** American Journal of Operations Research ** Vol. 2 No. 4 (2012) , Article ID: 25152 , 10 pages DOI:10.4236/ajor.2012.24063

Optimality Conditions and Second-Order Duality for Nondifferentiable Multiobjective Continuous Programming Problems

Department of Mathematics, Jaypee University of Engineering and Technology, Guna (M.P.), India

Email: ^{*}ihusain11@yahoo.com, ^{*}jainvikas13@yahoo.com

Received August 4, 2012; revised September 5, 2012; accepted September 18, 2012

**Keywords:** Nondifferentiable Multiobjective Programming; Second-Order Invexity; Second-Order Pseudoinvexity; Second-Order Quasi-Invexity; Second-Order Duality; Nonlinear Multiobjective Programming

ABSTRACT

Fritz John and Karush-Kuhn-Tucker type optimality conditions for a nondifferentiable multiobjective variational problem are derived. As an application of Karush-Kuhn-Tucker type optimality conditions, Mond-weir type second-order nondifferentiable multiobjective dual variational problems is constructed. Various duality results for the pair of Mond-Weir type second-order dual variational problems are proved under second-order pseudoinvexity and second-order quasi-invexity. A pair of Mond-Weir type dual variational problems with natural boundary values is formulated to derive various duality results. Finally, it is pointed out that our results can be considered as dynamic generalizations of their static counterparts existing in the literature.

1. Introduction

Second-order duality in mathematical programming has been extensively investigated in the literature. In [1], Chen formulated second order dual for a constrained variational problem and established various duality results under an involved invexity-like assumptions. Subsequently, Husain et al. [2], have presented Mond-Weir type second order duality for the problem of [1], by introducing continuous-time version of second-order invexity and generalized second-order invexity. Husain and Masoodi [3] formulated a Wolfe type dual for a nondifferentiable variational problem and proved usual duality theorems under second-order pseudoinvexity condition while Husain and Srivastav [4] presented a MondWeir type dual to the problem of [2] to study duality under second-order pseudo-invexity and second-order quasiinvexity.

The purpose of this research is to present multiobjective version of the nondifferentiable variational problems considered in [2,4] and study various duality in terms of efficient solutions. The relationship between these multiobjective variational problems and their static counterparts is established through problems with natural boundary values.

2. Definitions and Related Pre-Requisites

Let be a real interval, and be twice continuously differentiable functions. In order to consider where is differentiable with derivative, denoted by and the first order derivatives of with respect to and, respectively, that is,

Further denote by, and the Hessian and Jacobian matrices respectively.

The symbols and have analogous representations.

Designate by X the space of piecewise smooth functions with the norm, where the differentiation operator D is given by

Thus except at discontinuities.

We incorporate the following definitions which are required for the derivation of the duality results.

Definition 1. (Second-order Invex): If there exists a vector function where and with at and such that for a scalar function, the functional where satisfies

then is second-order invex with respect to

where and the space of n-dimensional continuous vector functions.

Definition 2. (Second-order Pseudoinvex): If the functional satisfies

then is said to be second-order pseudoinvex with respect to.

Definition 3. (Second-order strict-pseudoinvex): If the functional satisfies

then is said to be second-order pseudoinvex with respect to.

Definition 4. (Second-order Quasi-invex): If the functional satisfies

then is said to be second-order quasi-invex with respect to.

Remark 1. If does not depend explicitly on t, then the above definitions reduce to those for static cases.

The following inequality will also be required in the forthcoming analysis of the research:

Lemma: 1 (Schwartz inequality): It states that

with equality in (1) if for some

Throughout the analysis of this research, the following conventions for the inequalities will be used:

If with and

, then

3. Statement of the Problem and Necessary Optimality Conditions

Consider the following nondifferentiable Multiobjective variational problem:

(VCP): Minimize

subject to

(1)

(2)

where 1) denote the space of piecewise smooth functions x with norm, where differentiation operator D already defined.

are assumed to be continuously differentiable functions, and 3) for each is an positive semi definite (symmetric) matrix, with continuous on I.

