ISRN Applied Mathematics

Volume 2014 (2014), Article ID 904640, 11 pages

http://dx.doi.org/10.1155/2014/904640

Research Article

Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

Pallavi Kharbanda,¹ Divya Agarwal,² and Deepa Sinha³

¹Centre for Mathematical Sciences, Banasthali University, Rajasthan 304022, India

²Department of Applied Sciences and Humanities, ITM University, Gurgaon 122017, India

³Department of Mathematics, South Asian University, New Delhi 110021, India

Received 18 February 2014; Accepted 25 March 2014; Published 22 May 2014

Academic Editors: P.-y. Nie and S. Utyuzhnikov

Copyright © 2014 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.

We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of --V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems 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. Consequently, various generalizations of convex functions have been introduced in the literature (see Hanson [1], Vial [2], Hanson and Mond [3], Jeyakumar and Mond [4], Hanson et al. [5], Liang et al. [6], and Gulati et al. [7]).

Nonsmooth optimization provides analytical tools for studying optimization problems involving functions that are not differentiable in the usual sense. Several nonlinear analysis problems arise from areas of optimization theory, game theory, differential equations, mathematical physics, convex analysis, and nonlinear functional analysis. For a nondifferentiable multiobjective programming problem, there exists a generalization of invexity to locally Lipschitz functions with gradients replaced by the Clarke generalized subgradient. Instead of Clarke generalized subgradient, Ye [8] used the concept of directional derivative to define the class of invex functions. Also, he derived necessary and sufficient optimality conditions taking functions and to be convex. However, Antczak [9] considered the directional derivatives of objective and constraint functions to be preinvex and derived duality results for Wolfe type, Mond-Weir type, and mixed type dual programs. Mishra and Noor [10] extended the class of functions to --type I functions and obtained sufficient optimality and duality results for Mond-Weir type multiobjective dual program. Nahak and Mohapatra [11] obtained duality results for multiobjective programming problem under --- invexity assumptions. Slimani and Radjef [12] introduced a far wider class of nondifferentiable functions called --type I functions in which each component is directionally differentiable in its own direction instead of the same direction and established sufficient optimality and duality results.

On the other hand, Bector et al. [13] generalized the notion of convexity to univex functions. Rueda et al. [14] obtained optimality and duality results for several mathematical programs by combining the concepts of type I and univex functions. Mishra [15] obtained optimality results and saddle point results for multiobjective programs under generalized type I univex functions. Generalizing the functions, Mishra et al. [16] obtained duality results for a nondifferentiable multiobjective programming problem under generalized -univexity. As an extension, Ahmad [17] introduced a new class of --type I univex functions which was generalized to a class of ----type I univex functions by Kharbanda et al. [18].

In this paper, we introduce a new generalized class of ---type I univex functions which generalizes the class of functions introduced by Kharbanda et al. [18], Ahmad [17], Slimani and Radjef [12], Mishra and Noor [10], Mishra et al. [16], Antczak [9], Suneja and Srivastava [19], and Ye [8]. Further, we establish weak, strong, converse, and strict converse duality results for Mond-Weir type multiobjective dual program.

2. Preliminaries and Definitions

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

Definition 1 (Weir and Mond [20] and Weir and Jeyakumar [21]). The function is called preinvex on if, for all , there exists a vector function such that ,  , one has

Definition 2 (Mititelu [22]). The set is said to be invex at with respect to , if, for each ,

is said to be an invex set with respect to , if is invex at each with respect to same .

Definition 3 (Antczak [9]). Let be an invex set. An -dimensional vector-valued function is said to be preinvex with respect to , if each of its components is preinvex on with respect to the same function .

Definition 4 (Clarke [23]). The function is said to be locally Lipschitz at , if there exist a neighbourhood of and a constant such that

where denotes the Euclidean norm. Also, one says that is locally Lipschitz on if it is locally Lipschitz at every point of .

Definition 5 (Bector et al. [13]). A differentiable function is said to be univex at with respect to if, , one has

Definition 6 (Mishra et al. [16]). Let be a nonempty open set. The function is called -univex at with respect to if it is directionally differentiable at such that, for any ,

where denotes the directional derivative of at in the direction :

If the above inequalities are satisfied at any point , then is said to be -univex on with respect to .

Definition 7 (Slimani and Radjef [12]). The function is said to be semidirectionally differentiable at in the direction if its directional derivative exists finite for all .

Definition 8 (Slimani and Radjef [12]). Let be a nonempty open set. The function is called -invex at with respect to , if, for any ,

where is semidirectionally differentiable at in direction , for .

We consider the following nonlinear multiobjective programming problem:

where and the functions , , and is a nonempty open subset of . Let be the set of feasible solutions of (MP). For , if we denote

then

Now we define a new class of ---type I univex functions where is a functional which for any satisfies the following properties:

(i), for all , (subadditive in third argument);(ii), for all ,   and for all , (positive homogeneous in third argument);(iii), for all .

