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Advances in Operations Research

Volumeย 2012ย (2012), Article IDย 279215, 13 pages

http://dx.doi.org/10.1155/2012/279215

## Generalized -Type I Univex Functions in Multiobjective Optimization

^{1}Department of Mathematics, Dronacharya Government College (DGC), New Railway Road, Gurgaon 122001, India^{2}Department of Applied Sciences and Humanities, ITM University, Gurgaon 122017, India^{3}Centre for Mathematical Sciences, Banasthali University, Rajasthan 304022, India

Received 13 March 2012; Revised 5 June 2012; Accepted 11 June 2012

Academic Editor: J. J.ย Judice

Copyright ยฉ 2012 Pallavi Kharbanda et al. 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.

#### Abstract

A new class of generalized functions -type I univex is introduced for a nonsmooth multiobjective programming problem. Based upon these generalized functions, sufficient optimality conditions are established. Weak, strong, converse, and strict converse duality theorems are also derived for Mond-Weir-type multiobjective dual program.

#### 1. Introduction

Generalizations of convexity related to optimality conditions and duality for nonlinear single objective or multiobjective optimization problems have been of much interest in the recent past and thus explored the extent of optimality conditions and duality applicability in mathematical programming problems. Invexity theory was originated by Hanson [1]. Many authors have then contributed in this direction.

For a nondifferentiable multiobjective programming problem, there exists a generalisation of invexity to locally Lipschitz functions with gradients replaced by the Clarke generalized gradient. Zhao [2] extended optimality conditions and duality in nonsmooth scalar programming assuming Clarke generalized subgradients under type I functions. However, Antczak [3] used directional derivative in association with a hypothesis of an invex kind following Ye [4]. On the other hand, Bector et al. [5] generalized the notion of convexity to univex functions. Rueda et al. [6] obtained optimality and duality results for several mathematical programs by combining the concepts of type I and univex functions. Mishra [7] obtained optimality results and saddle point results for multiobjective programs under generalized type I univex functions which were further generalized to univex type I-vector-valued functions by Mishra et al. [8]. Jayswal [9] introduced new classes of generalized -univex type I vector valued functions and established sufficient optimality conditions and various duality results for Mond-Weir type dual program. Generalizing the work of Antczak [3], recently Nahak and Mohapatra [10] obtained duality results for multiobjective programming problem under invexity assumptions.

In this paper, by combining the concepts of Mishra et al. [8] and Nahak and Mohapatra [10], we introduce a new generalized class of -type I univex functions and establish weak, strong, converse, and strict converse duality results for Mond-Weir type dual.

#### 2. Preliminaries and Definitions

The following convention of vectors in will be followed throughout this paper: , ; , ; , . Let be a nonempty subset of , , be an arbitrary point of and ,โโ,โโ. Also, we denote and and and .

*Definition 2.1 (Ben-Israel and Mond [11]). *Let be an invex set. A function is called preinvex on with respect to , if for all ,

*Definition 2.2 (Clarke [12]). *The function is said to be locally Lipschitz at , if there exists a neighbourhood of and a constant such that
where denotes the euclidean norm. Also, we say that is locally Lipschitz on if it is locally Lipschitz at every point of .

*Definition 2.3 (Bector et al. [5]). *A differentiable function is said to be univex at if for all , we have

We consider the following nonlinear multiobjective programming problem:
where and the functions , . Let be a set of feasible solutions of . For , if we denote by
then

Since the objectives in multiobjective programming problems generally conflict with one another, an optimal solution is chosen from the set of efficient or weak efficient solution in the following sense by Miettinen [13].

*Definition 2.4. *A point is said to be an efficient solution of , if there exists no such that

*Definition 2.5. *A point is said to be a weak efficient solution of , if there exists no such that

Now we define a new class of -type I univex functions which generalize the work of Mishra et al. [8] and Nahak and Mohapatra [10]. Let functions and are directionally differentiable at , and are nonnegative functions defined on , and , while and be vector-valued functions.

*Definition 2.6. * is said to be -type I univex at if for all

If the inequalities in are strict (whenever ), then is said to be semistrictly -type I univex at .

*Remark 2.7. *(i) If ,โโ,โโ, then above definition becomes that of -type I function [14].

(ii) If in the above definition, the functions and are differentiable functions such that ,โโ;โโ, ; , , , then we obtain the definition of type I function [15].

*Definition 2.8. * is said to be aweak strictly pseudo-quasi -type I univex at if for all

*Definition 2.9. * is said to be strong pseudo-quasi -type I univex at if for all

*Definition 2.10. * is said to be weak strictly-pseudo -type I univex at if for all

*Definition 2.11. * is said to be aweak quasistrictly-pseudo -type I univex at if for all

*Remark 2.12. *In the above definitions, if and are differentiable functions such that ; ; , then we obtain the functions given in Mishra et al. [8].

