Multiobjective Fractional Programming Involving Generalized Semilocally V-Type I-Preinvex and Related Functions

Slimani, Hachem; Mishra, Shashi Kant

doi:https://doi.org/10.1155/2014/496149

International Journal of Mathematics and Mathematical Sciences

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Research Article | Open Access

Volume 2014 | Article ID 496149 | https://doi.org/10.1155/2014/496149

Multiobjective Fractional Programming Involving Generalized Semilocally V-Type I-Preinvex and Related Functions

Hachem Slimani¹and Shashi Kant Mishra²

Academic Editor: Ram U. Verma

Received12 Jan 2014

Revised05 Apr 2014

Accepted05 Apr 2014

Published05 May 2014

Abstract

We study a nonlinear multiple objective fractional programming with inequality constraints where each component of functions occurring in the problem is considered semidifferentiable along its own direction instead of the same direction. New Fritz John type necessary and Karush-Kuhn-Tucker type necessary and sufficient efficiency conditions are obtained for a feasible point to be weakly efficient or efficient. Furthermore, a general Mond-Weir dual is formulated and weak and strong duality results are proved using concepts of generalized semilocally V-type I-preinvex functions. This contribution extends earlier results of Preda (2003), Mishra et al. (2005), Niculescu (2007), and Mishra and Rautela (2009), and generalizes results obtained in the literature on this topic.

1. Introduction

Because of many practical optimization problems where the objective functions are quotients of two functions, multiobjective fractional programming has received much interest and has grown significantly in different directions in the setting of efficiency conditions and duality theory these later years. The field of multiobjective fractional optimization has been naturally enriched by the introductions and applications of various types of convexity theory, with and without differentiability assumptions, and in the framework of symmetric duality, variational problems, minimax programming, continuous time programming, and so forth. More specifically, works in the area of nonsmooth setting can be found in Chen [1], Kim et al. [2], Kuk et al. [3], Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], Preda [7], and Soleimani-damaneh [8]. Efficiency conditions and duality models for multiobjective fractional subset programming problems are studied by Preda et al. [9], Verma [10], and Zalmai [11–13]. Higher order duality in multiobjective fractional programming is discussed in Gulati and Geeta [14] and Suneja et al. [15]. Solving nonlinear multiobjective fractional programming problems by a modified objective function method is the subject matter of Antczak [16]. Further works on multiobjective fractional programming are established by Chinchuluun et al. [17], J.-C. Liu and C.-Y. Liu [18], Mishra et al. [19], Verma [20], Zhang and Wu [21], and others.

The common point in all of these developments is the convexity theory that does not stop extending itself in different directions with new variants of generalized convexity and various applications to nonlinear programming problems in different settings. The concept of invexity introduced by Hanson [22] is a generalization of convexity which has received much interest these later years, and many advances in the theory and practice have been established using this concept and its extensions. In practice, recently Dinuzzo et al. [23] have obtained some kernel function in machine learning which is not quasiconvex (and hence also neither convex nor pseudoconvex), but it is invex. Nickisch and Seeger [24] have studied a multiple kernel learning problem and have used the invexity to deal with the optimization which is nonconvex. The concept of semilocally convex functions was introduced by Ewing [25] and was further extended to semilocally quasiconvex, semilocally pseudoconvex functions by Kaul and Kaur [26, 27]. Other generalizations of semilocally convex functions and their properties were investigated in Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], Preda [7, 28], Preda and Stancu-Minasian [29], Preda et al. [30], and Stancu-Minasian [31].