In this section we will derive Fritz John and Karush-Kuhn-Tucker type necessary optimality conditions for (VCP).

Definition: A point is said to be efficient solution of (VCP) if there exist such that

for some and

for

The following result which is a recast of a result of Chankong and Haimes [5] giving a linkage between an efficient solution of (VCP) and an optimal solution of p-single objective variational problem:

Proposition 1. (Chankong and Haimes [5]): A point is an efficient solution of (VCP) if and only if is an optimal solution of for each

: Minimize

subject to

for obtaining the optimal conditions for (VCP) we will use the optimal conditions obtained by Chandra et al. [6] for a single-objective variational problem which does not contain integral inequality constraints of.

The validity of the following proposition is quite essential in obtaining the optimality conditions for (VCP)Proposition 2. If is an efficient solution of (VCP), then is an optimal solution of the following problem for each

: Minimize

subject to

Proof: Let be an efficient solution of (VCP). Suppose that is not optimal solution of, for some Then there exists an such that

(3)

and

The inequality (3) for

(4)

The inequalities (3) along with (4) contradicts the fact that is an efficient solution of (VCP).

Hence is an optimal solution of, for some

Theorem 1. (Fritz John Type necessary optimality condition): Let be an efficient solution of (VCP). Then there exist and piecewise smooth functions and such that

(5)

(6)

(7)

(8)

(9)

Proof: Since is an efficient solution of (VCP), by Proposition 2, is an efficient solution of

for each and hence in particular. So by the results of [6] there exist and piecewise smooth functions and such that

The above conditions yield the relations (5) to (9).

Theorem 2 (Kuhn-Tucker type necessary optimality condition):

Let be an efficient solution of (VCP) and let for each, the conditions of satisfy Slaters or Robinson condition [6] at. Then there exist and piecewise smooth functions and such that

(10)

(11)

(12)

(13)

(14)

(15)

Proof: Since is an efficient solution of (VCP) by Proposition 2, is an optimal solution of for each. Since for each, the contradicts of, satisfy Slaters or Robinson conditions [6] at, by Kuhn-Tucker necessary condition of [6], for each, there exist and piecewise smooth function such that

Summing over we obtain

where for each

These can be written as,

where

and

Setting

we get

That is

4. Mond-Weir Type Second Order Duality

In this section, we present the following Mond-Weir type second-order dual to (VCP) and validate duality results:

(M-WD): Maximize

subject to

(16)

(17)

(18)

(19)

where

and

We denote by C_{P} and C_{D} the sets of feasible solutions to (VCP) and (M-WD) respectively.

Theorem 3. (Weak Duality): Assume that

(A_{1}) and

.

(A_{2}) is second-order pseudoinvex.

(A_{3}) is second-order quasi-invex.

Then

(20)

and

(21)

cannot hold.

Proof. Suppose to the contrary, that (20) and (21) hold.

Since we have

Since

we have

Now, by the constraints (2), (18) and (19), we have

This by (A_{3}), yields

By integration by parts, we have

This, by using (4) gives

(22)

By hypothesis (A_{1}), it implies

Using in the above, we have

This contradicts (20) and (21). Hence the result.

Theorem 4 (Strong duality): Let be normal and is an efficient solution of (VP). Then there exist, a piecewise smooth function such that is feasible for (M-WD) and the two objective functions are equal. Furthermore, if the hypotheses of Theorem 3 hold for all feasible solutions of (VCP) and (M-WD) ,then is an efficient solution of (M-WD).

Proof: Since is normal and an efficient solution of (VP), by Proposition 2, there exist and piecewise smooth and satisfying

(23)

(24)

(25)

(26)

(27)

From (24) along with, we have

Hence

satisfies the constraints of (M-WD) and

That is, the two objective functionals have the same value.

Suppose that is not the efficient solution of (M-WD). Then there exists such that

As, we have

This contradicts the conclusion of Theorem 3. Hence is an efficient solution of (M-WD).

