# Extended Duality in Fuzzy Optimization Problems

Authors: Tingting Zou
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Duality theorem is an attractive approach for solving fuzzy optimization problems. However, the duality gap is generally nonzero for nonconvex problems. So far, most of the studies focus on continuous variables in fuzzy optimization problems. And, in real problems and models, fuzzy optimization problems also involve discrete and mixed variables. To address the above problems, we improve the extended duality theory by adding fuzzy objective functions. In this paper, we first define continuous fuzzy nonlinear programming problems, discrete fuzzy nonlinear programming problems, and mixed fuzzy nonlinear programming problems and then provide the extended dual problems, respectively. Finally we prove the weak and strong extended duality theorems, and the results show no duality gap between the original problem and extended dual problem.


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## Body

## 1. Introduction

Nonlinear programming problems (NLPs) play an important role in both manufacturing systems and industrial processes and have been widely used in the fields of operations research, planning and scheduling, optimal control, engineering designs, and production management [1–4]. Due to its significance in both academic and engineering applications, different kinds of approaches have been proposed to solve NLPs and obtained some achievements [5–9]. In [10], we present three algorithms using reverse bridge theorem (RBTH) for solving discrete nonlinear programming problems (DNLPs), continuous nonlinear programming problems (CNLPs), and mixed constrained nonlinear programming problems (MINLPs), respectively, and finally prove the soundness and completeness of these algorithms.In fact, many practical problems encountered by designers and decision makers would take place in an environment in which the statements might be vague or imprecise. Therefore, in 1970, Bellman and Zadeh first introduced fuzzy optimization problem, which combined the fuzzy decision and fuzzy goals [11]. Since then, there are many articles with regard to the fuzzy optimization problems [12–14]. In 2008, Wu proposed continuous and differentiable fuzzy-valued objective function with real constraints and presented the sufficient optimality conditions for obtaining the nondominated solution of fuzzy optimization problem [15]. Later, he adopted the Karush-Kuhn-Tucker optimality conditions to solve the fuzzy optimization problems [16]. Furthermore, Pathak and Pirzada presented the necessary and sufficient Kuhn-Tucker like optimality conditions for nonlinear fuzzy optimization problems with fuzzy-valued objective function and fuzzy-valued constraints [17]. Jameel and Sadeghi showed that the results solution of fuzzy optimization is a generalization of the solution of the crisp optimization problem [18]. Moreover, Baykasoğlu and Göçken gave the review of fuzzy mathematical programming models according to fuzzy components [19]. So far, most of the studies focus on treating continuous variables in fuzzy optimization problems. However, in real life problems and models, fuzzy optimization problems also involve discrete and mixed variables. Therefore, in this paper, we define continuous fuzzy nonlinear programming problems (CFNPs) with continuous variables, discrete fuzzy nonlinear programming problems (DFNPs) with discrete variables, and mixed fuzzy nonlinear programming problems (MFNPs) with continuous and discrete variables. Compared to previous formula of fuzzy optimization problems, the above three problems increase equality constraints of the variables.On the other hand, duality theorem has been proved to be an attractive approach for solving fuzzy optimization problems recently [20–27]. The most important aspect of duality is the existence of the duality gap, which is the difference between the optimal solution by solving the original problem and the lower bound of the dual problem. However, for nonconvex problems, the duality gap is generally nonzero and may be large value for some problems. Thus, the duality approach cannot be directly used for solving fuzzy optimization problems with nonconvex functions [28, 29]. Recently, Y. Chen and M. Chen proposed an extended duality theory for nonlinear optimization and proved that there was zero duality gap for general nonconvex optimization problems [30]. To deal with nonconvex fuzzy optimization problems with continuous, discrete, and mixed variable, we improve the extended duality theory by adding fuzzy objective functions. In this paper, we define extended duality theory of fuzzy nonlinear optimization with continuous, discrete, and mixed spaces and prove the weak and strong extended duality theorems, and the results show no duality gap between the original problem and extended dual fuzzy optimization problems.The remainder of this paper is organized as follows. After an introduction, we recall some basic notions and work related to fuzzy optimization problems in Section2. Then in Section 3, we define fuzzy nonlinear programming problems in continuous, discrete, and mixed spaces and the extended dual problem, respectively. In Section 4, we prove the weak and strong extended duality theorems. Last section is the conclusion of the paper.


## 2. Related Previous Works

In this section, we recall some basic definitions and work related to fuzzy optimization problems. LetU be a universal set. A fuzzy subset c~ of U is a mapping μc~:U→[0,1]. The α-level of c~ denoted by c~α is defined by c~α={x∈U:μc~(x)≥α} for all α∈(0,1]. The 0-level set c~0 is defined as the closure of the set {x∈U:μc~(x)>0}.Definition 1.
We denote byF(U) the set of all fuzzy subset c~ of U with membership function μc~ satisfying the following conditions:(1)
c~ is normal; that is, there exists an x∈U such that μc~(x)=1;(2)
μc~ is quasi concave; that is, μc~(λx+(1-λ)y)≥min⁡{μc~(x),μc~(y)} for all λ∈[0,1];(3)
μc~ is upper semicontinuous; that is, {x:μc~(x)≥α} is a closed subset of U for all α∈[0,1];(4)
the 0-level setc~0 is a compact subset of U.
Throughout this paper, the universal setU is the set of all real number R. The member c~ in F(R) is called a fuzzy number. For all α∈[0,1], we can denote the α-level of c~ by c~α=[c~αL,c~αU].
Letx~∈FnR≡FR×⋯×F(R); that is, x~=(x~1,…,x~n), where x~i∈F(R) for i=1,…,n. We also write x~αL=(x~1αL,…,x~nαL) and x~αU=(x~1αU,…,x~nαU), where x~iαL≡(x~i)αL and x~iαU≡(x~i)αU for all i=1,…,n.Definition 2.
The fuzzy scalar product of the fuzzy vectorsx~ and y~ in Fn(R) is defined by (1)x~,y~=x~1⊗y~1⊕⋯⊕x~n⊗y~n.
Letc~,d~∈F(R). We write d~≻_c~ if and only if d~αL≥c~αL and d~αU≥c~αU for all α∈[0,1]. The relation “≻_” on F(R) is a partial ordering.Definition 3.
LetA and B be two subsets of F(R). We write A≺_x~ if y~≺_x~ for all y~∈A. We write A≺_B if A≺_x~ for all x~∈B.
Letf~ and g~ be two fuzzy-valued functions defined on the same real vector space V, and let X be a subset of V. Then, (2)MIN⁡(f~,X)=f~x′:thereexistsnox∈XMIN⁡f~,X=1f~x′:suchthatf~x′≻f~x,ARG-MINf~x′,X=x∈X:f~x≻_f~x′,MAXg~,X=g~x′:thereexistsnox∈XMAXg~,X=1g~x′:suchthatg~x′≺g~x,ARG-MAXg~x′,X=x∈X:g~x≺_g~x′.Definition 4.
The primal fuzzy optimization problem(P) is defined as follows:(3)Pminimizef~x,subjecttog~ix≤0~,fori=1,…,m,x∈X.
The fuzzy-valued Lagrangian function for the primal problem(P) is defined as follows:(4)ϕ~(x,u)=f~(x)⊕u,g~(x)for all x∈X and all u=(u1,…,um)∈R+m; that is, ui≥0 for all i=1,…,m. We also write u≥0 if u∈R+m.Definition 5.
The dual fuzzy optimization problem(D) is defined as follows:(5)DmaximizeL~(u),subjecttou≥0,where the fuzzy-valued Lagrangian dual function is defined as(6)L~u=MIN⁡(ϕ~(·,u),X)=ϕ~x~,u:thereexistsnox∈Xlllllllllllllllllllsuchthatϕ~x~,u≻ϕ~x,u.