In Preda [7], necessary and sufficient efficiency conditions for a nonlinear fractional multiple objective programming problem are obtained involving -semidifferentiable functions. Furthermore, a general dual was formulated and duality results were proved using concepts of generalized semilocally preinvex functions. Thus, results of Preda [28], Preda and Stancu-Minasian [29], and Preda et al. [30] were generalized. Mishra et al. [5] extended the issues of Preda [7] to the case of semilocally type I and related functions, generalizing results of Preda [7] and Stancu-Minasian [31]. Niculescu [6] extended the work of Mishra et al. [5] by using concepts of generalized -semilocally type I-preinvex functions. Mishra and Rautela [4] extended the works of Mishra et al. [5] and Preda [7] to the case of semilocally type I univex and related functions.

By considering the invexity with respect to different (each function occurring in the studied problem is considered with respect to its own function instead of the same function ), Slimani and Radjef [32–34] have obtained necessary and sufficient optimality/efficiency conditions and duality results for nonlinear scalar and (nondifferentiable) multiobjective problems. Ahmad [35] has considered a nondifferentiable multiobjective problem and, by using generalized univexity with respect to different , he has obtained efficiency conditions and duality results. Arana-Jimnez et al. [36] have used the concept of semidirectionally differentiable functions introduced in [34] to derive characterizations of solutions and duality results by means of generalized pseudoinvexity for nondifferentiable multiobjective programming.

In this paper, motivated by the work of Slimani and Radjef [34], we define semilocally vector-type I problems, where each component of the objective and constraint functions is semidifferentiable along its own direction instead of the same direction. Then we consider necessary and sufficient efficiency conditions for a nonlinear fractional multiple objective programming problem involving semidifferentiable functions. Furthermore, we formulate a general Mond-Weir dual and prove duality results using concepts of generalized semilocally V-type I-preinvex and related functions. Thus, we extend the works of Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], and Preda [7] and generalize results obtained in the literature on this topic.

2. Preliminaries and Definitions

The following conventions for equalities and inequalities will be used. If and , then , ; , ; , ; and .

We also denote by (resp., or ) the set of vectors with (resp., or ).

Let be a set and a vector application. We say that is -invex at if for any and . We say that the set is -invex if is -invex at any .

Definition 1 (see [7]). We say that the set is an -locally star-shaped set at , if, for any , there exists such that for any .

Definition 2 (see [7]). Let be a function, where is an -locally star-shaped set at . We say that is -semidifferentiable at if exists for each , where (the right derivative of at along the direction ). If is -semidifferentiable at any , then is said to be -semidifferentiable on .

Note that a function which is -semidifferentiable at is not necessarily directionally differentiable at this point (see [34]). Slimani and Radjef [34] have considered the functions whose directional derivatives exist and are finite in some directions (not necessarily in all directions) and they called them semidirectionally differentiable functions. This class of functions, where no assumptions on continuity are needed, is an extension of locally Lipschitz functions.

Definition 3 (see [37]). A function is a convex-like function if, for any and , there exists such that
We consider the following multiobjective fractional optimization problem: where , with is a nonempty subset of , and , for all and each . Let , and .
We put as the set of all feasible solutions of VFP. For , we denote by the set , where is the cardinal of set , and by (resp., ) the set . We have and if .

For such optimization problems, minimization means obtaining (weakly) efficient solutions in the following sense.

Definition 4. A point is said to be a weakly efficient solution of the problem VFP, if there exists no such that

Definition 5. A point is said to be an efficient solution of the problem VFP, if there exists no such that for some

Slimani and Radjef [34] considered semidirectionally differentiable functions and extended the -invexity of Ye [38] by introducing new concepts of generalized -invexity in which each component of the objective and constraint functions is directionally differentiable in its own direction instead of the same direction . In the same way, we define semilocally vector-type I problems, where each component of the objective and constraint functions is semidifferentiable along its own direction , , or instead of the same direction .

Definition 6. Let , and , be vector functions. We say that the problem VFP is semilocally V-type I-preinvex at with respect to , , and , if, for all , we have If the inequalities in (6) are strict (whenever ), we say that VFP is semistrictly semilocally V-type I-preinvex at with respect to , , and .

