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

Authors: Nanxiang Yu; Dong Qiu
Journal: Complexity
 (2017)

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## 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.


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## Body

## 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]).
Letf:a,b→R be a function. f is said to be of bounded variation if there exists a C>0 such that (1)∑i=1nfxi-1-fxi-1≤Cfor every partition a=x0<x1<x2<⋯<xn=b on a,b. The set of all functions of bounded variation on [a,b] is denoted by BVa,b.Definition 2 (see [22]).
Letf:a,b→R be a function of bounded variation. The total variation of f on a,b, denoted by Vabf, is defined by (2)Vabf=supp∑i=1nfxi-1-fxi,where p represents all partitions of a,b.Lemma 3 (see [22]).
Letf,g∈BVa,b, and then we have the following:(1)
cf+dg∈BVa,b and (3)Vabcf+dg≤cVabf+dVabgfor any contentsc,d∈R.(2)
f·g∈BVa,b and (4)Vabf·g≤Vabfsupx∈a,bgx+Vabgsupx∈a,bfx.Lemma 4 (see [22]).
Every monotonic functionf:a,b→R is of bounded variation and (5)Vabf=fa-fb.Any mappingx~:R→0,1 will be called a fuzzy set x~ on R. Its α-level set of x~ is x~α=x∈R:x~x≥α for each α∈0,1. Specifically, for α=0, the set x~0 is defined by x~0=clx∈R:x~x>0, where clA denotes the closure of a crisp set A. 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.LetF be the set of all fuzzy numbers on R. Then for an x~∈F it is well known that the α-level set x~α=x~Lα,x~Rα is a nonempty bounded closed interval in R for all α∈0,1, where x~Lα denotes the left-hand end point of x~α and x~Rα denotes the right one. For any x~,y~∈F and λ∈R, owing to Zadeh’s extension principle [23], the addition and scalar multiplication can be, respectively, defined for any x∈R by (6)x~+y~x=supx1,x2:x1+x2=xmin⁡x~x1,x~x2,λx~x=x~xλ,λ≠0,0,λ=0.We say that a fuzzy number s~∈F is symmetric if s~=-s~ [16]. We denote the set of all symmetric fuzzy numbers by φ.Definition 5 (see [15]).
Letx~∈F, and we define a function x~M:0,1→R by assigning the midpoint of each α-level set to x~Mα for all α∈0,1; that is,(7)x~Mα=x~Lα+x~Rα2.Then the function x~M:0,1→R will be called the midpoint function of the fuzzy number x~.Lemma 6 (see [15]).
For anyx~∈F, the midpoint function x~M 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]).
Letx~,y~∈F, and we say that x~ is equivalent to y~, if there exist two symmetric fuzzy numbers s~1,s~2∈φ such that x~+s~1=y~+s~2 and then we denote this by x~~y~.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 F/φ.Definition 8 (see [17]).
Letx~∈F and let x^ be a fuzzy number such that x~=x^+s~ for some s~∈φ, and if x^=y~+s~1 for some y~∈F and s~1∈φ, then s~1=0~. Then the fuzzy number x^ will be called the Mareš core of the fuzzy number x~.Definition 9 (see [21]).
Letx~∈F/φ, and we define the midpoint function Mx~:0,1→R by (8)Mx~α=x^Mαfor all α∈0,1, where x^ is the Mareš core of x~.Definition 10 (see [21]).
Letx~,y~∈F/φ, and we define the sum of this two fuzzy number equivalence classes as a fuzzy equivalence class z~∈F/φ, which satisfies the condition (9)Mx~α+My~α=Mz~αfor all α∈0,1 and we denote this by (10)x~+y~=x~+y~=z~.Remark 11.
The addition operation defined by Definition10 is a group operation over the set of fuzzy number equivalence classes F/φ up to the equivalence relation in Definition 7. For the details of the discussion, please see [25, 26].Definition 12 (see [15]).
Letx~,y~∈F/φ, and we say that z~∈F/φ is the product of x~ and y~ if their midpoint functions satisfy (11)Mx~α·My~α=Mz~αfor all α∈0,1 and we denote this by (12)x~·y~=z~.Definition 13 (see [21]).
For anyx~∈F/φ and λ∈R, we define λ·x~=λx~ by (13)λx~=x~λ=λx~.It is obvious that Mλx~α=Mλx~α=λMx~α for all α∈0,1.Definition 14 (see [15]).