Theorem 5 (Converse duality):

(A_{1}): Assume that is an efficient solution of (M-WD)

(A_{2}): The vectors are linearly independent where the row of is and is the row of G,

(A_{3})

are linearly independent and

(A_{4}) for either a)

and

or b)

and

Then is feasible for (VCP) and the two objective functionals have the same value. Also, if Theorem 3 holds for all feasible solutions of (CP) and (M-WD), the is an efficient solution of (VCP).

Proof: Since is an efficient solution of (M-WD), there exist, and, and piecewise smooth, , and such that the following Fritz John optimality conditions (Theorem 1)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

From (31), we have

(39)

This, by the hypothesis (A_{2}) gives

(40)

and (41)

Using (40), (41) and (17), we have

(42)

Let Then (41) gives and (40) gives,

Using and, (42) implies

This, because of (A_{3}) yields

(43)

The relation (43) with gives

Since, (36) gives The relation (30) yields we have, from (32) and, , from (35). These yield, ,.

Consequently

contradicting (38).

Hence and from (43)

Multiplying (30) by, summing over j, and then using (34) and (41), we have

In view of the hypothesis (A_{4}), this gives,

, The relation (30) implies,

yielding the feasibility of for (VCP).

The relation (32) with and gives

(44)

This by Schwartz inequality gives

(45)

If then (35) give

,. and so (45 ) implies

If (44) gives. So we still get

Now suppose that is not an efficient of (VCP). Then, there exists such that

and

Using and

We have

for some and

This contradicts Theorem 3. Hence is an efficient solution for (VCP).

Theorem 6 (Strict converse duality): Assume that

is second-order strictly pseudoinvex, and

is second-order quasi-invex with respect to same. Assume also that (VCP) has an optimal solution which is normal [6]. If is an optimal solution of (M-WD), then is an efficient solution of (VCP) with

Proof: We assume that and exhibit a contradiction. Since is an efficient solution, it follows from Theorem, that there exist, , , , , and such that

is an efficient solution of (M-WD). Since is an optimal solution of (M-WD), it follows that

This, because of second-order strict-pseudoinvexity of

(46)

Also from the constraint of (VCP) and (M-WD), we have

Because of second-order quasi-invexity of

, this implies

(47)

Combining (46) and (47), we have

(By integrating by parts)

This, by using η = 0, at t = a and t = b, implies

contradicting the feasibility of

for (M-WD).

5. Problems with Natural Boundary Values

In this section, we formulate a pair of nondifferentiable Mond-Weir type dual variational problems with natural boundary values rather than fixed end points given bellow

: Minimize

Subject to

: Maximize

Subject to

and, at.

We shall not repeat the proofs of Theorems 3-6 for the above problems, as these follow on the lines of the analysis of the preceding section with slight modifications.

6. Non-Linear Multiobjective Programming Problem

If the time dependency of and is ignored, then these problems reduce to the following nondifferentiable second-order nonlinear problems already studied in the literature:

(VP_{1}): Minimize

subject to

(VD_{1}): Maximize

subject to

REFERENCES

- X. H. Chen, “Second Order Duality for the Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 286, No. 1, 2003, pp. 261-270. doi:10.1016/S0022-247X(03)00481-5
- I. Husain, A. Ahmed and M. Masoodi, “Second-Order Duality for Variational Problem,” European Journal of Pure and Applied Mathematics, Vol. 2, No. 2, 2009, pp. 278-295.
- I. Husain and M. Masoodi, “Second-Order Duality for a Class of Nondifferentiable Continuous Programming Problems,” European Journal of Pure and Applied Mathematics, Vol. 5, No. 3, 2012, pp. 390-400.
- I. Husain and S. K. Srivastav, “On Second-Order Duality in Nondifferentiable Continuous Programming,” American Journal of Operations Research, Vol. 2, No. 3, 2012, pp. 289-295.
- V. Chankong and Y. Y. Haimes, “Multiobjective Decision Making Theory and Methodology,” North-Holland, New York, 1983.
- S. Chandra, B. D. Craven and I. Husain, “A Class of Nondifferentiable Continuous Programming Problem,” Journal of Mathematical Analysis and Applications, Vol. 107, No. 1, 1985, pp. 122-131. doi:10.1016/0022-247X(85)90357-9

NOTES

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