Definition 7. Let , and , be vector functions. We say that the problem VFP is semilocally pseudo quasi-V-type I-preinvex at with respect to , , and , if, for some vectors , , and , we have If the second (implied) inequality in (9) is strict (), we say that VFP is semilocally strictly pseudo quasi-V-type I-preinvex at with respect to , , and .

Definition 8. Let , and , be vector functions. We say that the problem VFP is semilocally quasi pseudo-V-type I-preinvex at with respect to , , and , if, for some vectors , , and , we have If the second (implied) inequality in (12) is strict (), we say that VFP is semilocally quasi strictly pseudo-V-type I-preinvex at with respect to , , and .

Definition 9. Let , and , be vector functions. We say that the problem VFP is semilocally extendedly pseudo partially quasi-V-type I-preinvex at with respect to , , and such that for and , if, for some vectors , , and , we have If the second (implied) inequality in (13) is strict (), we say that VFP is semilocally strictly extendedly pseudo partially quasi-V-type I-preinvex at with respect to , and . If, for all , we say that VFP is semilocally (strictly) extendedly pseudo partially quasi-V-type I-preinvex at with respect to , and .

Remark 10. In Definition 9 if and , then the concept of semilocally (strictly) extendedly pseudo partially quasi-V-type I-preinvexity reduces to the concept of semilocally (strictly) pseudo quasi-V-type I-preinvexity given in Definition 7.

Definition 11. Let , and , , be vector functions. We say that the problem VFP is semilocally extendedly quasi partially pseudo-V-type I-preinvex at with respect to , , and such that for and , if, for some vectors , , and , we have If the second (implied) inequality in (16) is strict (), we say that VFP is semilocally extendedly quasi strictly partially pseudo-V-type I-preinvex at with respect to , , and . If, for all , we say that VFP is semilocally extendedly quasi (strictly) partially pseudo-V-type I-preinvex at with respect to , and .

Remark 12. In Definition 11 if and , then the concept of semilocally extendedly quasi (strictly) partially pseudo-V-type I-preinvexity reduces to the concept of semilocally quasi (strictly) pseudo-V-type I-preinvexity given in Definition 8.

3. Necessary Efficiency Conditions

To prove necessary conditions for VFP, we need to prove the following lemma.

Lemma 13. Suppose that (i) is a (local) weakly efficient solution for VFP;(ii) is continuous at for and there exist vector functions , , and , which satisfy at with respect to the following inequalities: Then the system of inequalities has no solution .

Proof. Let be a local weakly efficient solution for VFP and suppose there exists such that inequalities (20)–(22) are true.
For , let .
We observe that this function vanishes at and = = using (17) and (20).
It follows that, for all if is in some open interval . Thus, for all , Similarly, by using (18) with (21) and (19) with (22), we get where, for all and, for all .
Now, since, for and is continuous at , then there exists such that Let . Then where is a neighborhood of . Now, for all , we have By (26) and (29), we get , for all .
Using (27) and (28), for , we get which contradicts the assumption that is a (local) weakly efficient solution for VFP. Hence, there exists no satisfying the system (20)–(22). Thus the lemma is proved.

The following lemma given by Hayashi and Komiya [39] will be used.

Lemma 14. Let be a nonempty set in and let be a convex-like function. Then either or but both alternatives are never true (here the symbol denotes the transpose of matrix).

Preda [7], Mishra et al. [5], Niculescu [6], and Mishra and Rautela [4] have given necessary conditions for to be a weakly efficient solution for VFP by taking the functions , and , semidifferentiable along the same direction . Now we give necessary efficiency criteria by considering each function (resp., ) semidifferentiable along its own direction (resp., ).

In the next theorem, we obtain Fritz John type necessary efficiency conditions.