## 3. Extend Duality Problems

In this section, we define continuous fuzzy nonlinear problem, discrete fuzzy nonlinear problem and mixed fuzzy nonlinear problem, and the extended dual problems, respectively. Letf~,  h~i(i=1,…,m) and g~j(j=1,…,r) be fuzzy-valued functions defined on the same real vector space V, and let X, Y be two subsets of V.


### 3.1. Continuous Fuzzy Nonlinear Programming Problems

Definition 6.
A continuous fuzzy nonlinear programming problem (CFNP) is defined as(7)(Pc)min⁡xf~(x)x=x1,x2,…,xnT∈X,Subjecttoh~x=h~1x,…,h~mx=0~,llllllllllllllllllg~x=g~1x,…,g~rx≺_0~,where x is a continuous variable and f~ is a continuous and differentiable fuzzy-valued function.Definition 7.
Pointx* is a solution of Pc, if x* is a feasible solution of Pc and there exists no feasible solution x∈X such that f~(x*)≻f~(x).
LetX′={x∈X:h~(x)=0~,g~(x)≺_0~} be the feasible set of Pc and OPMc(f~,h~,g~,X) be the set of all solutions of Pc; then(8)MINc⁡f~,h~,g~,X=MINc(f~,X′)MINc⁡f~,h~,g~,X=f~x*:thereexistsnox∈X′MINc⁡f~,h~,g~,X=1f~x*:suchthatf~x*≻f~x,ARG-MINcf~x*,h~,g~,X=ARG-MINcf~x*,X′=x∈X′:f~x≻_f~x*.Definition 8.
The fuzzy-valuedl1m-penalty function for Pc in (7) is defined as follows:(9)ϕ~cx,α,β=f~x⊕α,h~x⊕β,max⁡⁡0~,g~x,where h~(x)=(h~1x,…,h~mx) and max⁡(0~,g~(x))=(max⁡(0~,g~1(x)),…,max⁡(0~,g~r(x))), and α∈Rm and β∈Rr are penalty multipliers.According to the fuzzy-valuedl1m-penalty function for Pc, we define the fuzzy-valued extended dual function as follows.Definition 9.
The fuzzy-valued extended dual function forPc is defined for α∈Rm and β∈Rr as(10)φ~cα,β=MINc⁡(ϕ~c(·,α,β),X)=ϕ~cx*,α,β:thereexistsnox∈Xlllllllllllllllllllllllllsuchthatϕ~cx*,α,β≻ϕ~cx,α,β,where φ~c is a point-to-set fuzzy-valued extended dual function; that is, for any fixed α and β,  φ~c(α,β) is a subset of F(R).Definition 10.
The extended dual continuous fuzzy nonlinear programming problem (EDCFNP) is defined as follows:(11)EDcmaximizeφ~cα,β,Subjecttoα≥0,β≥0.Definition 11.
Point(α*,β*) is a solution of EDc, if there exists a f(x~)∈φ~c(α*,β*) such that f(x~)≻_φ~c(α,β) for all α≠α*, β≠β*, α≥0, and β≥0.LetOPMEDc(φ~c,R+m,R+r)  denote the set of all solutions of extended dual continuous fuzzy nonlinear programming problem  EDc, MAXEDc(φ~c,R+m,R+r)={φ~cα,β:(α,β)isasolutionofEDc}. If α and β are fixed, then ARG-MAXEDc(φ~c(x*,α,β),X)={x∈X:φ~c(x,α,β)≺_φ~c(x*,α,β)}.


### 3.2. Discrete Fuzzy Nonlinear Programming Problems

Definition 12.
A discrete fuzzy nonlinear programming problem (DFNP) is defined as(12)(Pd)min⁡yf~(y),y=y1,y2,…,ynT∈Y,Subjecttoh~y=h~1y,…,h~my=0~,lllllllllllllllllllg~y=g~1y,…,g~ry≺_0~,where y is a discrete variable.Definition 13.
Pointy* is a solution of Pd, if y* is a feasible solution of Pd and there exists no feasible solution y∈Y such that f~(y*)≻f~(y).
LetY′={y∈Y:h~(y)=0~,g~(y)≺_0~} be the feasible set and let OPMd(f~,h~,g~,Y) be the set of all solutions of Pd; then,(13)MINd⁡f~,h~,g~,Y=MINdf~,Y′llllllllllllllllllllllllll=f~y*:thereexistsnoy∈Y′llllllllllllllllllllllllllllllllllllllllllllsuchthatf~y*≻f~y,ARG-MINdf~y*,h~,g~,Y=ARG-MINdf~y*,Y′ARG-MINdf~y*,h~,g~,Y=y∈Y′:f~y≻_f~y*.Definition 14.
The fuzzy-valuedl1m-penalty function for Pd in (12) is defined as follows:(14)ϕ~dy,α,β=f~y⊕α,h~y⊕β,max⁡⁡0~,g~y,where h~(y)=(h~1y,…,h~m(y)) and max⁡(0~,g~(y))=(max⁡(0~,g~1(y)),…,max⁡(0~,g~r(y))), and α∈Rm and β∈Rr are penalty multipliers.Definition 15.
The fuzzy-valued extended dual function forPd is defined for α∈Rm and β∈Rr as(15)φ~dα,β=MINd⁡ϕ~d·,α,β,Y=ϕ~dy*,α,β:thereexistsnoy∈Ylllllllllllllllllllllllllsuchthatϕ~dy*,α,β≻ϕ~dy,α,β.
Similarly,φ~d is a point-to-set fuzzy-valued extended dual function for Pd; that is, for any fixed α and β, φ~d(α,β) is a subset of F(R).Definition 16.
The extended dual discrete fuzzy nonlinear programming problem (EDDFNP) is defined as follows:(16)EDdmaximizeφ~dα,βSubjecttoα≥0,β≥0.Definition 17.
Point(α*,β*) is a solution of EDd, if there exists a f(y~)∈φ~d(α*,β*) such that f(y~)≻_φ~d(α,β) for all α≠α*, β≠β*, α≥0, and β≥0.
LetOPMEDd(φ~d,R+m,R+r) denote the set of all solutions of extended dual discrete fuzzy nonlinear programming problem EDd, MAXEDd(φ~d,R+m,R+r)={φ~d(α,β): (α,β)isasolutionofEDd}. If α and β are fixed, then(17)ARG-MAXEDd⁡φ~dy*,α,β,Y=y∈Y:φ~dy,α,β≺_φ~dy*,α,β.Definition 18.
A mixed fuzzy nonlinear programming problem (MFNP) is defined as(18)Pmmin⁡x,yf~x,y,x=x1,x2,…,xnT∈X,llllllllllllllllllllllllxisacontinuousvariablelllllllllllllllllllllllly=y1,y2,…,ynT∈Y,llllllllllllllllllllllllyisadiscretevaribleSubjecttoh~x,y=h~1x,y,…,h~mx,yllllllllllllllllllllllllllllll=0~,lllllllllllllllllllg~x,y=g~1x,y,…,g~rx,ylllllllllllllllllllllllllllllll≺_0~.Definition 19.
Point(x*,y*) is a solution of Pm, if (x*,y*) is a feasible solution of Pm and there exists no feasible solution (x,y)∈(X,Y) such that f~(x*,y*)≻f~(x,y).
Similarly,X,Y′={(x,y)∈(X,Y): h~(x,y)=0~, g~(x,y)≺_0~} denotes the feasible set of mixed fuzzy nonlinear programming problem Pm, OPMm(f~,h~,g~,(X,Y)) denotes the set of all solutions of mixed fuzzy nonlinear programming problem Pm, MINmf~,h~,g~,X,Y = MINmf~,X,Y′={f~x*,y*: thereexistsno(x,y)∈X,Y′ such that  f~(x*,y*)≻f~(x,y)},  and  ARG-MINm(f~(x*,y*),h~,g~,(X,Y)) = ARG-MINm(f~(x*,y*),X,Y′) = {(x,y)∈X,Y′:f~(x,y)≻_f~(x*,y*)}.Definition 20.
The fuzzy-valuedl1m-penalty function for Pm in (18) is defined as follows:(19)ϕ~mx,y,α,β=f~x,y⊕α,h~x,y⊕β,max⁡⁡0~,g~x,y,where h~(x,y)=(h~1x,y,…,h~mx,y) and max⁡(0~,g~(x,y))=(max⁡(0~,g~1(x,y)),…,max⁡(0~,g~r(x,y))), and α∈Rm and β∈Rr are penalty multipliers.Definition 21.
The fuzzy-valued extended dual function forPm is defined for α∈Rm and β∈Rr as(20)φ~mα,β=MINmϕ~m·,α,β,X,Y=ϕ~mx*,y*,α,β:thereexistsnox,y∈X,Yllllllllllllllllllllllllllllllllllsuchthatϕ~mx*,y*,α,βlllllllllllllllllllllllllllllllllllllllllllllllll≻ϕ~mx,y,α,β.φ~m is a point-to-set fuzzy-valued extended dual function for Pm; that is, for any fixed α and β, φ~m(α,β) is a subset of F(R).Definition 22.
The extended dual mixed fuzzy nonlinear programming problem (EDMFNP) is defined as follows:(21)EDmmaximizeφ~m(α,β)Subjecttoα≥0,β≥0.Definition 23.
Point(α*,β*) is a solution of EDm, if there exists a f(z~)∈φ~m(α*,β*) such that f(z~)≻_φ~m(α,β) for all α≠α*, β≠β*, α≥0, and β≥0.
LetOPMEDm(φ~m,R+m,R+r) denote the set of all solutions of extended dual mixed fuzzy nonlinear programming problem EDm, MAXEDm(φ~m,R+m,R+r)={φ~m(α,β):(α,β)isasolutionofEDm}. If α and β are fixed, then(22)ARG-MAXEDm⁡φ~mx*,y*,α,β,X,Y=x,y∈X,Y:φ~mx,y,α,β≺_φ~mx*,y*,α,β.


