Research Article | Open Access

Volume 2014 |Article ID 541524 | 8 pages | https://doi.org/10.1155/2014/541524

# Second Order Duality in Multiobjective Fractional Programming with Square Root Term under Generalized Univex Function

Accepted10 Apr 2014
Published07 Jul 2014

#### Abstract

Three approaches of second order mixed type duality are introduced for a nondifferentiable multiobjective fractional programming problem in which the numerator and denominator of objective function contain square root of positive semidefinite quadratic form. Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterization technique is used to establish duality results under generalized second order -univexity assumption.

#### 1. Introduction

A fractional programming problem arises in many types of optimization problems such as portfolio selection, production, information theory, and numerous decision making problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical function or both, such as cost/time, cost/volume, and cost/benefit, in order to measure the efficiency or productivity of the system. Many economic, noneconomic, and indirect applications of fractional programming problem have also been given by Bector , Bector and Chandra , Craven , Mond and Weir , Stancu-Minasian , Schaible and Ibaraki , Ahmad et al. , Ahmad and Sharma , and Gulati et al. .

The central concept in optimization is known as the duality theory which asserts that, given a (primal) minimization problem, the infimum value of the primal problem cannot be smaller than the supermom value of the associated (dual) maximization problem and the optimal values of primal and dual problems are equal. Duality in fractional programming is an important class of duality theory and several contributions have been made in the past [1, 5, 8, 1014]. Second order duality provides a tighter bound for the value of the objective function when approximations are used. For more details, one can consult [15, page 93]. Another advantage of second order duality when applicable is that if a feasible point in the primal is given and first order duality does not apply, then we can use second order duality to provide a lower bound of the value of the primal problem (see ).

Multiobjective fractional programming duality has been of much interest in the recent past. Schaible  and Bector et al.  derived Fritz John and Karush-Kuhn Tucker necessary and sufficient optimality condition for a class of nondifferentiable convex multiobjective fractional programming problems and established some duality theorems. Liang et al. [17, 18] discussed the optimality condition and duality for nonlinear fractional programming. Santos et al.  discussed the optimality and duality for nonsmooth multiobjective fractional programming with generalized convexity. Bector et al.  and Xu  gave a mixed type duality for fractional programming, established some sufficient conditions, and obtained various duality results between the mixed dual and primal problem. Zhou and Wang  introduced a class of mixed type dual for nonsmooth multiobjective fractional programming and established the duality results under (, ) invexity assumption.

Duality for various forms of mathematical problems involving square roots of positive semidefinite quadratic forms has been discussed by many authors [10, 2325]. Mond  considered a nonlinear fractional programming problem involving square roots of positive semidefinite quadratic form in the numerator and denominator and proved the necessary and sufficient condition for optimality. Kim et al. [26, 27] formulated a nondifferentiable multiobjective fractional problem in which numerators contain support function. One of the most known approaches used for solving nonlinear fractional programming problem is called parametric approach. Dinklebaeh  and Jagannathan  introduced this approach that was used later by Osuna-Gmez et al.  to characterize solution of a multiobjective fractional problem under generalized convexity. Tripathy  introduced three approaches given by Dinklebaeh  and Jagannathan  for both primal and mixed type dual of a nondifferentiable multiobjective fractional programming and established the duality results under generalized -univexity.

To relax convexity assumption imposed on the function in theorems on optimality and duality, various generalized convexity concepts have been proposed. Hanson  introduced the class of invex functions. Bector et al.  introduced univex function. Mishra  derived the optimality condition for multiobjective programming with generalized univexity. Jayswal  presented minimax fractional programming under generalized -univexity assumption.

Motivated by the earlier authors in this paper, we have introduced three approaches given by Dinklebaeh  and Jagannathan  for both primal and second order mixed type dual of a nondifferentiable multiobjective fractional programming problem in which the numerator and denominator of objective function contain square root of positive semidefinite quadratic form. Also we have established the necessary and sufficient optimality condition and used a parameterization technique to establish duality results under generalized -univexity assumption.

#### 2. Notations and Preliminaries

Let be the -dimensional Euclidean space and its nonnegative orthant. The following conventions for inequality will be used throughout this paper. For any , , we denote the following.(i), for all .(ii), for all .Throughout the paper, let be a nonempty open subset of .

Consider the following nondifferentiable multiobjective fractional programming problem.

