Sufficiency and Duality for Multiobjective Programming under New Invexity

Abstract

A class of multiobjective programming problems including inequality constraints is considered. To this aim, some new concepts of generalized -type I and -type II functions are introduced in the differentiable assumption by using the sublinear function . These new functions are used to establish and prove the sufficient optimality conditions for weak efficiency or efficiency of the multiobjective programming problems. Moreover, two kinds of dual models are formulated. The weak dual, strong dual, and strict converse dual results are obtained under the aforesaid functions.

1. Introduction

The field of multiobjective programming, also called vector programming, has grown remarkably in different directions since the 1980s. Many researchers have been interested in the optimality conditions and duality results for the weak efficient solution and efficient solution of the multiobjective programming problems. A large literature was developed around the sufficiency and duality in multiobjective optimization [1]. In [2], Jayswal obtained the Kuhn-Tucker type sufficient optimality conditions for a feasible solution to be an efficient solution and the Mond-Weir type duality results are also presented. More specifically, Gao [3] considered the nonsmooth multiobjective semi-infinite programming and obtained several sufficient conditions and duality results. Also, Bae et al. [4] established duality theorems for nondifferentiable multiobjective programming problems under generalized convexity assumptions. Recently, Kim and Lee [5] introduced the nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. For more descriptions of the multiobjective programming, we refer to [6–9].

Furthermore, various types of generalizations of convexity theory have played an important role in the evolution of the multiobjective programming. During the past decades, the generalizations of invexity were enriched with and without differentiability assumptions. For example, we can see [10–12]. In particular, Nahak and Mohapatra [13] introduced the concept of --invexity function and discussed a class of multiobjective programming problems by using the new generalized functions. Padhan and Nahak [14] introduced higher-order --invexity functions for studying two different pairs of higher-order symmetric dual programs. In [15], Antczak extended the concept of -invexity for differentiable optimization problems to the case of mathematical programming problems with locally Lipschitz functions. In [16], Antczak and Stasiak introduced the concept of -invexity for strong compact Lipschitz mappings in Banach spaces. Sufficient optimality conditions and Mond-Weir duality theorems are derived by the assumption of generalized nonsmooth -invexity between Banach spaces. In [17], based upon the -convexity and -convexity, the authors defined the -V-type I functions to consider a class of nonsmooth multiobjective programming problems. The invexity of functions is more useful in the research of optimization.

In this paper, we consider the multiobjective programming problems. The new class of generalized invexity functions, namely, pseudoinvex -type I (pseudoinvex -type II, etc.) are introduced. The sufficient optimality conditions are obtained. Then weak, strong, and strict converse dual results are also established for two types of dual models related to multiobjective programming problems involving the new generalized invex functions.

2. Notations and Preliminaries

Throughout the paper, we use the following conventions for vectors in : In this paper, we consider the following multiobjective programming problem:where is an open set and and are differentiable on . Let be the set of all feasible solutions of .

Definition 1. A feasible solution of is said to be a weakly efficient solution for , if there exists no other , such that

Definition 2. A feasible solution of is said to be an efficient solution for , if there exists no other , such that

Definition 3. A function is sublinear if, for any ,

Remark 4. It should be noted that .
Let and be differentiable at a given point ,  ,  ,  ,  .

Definition 5. is said to be pseudoinvex -type I at , if there exists functions and , such that each ; the following inequalities hold:

Example 6. Let . Let the functions , , and be defined byand the functions , be given by Moreover, the functions ,  , and are defined by Then, let Now, we haveThenHence is pseudoinvex -type I at , where

Definition 7. is said to be pseudoinvex -type II at , if there exists functions and , such that each ; the following inequalities hold:

Definition 8. is said to be pseudoquasi-invex -type I at , if there exists functions and , such that each ; the following inequalities hold:

Example 9. Let , , and be defined byand the functions and   be given by Moreover, the functions ,  , and are defined by Then, let It is easy to see that is pseudoquasi-invex -type I at the point with respect to , and .

Definition 10. is said to be pseudoquasi-invex -type II at , if there exists functions , and , such that each ; the following inequalities hold:

Definition 11. is said to be quasipseudo-invex -type I at , if there exists functions , and , such that each ; the following inequalities hold:

Definition 12. is said to be quasipseudo-invex -type II at , if there exists functions , and , such that each ; the following inequalities hold:

3. Sufficient Optimality Conditions

Now, we establish sufficient optimality conditions for the considered optimization problem under the new invexity.

Theorem 13. Let be a feasible solution in problem . Suppose that (i)there exists , , , , such that  (ii) is pseudoinvex -type I at ; (iii)Then is a weakly efficient solution for .

Proof. Suppose contrary to the result that is not a weakly efficient solution to . Then there exists such that With , the above inequality yieldsfrom hypothesis (ii), which impliesBy sublinearity of with and , inequalities (24) yieldUsing , , and ,, along with the sublinearity of , from inequality (25), we getBy the sublinearity of , we sum (26) to obtainfrom hypothesis (iii), which followsOn the other hand, the hypothesis (i) implieswhich contradicts (28). Hence the conclusion of the theorem is established.

