International Journal of Mathematics and Mathematical Sciences

Volume 2013 (2013), Article ID 284046, 10 pages

http://dx.doi.org/10.1155/2013/284046

Research Article

Optimality and Duality for Multiobjective Fractional Programming Involving Nonsmooth Generalized -Univex Functions

Jen-Chwan Liu¹ and Chun-Yu Liu²

¹Department of Mathematics and Science, National Taiwan Normal University, Linkou Campus, Renai Road, Section 1, Linkou Shiang, Taipei 24499, Taiwan

²Department of Mathematics, National Central University, Zhongli, Taoyuan, Taiwan

Received 11 July 2013; Accepted 23 October 2013

Academic Editor: Onesimo Hernandez Lerma

Copyright © 2013 Jen-Chwan Liu and Chun-Yu Liu. 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 establish properly efficient necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth generalized -univex functions. Utilizing the necessary optimality conditions, we formulate the parametric dual model and establish some duality results in the framework of generalized -univex functions.

1. Introduction

In this paper, we consider the following nondifferentiable nonconvex multiobjective fractional programming problem:

where

(a1), , and , are Lipschitz on , and is an open subset of ; (a2), , .

Minimize means obtaining efficient solution in the following sense. A point is said to be an efficient solution for (MFP) if there is no such that

with at least one strict inequality.

A point is said to be a  properly efficient solution for (MFP) which was introduced by Geoffrion [1] if and only if

(a) is an efficient solution for (MFP), (b) there exists a scalar such that for each , we have

 for some such that , whenever and .

An efficient point for (MFP) that is not properly efficient is said to be improperly efficient. Thus, for to be improperly efficient for (MFP) means that to every scalar , there is a point and an such that and

for all such that .

Many papers have been devoted to the multiobjective fractional programming problem in recent decades; see for example [1–13].

In [8], Preda introduced generalized -convexity, an extension of -convexity and generalized -convexity defined by Vial [14, 15]. Bhatia and Jain [2] defined generalized -convexity for nonsmooth functions, an extension of generalized -convexity defined by Preda [8], and they derived some duality theorems for nonsmooth multiobjective programs. In [5, 6], Liu also established the Kuhn-Tucker type necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth pseudoinvex functions in [5] or -convex functions in [6] and considered the parameter dual problem in the framework of generalized convex functions.

Recently, Zalmai [13, 16, 17] introduced generalized -univex -set functions and he also established sufficient efficiency conditions in multiobjective fractional subset programming [13] and sufficient optimality conditions in minimax fractional subset programming [16, 17] under various generalized -univexity assumptions. In [10], Preda et al. obtained duality results for a dual model of Zalmai [13] replacing the assumptions of sublinearity or convexity by that of quasiconvexity in the third argument.

In this paper, we are inspired to consider the optimality and duality of properly efficient for (MFP) containing nonsmooth generalized -univex functions. We organize this paper as follows. Some definitions and notations are given in Section 2. In Section 3, we establish some properly efficient sufficient optimality conditions for (MFP) involving nonsmooth generalized -univex functions. We also derive some duality results in Section 4.

2. Notations and Preliminary

Throughout the paper, let be the -dimensional Euclidean space and let be its nonnegative orthant.

Definition 1. The function is said to be Lipschitz on if there exists such that for all ,

where denotes any norm in .

For each in , is the generalized directional derivative of Clarke [18] defined by

It then follows that

where denotes Clarke’s generalized gradient [18].

Definition 2. A functional is said to be sublinear (superlinear) if for all , and for all and .

Let be an open subset of . We suppose that the function is Lipschitz on and such that for any .

Definition 3 (cf. [13, 16, 17, 19, 20]). (1) Let be a sublinear function, a function with positive values, a function, a real number, and a function such that . The function is said to be -univex at if

for all and .

(2) The function is said to be -pseudounivex at if

 for all and .

(3) The function is said to be -strictly-pseudounivex at if

 for all and .

(4) The function is said to be -prestrictly-pseudounivex at if

 for all and .

(5) The function is said to be -quasiunivex at if

 for all and .

(6) The function is said to be -strictly-quasiunivex at if

 for all and .

(7) The function is said to be -prestrictly-quasiunivex at if

 for all and .

In order to establish some Kuhn-Tucker conditions for efficient optimality of (MFP), we give the scalar minimization problem as follows:

In [6], we established the following necessary efficient optimality conditions for (MFP).

