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In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These conditions are responsible for the development of duality theory which is an extremely important feature for any class of problems, but the literature available so far lacks these necessary optimality conditions for the stated problem. A lemma is also proved to find the topological dual of as it is required to prove the desired result.

A Semi-infinite Programming Problem (SIP) [

Semi-infinite Programming Problem is tightly interwoven with Variational Problem [

In this article, we propose Semi-infinite Variational Problem for which necessary optimality conditions are established. These optimality conditions are further extended to Multi-objective Semi-infinite Variational Problem (MSVP). We also clarify, with proper reasoning, certain points which were left for later validation in [

Necessary optimality conditions are important because these conditions lay down foundation for many computational techniques in optimization problems as they indicate when a feasible point is not optimal. At the same time these conditions are useful in the development of numerical algorithms for solving certain optimization problems. Further, these conditions are also responsible for the development of duality theory on which there exists an extensive literature and a substantial use of which (duality theory) has been made in theoretical as well as computational applications in many diverse fields. While browsing the literature, we found that necessary optimality conditions were not proved for the class of semi-infinite variational problems.

The paper is organized as follows: In section 2 some basic definitions and preliminaries are given. Section 3 deals with necessary optimality conditions for semi-infinite variational problem; single objective as well as multi- objective. In section 4, we prove a lemma which is required to prove necessary optimality conditions of section 3, for semi-infinite variational problem.

Let E be a topological vector space over the field of real numbers and

For any

1)

2)

3)

4)

Let

Consider the following Multi-objective Semi-infinite Variational Problem (MSVP):

subject to

Let

Definition 1 A point

Let us first prove necessary optimality conditions for the following single objective Semi-infinite Variational Problem (SVP):

subject to

where

The problem (SVP) may be rewritten as Cone Constrained Problem (CCP):

subject to

where

where

Theorem 2 Let

Proof. Since

where

Also for every

Since

By Lemma 1 (proved in Section 4)

Let

for

Substituting

Now it follows from (11)

(13) along with (16) implies

On using (14), we get

Integrating by parts the following function and using boundary condition of h,

Using above equation in (20), we get

By fundamental theorem of calculus of variation [

Claim 1:

Without loss of generality assume that

Since

In particular

Claim 2:

Let if possible

Define

Then

Hence Claim 2 holds, that is,

Using the same argument

The relations (16) are generally valid only if

linear first order differential equation for

Theorem 3 (Necessary Optimality Conditions) Let

Proof. This theorem can be proved by using Theorem 2 and proceeding on the similar lines of ([

The following example illustrates the validity of Theorem 3.

Example 4 Consider the problem (P1):

Subject to

where

The following example illustrates that a feasible solution of (MSVP) fails to be a normal efficient solution if it does not satisfy any one of the necessary optimality conditions (27), (28) or (29).

Example 5 Consider the problem (P2):

Subject to

where

Let us summarizes some basic concepts and tools to find topological dual of

1)

2)

3) Order dual of

4)

5) Given

Motivated by the topological dual of

Lemma 1 The topological dual of

Proof. For any

Clearly

For the converse, assume that

So for each

For each

Then

By Riesz representation theorem, for

Now let

note that

That is

Now, if

Conversely, proceeding similarly as in claim 1 of Theorem 2, it can be shown that

if

This infers

Hence

In this paper, we have developed necessary optimality conditions for a Semi-Infinite Variational Problem. These optimality conditions are further extended to Multi-objective Semi-infinite Variational Problem (MSVP) as Theorem 3. The results proved in this article are significant for the growth of optimality and duality theory for the class of semi-infinite variational problems. An example is presented to demonstrate the validity of the theorem proved. Another example illustrates that a feasible solution of (MSVP) fails to be a normal efficient solution if it does not satisfy any one of the necessary optimality conditions stated in the theorem. Vital part of the result depends on the topological dual of

We thank the Editor and the referee for their comments. The first author was supported by Council of Scientific and Industrial Research, Junior Research Fellowship, India (Grant no 09/045(1350)/2014-EMR-1).

BhartiSharma,PromilaKumar, (2016) Necessary Optimality Conditions for Multi-Objective Semi-Infinite Variational Problem. American Journal of Operations Research,06,36-43. doi: 10.4236/ajor.2016.61006