Theorem 14. Let be a feasible solution in problem . Suppose that (i)there exists , , , , , such that  (ii) is pseudoinvex -type II at ; (iii)Then is an efficient solution for .

Proof. By the way of contradiction, suppose that is not an efficient solution for . Then there exists such that With , the above inequality yieldsby hypothesis (ii), which followsUsing , and ,  , , along with the sublinearity of , inequality (33) yieldsSumming inequalities (34) with the sublinearity of , we obtainFrom assumption (i), we havethat is,which contradicts hypothesis (iii). That completes the proof.

Theorem 15. Let be a feasible solution in problem . Suppose that (i)there exists , , , , , such that  (ii) is pseudoquasi-invex -type I at ; (iii)Then is a weakly efficient solution for .

Proof. The proof follows the lines of Theorem 13.

Theorem 16. Let be a feasible solution in problem . Suppose that (i)there exists , , , , , such that (ii) is pseudoquasi-invex -type II at ; (iii)Then is an efficient solution for .

Proof. The proof follows the lines of Theorem 14.

Theorem 17. Let be a feasible solution in problem . Suppose that (i)there exists , (at least one ), , , such that  (ii) is quasipseudo-invex -type I at ; (iii)Then is a weakly efficient solution for .

Proof. Suppose contrary to the result that is not a weakly efficient solution to . Then there exists such that with ; the above inequality yieldsfrom hypothesis (ii), which impliesBy sublinearity of with and , inequalities (43) yieldUsing ,   along with the sublinearity of , from inequality (44), we getWith (at least one ),  , and using the sublinearity of , inequality (45) followsBy the sublinearity of , we sum (46) and (47) to obtainfrom hypothesis (iii), which followsOn the other hand, the hypothesis (i) implieswhich contradicts (49). Hence the conclusion of theorem is established.

Example 18. We consider the following programming problem:The set of all feasible solutions of can be given by .
Again, let be the function defined by
It can be verified that is quasipseudo-invex -type I at with ,  ,  ,  ,  , and , ,  ,  
Clearly, is a feasible solution for problem and it satisfied the assumptions of Theorem 17, as there exist ,  , such thatWe observe that there exists no other , such that . Hence, is an efficient solution for .

Similarly, we can establish the following theorem.

Theorem 19. Let be a feasible solution in problem . Suppose that (i)there exists ,   (at least one , as ), , , such that  (ii) is quasipseudo-invex -type II at ; (iii)Then is an efficient solution for .

4. Mond-Weir Duality

In this section, a dual problem is considered for the class of multiobjective programming problem with the new invex functions.

Consider the following Mond-Weir dual problem related to problem :

Let ,  ,   be the set of all feasible solutions in problem .

Theorem 20 (weak duality). Let and be feasible solutions for and , respectively. Moreover, assume that If one of the following conditions is satisfied: (a) is pseudoinvex -type I at , (b) is pseudoquasi-invex -type I at ,then the following can not hold:

Proof. Suppose contrary to the result that hold.
By , the above inequality followsby assumption (a), is pseudoinvex -type I at , which yieldsBy sublinearity of together with and , the above inequalities yieldFrom the feasibility of in Mond-Weir dual problem , it follows that ,  , , . Multiplying inequalities (58) by and , respectively, together with the sublinearity of , we getAdding both sides of (59) with the sublinearity of , we obtainFrom assumption, ; inequality (60) impliesBy the constraint condition of dual problem and , we havewhich contradicts inequality (61). Thus, the conclusion of theorem holds.
The proof of part (b) is similar to the proof of part (a).

Theorem 21 (weak duality). Let and be feasible solutions for and , respectively. Moreover, assume that If one of the following conditions is satisfied: (a) is pseudoinvex -type II at , (b) is pseudoquasi-invex -type II at ,then the following can not hold:

Proof. Suppose contrary to the result that hold.
By , the above inequality yieldsby assumption (a), is pseudoinvex -type II at , which yieldsBy sublinearity of together with and , the above inequalities yieldFrom the feasibility of in Mond-Weir dual problem , we haveFor , ,  ,  . Multiplying inequalities (68) by and , respectively, together with the sublinearity of , we haveAdding both sides of (70) with the sublinearity of , we getCombining (69) and (71), we obtainwhich contradicts the assumptionThus, the conclusion of the theorem holds.
The proof of part (b) is similar to the proof of part (a).

Theorem 22 (strong duality). Assume that is a weakly efficient solution of . Suppose that there exists and , such that is feasible for . Furthermore, if the weak duality Theorem 20 holds for all feasible solutions of the problems and , then is a weakly efficient solution of .

Proof. Suppose that is not a weakly efficient solution of ; then there exists another feasible solution of such that which is a contradiction to Theorem 20. Hence is a weakly efficient solution of .