##### 2.1. Multiobjective Fractional Primal Problem

(i) MFP0. Minimize where (ii) MFP1. Minimize where are fixed parameters.(iii). Minimize ; is -dimensional strictly positive vector,all subject to same constraint where , , and ; , , are differentiable functions, and , are positive semidefinite matrices of order . In the sequel, we assume that and on for .

Let for all feasible solutions of MFP0, MFP1, and and denote , , , and . It is obvious that .

Throughout the paper, consider , , , .

Assume that satisfying or and , . For we can write .

Definition 1. The real differentiable function is said to be second order -univex at with respect to , , and , if

Definition 2. The real differentiable function is said to be second order -pseudounivex at with respect to , , and , if

Definition 3. The real differentiable function is said to be second order -quasiunivex at with respect to , , and , if

Remark 4. If , the above definitions reduce to the definitions of -univex, -pseudounivex, and -quasiunivex as introduced in .

Definition 5. A feasible point is said to be efficient for MFP1, if there exists no other feasible point in MFP1 such that , , and for some .

Definition 6 (see ). A feasible point is said to be properly efficient for MFP1, if it is efficient and there exist such that, for each and for all feasible point in MFP1 satisfying , we have for some such that .

We assume that , , for all .

Definition 7 (generalized Schwarz Inequality). Let be a positive semidefinite matrix of order . Then, for all , .

The equality holds if for some .

Let and .

Then define the set satisfying any one of the following conditions:(a), , , ;(b), , , ;(c), , , ;(d), , , .

Lemma 8 (see ). If is an optimal solution of , then is properly efficient for MFP1.

Lemma 9 (see ). is an efficient solution for MFP0 if and only if it is an efficient solution of MFP1 with .

Lemma 10 (see  necessary optimality condition). If is an optimal solution of () such that , then there exist , , and such that

Theorem 11 (sufficient optimality condition). Let be a feasible solution of MFP1 and there exist ; , , and satisfying the condition in Lemma 10 at . Furthermore suppose that the following conditions hold.(i) is second order -pseudounivex with respect to , and at , with , where , , , ; , , , , and satisfying .(ii).
Then is an efficient solution of MFP1.

Proof. Suppose that the hypothesis holds.
Since the conditions of Lemma 10 are satisfied, from (9) and (13), we have Also from hypothesis (i), we have
So we can write .
Now for , we can write .
For , we have .
Since is second order -pseudounivex with respect to , and at , we have , and using the properties of , it gives Since , the above inequality implies Suppose that is not efficient solution of MFP1; then there exist such that and , for some .
The above relation, together with the relation , implies that From the relations (5), (11), and (14), we get Consequently (20) and (21) yield This contradicts (18). Hence is an efficient solution for MFP1.

Theorem 12 (sufficient optimality condition). Let be a feasible solution of MFP1 and there exist ; , , and satisfying the condition in Lemma 10 at . Furthermore suppose that the following conditions hold.(i) is second order -pseudounivex with respect to , and at and is second order -quasiunivex with respect to , and at with and where , , , ; , , , and satisfying and .(ii). Then is an efficient solution of .

Proof. Suppose hypothesis holds.
From the relations (5), (11), and (14), we get Also from hypothesis (i), we get .
So we have the following: Hence, the -quasiunivexity of with respect to , and implies the following: From (9), we get Using (25) in (26), we get Since , we get So, we have Since is -pseudounivex with respect to , and , we obtained Using the property of and , we get If were not an efficient solution to MFP1, then, from (20), we have This contradicts (31).
Therefore, is an efficient solution for MFP1.

#### 3. Second Order Mixed Type Multiobjective Fractional Duality

(i) MMFD0. Maximize where (ii) MMFD1. Maximizewhere , ; are fixed parameters.(iii). Maximize ; is -dimensional strictly positive vector,all subject to same constraints where , , , ; are differentiable functions, , . , and , are positive semidefinite matrices of order .

For the following theorems, we assume that , , and satisfying and and .

Theorem 13 (weak duality). Let be a feasible solution for the primal MFP and let be feasible for dual SMMFD. If (i) is second order -pseudounivex with respect to , , , and for , is second order -quasiunivex with respect to , , and along with(ii), then .

