Duality of -Multiobjective Programming Involving Generalized Invex Functions
In the setting of Ben-Tal's generalized algebraic operations, this paper deals with Mond-Weir type dual theorems of multiobjective programming problems involving generalized invex functions. Two classes of functions, namely, -pseudoinvex and -quasi-invex, are defined for a vector function. By utilizing these two classes of functions, some dual theorems are established for conditionally proper efficient solution in -multiobjective programming problems.
The theory and applications of multiobjective programming problems have been closely tied with convex analysis. Optimality conditions and duality theorems were established for the class of problems involving the optimizations of convex objective functions over convex feasible regions. Such assumptions were very convenient because of the known separation theorems and the guarantee that necessary conditions for optimality were sufficient under convexity. However, not all practical problems, when formulated as multiobjective programs, fulfill the requirements of convexity. Fortunately, such problems were often found to have some characteristics in common with convex problems, and these properties could be exploited to establish theoretical results or develop algorithms. Many notions of generalized convexity having some useful properties shared with convexity have been defined by a sizeable number of researchers. A meaningful generalization of convex functions is the introduction of invex functions, which was given by Hanson , for the scalar case. Nowadays, with and without differentiability, the invex functions are extended to vector functions in finite dimensions or infinite dimensions abstract spaces, and sufficient optimality criteria and duality results are obtained for multiobjective programming or vector optimization, respectively, see [1–15].
In 1976, Ben-Tal  introduced certain generalized operations of addition and multiplication. This kind of generalized algebraic means has many applications in pure and applied mathematical fields, see [6, 7, 10–16]. The biggest advantage under Ben-Tal's generalized means is that the function has some transformable properties. As pointed out in literature  that a function is not convex or differentiable, however it may be transformed into convex function or differentiable function in the setting of Ben-Tal's generalized algebraic operations. In this way, Ben-Tal's generalized means provided a manner in extension of convexity. Recently, more and more interest has been paid on dealing with optimality and duality of multiobjective program problems involving generalized convexity under Ben-Tal's generalized means circumstances, for instance, see [10–16].
The properness of the efficient solution of the multiobjective programming problem is of importance. In 1991, Singh and Hanson  introduced conditionally properly efficiency for multiobjective programming problems. This kind of proper efficiency has specific significance in the optimal problem with multicriteria. In present paper, we first extend the notions of the conditionally proper efficiency for multiobjective programming problems, pseudoinvexity and quasi-invexity for vector functions in the setting of Ben-Tal's generalized means. Then, for a class of constraint multiobjective program problem, we will establish several duality results by using the new defined proper efficient solutions and generalized invex functions. This paper is organized as follows. In Section 2, we present some preliminaries and related results which will be used in the rest of the paper. In Section 3, some duality theorems are derived.
Let be the -dimensional Euclidean space and be the set of all positive real numbers. Throughout this paper, the following convention for vector in will be used:
We first present the generalized algebraic operations given by Ben-Tal .
Definition 2.1 (see [6, 8]). Let be a continuous vector function. Suppose that the inverse function of exists. Then the -vector addition of defined by
and the -scalar multiplication of and is defined by
Similarly, generalized algebraic operations for scalar-valued functions can be defined as follows.
Definition 2.3 (see [6, 8]). The -inner product of vector is defined as
In this paper, we denote
For the differentiability of a real-valued function in the setting of generalized algebraic means, Avriel  introduced the following important concept.
Definition 2.4 (see ). Let be a real-valued function defined on , denote . For simplicity, write . The function is said to be -differentiable at , if is differentiable at , and denoted by . In addition, It is said that is -differentiable on if it is -differentiable at each . A vector-valued function is called -differentiable on if each of its components is -differentiable at each .
We collect some basic properties concerning Ben-Tal's generalized means from the literatures [12, 14], which will be used in the squeal.
Lemma 2.7 (see [12, 14]). Suppose that function , which appears in Ben-Tal generalized algebraic operations, is strictly monotone with . Then, the following statements hold:(1)let , , and . Then ;(2)let , , and . Then ;(3)let , , and . Then ;(4)let , . If for any , then If for any , and there exists at least an index such that , then
Throughout the rest of this paper, one further assumes that is a continuous one-to-one and onto function with . Similarly, suppose that is a continuous one-to-one strictly monotone and onto function with . Under the above assumptions, it is clear that .
Let be a nonempty subset of and the functions and are -differentiable on the set with respect to the same . Consider the following -multiobjective programming problem:
Definition 2.9. A point is said to be an efficient solution for if and for all .
Singh and Hanson  introduced the concept of conditionally properly efficient for multiobjective optimization. Now, we extend this notion under Ben-Tal's generalized algebraic operations as follows.
Definition 2.10. The point is said to be -conditionally proper efficient solution for if is an efficient solution and there exists a positive function such that, for , one has for some such that , whenever and
Example 2.11. Consider the following multiobjective problem:
Taking , , it can be shown that every point of the feasible region is efficient. Let be an efficient solution. Choosing , where . For , we get
for such that whenever is feasible and
Thus, is -conditionally proper efficient solution.
Xu and Liu  introduced -Kuhn-Tucker constraint qualification and used it to establish Kuhn-Tucker necessary condition for -multiobjective programming problems, for more details concerning -Kuhn-Tucker constraint qualification, please see . We now state this result as the following (Lemma 2.12).