## 3.1. Continuous Fuzzy Nonlinear Programming Problems

Definition 6.
A continuous fuzzy nonlinear programming problem (CFNP) is defined as(7)(Pc)min⁡xf~(x)x=x1,x2,…,xnT∈X,Subjecttoh~x=h~1x,…,h~mx=0~,llllllllllllllllllg~x=g~1x,…,g~rx≺_0~,where x is a continuous variable and f~ is a continuous and differentiable fuzzy-valued function.Definition 7.
Pointx* is a solution of Pc, if x* is a feasible solution of Pc and there exists no feasible solution x∈X such that f~(x*)≻f~(x).
LetX′={x∈X:h~(x)=0~,g~(x)≺_0~} be the feasible set of Pc and OPMc(f~,h~,g~,X) be the set of all solutions of Pc; then(8)MINc⁡f~,h~,g~,X=MINc(f~,X′)MINc⁡f~,h~,g~,X=f~x*:thereexistsnox∈X′MINc⁡f~,h~,g~,X=1f~x*:suchthatf~x*≻f~x,ARG-MINcf~x*,h~,g~,X=ARG-MINcf~x*,X′=x∈X′:f~x≻_f~x*.Definition 8.
The fuzzy-valuedl1m-penalty function for Pc in (7) is defined as follows:(9)ϕ~cx,α,β=f~x⊕α,h~x⊕β,max⁡⁡0~,g~x,where h~(x)=(h~1x,…,h~mx) and max⁡(0~,g~(x))=(max⁡(0~,g~1(x)),…,max⁡(0~,g~r(x))), and α∈Rm and β∈Rr are penalty multipliers.According to the fuzzy-valuedl1m-penalty function for Pc, we define the fuzzy-valued extended dual function as follows.Definition 9.
The fuzzy-valued extended dual function forPc is defined for α∈Rm and β∈Rr as(10)φ~cα,β=MINc⁡(ϕ~c(·,α,β),X)=ϕ~cx*,α,β:thereexistsnox∈Xlllllllllllllllllllllllllsuchthatϕ~cx*,α,β≻ϕ~cx,α,β,where φ~c is a point-to-set fuzzy-valued extended dual function; that is, for any fixed α and β,  φ~c(α,β) is a subset of F(R).Definition 10.
The extended dual continuous fuzzy nonlinear programming problem (EDCFNP) is defined as follows:(11)EDcmaximizeφ~cα,β,Subjecttoα≥0,β≥0.Definition 11.
Point(α*,β*) is a solution of EDc, if there exists a f(x~)∈φ~c(α*,β*) such that f(x~)≻_φ~c(α,β) for all α≠α*, β≠β*, α≥0, and β≥0.LetOPMEDc(φ~c,R+m,R+r)  denote the set of all solutions of extended dual continuous fuzzy nonlinear programming problem  EDc, MAXEDc(φ~c,R+m,R+r)={φ~cα,β:(α,β)isasolutionofEDc}. If α and β are fixed, then ARG-MAXEDc(φ~c(x*,α,β),X)={x∈X:φ~c(x,α,β)≺_φ~c(x*,α,β)}.


