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Advances in Operations Research
Volume 2013 (2013), Article ID 708979, 9 pages
Optimality Conditions and Duality of Three Kinds of Nonlinear Fractional Programming Problems
Department of Mathematics, Chengdu University of Information Technology, Sichuan 610225, China
Received 5 April 2013; Accepted 24 October 2013
Academic Editor: Ching-Jong Liao
Copyright © 2013 Xiaomin Zhang and Zezhong Wu. 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.
Some assumptions for the objective functions and constraint functions are given under the conditions of convex and generalized convex, which are based on the -convex, -convex, and -convex. The sufficiency of Kuhn-Tucker optimality conditions and appropriate duality results are proved involving -convex, -convex, and generalized -convex functions.
Multiobjective optimization theory is a development of numerical optimization and related to many subjects, such as nonsmooth analysis, convex analysis, nonlinear analysis, and the theory of set value. It has a wide range of applications in the fields of industrial design, economics, engineering, military, management sciences, financial investment, transport, and so forth, and now it is an interdisciplinary science branch between applied mathematics and decision sciences. Convexity plays an important role in optimization theory, and it becomes an important theoretical basis and useful tool for mathematical programming and optimization theory.
Convex function theory can be traced back to the works of Holder, Jensen, and Minkowski in the beginning of this century, but the real work that caught the attention of people is the research on game theory and mathematical programming by von Neumann and Morgenstern , Dantzing, and Kuhn and Tucker in the forties to fifties, and people have done a lot of intensive research about convex functions from the fifties to sixties. In the middle of the sixties convex analysis was produced, and the concept of convex function is promoted in a variety of ways, and the notion of generalized convex is given.
Fractional programming has an important significance in the optimization problems; for instance, in order to measure the production or the efficiency of a system, we should minimize a ratio of functions between a given period of time and a utilized resource in engineering and economics.
Preda  has established the concept of -convex based on -convex  and -convex  and obtained some results, which are the expansion of -convex and -convex. Motivated by various concepts of convexity, Liang et al.  have put forward a generalized convexity, which was called -convex, which extended -convex, and Liang et al. , Weir and Mond , Weir , Jeyakumar and Mond , Egudo , Preda , and Gulati and Islam  obtained some corresponding optimality conditions and applied these optimality conditions to define dual problems and derived duality theorems for single objective fractional problems and multiobjective problems. Then the definition of generalized -convex is given under the condition of -convex. However, in general, fractional programming problems are nonconvex and the Kuhn-Tucker optimality conditions are only necessary. Under what conditions are the Kuhn-Tucker conditions sufficient for the optimality of problems? This question appeals to the interests of many researchers, and those are what we should probe. Based on the former conclusions, by adding conditions to objective functions and constraint functions and by changing K-T conditions , the optimality conditions and dual are given involving weaker convexity conditions. The main results in this paper are based on convex and generalized convex functions and the properties of sublinear functions.
In this paper, we will discuss sufficient optimality conditions and dual problems for three kinds of nonlinear fractional programming problems, and the paper is organized as follows.
In Sections 3.1 and 3.2, we present the Kuhn-Tucker sufficient optimality conditions and dual for nonlinear fractional programming problem and multiobjective fractional programming problem based on generalized -convex. Section 3.3 contains optimality conditions and dual for multiobjective fractional programming problem under -convex. In these sections, I present some assumptions for the objective functions and constraint functions such that the Kuhn-Tucker optimality conditions are sufficient and obtain the corresponding duality theorem.
Let be the -dimensional real vector space, that is, -dimensional Euclidean space, where , , and provides as follows, (see ),
Definition 1 (see ). Suppose that ; that is, if , such that , is an efficient solution of multiobjective programming problem.
Definition 2 (see ). Suppose that ; that is, if , such that , is a weakly efficient solution of multiobjective programming problem.
Definition 3 (see ). Given an open set , a functional is called sublinear if, for any , ,
It follows from the second equality that
Let be a sublinear function, and let , , , , and the function is differentiable at .
Definition 4 (see ). Let be a differentiable function defined on . The function is said to be -convex on with respect to , if .
Definition 5 (see [4, 12]). Let be a real-valued function defined on the convex set , if there exists a real number , such that
for any and any , then the function is said to be -convex on .
Especially, if , then we obtain the definition of convex.
If in the above definition, then we have strong convex (or weak convex).
Definition 6 (see ). The function is said to be -convex at , if for any , satisfies the following condition:
Definition 7 (see ). The function is said to be -convex at , if
The function is said to be -convex at , if each component of is -convex at .
The function is said to be -convex on , if it is -convex at every point in .
Definition 8. The function is said to be -quasiconvex at , if .
The function is said to be -quasiconvex at , if each component of is -quasiconvex at .
Definition 9. The function is said to be -pseudoconvex at , if for all , .
The function is said to be -pseudoconvex at , if each component of is -pseudoconvex at .
Definition 10. The function is said to be strictly -pseudoconvex at , if , where .
Further, is said to be weakly strictly -pseudoconvex at , if .
In order to prove our main result, we need a lemma which we present in this section.