Theorem 15 (Fritz John type necessary efficiency conditions). Suppose that (i) is a (local) weakly efficient solution for VFP;(ii) is continuous at for and there exist vector functions , and , which satisfy at with respect to inequalities (17)–(19);(iii)for all (for all ) is semidifferentiable at along the direction () and let be a convex-like function of on .Then there exists such that satisfies

Proof. If the conditions and are satisfied, then, by Lemma 13, system (20)–(22) has no solution for . Since, by hypothesis , , , is a convex-like function of on , therefore, by Lemma 14, there exists such that relation (33) is satisfied.

Remark 16. As particular case of Theorem 15, if , , , and (i.e., if we consider that all of the functions , and , are semidifferentiable at along the same direction ), we obtain Theorem 14 of Preda [7], Lemma 5 of Mishra et al. [5], Lemma 5 of Niculescu [6], and Lemma 2.5 of Mishra and Rautela [4].

Now, we define a constraint qualification given as follows.

Definition 17. Let be a feasible point of and let be vector functions. (i)The function is said to satisfy the semiconstraint qualification at with respect to , if there exist such that (ii)The function is said to satisfy the semiconstraint qualification at with respect to , if there exist such that

To prove Karush-Kuhn-Tucker type necessary efficiency conditions for VFP, we need to prove the following result.

Theorem 18. Suppose that (i) is a (local) weakly efficient solution for the following problem: where ;(ii) is continuous at for and there exist vector functions , and , which satisfy at with respect to the following inequalities: (iii)for all (for all ) is semidifferentiable at along the direction () and let be a convex-like function of on ;(iv) the function satisfies the semiconstraint qualification at with respect to . Then there exist and such that satisfies

Proof. In the same line as in the proof of Theorem 15, we prove that there exists such that relation (38) is satisfied. Now it is enough to prove that . We proceed by contradiction. If , then and (38) takes the following form: which contradicts semiconstraint qualification (34). Hence .

For each , where denotes the positive orthant of , we consider The following lemma can be proved without difficulty.

Lemma 19 (see [7]). If is a (local) weakly efficient solution for VFP, then is a (local) weakly efficient solution for (), where .

Using this lemma and Theorem 18, we can derive Karush-Kuhn-Tucker type necessary efficiency conditions for the problem VFP.

Theorem 20 (Karush-Kuhn-Tucker type necessary efficiency conditions). Suppose that(i) is a (local) weakly efficient solution for VFP;(ii) is continuous at for and there exist vector functions , and , which satisfy at with respect to inequalities (17) and (19) with (iii)for all (for all ) is semidifferentiable at along the direction () and let be a convex-like function of on , where ;(iv) the function satisfies the semiconstraint qualification at with respect to . Then there exist and such that satisfies

Proof. Let be a (local) weakly efficient solution for VFP. According to Lemma 19 we have that is a (local) weakly efficient solution for (). Now applying Theorem 18 to problem (), we get that there exist and such that relation (42) is satisfied, and the theorem is proved.

In the Karush-Kuhn-Tucker type necessary efficiency condition (42) of Theorem 20, the functions and are considered semidifferentiable at along the same direction , . To obtain a necessary condition with different directions and , we need to use the second variant of the semiconstraint qualification given in Definition 17.

Theorem 21 (Karush-Kuhn-Tucker type necessary efficiency conditions). Suppose that the hypotheses , and of Theorem 15 are satisfied and the function satisfies the semiconstraint qualification at with respect to . Then there exist , , and such that satisfies relation (33).

Proof. Based on Theorem 15, we obtain the existence of such that satisfies relation (33). Now it is enough to prove that . We proceed by contradiction. If , then and (33) takes the following form: which contradicts the semiconstraint qualification of Definition 17. Hence .

4. Sufficient Efficiency Criteria

In this section, we present some Karush-Kuhn-Tucker type sufficient efficiency conditions for a feasible solution to be efficient or weakly efficient for VFP under various types of generalized semilocally V-type I-preinvex assumptions.