Lemma 2.12 (Kuhn-Tucker-type necessary condition). Let for , for be -differentiable on , be an efficient solution of and the -Kuhn-Tucker constraint qualification be satisfied at . Then there exist and such that
Jeyakumar and Mond  introduced the notion of -invexity for a vector function and discussed its applications to a class of constrained multi-objective optimization problems. One now gives the definitions of generalized -invexity for a vector function in the setting of Ben-Tal's generalized algebraic operations as follows.
Definition 2.13. A vector function is said to be --invex at if there exist functions and such that for each and for ,
If we take and as the identity functions, the above definitions reduce to the -invex function given by Jeyakumar and Mond .
Example 2.14. The functions , . Let and . Then, is --invex function at with respect to any and , .
Definition 2.15. A vector function is said to be --pseudoinvex at if there exist functions and such that for each and for , If in the above definition and (2.16) is satisfied as then we say that is strictly --pseudoinvex at .
Example 2.16. The functions , . Let and . Then, is --quasi-invex function at with respect to and any (). In fact, observing that and , . In this case, we have and for , it follows that Thus, we get from Lemmas 2.6 and 2.7 that By Definition 2.15, we have shown that is --pseudoinvex at .
Definition 2.17. A vector function is said to be --quasi-invex at if there exist functions and such that for each and for ,
Example 2.18. The function is defined as . Taking and , then, is --quasi-invex at with respect to and any .
In this section, we will establish the weak and strong duality theorems under the generalized --invexity assumptions for Mond and Weir type dual model in relation to Considering the following dual problem:
Theorem 3.1 (weak duality). Let and be any feasible solutions for and , respectively. Let either (a) or (b) below hold:(a) is --pseudoinvex and is --quasi-invex at with respect to same ; (b) is --quasi-invex and is strictly --pseudoinvex at with respect to same . Then
Proof. Since is a feasible solution for , by Lemma 2.5 and (3.1), for all we obtain that (a) Let be feasible for and . Since and , for all , it follows from Lemmas 2.6 and 2.7 that and --pseudoinvexity at of implies Observing that and are feasible of and , respectively, we get from Lemma 2.7 that Again, since , for all , it follows from Lemma 2.7 that Now, --quasi-invexity at of implies that Together with (3.9) and (3.12), it yields from Lemma 2.7 that which contradicts to (3.7)(b) Let be feasible for and feasible for . Suppose that . Since , for all , and , we get from Lemmas 2.6 and 2.7 that The --quasi-invexity at of implies that By (3.7), we get from Lemmas 2.7 and 2.8 that and since is strictly --pseudoinvex, we have According to Lemma 2.7, this is a contradiction, since , and , for all .
Theorem 3.2. If is feasible for and feasible for such that . Let neither (a) or (b) bellow hold:(a) is --pseudoinvex and is --quasi-invex at with respect to same ; (b) is --quasi-invex and is strictly --pseudoinvex at with respect to same .
Proof. Suppose is not an efficient solution for , then there exists feasible for such that Using the assumption , a contradiction to Theorem 3.1 is obtained. Hence, is an efficient solution for . Similarly it can be ensured that is an efficient solution for .
Now suppose that is not -conditionally properly efficient solution for . Therefore, for every positive function , there exists feasible for and an index such that for all satisfying , whenever . This shows that can be made arbitrarily large and hence for and , for all , the inequality is obtained. Consequently, we ge from Lemmas 2.7 and 2.8 that Now from feasibility conditions, we have Since , for all , Suppose that the hypothesis (a) holds at , we can get from --quasi-invexity at of that Therefore, from (3.1), we get from Lemmas 2.5, 2.7, and 2.8 that Since is --pseudoinvex and is --quasi-invex at , we have On using the assumption in the above equation, we get which is a contradiction to (3.21). Hence is a -conditionally properly efficient solution for .
We now suppose that is not -conditionally properly efficient solution for . Therefore, for every positive function , there exists a feasible feasible for and an index such that for all satisfying whenever . This means can be made arbitrarily large and hence for and , for all , the inequality is obtained. Since and are feasible for and , respectively, it follows that as in first part: which contradicts (3.29). Hence, is -conditionally properly efficient solution for .
Assuming that the hypothesis holds, we can finish the proof with the similar argument.
Theorem 3.3 (strong duality). Let be an efficient solution for . If the -Kuhn-Tucker constraint qualification is satisfied, then there are such that is feasible for and the objective values of and are equal at . Furthermore, if the hypothesis or of Theorem 3.2 hold at , then is -conditionally properly efficient for the problem .
Proof. Since is an efficient solution for at which the -Kuhn-Tucker-type necessary conditions are satisfied, it follows from Lemma 2.12 that there exist such that is feasible for . Evidently, the objective values of and are equal at , since the objective functions for both problems are the same. The -conditionally proper efficiency of for the problem yields from Theorem 3.2.
This research is supported by Zizhu Science Foundation of Beifang University of Nationalities (no. 2011ZQY024); Natural Science Foundation for the Youth (no. 10901004); Natural Science Foundation of Ningxia (no. NZ12207); Ministry of Education Science and technology key projects (no. 212204).
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