## 3.2. Discrete Fuzzy Nonlinear Programming Problems

Definition 12.
A discrete fuzzy nonlinear programming problem (DFNP) is defined as(12)(Pd)min⁡yf~(y),y=y1,y2,…,ynT∈Y,Subjecttoh~y=h~1y,…,h~my=0~,lllllllllllllllllllg~y=g~1y,…,g~ry≺_0~,where y is a discrete variable.Definition 13.
Pointy* is a solution of Pd, if y* is a feasible solution of Pd and there exists no feasible solution y∈Y such that f~(y*)≻f~(y).
LetY′={y∈Y:h~(y)=0~,g~(y)≺_0~} be the feasible set and let OPMd(f~,h~,g~,Y) be the set of all solutions of Pd; then,(13)MINd⁡f~,h~,g~,Y=MINdf~,Y′llllllllllllllllllllllllll=f~y*:thereexistsnoy∈Y′llllllllllllllllllllllllllllllllllllllllllllsuchthatf~y*≻f~y,ARG-MINdf~y*,h~,g~,Y=ARG-MINdf~y*,Y′ARG-MINdf~y*,h~,g~,Y=y∈Y′:f~y≻_f~y*.Definition 14.
The fuzzy-valuedl1m-penalty function for Pd in (12) is defined as follows:(14)ϕ~dy,α,β=f~y⊕α,h~y⊕β,max⁡⁡0~,g~y,where h~(y)=(h~1y,…,h~m(y)) and max⁡(0~,g~(y))=(max⁡(0~,g~1(y)),…,max⁡(0~,g~r(y))), and α∈Rm and β∈Rr are penalty multipliers.Definition 15.
The fuzzy-valued extended dual function forPd is defined for α∈Rm and β∈Rr as(15)φ~dα,β=MINd⁡ϕ~d·,α,β,Y=ϕ~dy*,α,β:thereexistsnoy∈Ylllllllllllllllllllllllllsuchthatϕ~dy*,α,β≻ϕ~dy,α,β.
Similarly,φ~d is a point-to-set fuzzy-valued extended dual function for Pd; that is, for any fixed α and β, φ~d(α,β) is a subset of F(R).Definition 16.
The extended dual discrete fuzzy nonlinear programming problem (EDDFNP) is defined as follows:(16)EDdmaximizeφ~dα,βSubjecttoα≥0,β≥0.Definition 17.
Point(α*,β*) is a solution of EDd, if there exists a f(y~)∈φ~d(α*,β*) such that f(y~)≻_φ~d(α,β) for all α≠α*, β≠β*, α≥0, and β≥0.
LetOPMEDd(φ~d,R+m,R+r) denote the set of all solutions of extended dual discrete fuzzy nonlinear programming problem EDd, MAXEDd(φ~d,R+m,R+r)={φ~d(α,β): (α,β)isasolutionofEDd}. If α and β are fixed, then(17)ARG-MAXEDd⁡φ~dy*,α,β,Y=y∈Y:φ~dy,α,β≺_φ~dy*,α,β.Definition 18.
A mixed fuzzy nonlinear programming problem (MFNP) is defined as(18)Pmmin⁡x,yf~x,y,x=x1,x2,…,xnT∈X,llllllllllllllllllllllllxisacontinuousvariablelllllllllllllllllllllllly=y1,y2,…,ynT∈Y,llllllllllllllllllllllllyisadiscretevaribleSubjecttoh~x,y=h~1x,y,…,h~mx,yllllllllllllllllllllllllllllll=0~,lllllllllllllllllllg~x,y=g~1x,y,…,g~rx,ylllllllllllllllllllllllllllllll≺_0~.Definition 19.
Point(x*,y*) is a solution of Pm, if (x*,y*) is a feasible solution of Pm and there exists no feasible solution (x,y)∈(X,Y) such that f~(x*,y*)≻f~(x,y).
Similarly,X,Y′={(x,y)∈(X,Y): h~(x,y)=0~, g~(x,y)≺_0~} denotes the feasible set of mixed fuzzy nonlinear programming problem Pm, OPMm(f~,h~,g~,(X,Y)) denotes the set of all solutions of mixed fuzzy nonlinear programming problem Pm, MINmf~,h~,g~,X,Y = MINmf~,X,Y′={f~x*,y*: thereexistsno(x,y)∈X,Y′ such that  f~(x*,y*)≻f~(x,y)},  and  ARG-MINm(f~(x*,y*),h~,g~,(X,Y)) = ARG-MINm(f~(x*,y*),X,Y′) = {(x,y)∈X,Y′:f~(x,y)≻_f~(x*,y*)}.Definition 20.
The fuzzy-valuedl1m-penalty function for Pm in (18) is defined as follows:(19)ϕ~mx,y,α,β=f~x,y⊕α,h~x,y⊕β,max⁡⁡0~,g~x,y,where h~(x,y)=(h~1x,y,…,h~mx,y) and max⁡(0~,g~(x,y))=(max⁡(0~,g~1(x,y)),…,max⁡(0~,g~r(x,y))), and α∈Rm and β∈Rr are penalty multipliers.Definition 21.
The fuzzy-valued extended dual function forPm is defined for α∈Rm and β∈Rr as(20)φ~mα,β=MINmϕ~m·,α,β,X,Y=ϕ~mx*,y*,α,β:thereexistsnox,y∈X,Yllllllllllllllllllllllllllllllllllsuchthatϕ~mx*,y*,α,βlllllllllllllllllllllllllllllllllllllllllllllllll≻ϕ~mx,y,α,β.φ~m is a point-to-set fuzzy-valued extended dual function for Pm; that is, for any fixed α and β, φ~m(α,β) is a subset of F(R).Definition 22.
The extended dual mixed fuzzy nonlinear programming problem (EDMFNP) is defined as follows:(21)EDmmaximizeφ~m(α,β)Subjecttoα≥0,β≥0.Definition 23.
Point(α*,β*) is a solution of EDm, if there exists a f(z~)∈φ~m(α*,β*) such that f(z~)≻_φ~m(α,β) for all α≠α*, β≠β*, α≥0, and β≥0.
LetOPMEDm(φ~m,R+m,R+r) denote the set of all solutions of extended dual mixed fuzzy nonlinear programming problem EDm, MAXEDm(φ~m,R+m,R+r)={φ~m(α,β):(α,β)isasolutionofEDm}. If α and β are fixed, then(22)ARG-MAXEDm⁡φ~mx*,y*,α,β,X,Y=x,y∈X,Y:φ~mx,y,α,β≺_φ~mx*,y*,α,β.


## 4. Extended Duality Theorems

Duality theorem is an important approach for fuzzy optimization problems. However, the duality gap is generally nonzero for nonconvex fuzzy optimization problems. In this section, we prove the weak and strong extended duality theorems and show there is no duality gap between original problem and extended dual problem for fuzzy nonlinear problem with continuous or discrete, or mixed variables.


### 4.1. Extended Duality Theorem for CFNPs

Theorem 24.
Supposex and (α,β) are feasible solution of problem Pc and EDc, respectively; moreover(23)x∈⋂{x′∈X:ϕ~c(x′,α,β)∈φ~c(α,β)}ARG-MINc⁡ϕ~cx′,α,β,X.Then we have φ~c(α,β)≺_f~(x).Proof.
Letϕ~c(x′,α,β)∈φ~c(α,β). For any x∈ARG-MINc(ϕ~c(x′,α,β),X), we have (24)ϕ~cx′,α,β≺_ϕ~c(x,α,β)=f~x⊕α,h~x⊕β,max⁡⁡0~,g~x.Since x is a feasible solution of problem Pc, we obtain h~(x)=0~, max⁡(0~,g~(x))=0~. Thus, (25)ϕ~cx′,α,β≺_f~x.
This inequality is satisfied for allϕ~c(x′,α,β)∈φ~c(α,β). According to Definition 3, therefore we have (26)φ~cα,β≺_f~x.Theorem 25 (weak extended duality theorem for CFNPs).
Suppose that(27)OPMcf~,h~,g~,X⊆⋂{α∈Rm,β∈Rr:(α,β)∈OPMEDc(φ~c,R+m,R+r)}⋂{x′∈X:ϕ~c(x′,α,β)∈φ~c(α,β)}ARG-MINc⁡ϕ~cx′,α,β,X.Then MAXEDc(φ~c,R+m,R+r)≺_MINc(f~,h~,g~,X).Proof.
Ifx∈OPMc(f~,h~,g~,X), then f~(x)∈MINc(f~,h~,g~,X). According to Theorem 24, we have φ~c(α,β)≺_f~(x) if x satisfies formula (23). Therefore, from Definition 3, if(28)OPMc(f~,h~,g~,X)⊆⋂{x′∈X:ϕ~c(x′,α,β)∈φ~c(α,β)}ARG-MINcϕ~cx′,α,β,X,then(29)φ~cα,β≺_MINc⁡(f~,h~,g~,X).Moreover, if α,β∈OPMEDc(φ~c,R+m,R+r), then (30)φ~cα,β∈MAXEDc⁡φ~c,R+m,R+r.Therefore, according to Definition 3, we have (31)MAXEDcφ~c,R+m,R+r≺_MINcf~,h~,g~,X.Definition 26.
LetPc be a continuous fuzzy nonlinear programming problem and let EDc be an extended dual continuous fuzzy nonlinear programming problem. There is no duality gap between Pc and EDc if there exist f~(x*)∈MINc(f~,h~,g~,X) and φ~c(α*,β*)∈MAXEDc(φ~c,R+m,R+r), such that f~(x*)∈φ~c(α*,β*).Theorem 27.
Supposex*∈X is a solution of continuous fuzzy nonlinear programming problem Pc; then there exist finite α*≥0 and β*≥0 such that(32)f~x*∈MINcϕ~c·,α**,β**,X,llllllforanyα**>α*,β**>β*.Proof.
Sincex* is a solution of problem Pc, we have h~(x*)=0~, g~(x*)≺_0~, and there exists no x∈X such that f~(x*)≻f~(x). We set the following α* and β*:(33)αi*=max⁡x∈X,h~i(x)≻0~f~αL(x*)-f~αL(x)h~iαL(x),f~αU(x*)-f~αU(x)h~iαU(x),llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllli=1,…,m.βj*=max⁡x∈X,g~j(x)≻0~f~αL(x*)-f~αL(x)g~jαL(x),f~αU(x*)-f~αU(x)g~jαU(x),lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllj=1,…,r.Suppose X′ be the set of feasible solutions of Pc.(1)
For anyx∈X′, that is to say that x is a feasible solution of Pc, then h~(x)=0~, g~(x)≺_0~. Thus we have (34)ϕ~cx,α**,β**=f~(x)⊕α**,h~(x)⊕β**,max⁡(0~,g~(x))=f~x.Therefore there exists no x∈X′ such that f~(x*)≻f~(x)=ϕ~c(x,α**,β**).(2)
For anyx∈X but x∉X′, that is to say that x is an infeasible solution of Pc. Assume x violates an equality constraint h~i(·) (the case with an inequality constraint function is similar), so h~i(x)≠0~. We also have h~iαL(x)≠0 and h~iαU(x)≠0, for all α∈[0,1](35)ϕ~cx,α**,β**αL=0~,g~xf~x⊕α**,h~xlllllllllllll⊕β**,max⁡⁡0~,g~xαL=f~αL(x)+∑i=1mαi**×h~iαLx+∑j=1rβj**×max⁡0,g~iαLx≥f~αL(x)+αi**×h~iαL(x)>f~αLx+f~αLx*-f~αLxh~iαLx×h~iαLx=f~αLx*,ϕ~cx,α**,β**αU=0~,g~xf~x⊕α**,h~xlllllllllllll⊕β**,max⁡⁡0~,g~xαU=f~αU(x)+∑i=1mαi**×h~iαUx+∑j=1rβj**×max⁡0,g~iαUx≥f~αU(x)+αi**×h~iαU(x)>f~αUx+f~αUx*-f~αUxh~iαUx×h~iαUx=f~αUx*.Thus, (36)ϕ~c(x,α**,β**)=f~x⊕α**,h~x⊕β**,max⁡⁡0~,g~x≻f~x*.Therefore,(37)f~x*∈MINc⁡ϕ~c·,α**,β**,X.Theorem 28 (strong extended duality theorem for CFNPs).
Under the assumptions and results in Theorem27, we further assume(38)x*∈⋂{α∈Rm,β∈Rr:α≠α**,β≠β**}⋂{x∈X:ϕ~c(x,α,β)∈φ~c(α,β)}ARG-MINcϕ~cx,α,β,X.Then there is no duality gap between the problem Pc and EDc.Proof.
According to Theorem27, there exist finite α*≥0 and β*≥0 such that(39)f~x*∈MINcϕ~c·,α**,β**,X,lllllllforanyα**>α*,β**>β*.Then we have f~(x*)∈φ~c(α**,β**). From Theorem 24, we have f~(x*)≻_φ~c(α,β) if(40)x*∈⋂{x∈X:ϕ~c(x,α,β)∈φ~c(α,β)}ARG-MINcϕ~cx,α,β,X.Thus, according to the known condition, we have f~(x*)≻_φ~c(α,β) for all α≠α**, β≠β**, α≥0, and β≥0.
Therefore,(α**,β**) is a solution of extended dual continuous fuzzy nonlinear programming problem EDc; that is, φ~c(α**,β**)∈MAXEDc(φ~c,R+m,R+r). This shows that there is no duality gap between the problem Pc and EDc.


