^{1}

^{*}

^{2}

^{*}

In this paper we have considered a non convex optimal control problem and presented the weak, strong and converse duality theorems. The optimality conditions and duality theorems for fractional generalized minimax programming problem are established. With a parametric approach, the functions are assumed to be pseudo-invex and v-invex.

Parametric nonlinear programming problems are important in optimal control and design optimization problems. The objective functions are usually multi objective. The constraints are convex, concave or non convex in nature. In [

Consider the real scalar function

If

For a r-dimensional vector function ` the gradient with respect to x is

Gradient with respect to

Definition 1. A vector function

functions

Definition 2. We define the vector function

Definition 3. Let S be a non-empty subset of a normed linear space

core of S (denoted S^{+}) is defined by

Problem P (Primal):

Minimize

subject to

The corresponding dual problem is given by:

Problem D (Dual):

Maximize

subject to

where

Theorem 1: (Weak Duality)

If

to

Theorem 2: (Strong Duality)

Under the pseudo invexity condition of theorem 1, if

[

Theorem 3: (Converse duality)

If

for all

Sufficiency:

It can be shown that, pseudo-convex functions together with positive dual conditions are sufficient for optimality [

Optimality conditions and duality for generalized fractional minimax programming problem:

We consider the following generalized fractional minimax programming problem:

1)

2)

3)

4) If

Consider the following minimax nonlinear parametric programming problem.

Lemma 1: If

Lemma 2: In relation to

subject to

Lemma 3: If

then there exist

Lemma 4:

equal to zero i.e.

Theorem 4: (Necessary conditions)

Let

binding constraints. i.e.

Then

and

Hence from (4) and (5)

Then there exist

Theorem 5: (Sufficient conditions)

For some

satisfied. Then

Two duals

subject to

Weir and Mond type dual.

subject to

Proof of the corresponding duality results for the above two duals follow the same lines as the proofs of the theorems 2, 3, 4.

Here in this presentation we have considered a non convex optimal control problem in parametric form and established the weak duality theorem, the strong duality theorem and the converse duality theorem. The results which are available in literature for v-invex functions are hereby extended to v-pseudo invex functions in a minimax fractional non convex optimal control problem.

The authors are thankful to the reviewers for their valuable suggestions in the improvisation of this paper.