Lemma 4. Let . If    is an efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for , . Then there exist , , , and such that

where

If we choose , the result of Lemma 4 can be restated as follows.

Theorem 5 (necessary optimality conditions). Let . If is an (properly) efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for , . Then there exist , , , and such that

In order to simplify the complication of (MFP), we consider the following nonfractional multiobjective programming problem with a parameter :

We obtained the equivalence between (MFP) and as follows.

Lemma 6. Let and . Then is a properly efficient solution for (MFP) if and only of is a properly efficient solution for .

Proof. (a) Let be an efficient solution for . Assume that is not an efficient solution for (MFP). Then there exists such that

with at least one strict inequality. Thus, we have

or equivalently,

with at least one strict inequality which contradicts the efficient solution of for . Therefore, we have as an efficient solution for (MFP). Conversely, if is an efficient solution for   (MFP), along with the same lines of above, we can obtain that is an efficient solution for   .

(b) Let be a properly efficient solution for   (MFP). Assume that is not a properly efficient solution for   . Then every scalar , there is a point and an such that and

for all such that .

It follows that for some such that and

for all such that , or equivalently, for some such that and

for all such that , which contradicts the properly efficient solution of for (MFP). Therefore, we have is a properly efficient solution for   . Conversely, if is a properly efficient solution for   , along with the same lines of above, we can obtain that is a properly efficient solution for (MFP). Thus, the proof is complete.

Lemma 7. Let and , . If there exists such that

then is a properly efficient solution to (MFP).

Proof. Assume that is not an efficient for (MFP). Then there exists a point such that

with at least one strict inequality. Thus, we have

or equivalently,

with at least one strict inequality.

Thus, for any , we have

which contradicts the inequality (28).

It remains to show that is properly efficient. Assume that it is an improperly efficient point for the problem (MFP). Then, by Lemma 6, is an improperly efficient solution for  . Then every scalar , there is a point and an such that and

for all such that .

For any , we choose , (we may assume that such that

for all . These imply that

Summing over yield

or

which contradicts the inequality (28). Thus, the proof is complete.

3. Sufficient Optimality Conditions

Applying Lemmas 6 and 7, we can establish the following sufficient optimality conditions for properly efficient solution of (MFP).

Let

Theorem 8 (sufficient optimality conditions). Let , and let there exist , , , and such that (18), (19), and (20). Let

If any one of the following conditions holds:

(a)

(i) is -univex at , is -univex at , is superlinear, and ; (ii) for each ,    is -quasiunivex at ,    is increasing, and ; (iii);

(b)

(i) is -univex at , is -univex at , , is superlinear, and ; (ii) is -quasiunivex at ,     is increasing, and ; (iii);

(c)

(i) is -pseudounivex at and ;

(d)

(i) is -pseudounivex at , and ; (ii) is -quasiunivex at , is increasing, and ; (iii);

(e)

(i) is -prestrictly-quasiunivex at , and ; (ii) is -quasiunivex at , is increasing, and ; (iii);

(f)

(i) is -prestrictly-quasiunivex at , and ; (ii) is -strictly-pseudounivex at , is increasing, and ; (iii);

(g)

(i) is -pseudounivex at , and ; (ii) for each , is -quasiunivex at , is increasing, and ; (iii).

Then is a properly efficient solution for (MFP).

Proof. Let be any feasible solution of (MFP). By (18), there exist , , for , and , for such that

From here it results

If hypothesis (a) holds,

Now, multiplying (42) by , (43) by , and adding the resulting inequalities, and then using the superlinearity of and sublinearity of , we obtain

Since , it is clear that

Using the increasing property of and , we have

Using the -quasiunivexity of at , we get from (46)

Because and (20) holds, for each , and is sublinear, the above inequality yields

From the nonnegativity of , sublinearity of , and (41), it follows that

Now adding (44) and (48), and then using (49) and , we obtain

But , and so (50) yields

Thus, we have

Therefore, by Lemma 7, is a properly efficient solution for (MFP).

If hypothesis (b) holds, from (20), , and , we have

and so using the properties of , we obtain

Using the -quasiunivexity of at , we get from (54)

Now combining (44), (49), and (55), and using (iii), we obtain (52). Therefore, by Lemma 7, is a properly efficient solution for (MFP).

If hypothesis (c) holds, using the -pseudounivexity of , we get from (41)

In view of the properties of , we deduce from this inequality that

Equation (53) along with (57) yields

Therefore, by Lemma 7, is a properly efficient solution for (MFP).

If hypothesis (d) holds, using the properties of and (53), we obtain

which in view of (ii) implies that

Now combining (49), (60), and , we obtain

Along with the fact that the is -pseudounivex at , we get from (61) that

In view of the properties of , we deduce from this inequality that

Therefore, by Lemma 7, is a properly efficient solution for (MFP). (e), (f), and (g) follows along with the same lines of (d) and (a). Thus, the proof is complete.

4. Duality Theorem

From the optimality of properly efficient for the problem (MFP), we can formulate the following parametric dual problem:

We denote by the set of all feasible solutions of problem . The point is called an efficient for if there is no such that

with at least one strict inequality.

The point is said to be a properly efficient for if and only if

(a) is an efficient solution for , (b) there exists a scalar such that for each , we have

 for some such that , whenever and .

If was not properly efficient solution for , for each fixed real number , there would exist a feasible solution and an index such that and

for all such that .

Theorem 9 (duality theorem). Let , , and let

If any one of the following conditions holds:

(a)

(i) is -univex at , is -univex at , , is superlinear, and ; (ii) is -quasiunivex at , is increasing, and ; (iii);

(b)

(i) is -univex at , is -univex at , , is superlinear, and ; (ii) is -quasiunivex at , is increasing, and ; (iii);

(c)

(i) is -pseudounivex at and .

(d)

(i) is -pseudounivex at , and ; (ii) is -quasiunivex at , is increasing, and ; (iii);

(e)

(i) is -prestrictly-quasiunivex at , and ; (ii) is -quasiunivex at , is increasing, and ; (iii);

(f)

(i) is -prestrictly-quasiunivex at , and ; (ii) is -strictly-pseudounivex at , is increasing, and ; (iii);

(g)

(i) is -pseudounivex at , and ; (ii) is -quasiunivex at , is increasing, and ; (iii).

Assume that is a properly efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for , . Then there exist , , , and such that is a properly efficient solution for .

Proof. With Theorem 5, there exist , , , and such that is a feasible solution of . By (64), there exist , , for , and , for such that

From here it results

If hypothesis (a) holds, then

Now, multiplying (74) by , (75) by , and adding the resulting inequalities, and then using the superlinearity of and sublinearity of , we obtain

Since , , and (66), it is clear that

Using the increasing property of and , we have

Using the -quasiunivexity of at , we get from (78)

Because the nonnegativity of , and is sublinear, the above inequality yields

From the nonnegativity of , sublinearity of , and (73), it follows that

Now adding (76) and (80), and then using (81) and , we obtain

But , and so (82) yields

Equation (65) along with (83) yields

Assume that is not an efficient for . Then there exists a point such that

with at least one strict inequality.

Thus, for any , we have

which contradicts inequality (84).

It remains to show that is properly efficient. Assume that it is an improperly efficient point for the problem and choose (we may assume that ), there would exist a feasible solution and an index such that and

for all such that .

These imply that

Thus, we have

Summing over yield

or

which contradicts inequality (84). Thus, the proof of (a) is complete.

If hypothesis (b) holds, from (66), , and , then we have

and so using the properties of , we obtain

Using the -quasiunivexity of at , then we get from (93)

Now combining (76), (81), and (94), and using (iii), we obtain (84). Therefore, along with the same lines of (a), we obtain is a properly efficient solution for .

If hypothesis (c) holds, using the -pseudounivexity of , we get from (73)

In view of the properties of , we deduce from this inequality that

Equation (92) and (65) along with (96) yields

Thus, we obtain is a properly efficient solution for .

If hypothesis (d) holds, using the properties of and (92), we obtain

which in view of (ii) implies that

Now combining (81), (99), and , we obtain

Along with the fact that the is -pseudounivex at , we get from (100)

In view of the properties of , we deduce from this inequality that

Therefore, is a properly efficient solution for . (e), (f), and (g) follows along with the same lines of (d) and (a). Thus, the proof is complete.

Acknowledgments

The authors are thankful to the referees for their comments and suggestions in an earlier version of the paper. The research was partly supported by the National Science Council, Taiwan.

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