### 4.2. Extended Duality Theorem for DFNPs

Theorem 29.
Supposey and (α,β) are feasible solution of problem Pd and EDd, respectively; moreover(41)y∈⋂{y′∈Y:ϕ~d(y′,α,β)∈φ~d(α,β)}ARG-MINdϕ~dy′,α,β,Y.Then we have φ~d(α,β)≺_f~(y).Proof.
Letϕ~d(y′,α,β)∈φ~d(α,β). For any y∈ARG-MINd(ϕ~d(y′,α,β),Y), we have(42)ϕ~d(y′,α,β)≺_ϕ~d(y,α,β)=f~(y)⊕α,h~y⊕β,max⁡⁡0~,g~y.Since y is a feasible solution of problem Pd, we obtain h~(y)=0~, max⁡(0~,g~(y))=0~. Thus, (43)ϕ~dy′,α,β≺_f~y.This inequality is satisfied for all ϕ~d(y′,α,β)∈φ~d(α,β). According to Definition 3, therefore we have (44)φ~dα,β≺_f~y.Theorem 30 (weak extended duality theorem for DFNPs).
Suppose that(45)OPMdf~,h~,g~,Y⊆⋂{α∈Rm,β∈Rr:(α,β)∈OPMEDd(φ~d,R+m,R+r)}⋂{y′∈Y:ϕ~d(y′,α,β)∈φ~d(α,β)}ARG-MINdϕ~dy′,α,β,Y.Then, MAXEDdφ~d,R+m,R+r≺_MINd(f~,h~,g~,Y).Proof.
The proof is similar to the proof of Theorem25.Definition 31.
LetPd be a discrete fuzzy nonlinear programming problem and let EDd be an extended dual discrete fuzzy nonlinear programming problem. There is no duality gap between Pd and EDd if there exist f~(y*)∈MINd(f~,h~,g~,Y) and φ~d(α*,β*)∈MAXEDd(φ~d,R+m,R+r), such that f~(y*)∈φ~d(α*,β*).Theorem 32.
Supposey*∈Y is a solution of discrete fuzzy nonlinear programming problem Pd; then there exist finite α*≥0 and β*≥0 such that(46)f~y*∈MINdϕ~d·,α**,β**,Y,lllllllforanyα**>α*,β**>β*.The proof of Theorem 32 is similar to Theorem 27, so we do not repeat it again.Theorem 33 (strong extended duality theorem for DNLPs).
Under the assumptions and results in Theorem32, assume(47)y*∈⋂{α∈Rm,β∈Rr:α≠α**,β≠β**}⋂{y∈Y:ϕ~d(y,α,β)∈φ~d(α,β)}ARG-MINdϕ~dy,α,β,Y.Then there is no duality gap between the problem Pd and EDd.Proof.
The proof is similar to the proof of Theorem28.


