﻿Nondifferentiable Multiobjective Programming with Equality and Inequality Constraints

Open Journal of Modelling and Simulation
Vol.1 No.2(2013), Article ID:30709,7 pages DOI:10.4236/ojmsi.2013.12002

Nondifferentiable Multiobjective Programming with Equality and Inequality Constraints

Iqbal Husain, Vikas K. Jain

Department of Mathematics, Jaypee University of Engineering and Technology, Guna, India

Email: ihusain11@yahoo.com, jainvikas13@yahoo.com

Copyright © 2013 Iqbal Husain, Vikas K. Jain. 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.

Received February 10, 2013; revised March 18, 2013; accepted March 31, 2013

Keywords: Nondifferentiable Multiobjective Programming Problem; Efficient Solution; Generalized Invexity; Duality; Nondifferentiable Multiobjective Programming Problems with Equality and Inequality

ABSTRACT

In this paper, we derive optimality conditions for a nondifferentiable multiobjective programming problem containing a certain square root of a quadratic form in each component of the objective function in the presence of equality and inequality constraints. As an application of Karush-Kuhn-Tucker type optimality conditions, a Mond-Weir type dual to this problem is formulated and various duality results are established under generalized invexity assumptions. Finally, a special case is deduced from our result.

1. Introduction

A number of researchers have discussed optimality and duality for a class of nondifferentiable problem containing the square root of a positive semi-definite quadratic form. Mond [1] presented Wolfe type duality while Chandra et al. [2] investigated Mond-Weir type duality for this class of problems. Later, Zhang and Mond [3] validated various duality results for the problem under generalized invexity conditions, it is observed that the popularity of this kind of problems seems to originate from the fact that, even through the objective functions, and/or constraint function are non-smooth, a simple and elegant representation for the dual to this type of problems may be obtained. Obviously non-smooth mathematical programming with more general type functions by means of generalized sub differentials. However, the square root of positive semi-definite quadratic form is one of some of a nondifferentiable function for which sub differentials can be explicitly be written.

Multiobjective optimization problems have been applied in various field of science, where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Researchers study multiobjective optimization problems from different viewpoints and, then there exist different goals when setting and solving them. The goal may be finding a representation set of Pareto optimal solutions, and/or qualifying the trade-offs in satisfying the different objectivesand/or finding a single solution that satisfies the preferences of a human decisions making. Motivated with these observations, there has been an increasing interest in studying optimality and duality for nondifferentiable multiobjective programming problems. Duality results for nondifferentiable multiobjective programming problems with square root term appearing in each component of the vector objective derived by Lal et al. [4]. In nondifferentiable multiobjective programming problems, having a support function in each component of the vector objective, further developments for duality results are found in Kim et al. [5] and Yang et al. [6].

In this paper, we obtain optimality conditions for a class of nondifferentiable multiobjective programming problems with equality and inequality involving a square root terms in each component of the objective. For this class of problems, Mond-Weir type dual is formulated and usual duality results are obtained. In the end a special case is generated.

2. Related Pre-Requisites and Expression of the Problem

In [1], the following problem is considered:

Problem (EP): Minimize

subject to

where 1), and are continuously differentiable.

2) B is an symmetric positive semi definite matrix.

The following generalized Schwartz inequality [7] will be needed in the present analysis:

The equality in the above holds if, for

.

The function, being convex and everywhere finite, has a subdifferential in the sense of convex analysis. The subdifferential of is given by

We also require the Mangasarian-Fromovitz constraint qualification which is described as the following:

Let be the set of feasible solution of the problem (EP), that is,

and by the set of inequality active constraint indices, that is,

where. We say the Mangasarian-Fromovitz constraint qualification holds at when the equality constraint gradients are linearly independent and there exist a vector such that

The following theorems (Theorem 2.1 and Theorem 2.2) give Fritz John and Karush-Kuhn-Tucker type optimality conditions using the concept of sub differential obtained by Husain and Srivastav [8] using the concept of subdifferential:

Theorem 2.1 (Fritz John Optimality Conditions): If is an optimal solution of (EP), then there exist Lagrange multipliers such that

If Mangasarian-Fromovitz constrain qualification (MFCQ) holds at, then the above theorem reduces to the following theorem giving Karush-Kuhn-Tucker optimality conditions:

Theorem 2.2 (Karush-Kuhn-Tucker optimality conditions): If is an optimal solution of (EP) and MFCQ holds at, then there exist such that

The following conventions for inequalities will be used in the subsequent analysis: If, then

Consider the following multiobjective programming problem containing square root of a certain quadratic form in each component of the objective.