Theorem 23 (strong duality). Assume that is an efficient solution of . Suppose that there exist and , such that is feasible for . Furthermore, if the weak duality Theorem 21 holds for all feasible solutions of the problems and , then is an efficient solution of .

Proof. Suppose that is not an efficient solution of ; then there exists another feasible solution of such that which is a contradiction to Theorem 21. Hence is an efficient solution of .

Theorem 24 (strict converse duality). Let and be feasible solutions for and , respectively. Suppose that , and . If one of the following conditions is satisfied: (a) is pseudoinvex -type II at , (b) is pseudoquasi-invex -type II at ,then .

Proof. Suppose that .
By , the condition yieldsusing assumption (a), is pseudoinvex -type II at , which yieldsBy sublinearity of together with and , the above inequalities yieldBecause is feasible for , thenFor . Multiplying inequalities (78) by and , respectively, together with the sublinearity of , we haveAdding both sides of (80) with the sublinearity of , we getUsing the assumption , inequality (81) follows that Using inequality (79) together with the sublinearity of , we obtain which is a contradiction to (82). Then .
The proof of part (b) is similar to the proof of part (a).

Theorem 25 (strict converse duality). Let and be feasible solutions for and , respectively. Suppose that , and . If one of the following conditions is satisfied: (a) is pseudoinvex -type I at , (b) is pseudoquasi-invex -type I at ,then .

Proof. Suppose that .
By , the condition yieldsusing assumption (a), is pseudoinvex -type I at , which yieldsBy using the sublinearity of with and and , inequalities (85) implyAdding both sides of (86) with the sublinearity of , we haveUsing the condition , inequality (87) yields According to the constraint condition of with , we get which is a contradiction to (88). Then .
The proof of part (b) is similar to the proof of part (a).

5. Wolfe Type Duality

In this section, we consider the Wolfe type dual for and establish various duality theorems. Let be the vector of whose components are all ones.

Let ,  ,   be the set of all feasible solutions in problem .

Theorem 26 (weak duality). Let and . Suppose that If one of the following conditions is satisfied: (a) is pseudoinvex -type I at , (b) is pseudoquasi-invex -type I at ,then the following can not hold:

Proof. If holds, then we haveSince and , the above inequality yieldsthat is,by using , we haveby assumption (a), is pseudoinvex -type I at , which implies By sublinearity of together with , , and the feasibility of in , the above inequalities yieldAdding both sides of (98) with the sublinearity of , we getFrom assumption, ; inequality (99) impliesBy the constraint condition of and , we havewhich contradicts inequality (100). Thus, the conclusion of theorem holds.
The proof of part (b) is similar to the proof of part (a).

Theorem 27 (weak duality). Let and . Suppose that If one of the following conditions is satisfied: (a) is pseudoinvex -type II at , (b) is pseudoquasi-invex -type II at ,then the following can not hold:

Proof. Suppose contrary to the result that hold. That is,Since and , the above inequality yieldsthat is,By using , we getBy assumption (a), is pseudoinvex -type II at , which yields By sublinearity of together with ,, and the feasibility of in , the above inequalities yieldAdding both sides of (110) with the sublinearity of , we getBy using the constraint condition of and , inequality (111) implieswhich contradicts the assumption .
Thus, the conclusion of theorem holds.
The proof of part (b) is similar to the proof of part (a).

Theorem 28 (strong duality). Assume that is a weakly efficient solution of . Suppose that there exist , , and , such that is feasible for . Furthermore, if the weak duality Theorem 26 holds for all feasible solutions of the problems and , then is a weakly efficient solution of .

Proof. Suppose that is not a weakly efficient solution of ; then there exists another feasible solution of such that that is,For and , the above inequality yieldsthat is,which is a contradiction to Theorem 26. Hence is a weakly efficient solution of .

Theorem 29 (strict converse duality). Let and be feasible solutions for and , respectively. Suppose that , and . If one of the following conditions is satisfied: (a) is pseudoinvex -type I at , (b) is pseudoquasi-invex -type I at ,then .

Proof. Suppose that .
From and , the condition yieldsthat is,by , which impliesusing assumption (a), is pseudoinvex -type I at , which yieldsBy sublinearity of together with ,, and , the above inequalities yieldAdding both sides of (121) with the sublinearity of , we haveUsing the assumption , inequality (122) implies By using the constraint condition of and , we obtain which is a contradiction to (123). Then .
The proof of part (b) is similar to the proof of part (a).

6. Discussion and Conclusion

In this paper, we study the multiobjective programming problems and two kinds of dual models. Then the sufficient optimality conditions, weak dual, strong dual, and strict converse dual, results are obtained and proved under a class of new generalized invex functions assumptions for the multiobjective programming.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grants no. 71103143 and no. 71473194; Science and Technology Foundation of Shaanxi Province of China under Grant no. 2013XJXX-40; Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 15JK1456).