 14 (see [15]). Let , and we define by It is easy to see that is a metric space [15].

3. The Karush-Kuhn-Tucker Optimality Conditions

In this paper, we always suppose that the range of fuzzy mappings is the set of all fuzzy number equivalence classes.

Definition 15 (see [21]). Let be a fuzzy mapping, where Then is said to be differentiable at if there exists an such that If (or ), then we consider only (or ).

Lemma 16 (see [21]). is differentiable on if and only if (1) is differentiable with respect to for all That is, exists and is of bounded variation with respect to for all ;(2)the mappings are uniformly differentiable with the derivatives That is, for each and , there exists a such that for all and

Definition 17 (see [27]). Let and be an -dimensional fuzzy number equivalence class vector and -dimensional real vector, respectively. We define their product as which is a fuzzy number equivalence class.

Definition 18 (see [27]). Let be a fuzzy mapping, where is an open subset in . We say that has a partial derivative at with respect to the th variable if there exists an such that where stands for the unit vector that the th component is and the others are

Definition 19 (see [27]). Let be a fuzzy mapping, where is an open subset in We say that is differentiable at if has continuous partial derivatives with respect to th variable and satisfies where is an -dimensional fuzzy number equivalence class vector defined by and is the usual Euclid norm of and is a fuzzy mapping that satisfies Then we call the gradient of the fuzzy mappings at

Definition 20 (see [27]). Let (1)We say that if for all (2)We say that if and there exists at least one such that (3)If and then Sometimes we may write instead of and write instead of Note that is a partial order relation on

Definition 21. Let , and we say that is nonnegative if ; that is, for all
Let be a fuzzy mapping. Consider the following optimization problem: where the feasible set is assumed to be convex subset of . Since is a partial order relation on , we may follow the similar solution concept (the nondominated solution) used in multiobjective programing problems to interpret the meaning of minimization in problem (22).

Definition 22. Let be a feasible solution of problem (22); that is, (1)We say that is a local nondominated solution of problem (22) if there exists an and there does not exist any such that , where is an -neighborhood around (2)We say that is a (global) nondominated solution of problem (22) if there exists no such that

Definition 23. Let be a fuzzy mapping, where is a nonempty convex subset in . is said to be convex on if, for any and , we always have . is said to be concave if is convex.

Theorem 24. Let be a fuzzy mapping, where is a nonempty convex subset in Then is convex on if and only if is convex with respect to for all

Proof. The result follows from Definitions 20 and 23 immediately.
Let be real-valued functions. Consider the following optimization problem: Suppose that the constraint functions are convex on for all , and then the feasible set is a convex subset of The well-known KKT optimality conditions for problem (23) are stated as below.

Theorem 25 (see [28, 29]). Let be the convex feasible set and be a feasible solution of problem (23). Suppose that the objective function and constraint functions are convex on and continuously differentiable at for all If there exist nonnegative Lagrange multipliers , such that (1),(2) for all ,then is nondominated solution of problem (23).
Let be a fuzzy mapping and be real-valued functions, Now we consider the following optimization problem: If we suppose that the constraint functions are convex on for all , then we can see that problem (24) follows from problem (22) by taking the convex feasible set as
Now we are in a position to present the KKT optimality conditions for nondominated solutions of problem (24).

Theorem 26. Let be the convex feasible set and be a feasible solution of problem (24). Suppose that the fuzzy-valued objective function and real-valued constraint functions are convex on and continuously differentiable at for all If there exist nonnegative real-valued Lagrange function multipliers for defined on such that (1) for all ,(2) for all and ,then is a nondominated solution of problem (24).

Proof. Suppose that conditions (1) and (2) are satisfied and is not a nondominated solution of problem (24). Then there exists a such that ; that is, for some we have that We now define a real-valued function by Then we have Since the fuzzy mapping is convex on and continuously differentiable at , by Theorem 24 and Lemma 16 we see that is also convex on and continuously differentiable at Furthermore, we have Since conditions (1) and (2) are satisfied, we can obtain the following two new conditions for any fixed :(1′);(2′) for all ,where for Now we consider the following constrained optimization problem: which has the same constraints of problem (24). By Theorem 25, conditions and are the KKT conditions of problem (26). Therefore, we have that is an optimal solution of problem (26) with the real-valued objective function ; that is, for all , which contradicts inequality (25). Then we get that is indeed a nondominated solution of problem (24).