Theorem 22. Let and suppose that there exist vector functions and , such that VFP is semilocally V-type I-preinvex at with respect to , , and . If there exist vectors , and such that then is a weakly efficient solution for VFP.

Proof. Suppose that is not a weakly efficient solution of VFP. Then there exists a feasible solution of VFP such that Since VFP is semilocally V-type I-preinvex at with respect to , , and , we get that inequalities (6), (7), and (8) are true for .
Multiplying (6) by , and (8) by , then summing the obtained relation and using (44), we get From the above inequality and the fact that , it follows that Since , from (48) we obtain that there exists such that By (45) and (7) it follows that Now using (49), (50), and , we obtain which is a contradiction to (46). Thus is a weakly efficient solution for VFP and the theorem is proved.

Remark 23. As particular case of Theorem 22, if , and , we obtain Theorem 1 of Mishra et al. [5].

Theorem 24. Let and suppose that there exist vector functions , and , such that VFP is semilocally V-type I-preinvex at with respect to , , and . If there exist vectors , and such that then is a weakly efficient solution for VFP.

Proof. Suppose that is not a weakly efficient solution of VFP. Then there exists a feasible solution of VFP such that Since VFP is semilocally V-type I-preinvex at with respect to , , and , we get that inequalities (6), (7), and (8) are true for .
Using these inequalities, , and (52), we get Therefore, Since , and , we obtain Since , from (57) we obtain that there exists such that that is, which is a contradiction to (53). Thus is a weakly efficient solution for VFP and the theorem is proved.

Remark 25. It is easy to see that, in Theorem 24, if the problem VFP is semistrictly semilocally V-type I-preinvex at with respect to , , and (instead of semilocally V-type I-preinvex), the point will be efficient for VFP.

Remark 26. As a particular case of Theorem 24, if , and , we obtain Theorem 2 of Mishra et al. [5].

Preda [7], Mishra et al. [5], Niculescu [6], and Mishra and Rautela [4] have given sufficient efficiency conditions for a feasible solution to be weakly efficient under various types of (generalized) -semilocally (type I-) preinvex (univex) assumptions. In the following theorem, we give sufficient efficiency conditions for a feasible solution to be efficient involving generalized semilocally V-type I-preinvex functions.

Theorem 27. Let and suppose that there exist vector functions , and vectors , and such that relation (52) is satisfied. If any one of the following conditions holds, (a)VFP is semilocally strictly pseudo quasi-V-type I-preinvex at with respect to , and and for , and ;(b)VFP is semilocally quasi strictly pseudo-V-type I-preinvex at with respect to , and and for , and ,then is an efficient solution for VFP.

Proof. Suppose that is not an efficient solution of VFP. Then there exists a feasible solution of VFP such that for some which is equivalent to Since , from (61) and (62) we obtain With condition , by using the reverse implication in (9) (in view of Definition 7), we deduce that From (in view of Definition 7), we get Now, by (64) and (65), we obtain which is a contradiction to (52).
The proof of part is very similar to the proof of part , except that, for this case, by using the implication in (11) in view of Definition 8, inequality (64) becomes not strict and, by using the reverse implication in (12), inequality (65) becomes strict (). Thus we get the contradiction again, and the theorem will be proved.

Remark 28. In Theorems 24 and 27, we do not need to require that . Thus, in Example 29 that uses Theorem 27, it is sufficient to have .

In order to illustrate the obtained results, we will give an example of multiobjective fractional optimization problem in which the efficient solution will be obtained by the application of Theorem 27, whereas it will be impossible to apply for this purpose the sufficient efficiency conditions devoted to locally Lipschitz functions or to -semidifferentiable functions (with the same ). In particular, the sufficient efficiency conditions given in Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], and Preda [7] are not applicable.