(VEP): Minimize

Subject to

(1)

(2)

where f, g and h are the same as in (EP).

Let

Definition 2.1 A point is said to be an efficient solution of (EP) if there exists no such that

for some and

for.

The following results relate an efficient solution of (EP) of k-scalar objective programming problems.

Lemma 2.2 (Chankong and Haimes [9]): A point is an efficient solution of (EP) if and only if is an optimal solution of for each.

: Minimize

subject to

We recall the following definitions of generalized invexity which will be used to derive various duality results.

Definitions 2.2: 1) A function is said to be quasi-invex with respect to a vector function if

2) A function is said to be pseudo-invex with respect to a vector functionif

.

3) is said to be the strictly pseudoinvex with respect to if

.

Equivalently, if

3. Optimality Conditions

In this section, the optimality conditions for the problem (EP) are obtained.

Theorem 3.1 (Fritz John Type Optimality Conditions): If be an efficient solution of (EP), then there exist such that

Proof: Since is an efficient solution of (EP), by Lemma 2.1is an optimal solution of for each and hence in particular of. Therefore by Theorem 2.1 there exist

such that

The theorem follows.

Theorem 3.2 (Kuhn-Tucker type necessary optimality conditions): If be an optimal solution of (VEP) and let for, the constraints of satisfy MECQ. Then there exist and such that

Proof: Since is an optimal solution of (VEP), by Lemma 3.1, is an optimal solution of for each r. As for some r, the constraint of satisfy MFCQ at, by Theorem 2.2 of their exist such that

yielding

From the above relation it is obvious that the theorem follows.

In Theorem 3.2, we assume MFCQ for some, which implies. In the following theorem, we assume MFCQ for every and obtain.

Theorem 3.3 (Kuhn-Tucker type optimality conditions): If be an efficient of (VEP) and let for each, the constraints of satisfy MECQ at. Then there exist andsuch that

Proof: Since is an efficient of (VEP) by Lemma 3.1, is an optimal solution of, by Kuhn-Tucker type necessary optimality conditions, for each, there exist

such that

Summing over, we get

where

where

Dividing throughout the above relation and setting, by

We obtain,

or

4. Mond-Weir Type Duality

We formulate the following differentiable multiobjective dual nonlinear problem for (VEP):

(M-WED): Maximize

subject to

(3)

(4)

(5)

(6)

(7)

(8)

In the following, we shall use for the set of feasible solutions of (M-WED)

Theorem 4.1 (Weak Duality): Let and such that with respect to the same

1) is pseudoinvex 2) is quasi-invex, and 3) is quasi-invex.

Then

(9)

(10)

cannot hold.

Proof: Suppose the contrary that (9) and (10) hold. Since the above inequalities (9) and (10) give

These inequalities because of quasi-invexity of and imply

Combining these, we give

Using the equality constraint of (M-WED), this yields,

This, because of 1), implies

Using this yields,

Hence the result follows.

Theorem 4.2 (Strong Duality): Letsatisfy MFCQ and be an efficient solution (VEP). Then there exist such that is feasible for (M-WED) and the two objective functions are equal. Furthermore, if the weak duality holds for all feasible solution of (VEP) and (M-WED), then is an efficient solution of the (M-WED).

Proof: Sinceis an efficient solution (VEP) satisfy MFCQ, therefore by Theorem (3.3), there exist satisfy

Hence satisfies the constraints of (M-WED) and

i.e. the two objective functions have the same value.

Now we claim that is an efficient solution of (M-WED). If not, then there exist

As, we have

This contradicts Theorem 4.1 Hence is an efficient solution.

Theorem 4.3 (Strict-converse duality): Let and be an efficient solution of (VEP) and (M-WED), such that

(11)

If with respect to the same

(A1) is strictly pseudoinvex(A2) is quasi-invex, and

(A3) is quasi-invex then

Proof: Let By hypothesis (A1), we have from (11)

(12)

By hypothesis (A2) and (A3) we have

(13)

(14)

Combining (12), (13) and (14), we have

which contradict the equality constraint of (M-WED). Hence

Theorem 4.4 (Converse duality):

Let be an efficient solution of

(M-WED) at which 1) the matrix is positive or negative definite and 2) the vectors are linearly independent .