### 4.3. Extended Duality Theorem for MFNPs

Theorem 34.
Suppose(x,y) and (α,β) are feasible solution of problem Pm and EDm, respectively; moreover(48)x,y∈⋂{(x′,y′)∈(X,Y):ϕ~m(x′,y′,α,β)∈φ~m(α,β)}ARG-MINmϕ~mx′,y′,α,β,X,Y.Then we have φ~m(α,β)≺_f~(x,y).Proof.
Letϕ~m(x′,y′,α,β)∈φ~m(α,β). For any (x,y)∈ARG-MINm(ϕ~m(x′,y′,α,β),(X,Y)), we have(49)ϕ~mx′,y′,α,β≺_ϕ~m(x,y,α,β)=f~(x,y)⊕α,h~x,y⊕β,max⁡⁡0~,g~x,y.Since (x,y) is a feasible solution of problem Pm, we obtain h~(x,y)=0~, max⁡(0~,g~(x,y))=0~. Thus, (50)ϕ~mx′,y′,α,β≺_f~x,y.This inequality is satisfied for all ϕ~m(x′,y′,α,β)∈φ~m(α,β). According to Definition 3, therefore we obtain(51)φ~mα,β≺_f~x,y.Theorem 35 (weak extended duality theorem for MFNPs).
Suppose that(52)OPMmf~,h~,g~,X,Y⊆⋂α∈Rm,β∈Rr:α,β∈OPMEDmφ~m,R+m,R+r⋂x′,y′∈X,Y:ϕ~mx′,y′,α,β∈φ~mα,βARG-MINmϕ~mx′,y′,α,β,X,Y.Then MAXEDm(φ~m,R+m,R+r)≺_MINm(f~,h~,g~,(X,Y)).Proof.
If(x,y)∈OPMm(f~,h~,g~,(X,Y)), then f~(x,y)∈MINm(f~,h~,g~,(X,Y)). According to Theorem 34, we have φ~m(α,β)≺_f~(x,y) if x satisfies formula (48). Therefore, from Definition 3, if(53)OPMmf~,h~,g~,X,Y⊆⋂{(x′,y′)∈(X,Y):ϕ~m(x′,y′,α,β)∈φ~m(α,β)}ARG-MINmϕ~mx′,y′,α,β,X,Y,then(54)φ~mα,β≺_MINmf~,h~,g~,X,Y.Moreover, if (α,β)∈OPMEDm(φ~m,R+m,R+r), then (55)φ~mα,β∈MAXEDm⁡φ~m,R+m,R+r.Therefore, according to Definition 3, we have (56)MAXEDmφ~m,R+m,R+r≺_MINmf~,h~,g~,X,Y.Definition 36.
LetPm be a mixed fuzzy nonlinear programming problem and let EDm be an extended dual mixed fuzzy nonlinear programming problem. There is no duality gap between Pm and EDm if there exist f~(x*,y*)∈MINm(f~,h~,g~,(X,Y)) and φ~m(α*,β*)∈MAXEDm(φ~m,R+m,R+r), such that f~(x*,y*)∈φ~m(α*,β*).Theorem 37.
Suppose(x*,y*)∈(X,Y) is a solution of continuous fuzzy nonlinear programming problem Pm; then there exist finite α*≥0 and β*≥0 such that(57)f~x*,y*∈MINmϕ~m·,α**,β**,X,Y,llllllforanyα**>α*,β**>β*.Proof.
Since(x*,y*) is a solution of problem Pm, we have h~(x*,y*)=0~,  g~(x*,y*)≺_0~, and there exist no (x,y)∈(X,Y) such that f~(x*,y*)≻f~(x,y). We set the following α* and β*:(58)αi*=max⁡(x,y)∈(X,Y),|h~i(x,y)|≻0~f~αL(x*,y*)-f~αL(x,y)h~iαL(x,y),llllllllllllllllllllllllllllllllf~αUx*,y*-f~αUx,yh~iαUx,y,llllllllllllllllllllllllllllllllllllllllllllllllllllllli=1,…,m.βj*=max⁡(x,y)∈(X,Y),g~j(x,y)≻0~f~αLx*,y*-f~αLx,yg~jαLx,y,llllllllllllllllllllllllllllllllf~αU(x*,y*)-f~αU(x,y)g~jαU(x,y),lllllllllllllllllllllllllllllllllllllllllllllllllllllllj=1,…,r.Suppose (X′,Y′) be the set of feasible solutions of Pm.(1)
For any(x,y)∈(X′,Y′), that is to say that (x,y) is a feasible solution of Pm, then h~(x,y)=0~, g~(x,y)≺_0~. Thus we have(59)ϕ~mx,y,α**,β**=f~x,y⊕α**,h~x,y⊕β**,max⁡0~,g~x,y=f~x,y.Therefore there exists no (x,y)∈(X′,Y′) such that f~(x*,y*)≻f~(x,y)=ϕ~m(x,y,α**,β**).(2)
For any(x,y)∈(X,Y) but (x,y)∉(X′,Y′), that is to say that (x,y) is an infeasible solution of Pm. Assume (x,y) violates an equality constraint h~i(·) (the case with an inequality constraint function is similar), so h~i(x,y)≠0~. We also have h~iαL(x,y)≠0 and h~iαU(x,y)≠0, for all α∈[0,1](60)ϕ~mx,y,α**,β**αL=0~,g~x,yf~x,y⊕α**,h~x,ylllllllllllll⊕β**,max⁡⁡0~,g~x,yαL=f~αL(x,y)+∑i=1mαi**×h~iαLx,y+∑j=1rβj**×max⁡0,g~iαLx,y≥f~αL(x,y)+αi**×h~iαL(x,y)>f~αL(x,y)+f~αLx*,y*-f~αLx,yh~iαLx,y×h~iαLx,y=f~αLx*,y*,ϕ~mx,y,α**,β**αU=0~,g~x,yf~x,y⊕α**,h~x,yllllllllllll⊕β**,max⁡⁡0~,g~x,yαU=f~αUx,y+∑i=1mαi**×h~iαUx,y+∑j=1rβj**×max⁡⁡0,g~iαUx,y≥f~αU(x,y)+αi**×h~iαU(x,y)>f~αU(x,y)+f~αUx*,y*-f~αUx,yh~iαUx,y×h~iαUx,y=f~αUx*,y*.Thus, (61)ϕ~mx,y,α**,β**=f~x,y⊕α**,h~x,y⊕β**,max⁡⁡0~,g~x,y≻f~x*,y*.Therefore,(62)f~x*,y*∈MINm⁡ϕ~m·,α**,β**,X,Y.Theorem 38 (strong extended duality theorem for MFNPs).
Under the assumptions and results in Theorem37, assume(63)x*,y*∈⋂{α∈Rm,β∈Rr:α≠α**,β≠β**}⋂{(x,y)∈(X,Y):ϕ~m(x,y,α,β)∈φ~m(α,β)}ARG-MINmϕ~mx,y,α,β,X,Y.Then there is no duality gap between the problem Pm and EDm.Proof.
According to Theorem37, there exist finite α*≥0 and β*≥0 such that (64)f~x*,y*∈MINmϕ~m·,α**,β**,X,Y,lllllforanyα**>α*,β**>β*.Then we have f~(x*,y*)∈φ~m(α**,β**). From Theorem 34, we have f~(x*,y*)≻_φ~m(α,β) if(65)x*,y*∈⋂{(x,y)∈(X,Y):ϕ~m(x,y,α,β)∈φ~m(α,β)}ARG-MINmϕ~mx,y,α,β,X,Y.Thus, according to the known condition, we have f~(x*,y*)≻_φ~m(α,β) for all α≠α**, β≠β**, α≥0, and β≥0.
Therefore,(α**,β**) is a solution of extended dual mixed fuzzy nonlinear programming problem EDm; that is, φ~m(α**,β**)∈MAXEDm(φ~m,R+m,R+r). This shows that there is no duality gap between the problem Pm and EDm.