#### 3. Sufficient Optimality Conditions

In this section, we discuss sufficient optimality conditions for a point to be an efficient solution of under generalized -type I univex assumptions. In the following theorems, and .

Theorem 3.1. *Suppose there exists a feasible solution for , vector functions and vectors and , such that *(i)*,
*(ii)* is a strong pseudo-quasi -type I univex at ,*(iii)*for any , and , ,*(iv)*,
** then is an efficient solution of .*

* Proof. *Suppose is not an efficient solution of , then there exists such that .

Since , therefore by hypothesis (iii), we get
which using hypothesis (ii) yields
Also and , so, we get
Adding the above inequalities, we obtain
which contradicts hypothesis (i). Hence the proof.

Theorem 3.2. *Suppose there exists a feasible solution for , vector functions and vectors and , such that *(i)*,
*(ii)* is a weak strictly-pseudo-quasi -type I univex at , *(iii)*for any and , ; , ,*(iv)*,
**โthen is an efficient solution of .*

*Proof. *Suppose is not an efficient solution of , then there exists such that .

As , , so, hypothesis (iii) yields
By hypothesis (ii), the above inequalities imply
Since and , we get
Adding the above inequalities, we obtain
which contradicts hypothesis (i). Hence the proof.

Theorem 3.3. *Suppose there exists a feasible solution for , vector functions and vectors and , such that *(i)*,
*(ii)* is a weak strictly-pseudo -type I univex at ,*(iii)*for any and , ; , ,*(iv)*,
**then is an efficient solution of .*

* Proof. *Suppose is not an efficient solution of , then there exists such that .

As , , so hypothesis (iii) implies
Since hypothesis (ii) holds, above inequalities imply
Also and , so we obtain
On adding and using hypothesis (iv), above inequalities yield
which contradicts hypothesis (i). Hence the proof.

Theorem 3.4. *Suppose there exists a feasible solution for , vector functions and vectors and , such that *(i)*,
*(ii)* is weak quasi-strictly-pseudo -type I univex at , *(iii)*for any and ,*(iv)*,
**then is an efficient solution of .*

*Proof. *Suppose is not an efficient solution of , then there exists such that .

Since , , therefore hypothesis (iii) yields
By hypothesis (ii), we get
Also and , so, we obtain

On adding and using hypothesis (iv), above inequalities yield
which contradicts hypothesis (i). Hence the proof.

Now, following Antczak [3], we state following necessary optimality conditions.

Theorem 3.5 (Karush-Kuhn-Tucker type necessary optimality conditions). *If *(i)* is a weakly efficient solution of , *(ii)* is continuous at for ,*(iii)*there exists a vector functions ,*(iv)*for all and , and are directionally differentiable at and the functions and are preinvex functions of on , *(v)*the function satisfies the generalized Slater's constraint qualification at , **then there exists and such that
*

#### 4. Mond-Weir Type Duality

In this section, we consider Mond-Weir type dual of and establish weak, strong, converse, and strict converse duality theorems. In this section, we denote . where ,โโ,โโ,โโ. Let be the set of feasible points of .

Theorem 4.1 (Weak Duality). *Let and be the feasible solutions for and respectively. If *(i)* is a weak strictly-pseudo-quasi -type I univex at , *(ii)*for any and ,*(iii)*,
** then
*

*Proof. *Suppose to the contrary that
Since , , hypothesis (ii) yields
As hypothesis (i) holds, therefore the above inequalities imply
Also , so, we obtain
On adding above inequalities and using hypothesis (iii), we get
which is a contradiction to the dual constraint. Hence the proof.

The proofs of the following weak duality theorems are similar to Theorem 4.1 and hence are omitted.

Theorem 4.2 (Weak Duality). *Let and be the feasible solutions for and , respectively, with , . If *(i)* is a strong pseudo-quasi -type I univex at , *(ii)*for any and ,*(iii)*,
** then .*

Theorem 4.3 (Weak Duality). *Let and be the feasible solutions for and , respectively. If *(i)* is weak strictly-pseudo -type I univex at , *(ii)*for any and ,*(iii)*,
** then .*

Corollary 4.4. *Let and be the feasible solutions for and , respectively, such that . If the weak duality holds between and for all feasible solutions of two problems, then is efficient for and is efficient for .*

* Proof. *Suppose that is not efficient for , then for some
which contradicts weak duality theorems as is feasible for and is feasible for . So, is efficient for . Similarly is efficient for .