Letx~,y~∈F/φ, and we define dsup:F/φ×F/φ→R+∪0 by (14)dsupx~,y~=supα∈0,1Mx~α-My~α.It is easy to see that F/φ,dsup 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]).
LetF:T→F/φ be a fuzzy mapping, where T=a,b⊆R. Then F is said to be differentiable at t∈T if there exists an F′t∈F/φ such that (15)limh→0⁡dsupFt+h-Fth,F′t=0.If t=a (or b), then we consider only h→0+ (or h→0-).Lemma 16 (see [21]).
F
:
T
→
F
/
φ is differentiable on T if and only if (1)
MFtα is differentiable with respect to t∈T for all α∈0,1. That is, ∂/∂tMFtα exists and is of bounded variation with respect to α∈0,1 for all t∈T;(2)
the mappingsMFtαα∈0,1 are uniformly differentiable with the derivatives ∂/∂tMFtα. That is, for each t∈T and ε>0, there exists a δ>0 such that(16)MFt+hα-MFtαh-∂∂tMFtα<εfor allh∈0,δ and α∈0,1.Definition 17 (see [27]).
Leta~=a~1,a~2,…,a~nT∈F/φn and t=t1,t2,…,tnT∈Rn be an n-dimensional fuzzy number equivalence class vector and n-dimensional real vector, respectively. We define their product as (17)a~Tt=∑i=1na~iti=a~1t1+a~2t2+⋯+a~ntn,which is a fuzzy number equivalence class.Definition 18 (see [27]).
LetF:Ω→F/φ be a fuzzy mapping, where Ω is an open subset in Rn. We say that F has a partial derivative at t=t1,t2,…,tnT∈Ω with respect to the ith variable ti if there exists an ∂/∂tiFt∈F/φ such that (18)limh→o⁡dsupFt+hei-Fth,∂∂tiFt=0,where ei stands for the unit vector that the ith component is 1 and the others are 0.Definition 19 (see [27]).
LetF:Ω→F/φ be a fuzzy mapping, where Ω is an open subset in Rn. We say that F is differentiable at t=t1,t2,…,tnT∈Ω if F has continuous partial derivatives ∂/∂tiFt with respect to ith variable tii=1,2,…,n and satisfies (19)Ft+h=Ft+∇~FtTh+oh,h=h1,h2,…,hnT∈Rn,where ∇~Ft∈F/φn is an n-dimensional fuzzy number equivalence class vector defined by (20)∇~Ft=∂Ft∂t1,∂Ft∂t2,…,∂Ft∂tnT,and h is the usual Euclid norm of h and o:0,+∞→F/φ is a fuzzy mapping that satisfies (21)limt→0⁡dsupott,0~=0.Then we call ∇~Ft the gradient of the fuzzy mappings F at t.Definition 20 (see [27]).
Letx~,y~∈F/φ.(1)
We say thatx~⪯y~ if Mx~α≤My~α for all α∈0,1.(2)
We say thatx~≺y~ if x~⪯y~ and there exists at least one α0∈0,1 such that Mx~α0<My~α0.(3)
Ifx~⪯y~ and y~⪯x~ then x~=y~.
Sometimes we may writey~⪰x~ instead of x~⪯y~ and write y~≻x~ instead of x~≺y~. Note that ⪯ is a partial order relation on F/φ.Definition 21.
Leta~∈F/φ, and we say that a~ is nonnegative if a~⪰0~; that is, Ma~α≥0 for all α∈0,1.
LetF:Rn→F/φ be a fuzzy mapping. Consider the following optimization problem: (22)minFt=Ft1,t2,…,tn,subjecttot=t1,t2,…,tnT∈Ω⊆Rn,where the feasible set Ω is assumed to be convex subset of Rn. Since ⪯ is a partial order relation on F/φ, 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.
Lett∗ be a feasible solution of problem (22); that is, t∗∈Ω.(1)
We say thatt∗ is a local nondominated solution of problem (22) if there exists an ε>0 and there does not exist any t∈Nεt∗∩Ω such that Ft≺Ft∗, where Nεt∗ is an ε-neighborhood around t∗.(2)
We say thatt∗ is a (global) nondominated solution of problem (22) if there exists no t∈Ω such that Ft≺Ft∗.Definition 23.