Let the functions   and   where and are semidirectionally differentiable functions in the directions and for and . Also, let be the vectors in whose components are the functions , respectively, for , , while and whose components are in and ;   and are nonnegative functions defined on , , and .

Definition 9. is said to be ---type I univex at with respect to and if for all

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

Remark 10. (i) If in the above definition,   = , = , for all and , then we obtain the definition of ----type I univex function given by Kharbanda et al. [18]. Also, if in addition, we take , then the above definition reduces to definition of --type I univex function introduced by Ahmad [17].

(ii) If is same as in (i) and ,  , for all and   and , , and , then the above definition becomes definition of --type I function introduced by Slimani and Radjef [12].

(iii) If and are differentiable functions and = , = , and ,   and    and  ,  ,  , then above definition reduces to -type I functions given by Hanson et al. [5]. Also, if, in addition, we take , then we get the definition of type I function defined by Hanson and Mond [3].

(iv) If, in the above definition,   = , = , and , , and and , , and , then we obtain the definition of -type I function introduced by Suneja and Srivastava [19].

Definition 11. is said to be quasi ---type I univex at with respect to and , if for some vectors ,   and for all

If the second (implied) inequality in is strict , then is said to be semistrictly quasi ---type I univex at with respect to and .

Definition 12. is said to be pseudo ---type I univex at with respect to and , if for some vectors ,   and for all

If the second (implied) inequality in (resp., ) is strict , then is said to be semistrictly pseudo ---type I univex in (resp., ) and if the second (implied) inequalities in and are both strict, then is said to be strictly pseudo ---type I univex at with respect to and .

Definition 13. is said to be quasi-pseudo ---type I univex at with respect to and , if for some vectors ,   and for all

If the second (implied) inequality in is strict , then is said to be quasistrictly pseudo ---type I univex at with respect to and .

Definition 14. is said to be pseudo-quasi ---type I univex at with respect to and , if for some vectors ,   and for all

If the second (implied) inequality in is strict , then is said to be strictly pseudo-quasi ---type I univex at with respect to and .

3. Illustration

In this section, we give some nontrivial examples which illustrate that the class of functions introduced in this paper is nonempty.

Example 15. Let and be defined by

Let , , , , and ,   where , .

Also, let , , , , , and .

The set of feasible solutions of problem is nonempty. Clearly, and are semidirectionally differentiable at with and .

It is easy to see that for all

Therefore is ---type I function at .

However, if we take , then

Thus is neither ----type I univex function given by Kharbanda et al. [18] nor --type I univex function at as given by Ahmad [17].

Hence the above example clearly illustrates that the class of ---type I univex functions is more generalized than the class of ----type I univex functions and the class of --type I univex functions.

Next we show that is pseudo-quasi ---type I univex function but not ---type I univex function.

Example 16. Let and be defined by

Let , , , , , , , , ,  , and .

Also, let , , , , , , , , , , , , ,  , and ,   where , , , and , , .

The set of feasible solutions of problem is nonempty. Clearly, and are semidirectionally differentiable at with ,  , , , and .

It is easy to see that for all

Therefore is pseudo-quasi ---type I univex function at .

However, for the above defined problem, if we take

(i),

(ii),

So is not ---type I univex function at .

4. Sufficient Optimality Conditions

In this section, we discuss some sufficient optimality conditions for a point to be an efficient solution of (MP) under newly defined class of ---type I univex functions.

Theorem 17. Suppose there exist a feasible solution of (MP) and vector functions ,  ,  ,  , and scalars ,  , and ,   such that

(i), ,(ii)for any , and , , ,(iii) is pseudo-quasi ---type I univex at with respect to and , and(iv),

then is an efficient solution of (MP).

Proof. Suppose that is not an efficient solution of (MP). Then there exists an of (MP) such that .

As ,  ,  , therefore

Also ,  ,  ,   imply

Since hypothesis (ii) holds, therefore inequality (24) and equality (25) become

Using hypothesis (iii), we obtain

The above inequalities along with subadditivity of yield

But as hypothesis (i) holds and and for , therefore

Thus we get a contradiction and hence the proof.

Theorem 18. Suppose there exist a feasible solution of (MP) and vector functions ,  , and ,  , and scalars , , and ,   satisfying

(i), for all ,(ii)for any , and , , ,(iii) is strictly pseudo ---type I univex at with respect to and , and(iv),

then is an efficient solution of (MP).

Proof. Suppose that is not an efficient solution of (MP). Then there exists of (MP) such that .

As , , , , and , ,  , , therefore

Using hypothesis (ii), we obtain

Since is strictly pseudo ---type I univex at , therefore, the above inequalities yield

Using subadditivity of and hypothesis (iv), we get

But hypothesis (i) and properties of imply

which leads to a contradiction. Hence is an efficient solution of (MP).

In order to illustrate the result obtained, we will give an example of a multiobjective optimization problem in which the efficient solution will be obtained by the application of Theorem 18.

Example 19. Let

where and be defined as

Let , , , , , , , , , and .