Theorem 4.5 (Strong Duality). *Let be a weakly efficient solution of , is continuous at for , are directionally differentiable at with , and as preinvex functions on . Also if satisfies the generalized Slater's constraint qualification at , then such that is feasible for and the objective function values of and are equal. Moreover, if any of weak duality theorem holds, then is an efficient solution of .*

* Proof. *Since is a weakly efficient solution of , therefore by Theorem 3.5, there exists such that

It follows that and therefore feasible for . Clearly objective function values of and are equal at optimal points.

Suppose is not an efficient solution for . Then such that , which contradicts weak duality theorems. Therefore is an efficient solution of . Hence the proof.

Theorem 4.6 (Converse Duality). *Let be a feasible solution of . If *(i)* is a weak strictly-pseudo-quasi -type I univex at , *(ii)*for any and ; โโ,*(iii)*,
**then is an efficient solution of .*

* Proof. *Suppose that is not an efficient solution of . Then such that
Now proceeding as in Theorem 4.1 (Weak Duality), we obtain a contradiction. Hence is an efficient solution of .

Theorem 4.7 (Strict Converse Duality). *Let and be the feasible solutions of and , respectively. If *(i)*,
*(ii)* is a weak quasi-strictly-pseudo -type I univex at , *(iii)*for any and ,*(iv)*,
**then .*

* Proof. *Suppose .

Since is a feasible solution of , therefore by hypothesis (i) and hypothesis (iii), we get
By hypothesis (ii), we obtain
Since , therefore the above inequalities yield
which on adding gives
which is a contradiction to feasibility of . Hence .

#### References

- M. A. Hanson, โOn sufficiency of the Kuhn-Tucker conditions,โ
*Journal of Mathematical Analysis and Applications*, vol. 80, no. 2, pp. 545โ550, 1981. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - F. A. Zhao, โOn sufficiency of the Kuhn-Tucker conditions in nondifferentiable programming,โ
*Bulletin of the Australian Mathematical Society*, vol. 46, no. 3, pp. 385โ389, 1992. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - T. Antczak, โMultiobjective programming under $d$-invexity,โ
*European Journal of Operational Research*, vol. 137, no. 1, pp. 28โ36, 2002. View at Publisher ยท View at Google Scholar - Y. L. Ye, โ$d$-invexity and optimality conditions,โ
*Journal of Mathematical Analysis and Applications*, vol. 162, no. 1, pp. 242โ249, 1991. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - C. R. Bector, S. K. Suneja, and S. Gupta, โUnivex functions and univex nonlinear programming,โ in
*Proceeding of the Administrative Sciences Association of Canada*, pp. 115โ124, 1992. - N. G. Rueda, M. A. Hanson, and C. Singh, โOptimality and duality with generalized convexity,โ
*Journal of Optimization Theory and Applications*, vol. 86, no. 2, pp. 491โ500, 1995. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - S. K. Mishra, โOn multiple-objective optimization with generalized univexity,โ
*Journal of Mathematical Analysis and Applications*, vol. 224, no. 1, pp. 131โ148, 1998. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - S. K. Mishra, S.-Y. Wang, and K. K. Lai, โOptimality and duality for multiple-objective optimization under generalized type I univexity,โ
*Journal of Mathematical Analysis and Applications*, vol. 303, no. 1, pp. 315โ326, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - A. Jayswal, โOn sufficiency and duality in multiobjective programming problem under generalized $\alpha $-type I univexity,โ
*Journal of Global Optimization*, vol. 46, no. 2, pp. 207โ216, 2010. View at Publisher ยท View at Google Scholar - C. Nahak and R. N. Mohapatra, โ$d-\rho -\eta -\theta $-invexity in multiobjective optimization,โ
*Nonlinear Analysis*, vol. 70, no. 6, pp. 2288โ2296, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - A. Ben-Israel and B. Mond, โWhat is invexity?โ
*Journal of the Australian Mathematical Society B*, vol. 28, no. 1, pp. 1โ9, 1986. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - F. H. Clarke,
*Optimization and Nonsmooth Analysis*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983. - K. Miettinen,
*Nonlinear Multiobjective Optimization*, vol. 12 of*International Series in Operations Research & Management Science*, Kluwer Academic, Boston, Mass, USA, 1999. - S. K. Mishra, S. Y. Wang, and K. K. Lai, โOptimality and duality in nondifferentiable and multiobjective programming under generalized $d$-invexity,โ
*Journal of Global Optimization*, vol. 29, no. 4, pp. 425โ438, 2004. View at Publisher ยท View at Google Scholar - M. A. Hanson and B. Mond, โNecessary and sufficient conditions in constrained optimization,โ
*Mathematical Programming*, vol. 37, no. 1, pp. 51โ58, 1987. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH