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

. 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


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][4][5][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][19][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 2 Complexity 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.

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  : [, ] → R be a function. is said to be of bounded variation if there exists a  > 0 such that for every partition Definition 2 (see [22]).Let  : [, ] → R be a function of bounded variation.The total variation of  on [, ], denoted by    (), is defined by where  represents all partitions of [, ].
(2)  ⋅  ∈ BV[, ] and Lemma 4 (see [22]).Every monotonic function  : [, ] → R is of bounded variation and Any mapping x : R → [0, 1] will be called a fuzzy set x on R. Its -level set of x is [ x]  = { ∈ R : x() ≥ } for each  ∈ (0, 1].Specifically, for  = 0, the set [ x] 0 is defined by [ x] 0 = cl{ ∈ R : x() > 0}, where cl denotes the closure of a crisp set .A fuzzy set x is said to be a fuzzy number if it is normal, fuzzy convex, and upper semicontinuous and the set [ x] 0 is compact.
Let  be the set of all fuzzy numbers on R. Then for an x ∈  it is well known that the -level set [ x]  = [x  (), x ()] is a nonempty bounded closed interval in R for all  ∈ [0, 1], where x () denotes the left-hand end point of [ x]  and x () denotes the right one.For any x, ỹ ∈  and  ∈ R, owing to Zadeh's extension principle [23], the addition and scalar multiplication can be, respectively, defined for any  ∈ R by We say that a fuzzy number s ∈  is symmetric if s = −s [16].
We denote the set of all symmetric fuzzy numbers by .
Definition 5 (see [15]).Let x ∈ , and we define a function x : [0, 1] → R by assigning the midpoint of each -level set to x () for all  ∈ [0, 1]; that is, Then the function x : [0, 1] → R will be called the midpoint function of the fuzzy number x.
Lemma 6 (see [15]).For any x ∈ , the midpoint function x is continuous from the right at 0 and continuous from the left on [0, 1].Furthermore, it is a function of bounded variation on [0, 1].
Definition 7 (see [24]).Let x, ỹ ∈ , and we say that x is equivalent to ỹ, if there exist two symmetric fuzzy numbers s1 , s2 ∈  such that x + s1 = ỹ + s2 and then we denote this by x ∼ ỹ.
It is easy to verify that the equivalence relation defined above is reflexive, symmetric, and transitive [16].Let ⟨ x⟩ denote the fuzzy number equivalence class containing the element x and denote the set of all fuzzy number equivalence classes by /.Definition 8 (see [17]).Let x ∈  and let x be a fuzzy number such that x = x + s for some s ∈ , and if x = ỹ + s1 for some ỹ ∈  and s1 ∈ , then s1 = 0. Then the fuzzy number x will be called the Mareš core of the fuzzy number x. Definition 9 (see [21]).Let ⟨ x⟩ ∈ /, and we define the midpoint function for all  ∈ [0, 1], where x is the Mareš core of x.
Definition 10 (see [21]).Let ⟨ x⟩, ⟨ ỹ⟩ ∈ /, and we define the sum of this two fuzzy number equivalence classes as a fuzzy equivalence class ⟨z⟩ ∈ /, which satisfies the condition for all  ∈ [0, 1] 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].

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 18 (see [27]).Let  : Ω → / be a fuzzy mapping, where Ω is an open subset in R  .We say that  has a partial derivative at  = ( where   stands for the unit vector that the th component is 1 and the others are 0.
Definition 19 (see [27]).Let  : Ω → / be a fuzzy mapping, where Ω is an open subset in R  .We say that  is differentiable at  = ( where ∇() ∈ (/)  is an -dimensional fuzzy number equivalence class vector defined by and ‖ℎ‖ is the usual Euclid norm of ℎ and  : [0, +∞) → / is a fuzzy mapping that satisfies lim Then we call ∇() the gradient of the fuzzy mappings  at .

Complexity
Let  : R  → / be a fuzzy mapping.Consider the following optimization problem: min  () =  ( 1 ,  2 , . . .,   ) , where the feasible set Ω is assumed to be convex subset of R  .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).
Proof.Suppose that conditions (1) and ( 2) are satisfied and  * is not a strongly nondominated solution of problem (24).Then there exists a  ∈ Ω with  ̸ =  * such that () ⪯ ( * ).Since  is differentiable and strictly pseudoconvex on Ω, we have that is, Let  =  −  * .Since Ω is a convex set and ,  * ∈ Ω, we have for any  ∈ (0, 1).By Lemma 28 we get that  ∈ , which means that where  is the cone of feasible directions of Ω at  * and  = { :   () = 0} 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:  ≤ 0,  ̸ = 0,  ≤ 0 for some  ∈ R  .

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.