Also, let , , , , , , , , , , , , and where ,  ,, and  , ,  .

The set of feasible solutions of problem is nonempty. Clearly, and are semidirectionally differentiable at with ,  ,  , and .

It is easy to see that for all

Hence is strictly pseudo ---type I univex function at .

Also, hypotheses (i), (ii), and (iv) of Theorem 18 are clearly satisfied and it follows that is an efficient solution of the above defined multiobjective optimization problem, whereas it will be impossible to apply for this purpose the sufficient optimality conditions given in Kharbanda et al. [18], Ahmad [17], Slimani and Radjef [12], Mishra and Noor [10], Mishra et al. [16], Antczak [9], Suneja and Srivastava [19], and Ye [8].

Theorem 20. Suppose there exist a feasible solution of (MP) and vector functions ,  , and ,  , and scalars ,  , and ,  , satisfying

(i), for all ,(ii)for any , and ,  , , and(iii).

Also, if either the fact that

(a) is semistrictly quasi ---type I univex at with respect to and , or(b) is quasistrictly pseudo ---type I univex at with respect to and , or(c) is strictly pseudo-quasi ---type I univex at with respect to and holds,

then is an efficient solution of (MP).

Proof. If (a) or (c) holds, then proceeding as in previous theorem we get

And if (b) holds, we get

The remaining part of proof runs on the lines of the proof of Theorem 18.

Now, following Antczak [9] and Slimani and Radjef [12], we state the following necessary optimality conditions.

Theorem 21 (Karush-Kuhn-Tucker type necessary optimality conditions). If

(i) is a weakly efficient solution of (MP),(ii) is continuous at for ,(iii)there exist vector functions , , and ,  , such that at the following inequalities are satisfied with respect to :

(iv)for all and , and are semidirectionally differentiable at and the functions , , and , , are preinvex functions of on ,(v)the function satisfies -constraint qualification at with respect to , then there exist and such that

5. Mond-Weir Type Duality

In this section, we consider Mond-Weir type dual of (MP) and establish weak, strong, converse, and strict converse duality theorems. Consider

where , , , , for all , and , for all . Let be the set of feasible points of (MWD).

Theorem 22 (weak duality). Let and be the feasible solutions of (MP) and (MWD), respectively, with . If

(i) is pseudo-quasi ---type I univex at with respect to and ,(ii)for any ,   and , , , and(iii),

then

Proof. Suppose to the contrary that

As , , , we get

Since , , , therefore

Using hypothesis (ii), we get

Since hypothesis (i) holds, therefore the above inequalities yield

Using subadditivity of and hypothesis (iii), we get

However, the feasible condition (42) and properties of imply

Thus we get a contradiction and hence the proof.

The proof of the following theorems runs on the lines of the proof of Theorem 22.

Theorem 23 (weak duality). Let and be the feasible solutions of (MP) and (MWD), respectively. If

(i) is pseudo-quasi ---type I univex at with respect to and ,(ii)for any , and , , , and(iii),

then

Theorem 24 (weak duality). Let and be the feasible solutions of (MP) and (MWD), respectively. If

(i) is quasistrictly pseudo ---type I univex at with respect to and ,(ii)for any , and ,  , , and(iii),

then

Theorem 25 (strong duality). Let be a weakly efficient solution of (MP) and is continuous at for . Also, the vector functions ,   and ,   exist for which

at and are semidirectionally differentiable at with and as preinvex functions on . Also if satisfies -constraint qualification at , then and such that is feasible for (MWD) and the objective function values of (MP) and (MWD) are equal. Moreover, if any weak duality holds, then is a weakly efficient solution of (MWD).

Proof. Since is a weakly efficient solution of (MP), therefore, by Theorem 21, there exist and such that

It follows that and therefore feasible for (MWD). Clearly, objective function values of (MP) and (MWD) are equal at these points.

Suppose is not a weakly efficient solution for (MWD). Then such that which contradicts weak duality theorems. Therefore is a weakly efficient solution of (MWD).

Theorem 26 (converse duality). Let be a feasible solution of (MWD). Suppose hypotheses of Theorem 24 hold at , then is an efficient solution of (MP).

Proof. Suppose that is not an efficient solution of (MP). Then such that

Hence, by Theorem 24 (weak duality), we obtain a contradiction. Therefore is an efficient solution of (MP).

Theorem 27 (strict converse duality). Let and be the feasible solutions of (MP) and (MWD), respectively. If

(i), (ii) is semistrictly quasi ---type I univex at with respect to and ,(iii)for any ,   and , , , and(iv),

then .

Proof. Suppose .

Since , , , and hypothesis (i) holds, therefore

As ,  , and is feasible solution of (MWD), therefore

Using hypothesis (iii), we obtain

Applying hypothesis (ii) to the above inequalities, we get

Using subadditivity of and hypothesis (iv), we get

which is a contradiction as feasible condition (42) holds and and for yield

Hence .

Conflict of Interests

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

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