LetF:Ω→F/φ be a fuzzy mapping, where Ω is a nonempty convex subset in Rn. F is said to be convex on Ω if, for any s,t∈Ω and λ∈0,1, we always have Fλs+1-λt⪯λFs+1-λFt. F is said to be concave if -F is convex.Theorem 24.
LetF:Ω→F/φ be a fuzzy mapping, where Ω is a nonempty convex subset in Rn. Then F is convex on Ω if and only if MFtα is convex with respect to t∈Ω for all α∈0,1.Proof.
The result follows from Definitions20 and 23 immediately.
Letf,gj:Rn→R be real-valued functions. Consider the following optimization problem: (23)minft=ft1,t2,…,tn,subjecttogjt≤0,j=1,2,…,m.Suppose that the constraint functions gj are convex on Rn for all j=1,2,…,m, and then the feasible set Ω=t∈Rn:gjt≤0,j=1,2,…,m is a convex subset of Rn. The well-known KKT optimality conditions for problem (23) are stated as below.Theorem 25 (see [28, 29]).
LetΩ=t∈Rn:gjt≤0,j=1,2,…,m be the convex feasible set and t∗∈Ω be a feasible solution of problem (23). Suppose that the objective function f and constraint functions gj are convex on Rn and continuously differentiable at t∗ for all j=1,2,…,m. If there exist nonnegative Lagrange multipliers uj∈R,j=1,2,…,m, such that (1)
∇ft∗+∑j=1muj∇gjt∗=0,(2)
ujgjt∗=0 for all j=1,2,…,m,then t∗ is nondominated solution of problem (23).
LetF:Rn→F/φ be a fuzzy mapping and gj:Rn→R be real-valued functions, j=1,2,…,m. Now we consider the following optimization problem: (24)minFt=Ft1,t2,…,tn,subjecttogjt≤0,j=1,2,…,m.If we suppose that the constraint functions gj are convex on Rn for all j=1,2,…,m, then we can see that problem (24) follows from problem (22) by taking the convex feasible set as Ω=t∈Rn:gjt≤0,j=1,2,…,m.
Now we are in a position to present the KKT optimality conditions for nondominated solutions of problem (24).Theorem 26.
LetΩ=t∈Rn:gjt≤0,j=1,2,…,m be the convex feasible set and t∗∈Ω be a feasible solution of problem (24). Suppose that the fuzzy-valued objective function F and real-valued constraint functions gj are convex on Rn and continuously differentiable at t∗ for all j=1,2,…,m. If there exist nonnegative real-valued Lagrange function multipliers uj for j=1,2,…,m defined on 0,1 such that (1)
M∇~Ft∗α+∑j=1mujα∇gjt∗=0 for all α∈0,1,(2)
ujαgjt∗=0 for all α∈0,1 and j=1,2,…,m,then t∗is a nondominated solution of problem (24).Proof.
Suppose that conditions (1) and (2) are satisfied andt∗ is not a nondominated solution of problem (24). Then there exists a t-∈Ω such that Ft-≺Ft∗; that is, for some α∗∈0,1 we have that MFt-α∗<MFt∗α∗. We now define a real-valued function f by ft=MFtα∗. Then we have (25)ft-<ft∗.Since the fuzzy mapping F is convex on Rn and continuously differentiable at t∗, by Theorem 24 and Lemma 16 we see that f is also convex on Rn and continuously differentiable at t∗. Furthermore, we have ∇ft=∇MFtα∗=M∇~Ftα∗. Since conditions (1) and (2) are satisfied, we can obtain the following two new conditions for any fixed α∗∈0,1:(1′)
∇ft∗+∑j=1mujα∗·∇gjt∗=0;(2′)
ujα∗·gjt∗=0 for all j=1,2,…,m,where ujα∗=ujα∗≥0 for j=1,2,…,m. Now we consider the following constrained optimization problem: (26)minft=ft1,t2,…,tn,subjecttogjt≤0,j=1,2,…,mwhich has the same constraints of problem (24). By Theorem 25, conditions 1′ and 2′ are the KKT conditions of problem (26). Therefore, we have that t∗ is an optimal solution of problem (26) with the real-valued objective function f; that is, ft∗≤ft for all t∈Ω, which contradicts inequality (25). Then we get that t∗ is indeed a nondominated solution of problem (24).Theorem 27.