Example 29. We consider the following multiobjective fractional optimization problem: where , , and defined by The set of feasible solutions of problem is nonempty. Observe that , and are not continuous at , and, consequently, they are not locally Lipschitz at this point. In addition, clearly, the functions , and are not differentiable at but only semidifferentiable functions at that point.
There exists no function , , and such that Then, the sufficient efficiency conditions given in Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], and Preda [7] are not applicable. However, there exist vector functions , , , = , and scalars , such that relation (52) is satisfied and problem (67) is semilocally strictly pseudo quasi-V-type I-preinvex at with respect to , and and for , and . It follows that, by Theorem 27, is an efficient solution for the given multiobjective fractional optimization problem.

5. General Mond-Weir Type Duality

We associate, for VFP, a general Mond-Weir dual GMWD given as follows: where for and . Here . Let be the set of all feasible solutions of problem GMWD.

Now, we establish certain duality results between VFP and GMWD by considering that, for all (for all ) is semidifferentiable on along its own direction () instead of the same direction .

Theorem 30 (weak duality). Assume that, for all feasible for VFP and all feasible for GMWD, any of the following holds: (a)the problem VFP is semilocally extendedly pseudo partially quasi-V-type I-preinvex at with respect to , and and for , and with ;(b)the problem VFP is semilocally strictly extendedly pseudo partially quasi-V-type I-preinvex at with respect to , and and for , and ;(c)the problem VFP is semilocally extendedly quasi strictly partially pseudo-V-type I-preinvex at with respect to , and and for , and .Then the following cannot hold:

Proof. By condition , since the problem VFP is semilocally extendedly pseudo partially quasi-V-type I-preinvex at with respect to , and and for , and , then From (73) and (80), we get Now we suppose contrary to the result of the theorem that (77) and (78) hold. Hence if (77) and (78) hold for some feasible for VFP and feasible for GMWD, we obtain According to (75) and the feasibility of for VFP, we have Combining (82), (83), (84), and (72), we get Since for any , by (85), (86) and (74), we obtain where by using the reverse implication in (79) it follows that Now, from (88) and (71) we obtain which is a contradiction to (81).
The proofs of parts and are very similar to the proof of part , except that, for part , since , then inequality (87) becomes nonstrict and it follows that inequalities (88) and (89) remain true and strict and , respectively.
For part , inequality (81) becomes strict . Since , then inequality (87) becomes nonstrict and it follows that the inequalities (88) and (89) become nonstrict and , respectively. In the two cases, inequalities (89) and (81) contradict each other always. This completes the proof of the theorem.

Now we establish the following strong duality result between VFP and GMWD.

Theorem 31 (strong duality). Let be a weakly efficient solution for VFP and suppose that conditions and of Theorem 20 are satisfied. Assume also that the function satisfies the semiconstraint qualification at with respect to . Then there exist , and such that and the objective functions of VFP and GMWD have the same values at and , respectively. If, further, the weak duality between VFP and GMWD in Theorem 30 holds, then is an efficient solution of GMWD.

Proof. By Theorem 20, there exist , and such that with . The vector may be normalized according to . By setting for all and , we obtain for all and it follows that . Clearly the objective functions of VFP and GMWD have the same values at and , respectively.
Next, suppose that is not an efficient solution of GMWD. Then there exists such that , which contradicts the weak duality Theorem 30. Hence is indeed an efficient solution of GMWD.

6. Conclusion

In this paper, we have defined new concepts of semilocally V-type I-preinvex functions to study efficiency and duality for constrained fractional multiobjective programming. New Fritz John type necessary and Karush-Kuhn-Tucker type necessary and sufficient efficiency conditions are obtained for a feasible point to be weakly efficient or efficient under various types of generalized semilocally V-type I-preinvex requirements. Furthermore, a general Mond-Weir dual is formulated and weak and strong duality results are proved. The results obtained in this paper generalize and extend previously known results in this area.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the anonymous referee for his/her careful reading of the paper and for his/her valuable suggestions and comments which have helped to improve the paper.

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Copyright © 2014 Hachem Slimani and Shashi Kant Mishra. 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.

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