Proof. Now from the primal and dual constraints, we have So Since is second order -quasiunivex with respect to , and and in view of (42), for , we have
Again from the dual constraint (36), we have Since , we have Using (43) in above equation, we get .
Since , we get .
So, we have Since is second order -pseudounivex with respect to , and , by Definition 2 and (46), we get .
Using the property of and , we get Equation (5) gives , for .
So (47) implies that Now by Schwarz Inequality and (39), we have So both (48) and (49) imply that

Theorem 14 (strong duality). Let be optimal solution for MFP and let . Then there exist ; , , and such that is a feasible solution for dual and the objective values of both primal and dual are equal to zero. Furthermore if (i) is feasible for dual, (ii) is second order -pseudounivex with respect to , and and for , is second order -quasiunivex with respect to , and along with (iii) , then is properly efficient for SMMFD0.

Proof. Since is optimal solution for (MFP), by Lemma 10, there exist ; , , and such that which can be written as These are nothing but the dual constraints.
So is feasible solution for dual problem.
And the objective values of MFP and SMMFD are equal to zero. It follows from Theorem 13 and for any feasible solution of dual that .
So is optimal solution of SMMFD. Then applying Lemmas 8 and 9, we conclude that is properly efficient for (SMMFD0).

#### 4. Special Case

If , , then our dual programming reduces to the dual programming proposed by Tripathy .

#### 5. Conclusion

In this paper, three approaches given by Dinklebaeh  and Jagannathan  for both primal and second order mixed type dual of a nondifferentiable multiobjective fractional programming problem are introduced and the necessary and sufficient optimality conditions are established and a parameterization technique is used to establish duality results under generalized second order -univexity assumption. The results developed in this paper can be further extended to higher order mixed type fractional problem containing square root term. Also the present work can be further extended to a class of nondifferentiable minimax mixed fractional programming problems.