Lemma 11 (see ). Suppose that differentiable real-valued functions are -quasiconvex at ; then is -quasiconvex at , where , and denote the transpose of the -dimensional column vector ; that is, .
3. Optimality Conditions and Duality
3.1. Nonlinear Fractional Programming Problem Involved Inequality and Equality Constraints Based on Generalized -Convex
Consider the nonlinear fractional programming problem where is an open set of , and are real-valued functions defined on , is an -dimensional vector-valued functions defined also on , and a -dimensional vector-valued function.
Let denotes the set of all feasible solutions for and assume that , , , and are continuously differentiable over and that , , for all .
If is a solution for problem and if a constraint qualification  holds, then the Kuhn-Tucker necessary conditions are given below: there exists and such that
Theorem 12. Suppose that is a feasible solution of , that the Kuhn-Tucker conditions hold at , that in problem is -pseudoconvex on , are -quasiconvex on , and that are -quasiconvex over , , , , , where is the inner product about and and the inner product about and . Then, is an optimality solution for problem .
Proof. Suppose that is not an optimality solution of . Then, there exists a feasible solution such that .
By the -pseudoconvexity assumption of , we have For each , by the -quasiconvexity assumption of and Lemma 11, we have that is -quasiconvex on . Therefore, that is, .
By the -quasiconvexity of over , we have that is -quasiconvexity on .
Then we obtain , that is, By (10), (11), and (12), and based on the sublinearity of , we have Considering that , we get By the K-T conditions, we have .
Hence, based on the sublinearity of , we obtain which contradicts (14). The proof is complete.
Consider the dual problem of :
Theorem 13. Suppose that is -pseudoconvex at , is -quasiconvex at in problem and , and that ; then , for any feasible solution of and of .
Proof. Assume that the conclusion is not true; that is, .
By the -pseudoconvex of at , we get Using , , , , we have By the -quasiconvex of , we get By (16) and (18) and based on the sublinearity of , we have Since is the feasible solution of , so Hence, which contradicts the known condition . The proof is complete.
3.2. Nonlinear Multiobjective Fractional Programming Problem Involved Inequality and Equality Constraints Based on -Convex and Generalized -Convex
Consider the nonlinear multiobjective fractional programming problem where is an open set of , , , are real-valued functions defined on , are real-valued functions defined also on , , , and .
Let denotes the set of all feasible solutions of and assume that , , , and are continuously differentiable over and that , for all .
Theorem 14. Suppose that is weakly strictly -pseudoconvex at , are -quasiconvex with respect to , are -convex with respect to , and that there exists (or ), , satisfying and . Then, is an efficient solution of , where
Proof. Suppose that is not an efficient solution of ; then there exists a feasible solution such that , that is, .
By the weakly strict -pseudoconvexity of at , we get Using (or ), then we have Based on the sublinearity of , we obtain Since , and , we have ; that is, .
Since are -quasiconvex at , by Lemma 11, we have is -quasiconvex at .
Hence, we obtain the following inequality: By the -convexity of at , we have Since , we have By the sublinearity of , we obtain By the known conditions, we have By (28), (29), and (32) and by the sublinearity of , we obtain which contradicts the fact of (33). Therefore, is an efficient solution of . The proof is complete.
Consider the dual problem of
Theorem 15. is -convex at , is -convex at , is -convex at , and , then
Proof. By the -convex of at , we get By the -convex of at , we get By the -convex of at , we get Since , , , and by the previous three inequalities, we have that By the sublinearity of , we obtain By the feasibility of , we have Since and by (41), we get Since , , we obtain The proof is complete.
3.3. Nonlinear Multiobjective Fractional Programming Problem Involved Inequality and Equality Constraints under -Convex
Consider the multiobjective fractional programming problem where is an open set of , , , , , and , are continuously differentiable over .
Denote by the set of all feasible solutions for ; that is, and let , .
Theorem 16. Assume that there exists and , , such that(i), , , , , , , where ;(ii) and are -convex at , and ;(iii) are -convex at for all , , and ;(iv) are -convex at for all , , and .Then is a Pareto optimality solution of .
Proof. Suppose that is not a Pareto optimality solution of ; then there exists a feasible solution such that , , that is, , that is, ; it follows that
By the -convexity of and , , we have
Using the conditions , , we see that .
By the properties of the sublinear functional , we obtain By (48) and (49) and based on the sublinearity of , we have By (47), we have If we sum up after multiplying by in the above inequality and by using the sublinearity of , we have Since , we get On the other hand, for , by the -convexity of at , we have On multiplying the inequality (54) by and using the sublinearity of , we have which together with and yields By accumulating the inequality (56) with , we have that is, For , by the -convexity of at , we have that On multiplying the inequality (59) with , we get which together with and yields By accumulating the inequality (61) with , we have The inequality (62) along with the sublinearity of implies The sublinearity of , (53), (58), and (63) yields According to the assumption and the sublinearity of , we obtain which contradicts (64) obviously.
Therefore, is a Pareto optimality solution of .
The proof is complete.
Consider the dual problem of
Theorem 17. is -convex at , is -convex at , is -convex at , and , , , , then .
Proof. By the -convexity of , , and , the sublinearity of , and since , , , we have that