If, for all feasible,

is pseudoinvex, is qausi-invex and is quasi-invex with respect to the same, then is an efficient solution (EP).

Proof: By Theorem 3.3, there exist and such that

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

Multiplying (15) by and summing over i, we have

(27)

Using the equality constraint, we have

(28)

(29)

From (18) and (19), we have

Using these in (15), we have

which because of the hypothesis 1) gives

Using and the hypothesis 2), we have

(30)

Let, i = 1, 2, ∙∙∙, k. Then (30) implies The relations (17) and (18) implies and. Using and along with (24) in (16), we get.

Thus a contradiction to (26). Hence Consequently Using, , in (18) and (19), we have

This implies that.

If in (16), we have

(31)

Hence by Schwartz inequality

(32)

If then (24) implies

Consequently (32) yields

If then (31) implies So we still get

(33)

Thus by (33), we have

implying the two objective functions have the same value.

Now, assume that is not an efficient solution of (VEP), then there exists such that

for some and

Using, we have

for some

This contradicts Theorem 4.1. Henceis an efficient solution of (VEP).

5. A Special Case

If and our problems reduce to the following problems recently studied by Husain and Srivastav [8]:

(EP): Minimize

subject to

(M-WED): Maximize

subject to

6. Conclusion

In this research optimality conditions are derived for a nondifferentiable multiobjective programming problem containing a certain square root of a quadratic form in each component of the objective function in the presence of equality and inequality constraints embodying many realistic problems. A Mond-Weir type dual to this problem is formulated and usual duality theorems are proved under appropriate generalized invexity conditions. A special case is also obtained from our duality results. Our results can be revisited in the multiobjective setting of a nondifferentiable control problem.

7. Acknowledgements

The authors acknowledge anonymous referees for their valuable comments which have improved the presentation of this research paper.

REFERENCES

1. B. Mond, “A Class of Nondifferentiable Mathematical Programming Problems,” Journal of Mathematical Analysis and Applications, Vol. 46, No. 1, 1974, pp. 169-174. doi:10.1016/0022-247X(74)90289-3
2. S. Chandra, B. D. Craven and B. Mond, “Generalized Concavity and Duality with a Square Root Term,” Optimization, Vol. 16, No. 5, 1985, pp. 654-662. doi:10.1080/02331938508843062
3. J. Zhang and B. Mond, “Duality for a Nondifferentiable Programming Problem,” Bulletin of the Australian Mathematical Society, Vol. 55, No. 1, 1997, pp. 29-44. doi:10.1017/S0004972700030513
4. S. N. Lal, B. Nath and A. Kumar, “Duality for Some Nondifferentiable Static Multiobjective Programming Problems,” Journal of Mathematical Analysis and Applications, Vol. 186, No. 3, 1994, pp. 862-867. doi:10.1006/jmaa.1994.1337
5. S. J. Kim, M. H. Kim and D. S. Kim, “Optimality and Duality for a Class of Nondifferentiable Multiobjective Fractional Programming Problems,” Journal of Optimization Theory and Applications, Vol. 129, No. 1, 2006, pp. 131-146. doi:10.1007/s10957-006-9048-1
6. X. M. Yang, K. L. Teo and X. Q. Yang, “Duality for a Class of Nondifferentiable Multiobjective Programming Problem,” Journal of Mathematical Analysis and Applications, Vol. 252, No. 2, 2000, pp. 999-1005. doi:10.1006/jmaa.2000.6991
7. F. Riesz and B. Sz.-Nagy, “Functional Analysis,” Fredrick Ungar Publishing Co., New York, 1955.
8. I. Husain and S. K. Srivastav, “On Nondifferentiable Nonlinear Programming,” Journal of Applied Mathematics and Bioinformatics, Vol. 3, No. 2, 2013, pp. 45-64.
9. V. Chankong and Y. Y. Haimes, “Multiobjective Decision Making Theory and Methodology,” North-Holland, New York, 1983.