LetΩ=t∈Rn:gjt≤0,j=1,2,…,m be the convex feasible set and t∗∈Ω be a feasible solution of problem (24). Suppose that the fuzzy-valued objective function F and real-valued constraint functions gj are convex on Rn and continuously differentiable at t∗ for all j=1,2,…,m. If there exist nonnegative fuzzy number equivalent class Lagrange multipliers x~j∈F/φ for j=1,2,…,m such that (1)
∇~Ft∗+∑j=1mx~j·∇gjt∗=0~,(2)
x~j·gjt∗=0~ for all j=1,2,…,m,then t∗ 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′)
M∇~Ft∗α+∑j=1mMx~jα·∇gjt∗=0 for all α∈0,1.(2′)
Mx~jα·gjt∗=0 for all α∈0,1 and j=1,2,…,m.Since the fuzzy number equivalence classes x~j are nonnegative for all j=1,2,…,m, then we can get that Mx~j are nonnegative real-valued functions defined on 0,1 for all j=1,2,…,m. So, 1′ and 2′ verify the KKT optimality conditions (1) and (2) of Theorem 26, respectively. Therefore, we get that t∗ is a nondominated solution of problem (24).Lemma 28 (see [28]).
LetΩ=t∈Rn:gjt≤0,j=1,2,…,m be a feasible set and t∗∈Ω. Assume that gj are differentiable at t∗ for all j=1,2,…,m. Let J=j:gjt=0 be the index set for the active constraints. Then we have (27)D⊆d∈Rn:∇gjt∗Td≤0∀j∈J,where D is the cone of feasible directions of Ω at t∗ defined by (28)D=d∈Rn:d≠0,there  exists  aδ>0such  thatt∗+ηd∈Ω∀η∈0,δ.Lemma 29 (see [28]).
LetA and C be two matrices. Exactly one of the following systems has a solution: System I:Ax≤0,Ax≠0,Cx≤0 for some x∈Rn.System II:ATλ+CTu=0 for some λ,u,λ>0,u≥0.Theorem 30.
LetΩ=t∈Rn:gjt≤0,j=1,2,…,m be the convex feasible set and t∗∈Ω be a feasible solution of problem (24). Suppose that the fuzzy-valued objective function F is differentiable and strictly pseudoconvex on Ω, and the real-valued constraint functions gj are convex on Rn and continuously differentiable at t∗ for all j=1,2,…,m. If there exist a α∗∈0,1 and nonnegative Lagrange multipliers uj∈R for j=1,2,…,m such that (1)
M∇~Ft∗α∗+∑j=1muj·∇gjt∗=0,(2)
uj·gjt∗=0 for all j=1,2,…,m,then t∗ is a strongly nondominated solution of problem (24).Proof.
Suppose that conditions (1) and (2) are satisfied andt∗ is not a strongly nondominated solution of problem (24). Then there exists a t-∈Ω with t-≠t∗ such that Ft-⪯Ft∗. Since F is differentiable and strictly pseudoconvex on Ω, we have (29)∇~Ft∗Tt--t∗≺0~;that is,(30)M∇~Ft∗α∗Tt--t∗<0.Let d=t--t∗. Since Ω is a convex set and t-,t∗∈Ω, we have(31)t∗+ηd=t∗+ηt--t∗=ηt-+1-ηt∗∈Ωfor any η∈0,1. By Lemma 28 we get that d∈D, which means that (32)∇gjt∗Td≤0∀i∈J,where D is the cone of feasible directions of Ω at t∗ and J=j:gjt=0 is the index set for the active constraints. Now let A=M∇~Ft∗α∗T and C be the matrix whose rows are ∇gjt∗T for j∈J. We consider the following two systems: System I:Ax≤0,Ax≠0,Cx≤0 for some x∈Rn.System II:ATλ+CTu=0 for some λ,u,λ>0,u≥0.Then by (30) and (32) we get that System I has a solution d=t--t∗. Further, by Lemma 29 System II has no solutions, which means that there exist no multipliers 0<λ∈R and 0≤uj∈R for j∈J such that (33)λM∇~Ft∗α∗+∑j∈Juj·∇gjt∗=0.Since λ>0, dividing (33) by λ and denoting ηj=uj/λ for j∈J, we have that (34)M∇~Ft∗α∗+∑j∈Jηj·∇gjt∗=0.Since J is the index set for the active constraints, we have gjt∗<0 for j∉J. Further, if uj·gjt∗=0 for all j=1,2,…,m, we can get that ηj=0 for j∉J; that is, (35)∑j∈Jηj·∇gjt∗=∑j=1mηj·∇gjt∗.From (34) and (35), there exist no multipliers 0≤ηj∈R for j=1,2,…,m such that (1′)
M∇~Ft∗α∗+∑j=1mηj∇gjt∗=0,(2′)
uj·gjt∗=0 for all j=1,2,…,m,which contradicts conditions (1) and (2) for the existence of multipliers 0≤uj∈R for j=1,2,…,m. Hence, we have that t∗ is indeed a strongly nondominated solution of problem (24).Example 31.
