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.

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 Section 2. 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.

In this section, we recall some basic definitions and work related to fuzzy optimization problems. Let be a universal set. A fuzzy subset of is a mapping . The -level of denoted by is defined by for all . The 0-level set is defined as the closure of the set .

Definition 1. We denote by the set of all fuzzy subset of with membership function satisfying the following conditions:(1) is normal; that is, there exists an such that ;(2) is quasi concave; that is, for all ;(3) is upper semicontinuous; that is, is a closed subset of for all ;(4)the 0-level set is a compact subset of .
Throughout this paper, the universal set is the set of all real number . The member in is called a fuzzy number. For all , we can denote the -level of by .
Let ; that is, , where for . We also write and , where and for all .

Definition 2. The fuzzy scalar product of the fuzzy vectors and in is defined by Let . We write if and only if and for all . The relation “” on is a partial ordering.

Definition 3. Let and be two subsets of . We write if for all . We write if for all .
Let and be two fuzzy-valued functions defined on the same real vector space , and let be a subset of . Then,

Definition 4. The primal fuzzy optimization problem is defined as follows:The fuzzy-valued Lagrangian function for the primal problem is defined as follows:for all and all ; that is, for all . We also write if .

Definition 5. The dual fuzzy optimization problem is defined as follows:where the fuzzy-valued Lagrangian dual function is defined as

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. Let ,   and be fuzzy-valued functions defined on the same real vector space , and let , be two subsets of .

3.1. Continuous Fuzzy Nonlinear Programming Problems

Definition 6. A continuous fuzzy nonlinear programming problem (CFNP) is defined aswhere is a continuous variable and is a continuous and differentiable fuzzy-valued function.

Definition 7. Point is a solution of , if is a feasible solution of and there exists no feasible solution such that .
Let be the feasible set of and be the set of all solutions of ; then

Definition 8. The fuzzy-valued -penalty function for in (7) is defined as follows:where and , and and are penalty multipliers.

According to the fuzzy-valued -penalty function for , we define the fuzzy-valued extended dual function as follows.

Definition 9. The fuzzy-valued extended dual function for is defined for and aswhere is a point-to-set fuzzy-valued extended dual function; that is, for any fixed and ,   is a subset of .

Definition 10. The extended dual continuous fuzzy nonlinear programming problem (EDCFNP) is defined as follows:

Definition 11. Point is a solution of , if there exists a such that for all , , , and .

Let   denote the set of all solutions of extended dual continuous fuzzy nonlinear programming problem  , . If and are fixed, then .

3.2. Discrete Fuzzy Nonlinear Programming Problems

Definition 12. A discrete fuzzy nonlinear programming problem (DFNP) is defined aswhere is a discrete variable.

Definition 13. Point is a solution of , if is a feasible solution of and there exists no feasible solution such that .
Let be the feasible set and let be the set of all solutions of ; then,

Definition 14. The fuzzy-valued -penalty function for in (12) is defined as follows:where and , and and are penalty multipliers.

Definition 15. The fuzzy-valued extended dual function for is defined for and asSimilarly, is a point-to-set fuzzy-valued extended dual function for ; that is, for any fixed and , is a subset of .

Definition 16. The extended dual discrete fuzzy nonlinear programming problem (EDDFNP) is defined as follows:

Definition 17. Point is a solution of , if there exists a such that for all , , , and .
Let denote the set of all solutions of extended dual discrete fuzzy nonlinear programming problem , : . If and are fixed, then

Definition 18. A mixed fuzzy nonlinear programming problem (MFNP) is defined as

Definition 19. Point is a solution of , if is a feasible solution of and there exists no feasible solution such that .
Similarly, : , denotes the feasible set of mixed fuzzy nonlinear programming problem , denotes the set of all solutions of mixed fuzzy nonlinear programming problem , = : such that  ,  and   = = .

Definition 20. The fuzzy-valued -penalty function for in (18) is defined as follows:where and , and and are penalty multipliers.

Definition 21. The fuzzy-valued extended dual function for is defined for and as is a point-to-set fuzzy-valued extended dual function for ; that is, for any fixed and , is a subset of .

Definition 22. The extended dual mixed fuzzy nonlinear programming problem (EDMFNP) is defined as follows:

Definition 23. Point is a solution of , if there exists a such that for all , , , and .
Let denote the set of all solutions of extended dual mixed fuzzy nonlinear programming problem , . If and are fixed, then

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. Suppose and are feasible solution of problem and , respectively; moreoverThen we have .