#### Conflict of Interests

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

1. C. R. Bector, “Duality in nonlinear fractional programming,” Zeitschrift für Operations Research, vol. 17, no. 5, pp. 183–193, 1973. View at: Publisher Site | Google Scholar
2. C. R. Bector and S. Chandra, “First and Second order duality for a class of non differentiable fractional programming,” Journal of Information & Optimization Sciences, vol. 7, pp. 335–348, 1986. View at: Google Scholar
3. B. D. Craven, “Fractional programming,” in Sig Ma Series in Applied Mathematics, vol. 4, pp. 145–152, Heldermann, Berlin, Germany, 1998. View at: Google Scholar
4. B. Mond and T. Weir, “Duality for fractional programming with generalized convexity condition,” Journal of Information & Optimization Sciences, vol. 3, pp. 105–124, 1982. View at: Google Scholar
5. I. M. Stancu-Minasian, Fractional Programming; Theory, Method and Application, Kluwer Academic Publishers, Dordrecht, The Netherland, 1997.
6. S. Schaible and T. Ibaraki, “Fractional programming,” European Journal of Operational Research, vol. 12, no. 4, pp. 325–338, 1983. View at: Google Scholar
7. I. Ahmad, S. K. Gupta, N. Kailey, and R. P. Agarwal, “Duality in nondifferentiable minimax fractional programming with B-(p,r)-invexity,” Journal of Inequalities and Applications, vol. 2011, article 75, 15 pages, 2011. View at: Publisher Site | Google Scholar
8. I. Ahmad and S. Sharma, “Second-order duality for nondifferentiable multiobjective programming problems,” Numerical Functional Analysis and Optimization, vol. 28, no. 9-10, pp. 975–988, 2007. View at: Publisher Site | Google Scholar
9. T. R. Gulati, I. Ahmad, and D. Agarwal, “Sufficiency and duality in multiobjective programming under generalized type i functions,” Journal of Optimization Theory and Applications, vol. 135, no. 3, pp. 411–427, 2007. View at: Publisher Site | Google Scholar
10. I. Ahmad, Z. Husain, and S. Al-Homidan, “Second-order duality in nondifferentiable fractional programming,” Nonlinear Analysis: Real World Applications, vol. 12, no. 2, pp. 1103–1110, 2011. View at: Publisher Site | Google Scholar
11. C. R. Bector, S. Chandra, and I. Husain, “Optimality conditions and duality in subdifferentiable multiobjective fractional programming,” Journal of Optimization Theory and Applications, vol. 79, no. 1, pp. 105–125, 1993. View at: Publisher Site | Google Scholar
12. R. Jagannathan, “Duality for nonlinear fractional programs,” Zeitschrift für Operations Research, vol. 17, no. 1, pp. 1–3, 1973. View at: Publisher Site | Google Scholar
13. R. Osuna-Gómez, A. Ruíz-Canales, and P. Rufián-Lizana, “Multiobjective fractional programming with generalized convexity,” Trabajos de Investigación Operativa, vol. 8, no. 1, pp. 97–110, 2000. View at: Publisher Site | Google Scholar
14. A. K. Tripathy, “Mixed type duality in multiobjective fractional programming under generalized ρ-univex function,” Journal of Mathematical Modelling and Algorithms in Operations Research, 2013. View at: Publisher Site | Google Scholar
15. S. K. Mishra, S. Wang, and K. K. Lai, “Role of α-pseudo-univex functions in vector variational-like inequality problems,” Journal of Systems Science and Complexity, vol. 20, no. 3, pp. 344–349, 2007. View at: Publisher Site | Google Scholar
16. S. Schaible, “Fractional programming,” in Handbook of Global Optimization, R. Horst and P. M. Pardalos, Eds., pp. 495–608, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. View at: Google Scholar
17. Z. A. Liang, H. X. Huang, and P. M. Pardalos, “Optimality conditions and duality for a class of nonlinear fractional programming problems,” Journal of Optimization Theory and Applications, vol. 110, no. 3, pp. 611–619, 2001. View at: Publisher Site | Google Scholar
18. Z.-A. Liang, H.-X. Huang, and P. M. Pardalos, “Efficiency conditions and duality for a class of multiobjective fractional programming problems,” Journal of Global Optimization, vol. 27, no. 4, pp. 447–471, 2003. View at: Publisher Site | Google Scholar
19. L. B. Santos, R. Osuna-Gomez, and A. Rojas-Medar, “Nonsmooth multiobjective fractional programming with generalized convexity,” Revista Integración, vol. 26, no. 1, pp. 1–12, 2008. View at: Google Scholar
20. C. R. Bector, S. Chandra, and A. Abha, “On mixed duality in mathematical programming,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 346–356, 2001. View at: Publisher Site | Google Scholar
21. Z. Xu, “Mixed type duality in multiobjective programming problems,” Journal of Mathematical Analysis and Applications, vol. 198, no. 3, pp. 621–635, 1996. View at: Publisher Site | Google Scholar
22. H. Zhou and Y. Wang, “Optimality and mixed duality for nonsmooth multi-objective fractional programming,” Pure Mathematics and Applications, vol. 14, no. 3, pp. 263–274, 2003. View at: Google Scholar
23. I. Ahmad and Z. Husain, “Duality in nondifferentiable minimax fractional programming with generalized convexity,” Applied Mathematics and Computation, vol. 176, no. 2, pp. 545–551, 2006. View at: Publisher Site | Google Scholar
24. I. Ahmad and Z. Husain, “Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity,” Journal of Optimization Theory and Applications, vol. 129, no. 2, pp. 255–275, 2006. View at: Publisher Site | Google Scholar
25. B. Mond, “A class of nondifferentiable fractional programming problem,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 58, no. 8, pp. 337–341, 1978. View at: Google Scholar
26. D. S. Kim, S. J. Kim, and M. H. Kim, “Optimality and duality for a class of nondifferentiable multiobjective fractional programming problems,” Journal of Optimization Theory and Applications, vol. 129, no. 1, pp. 131–146, 2006. View at: Publisher Site | Google Scholar
27. D. S. Kim, Y. J. Lee, and K. D. Bae, “Duality in nondifferentiable multiobjective fractional programs involving cones,” Taiwanese Journal of Mathematics, vol. 13, no. 6 A, pp. 1811–1821, 2009. View at: Google Scholar
28. W. Dinklebaeh, “On nonlinear fractional programming,” Management Science, vol. 13, no. 7, pp. 492–498, 1967. View at: Google Scholar
29. M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545–550, 1981. View at: Google Scholar
30. C. R. Bector, S. K. Suneja, and S. Gupta, “Univex functions and univex nonlinear programming,” in Proceeding of the Administrative Science Association of Canada, pp. 115–124, 1992. View at: Google Scholar
31. S. K. Mishra, “On multiple-objective optimization with generalized univexity,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 131–148, 1998. View at: Publisher Site | Google Scholar
32. A. Jayswal, “Non-differentiable minimax fractional programming with generalized α-univexity,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 121–135, 2008. View at: Publisher Site | Google Scholar
33. A. M. Geoffrion, “Proper efficiency and the theory of vector maximization,” Journal of Mathematical Analysis and Applications, vol. 22, no. 3, pp. 618–630, 1968. View at: Google Scholar

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