## 4.1. Extended Duality Theorem for CFNPs

Theorem 24.
Supposex and (α,β) are feasible solution of problem Pc and EDc, respectively; moreover(23)x∈⋂{x′∈X:ϕ~c(x′,α,β)∈φ~c(α,β)}ARG-MINc⁡ϕ~cx′,α,β,X.Then we have φ~c(α,β)≺_f~(x).Proof.
Letϕ~c(x′,α,β)∈φ~c(α,β). For any x∈ARG-MINc(ϕ~c(x′,α,β),X), we have (24)ϕ~cx′,α,β≺_ϕ~c(x,α,β)=f~x⊕α,h~x⊕β,max⁡⁡0~,g~x.Since x is a feasible solution of problem Pc, we obtain h~(x)=0~, max⁡(0~,g~(x))=0~. Thus, (25)ϕ~cx′,α,β≺_f~x.
This inequality is satisfied for allϕ~c(x′,α,β)∈φ~c(α,β). According to Definition 3, therefore we have (26)φ~cα,β≺_f~x.Theorem 25 (weak extended duality theorem for CFNPs).
Suppose that(27)OPMcf~,h~,g~,X⊆⋂{α∈Rm,β∈Rr:(α,β)∈OPMEDc(φ~c,R+m,R+r)}⋂{x′∈X:ϕ~c(x′,α,β)∈φ~c(α,β)}ARG-MINc⁡ϕ~cx′,α,β,X.Then MAXEDc(φ~c,R+m,R+r)≺_MINc(f~,h~,g~,X).Proof.
Ifx∈OPMc(f~,h~,g~,X), then f~(x)∈MINc(f~,h~,g~,X). According to Theorem 24, we have φ~c(α,β)≺_f~(x) if x satisfies formula (23). Therefore, from Definition 3, if(28)OPMc(f~,h~,g~,X)⊆⋂{x′∈X:ϕ~c(x′,α,β)∈φ~c(α,β)}ARG-MINcϕ~cx′,α,β,X,then(29)φ~cα,β≺_MINc⁡(f~,h~,g~,X).Moreover, if α,β∈OPMEDc(φ~c,R+m,R+r), then (30)φ~cα,β∈MAXEDc⁡φ~c,R+m,R+r.Therefore, according to Definition 3, we have (31)MAXEDcφ~c,R+m,R+r≺_MINcf~,h~,g~,X.Definition 26.
LetPc be a continuous fuzzy nonlinear programming problem and let EDc be an extended dual continuous fuzzy nonlinear programming problem. There is no duality gap between Pc and EDc if there exist f~(x*)∈MINc(f~,h~,g~,X) and φ~c(α*,β*)∈MAXEDc(φ~c,R+m,R+r), such that f~(x*)∈φ~c(α*,β*).Theorem 27.
Supposex*∈X is a solution of continuous fuzzy nonlinear programming problem Pc; then there exist finite α*≥0 and β*≥0 such that(32)f~x*∈MINcϕ~c·,α**,β**,X,llllllforanyα**>α*,β**>β*.Proof.
Sincex* is a solution of problem Pc, we have h~(x*)=0~, g~(x*)≺_0~, and there exists no x∈X such that f~(x*)≻f~(x). We set the following α* and β*:(33)αi*=max⁡x∈X,h~i(x)≻0~f~αL(x*)-f~αL(x)h~iαL(x),f~αU(x*)-f~αU(x)h~iαU(x),llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllli=1,…,m.βj*=max⁡x∈X,g~j(x)≻0~f~αL(x*)-f~αL(x)g~jαL(x),f~αU(x*)-f~αU(x)g~jαU(x),lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllj=1,…,r.Suppose X′ be the set of feasible solutions of Pc.(1)
For anyx∈X′, that is to say that x is a feasible solution of Pc, then h~(x)=0~, g~(x)≺_0~. Thus we have (34)ϕ~cx,α**,β**=f~(x)⊕α**,h~(x)⊕β**,max⁡(0~,g~(x))=f~x.Therefore there exists no x∈X′ such that f~(x*)≻f~(x)=ϕ~c(x,α**,β**).(2)
For anyx∈X but x∉X′, that is to say that x is an infeasible solution of Pc. Assume x violates an equality constraint h~i(·) (the case with an inequality constraint function is similar), so h~i(x)≠0~. We also have h~iαL(x)≠0 and h~iαU(x)≠0, for all α∈[0,1](35)ϕ~cx,α**,β**αL=0~,g~xf~x⊕α**,h~xlllllllllllll⊕β**,max⁡⁡0~,g~xαL=f~αL(x)+∑i=1mαi**×h~iαLx+∑j=1rβj**×max⁡0,g~iαLx≥f~αL(x)+αi**×h~iαL(x)>f~αLx+f~αLx*-f~αLxh~iαLx×h~iαLx=f~αLx*,ϕ~cx,α**,β**αU=0~,g~xf~x⊕α**,h~xlllllllllllll⊕β**,max⁡⁡0~,g~xαU=f~αU(x)+∑i=1mαi**×h~iαUx+∑j=1rβj**×max⁡0,g~iαUx≥f~αU(x)+αi**×h~iαU(x)>f~αUx+f~αUx*-f~αUxh~iαUx×h~iαUx=f~αUx*.Thus, (36)ϕ~c(x,α**,β**)=f~x⊕α**,h~x⊕β**,max⁡⁡0~,g~x≻f~x*.Therefore,(37)f~x*∈MINc⁡ϕ~c·,α**,β**,X.Theorem 28 (strong extended duality theorem for CFNPs).
Under the assumptions and results in Theorem27, we further assume(38)x*∈⋂{α∈Rm,β∈Rr:α≠α**,β≠β**}⋂{x∈X:ϕ~c(x,α,β)∈φ~c(α,β)}ARG-MINcϕ~cx,α,β,X.Then there is no duality gap between the problem Pc and EDc.Proof.
According to Theorem27, there exist finite α*≥0 and β*≥0 such that(39)f~x*∈MINcϕ~c·,α**,β**,X,lllllllforanyα**>α*,β**>β*.Then we have f~(x*)∈φ~c(α**,β**). From Theorem 24, we have f~(x*)≻_φ~c(α,β) if(40)x*∈⋂{x∈X:ϕ~c(x,α,β)∈φ~c(α,β)}ARG-MINcϕ~cx,α,β,X.Thus, according to the known condition, we have f~(x*)≻_φ~c(α,β) for all α≠α**, β≠β**, α≥0, and β≥0.
Therefore,(α**,β**) is a solution of extended dual continuous fuzzy nonlinear programming problem EDc; that is, φ~c(α**,β**)∈MAXEDc(φ~c,R+m,R+r). This shows that there is no duality gap between the problem Pc and EDc.


## 4.2. Extended Duality Theorem for DFNPs

Theorem 29.
Supposey and (α,β) are feasible solution of problem Pd and EDd, respectively; moreover(41)y∈⋂{y′∈Y:ϕ~d(y′,α,β)∈φ~d(α,β)}ARG-MINdϕ~dy′,α,β,Y.Then we have φ~d(α,β)≺_f~(y).Proof.
Letϕ~d(y′,α,β)∈φ~d(α,β). For any y∈ARG-MINd(ϕ~d(y′,α,β),Y), we have(42)ϕ~d(y′,α,β)≺_ϕ~d(y,α,β)=f~(y)⊕α,h~y⊕β,max⁡⁡0~,g~y.Since y is a feasible solution of problem Pd, we obtain h~(y)=0~, max⁡(0~,g~(y))=0~. Thus, (43)ϕ~dy′,α,β≺_f~y.This inequality is satisfied for all ϕ~d(y′,α,β)∈φ~d(α,β). According to Definition 3, therefore we have (44)φ~dα,β≺_f~y.Theorem 30 (weak extended duality theorem for DFNPs).
Suppose that(45)OPMdf~,h~,g~,Y⊆⋂{α∈Rm,β∈Rr:(α,β)∈OPMEDd(φ~d,R+m,R+r)}⋂{y′∈Y:ϕ~d(y′,α,β)∈φ~d(α,β)}ARG-MINdϕ~dy′,α,β,Y.Then, MAXEDdφ~d,R+m,R+r≺_MINd(f~,h~,g~,Y).Proof.
The proof is similar to the proof of Theorem25.Definition 31.
LetPd be a discrete fuzzy nonlinear programming problem and let EDd be an extended dual discrete fuzzy nonlinear programming problem. There is no duality gap between Pd and EDd if there exist f~(y*)∈MINd(f~,h~,g~,Y) and φ~d(α*,β*)∈MAXEDd(φ~d,R+m,R+r), such that f~(y*)∈φ~d(α*,β*).Theorem 32.
Supposey*∈Y is a solution of discrete fuzzy nonlinear programming problem Pd; then there exist finite α*≥0 and β*≥0 such that(46)f~y*∈MINdϕ~d·,α**,β**,Y,lllllllforanyα**>α*,β**>β*.The proof of Theorem 32 is similar to Theorem 27, so we do not repeat it again.Theorem 33 (strong extended duality theorem for DNLPs).
Under the assumptions and results in Theorem32, assume(47)y*∈⋂{α∈Rm,β∈Rr:α≠α**,β≠β**}⋂{y∈Y:ϕ~d(y,α,β)∈φ~d(α,β)}ARG-MINdϕ~dy,α,β,Y.Then there is no duality gap between the problem Pd and EDd.Proof.
The proof is similar to the proof of Theorem28.