Define a fuzzy mappingF:R3→F/φ by (36)Ft=a~Tt+t2=∑i=13a~iti+ti2=a~1t1+a~2t2+a~3t3+t12+t22+t32for all t=t1,t2,t3T∈R3, where a~=a~1,a~2,a~3T∈F/φ3 and we define a~i by the level sets of its Mareš core a^1α=-6,-12α+6, a^2α=-1,-2α+1, and a^3α=-4,-8α+4 for all α∈0,1 and i=1,2,3, respectively. Thus, we have (37)MFt1,t2,t3α=-6αt1+t12-αt2+t22-4αt3+t32=α-6t1-t2-4t3+t12+t22+t32for all α∈0,1 and t=t1,t2,t3T∈R3. It is obvious that MFtα is continuous from the right at 0 and continuous from the left on 0,1 with respect to α. Now we consider the following optimization problem:(38)minFt=Ft1,t2,t3,subjecttog1t1,t2,t3=4t1-t2+2t3-8≤0,g2t1,t2,t3=3t1+2t2-t3-1≤0,gjt1,t2,t3=-tj-2≤0,forj=3,4,5,gjt1,t2,t3=tj-5-2≤0,forj=6,7,8.It is obvious that the constraint functions gj are convex on R3 for all j=1,2,…,8, and then we know that the feasible set Ω=t∈R3:gjt≤0,j=1,2,…,8 is convex. Since MFtα is decreasing with respect to α for all t∈Ω, we get that (39)V01MFt=MFt1-MFt0=6t1+t2+4t3≤6t1+t2+4t3≤22.Thus, we find that MFtα is of bounded variation with respect to α for all t=t1,t2,t3T∈Ω⊆R3. It is easy to verify that F is differentiable and strictly pseudoconvex on Ω, and gj are convex on R3 and continuously differentiable at t∗ for all j=1,2,…,8. Then we obtain (40)M∇~Ft1,t2,t3α=2t1-6α,2t2-α,2t3-4αT,∇g1t1,t2,t3=4,-1,2T,∇g2t1,t2,t3=3,2,-1T,∇g3t1,t2,t3=-1,0,0T,∇g4t1,t2,t3=0,-1,0T,∇g5t1,t2,t3=0,0,-1T,∇g6t1,t2,t3=1,0,0T,∇g7t1,t2,t3=0,1,0T,∇g8t1,t2,t3=0,0,1T,for all α∈0,1 and t=t1,t2,t3T∈Ω. Now we consider the point t∗=t1∗,t2∗,t3∗T=1,0,2T∈Ω. Since (41)g3t∗≠0,g5t∗≠0,g6t∗≠0,g7t∗≠0,from condition (2) in Theorem 30, we get that (42)u3=u5=u6=u7=0.Now, applying condition (2) of Theorem 30 at the point t∗, we obtain (43)M∇~Ft∗α∗+∑j=18uj·∇gjt∗=2-6α∗+4u1+3u2-α∗-u1+2u2-u44-4α∗+2u1-u2+u8=0.After these algebraic calculations, we obtain that there exist a α∗=1∈0,1 and nonnegative Lagrange multipliers (44)u1=14,u2=1,u4=34,u8=12,uj=0,j=3,5,6,7,which satisfied conditions (1) and (2) of Theorem 30. Hence, we get that t∗=t1∗,t2∗,t3∗T=1,0,2T∈Ω 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.



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