Proof. Let . For any , we have Since is a feasible solution of problem , we obtain , . Thus, This inequality is satisfied for all . According to Definition 3, therefore we have

Theorem 25 (weak extended duality theorem for CFNPs). Suppose thatThen .

Proof. If , then . According to Theorem 24, we have if satisfies formula (23). Therefore, from Definition 3, ifthenMoreover, if , then Therefore, according to Definition 3, we have

Definition 26. Let be a continuous fuzzy nonlinear programming problem and let be an extended dual continuous fuzzy nonlinear programming problem. There is no duality gap between and if there exist and , such that .

Theorem 27. Suppose is a solution of continuous fuzzy nonlinear programming problem ; then there exist finite and such that

Proof. Since is a solution of problem , we have , , and there exists no such that . We set the following and :Suppose be the set of feasible solutions of .(1)For any , that is to say that is a feasible solution of , then , . Thus we have Therefore there exists no such that .(2)For any but , that is to say that is an infeasible solution of . Assume violates an equality constraint (the case with an inequality constraint function is similar), so . We also have and , for all Thus, Therefore,

Theorem 28 (strong extended duality theorem for CFNPs). Under the assumptions and results in Theorem 27, we further assumeThen there is no duality gap between the problem and .

Proof. According to Theorem 27, there exist finite and such thatThen we have . From Theorem 24, we have ifThus, according to the known condition, we have for all , , , and .
Therefore, is a solution of extended dual continuous fuzzy nonlinear programming problem ; that is, . This shows that there is no duality gap between the problem and .

4.2. Extended Duality Theorem for DFNPs

Theorem 29. Suppose and are feasible solution of problem and , respectively; moreoverThen we have .

Proof. Let . For any , we haveSince is a feasible solution of problem , we obtain , . Thus, This inequality is satisfied for all . According to Definition 3, therefore we have

Theorem 30 (weak extended duality theorem for DFNPs). Suppose thatThen, .

Proof. The proof is similar to the proof of Theorem 25.

Definition 31. Let be a discrete fuzzy nonlinear programming problem and let be an extended dual discrete fuzzy nonlinear programming problem. There is no duality gap between and if there exist and , such that .

Theorem 32. Suppose is a solution of discrete fuzzy nonlinear programming problem ; then there exist finite and such thatThe 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 Theorem 32, assumeThen there is no duality gap between the problem and .

Proof. The proof is similar to the proof of Theorem 28.

4.3. Extended Duality Theorem for MFNPs

Theorem 34. Suppose and are feasible solution of problem and , respectively; moreoverThen we have .

Proof. Let . For any , we haveSince is a feasible solution of problem , we obtain , . Thus, This inequality is satisfied for all . According to Definition 3, therefore we obtain

Theorem 35 (weak extended duality theorem for MFNPs). Suppose thatThen .

Proof. If , then . According to Theorem 34, we have if satisfies formula (48). Therefore, from Definition 3, ifthenMoreover, if , then Therefore, according to Definition 3, we have

Definition 36. Let be a mixed fuzzy nonlinear programming problem and let be an extended dual mixed fuzzy nonlinear programming problem. There is no duality gap between and if there exist and , such that .

Theorem 37. Suppose is a solution of continuous fuzzy nonlinear programming problem ; then there exist finite and such that

Proof. Since is a solution of problem , we have ,  , and there exist no such that . We set the following and :Suppose be the set of feasible solutions of .(1)For any , that is to say that is a feasible solution of , then , . Thus we haveTherefore there exists no such that .(2)For any but , that is to say that is an infeasible solution of . Assume violates an equality constraint (the case with an inequality constraint function is similar), so . We also have and , for all Thus, Therefore,

Theorem 38 (strong extended duality theorem for MFNPs). Under the assumptions and results in Theorem 37, assumeThen there is no duality gap between the problem and .

Proof. According to Theorem 37, there exist finite and such that Then we have . From Theorem 34, we have ifThus, according to the known condition, we have for all , , , and .
Therefore, is a solution of extended dual mixed fuzzy nonlinear programming problem ; that is, . This shows that there is no duality gap between the problem and .

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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant nos. 61272171 and 61371090.