## 4.3. Extended Duality Theorem for MFNPs

Theorem 34.
Suppose(x,y) and (α,β) are feasible solution of problem Pm and EDm, respectively; moreover(48)x,y∈⋂{(x′,y′)∈(X,Y):ϕ~m(x′,y′,α,β)∈φ~m(α,β)}ARG-MINmϕ~mx′,y′,α,β,X,Y.Then we have φ~m(α,β)≺_f~(x,y).Proof.
Letϕ~m(x′,y′,α,β)∈φ~m(α,β). For any (x,y)∈ARG-MINm(ϕ~m(x′,y′,α,β),(X,Y)), we have(49)ϕ~mx′,y′,α,β≺_ϕ~m(x,y,α,β)=f~(x,y)⊕α,h~x,y⊕β,max⁡⁡0~,g~x,y.Since (x,y) is a feasible solution of problem Pm, we obtain h~(x,y)=0~, max⁡(0~,g~(x,y))=0~. Thus, (50)ϕ~mx′,y′,α,β≺_f~x,y.This inequality is satisfied for all ϕ~m(x′,y′,α,β)∈φ~m(α,β). According to Definition 3, therefore we obtain(51)φ~mα,β≺_f~x,y.Theorem 35 (weak extended duality theorem for MFNPs).
Suppose that(52)OPMmf~,h~,g~,X,Y⊆⋂α∈Rm,β∈Rr:α,β∈OPMEDmφ~m,R+m,R+r⋂x′,y′∈X,Y:ϕ~mx′,y′,α,β∈φ~mα,βARG-MINmϕ~mx′,y′,α,β,X,Y.Then MAXEDm(φ~m,R+m,R+r)≺_MINm(f~,h~,g~,(X,Y)).Proof.
If(x,y)∈OPMm(f~,h~,g~,(X,Y)), then f~(x,y)∈MINm(f~,h~,g~,(X,Y)). According to Theorem 34, we have φ~m(α,β)≺_f~(x,y) if x satisfies formula (48). Therefore, from Definition 3, if(53)OPMmf~,h~,g~,X,Y⊆⋂{(x′,y′)∈(X,Y):ϕ~m(x′,y′,α,β)∈φ~m(α,β)}ARG-MINmϕ~mx′,y′,α,β,X,Y,then(54)φ~mα,β≺_MINmf~,h~,g~,X,Y.Moreover, if (α,β)∈OPMEDm(φ~m,R+m,R+r), then (55)φ~mα,β∈MAXEDm⁡φ~m,R+m,R+r.Therefore, according to Definition 3, we have (56)MAXEDmφ~m,R+m,R+r≺_MINmf~,h~,g~,X,Y.Definition 36.
LetPm be a mixed fuzzy nonlinear programming problem and let EDm be an extended dual mixed fuzzy nonlinear programming problem. There is no duality gap between Pm and EDm if there exist f~(x*,y*)∈MINm(f~,h~,g~,(X,Y)) and φ~m(α*,β*)∈MAXEDm(φ~m,R+m,R+r), such that f~(x*,y*)∈φ~m(α*,β*).Theorem 37.
Suppose(x*,y*)∈(X,Y) is a solution of continuous fuzzy nonlinear programming problem Pm; then there exist finite α*≥0 and β*≥0 such that(57)f~x*,y*∈MINmϕ~m·,α**,β**,X,Y,llllllforanyα**>α*,β**>β*.Proof.
Since(x*,y*) is a solution of problem Pm, we have h~(x*,y*)=0~,  g~(x*,y*)≺_0~, and there exist no (x,y)∈(X,Y) such that f~(x*,y*)≻f~(x,y). We set the following α* and β*:(58)αi*=max⁡(x,y)∈(X,Y),|h~i(x,y)|≻0~f~αL(x*,y*)-f~αL(x,y)h~iαL(x,y),llllllllllllllllllllllllllllllllf~αUx*,y*-f~αUx,yh~iαUx,y,llllllllllllllllllllllllllllllllllllllllllllllllllllllli=1,…,m.βj*=max⁡(x,y)∈(X,Y),g~j(x,y)≻0~f~αLx*,y*-f~αLx,yg~jαLx,y,llllllllllllllllllllllllllllllllf~αU(x*,y*)-f~αU(x,y)g~jαU(x,y),lllllllllllllllllllllllllllllllllllllllllllllllllllllllj=1,…,r.Suppose (X′,Y′) be the set of feasible solutions of Pm.(1)
For any(x,y)∈(X′,Y′), that is to say that (x,y) is a feasible solution of Pm, then h~(x,y)=0~, g~(x,y)≺_0~. Thus we have(59)ϕ~mx,y,α**,β**=f~x,y⊕α**,h~x,y⊕β**,max⁡0~,g~x,y=f~x,y.Therefore there exists no (x,y)∈(X′,Y′) such that f~(x*,y*)≻f~(x,y)=ϕ~m(x,y,α**,β**).(2)
For any(x,y)∈(X,Y) but (x,y)∉(X′,Y′), that is to say that (x,y) is an infeasible solution of Pm. Assume (x,y) violates an equality constraint h~i(·) (the case with an inequality constraint function is similar), so h~i(x,y)≠0~. We also have h~iαL(x,y)≠0 and h~iαU(x,y)≠0, for all α∈[0,1](60)ϕ~mx,y,α**,β**αL=0~,g~x,yf~x,y⊕α**,h~x,ylllllllllllll⊕β**,max⁡⁡0~,g~x,yαL=f~αL(x,y)+∑i=1mαi**×h~iαLx,y+∑j=1rβj**×max⁡0,g~iαLx,y≥f~αL(x,y)+αi**×h~iαL(x,y)>f~αL(x,y)+f~αLx*,y*-f~αLx,yh~iαLx,y×h~iαLx,y=f~αLx*,y*,ϕ~mx,y,α**,β**αU=0~,g~x,yf~x,y⊕α**,h~x,yllllllllllll⊕β**,max⁡⁡0~,g~x,yαU=f~αUx,y+∑i=1mαi**×h~iαUx,y+∑j=1rβj**×max⁡⁡0,g~iαUx,y≥f~αU(x,y)+αi**×h~iαU(x,y)>f~αU(x,y)+f~αUx*,y*-f~αUx,yh~iαUx,y×h~iαUx,y=f~αUx*,y*.Thus, (61)ϕ~mx,y,α**,β**=f~x,y⊕α**,h~x,y⊕β**,max⁡⁡0~,g~x,y≻f~x*,y*.Therefore,(62)f~x*,y*∈MINm⁡ϕ~m·,α**,β**,X,Y.Theorem 38 (strong extended duality theorem for MFNPs).
Under the assumptions and results in Theorem37, assume(63)x*,y*∈⋂{α∈Rm,β∈Rr:α≠α**,β≠β**}⋂{(x,y)∈(X,Y):ϕ~m(x,y,α,β)∈φ~m(α,β)}ARG-MINmϕ~mx,y,α,β,X,Y.Then there is no duality gap between the problem Pm and EDm.Proof.
According to Theorem37, there exist finite α*≥0 and β*≥0 such that (64)f~x*,y*∈MINmϕ~m·,α**,β**,X,Y,lllllforanyα**>α*,β**>β*.Then we have f~(x*,y*)∈φ~m(α**,β**). From Theorem 34, we have f~(x*,y*)≻_φ~m(α,β) if(65)x*,y*∈⋂{(x,y)∈(X,Y):ϕ~m(x,y,α,β)∈φ~m(α,β)}ARG-MINmϕ~mx,y,α,β,X,Y.Thus, according to the known condition, we have f~(x*,y*)≻_φ~m(α,β) for all α≠α**, β≠β**, α≥0, and β≥0.
Therefore,(α**,β**) is a solution of extended dual mixed fuzzy nonlinear programming problem EDm; that is, φ~m(α**,β**)∈MAXEDm(φ~m,R+m,R+r). This shows that there is no duality gap between the problem Pm and EDm.


## 5. Conclusions

In this paper, we proposed the fuzzy optimization problems in continuous, discrete, and mixed spaces and defined the extended duality problems, respectively. Moreover, we presented the theory of extended duality for fuzzy nonlinear programming problems. This approach overcomes the limitation of conventional duality theory by providing a duality condition that leads to no duality gap for general nonconvex fuzzy optimization problems in continuous, discrete, and mixed spaces. Based on the definition of new penalty function, our theory first transforms the fuzzy nonlinear programming to an equivalent extended dual problem and then solves the dual problem and finally shows no duality gap between the original problem and the extended dual problem.



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*Source: 826752-2015-03-23.xml*