Theorem 27. Let be the convex feasible set and be a feasible solution of problem (24). Suppose that the fuzzy-valued objective function and real-valued constraint functions are convex on and continuously differentiable at for all If there exist nonnegative fuzzy number equivalent class Lagrange multipliers for such that (1),(2) for all ,then is a nondominated solution of problem (24).

Proof. Since conditions (1) and (2) are satisfied, taking the midpoint function of (1) and (2), we obtain the following new conditions: (1′) for all .(2′) for all and Since the fuzzy number equivalence classes are nonnegative for all , then we can get that are nonnegative real-valued functions defined on for all So, and verify the KKT optimality conditions (1) and (2) of Theorem 26, respectively. Therefore, we get that is a nondominated solution of problem (24).

Lemma 28 (see [28]). Let be a feasible set and Assume that are differentiable at for all Let be the index set for the active constraints. Then we have where is the cone of feasible directions of at defined by

Lemma 29 (see [28]). Let and be two matrices. Exactly one of the following systems has a solution: System I: for some . System II: for some

Theorem 30. Let be the convex feasible set and be a feasible solution of problem (24). Suppose that the fuzzy-valued objective function is differentiable and strictly pseudoconvex on , and the real-valued constraint functions are convex on and continuously differentiable at for all If there exist a and nonnegative Lagrange multipliers for such that (1),(2) for all ,then is a strongly nondominated solution of problem (24).

Proof. Suppose that conditions (1) and (2) are satisfied and is not a strongly nondominated solution of problem (24). Then there exists a with such that Since is differentiable and strictly pseudoconvex on , we have that is,Let . Since is a convex set and , we havefor any By Lemma 28 we get that , which means that where is the cone of feasible directions of at and is the index set for the active constraints. Now let and be the matrix whose rows are for We consider the following two systems: System I: for some . System II: for some Then by (30) and (32) we get that System I has a solution . Further, by Lemma 29 System II has no solutions, which means that there exist no multipliers and for such that Since , dividing (33) by and denoting for , we have that Since is the index set for the active constraints, we have for Further, if for all , we can get that for ; that is, From (34) and (35), there exist no multipliers for such that (1′),(2′) for all ,which contradicts conditions (1) and (2) for the existence of multipliers for Hence, we have that is indeed a strongly nondominated solution of problem (24).

Example 31. Define a fuzzy mapping by for all , where and we define by the level sets of its Mareš core , , and for all and , respectively. Thus, we have for all and It is obvious that is continuous from the right at and continuous from the left on with respect to Now we consider the following optimization problem:It is obvious that the constraint functions are convex on for all , and then we know that the feasible set is convex. Since is decreasing with respect to for all , we get that Thus, we find that is of bounded variation with respect to for all It is easy to verify that is differentiable and strictly pseudoconvex on , and are convex on and continuously differentiable at for all Then we obtain for all and Now we consider the point . Since from condition (2) in Theorem 30, we get that Now, applying condition (2) of Theorem 30 at the point , we obtain After these algebraic calculations, we obtain that there exist a and nonnegative Lagrange multipliers which satisfied conditions (1) and (2) of Theorem 30. Hence, we get that is a strongly nondominated solution of problem (38).

4. Conclusions

In this present investigation, the KKT optimality conditions are elicited naturally by introducing the Lagrange function multipliers, and we also provided some examples to illustrate the main results. The research on the quotient space of fuzzy numbers can be traced back to the works of Mareš [16, 17]. Hong and Do [24] improved this result and proposed a more refined equivalence relation. This equivalence relation can be used to partition the set of fuzzy numbers into equivalence class having the desired group properties for the addition operation. Since the quotient space of fuzzy numbers is characterized by the midpoint functions, there are more differentiable fuzzy mappings. As a matter of fact, there are still many other types of the KKT optimality conditions that can be derived using the similar techniques discussed in this paper on the quotient space of fuzzy numbers. However, for the nondifferentiable fuzzy optimization problem, we can follow the approach proposed by Ruziyeva and Dempe [30] to derive the necessary and sufficient optimality conditions in the quotient space of fuzzy numbers. In addition, Fuzzy sets and fuzzy optimization problems have several appropriate applications to today’s world. But there are no sufficient examples and applications of the topics discussed in this paper. Therefore, we will develop the contribution of this research to practical problems in future studies.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by The National Natural Science Foundations of China (Grants nos. 11671001 and 61472056), The Natural Science Foundation Project of CQ CSTC (cstc2015jcyjA00034, cstc2014jcyjA00054), and The Graduate Teaching Reform Research Program of Chongqing Municipal Education Commission (no. YJG143010).

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Copyright © 2017 Nanxiang Yu and Dong Qiu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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