Differentiable Multiobjective Fractional Programming Problem under V-Convex Functions

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33

Online version available at: www.crdeep.com International Journal of Basic and Applied Sciences Vol. 3. No.2. 2014. Pp. 33-37

Full Length Research Paper

Differentiable Multiobjective Fractional Programming Problem under V-Convex Functions

O.Chandra Sekhara Reddy

¹

, G.Varalakshmi

²

, S. Chandra Babu

³

and T.Ramamohana Rao

⁴

1.

Associate Professor, Department of Statistics , Ambo University, Ethiopia

2.

Statistics Lecturer in P.V.K.N. Govt. College,Chittoor

3.

Statistics Lecturer in P.V.K.N. Govt. College,Chittoor

4.

Research Scholar in Dravidian University, Kuppam

*Corresponding Author: O. Chandra Sekhara Reddy

Abstract

The concept of Differentiable multiobjective fractional programming problem under V-convex functions and necessary and sufficient theorems on differentiable multiobjective fractional problem and also some other classes of non linear Composite problems as a special case.

Keywords: Generalized convex vector function, Differentiable and V-convex with positive derivative , Quasi-convexity, V-pseudo convex, weak and strong duality of multiobjective fractional problem.

Introduction

In the differentiable case, Defined a vector invexity that avoids the major difficulty of verifying that the inequality holds for the same function



for invex functions [ Jeyakumar and Mond ]. Established sufficient optimality criteria under V-pseudo invexity and obtained duality results under these assumptions [ Jeyakumar and Mond ]. To generalize, the concept of V-invexity to the non smooth case by replacing the gradients [Egudo & Hanson] . Defined on generalized convex mathematical programming [Clarke V.

Jeyakumar and Mond ]. But they not consider this in multiobjective fractional programming problem. Hence, in this chapter is an attempt is mode to fill the gap in the aim of research we extended the concept of Differentiable multiobjective fractional

programming problem under V-convex functions and some other classes of non linear Composite problems as a special case.

Notations & Preliminaries

Consider constrained multiobjective fractional optimization problem.

(VFP) V- Minimize

 





 



   

(x) g

(x) , f (x) ,

g (x) f

p p i

i

Subject to

h

_j

( x )  0 , j  1 , 2 ..., m

Where

: X R. and h : X R

^m

g f

j i

i _



_



are differential functions and

X

_o is an open set in

R

ⁿ. Here the symbol V- minimize stands for vector minimization. This is the problem of finding the set of weak minimum for points (VFP). When P=1, the problem (VPF) reduces to a Scalar optimization problem and it is denoted by (FP). Convexity of the Scalar problem (FP) is characterized by the inequalities.

 ^x ^a  ⁰

(a) g

(a) f (a) g

(a) f (x) g

(x) f

1 i

1 i i

i i

i

   

 x a  0 x, a x . (x)

h (a) h (x)

h

_j



_j



_j¹

   

_

The functional form

( x  a )

here plays no role in establishing the following two well-known properties in scalar convex programming:

(s). Every feasible Kuhn - Tucker point is a global minimum (w) weak duality holds between (FP) and its associated dual problem. Having this in mind, considered problem (FP) for which there exists a function

 : X

_

* X

_

 R

ⁿ such that

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(I)

η  x, a  0

(a) g

(a) f (a) g

(a) f (x) g

(x) f

1 i

1 i i

i i

i

  

,

h

_j

(x)  h

_j

(a)  h

_j¹

(a)μ  x, a   0,  x, a  x

_

.

and showed that such problems (known now as invex problem) also. Possess properties (s) and (w). Since then, various generalization of conditions ( I ) to multiobjective problems and many properties of functions that satisfy ( I ) have been established in the literature. However, the major difficulty is that the invex problems require the same function

η  x, a 

for the objective function and the constraints. This requirement turns out to be a severe restriction in applications. Because of this restriction, pseudo liner multiobjective, problems and certain non-linear multiobjective fractional programming problems require separate treatment as far as optimality and duality properties are concerned. In this chapter we show how this situation can be improved and how the properties (s) and (w) can be extended to hold for generalized convex multiobjective problems and certain multiobjective fraction problems. To this, we modify the condition ( I ) in the next section as follows:

New Classes & generalized convex vector functions A vector function ^p

i

: X R

g

f

_



is said to be V-Convex if there exist functions

μ : X

_

* X

_

 R

^p and

 0

i

: X * X R \

α

_ _



_ such that for each

x , a  X

_, and for i=1,2,……,p,

 x a  0 (a)

g (a) a) f (a) α(x,

g (a) f (x) g

(x) f

1 i

1 i i

i i

i

   

,

When p=1, the definition of V-Convexity reduces to the notion of convexity. The problem of (VPF) is said to be V-Convex if the

vector function



₁ ₂ _m



p p 2 2 2

1

& h , h ,..., h g

... f g , f g

f

 





 





is V- convex. Equivalently, V-convexity of (VFP) means that there exists function

μ : X

_

* X

_

 R

ⁿand

α

_i

, β

_j

: X

_

* X

_

 R

_

\ {0 }, i  1 , 2 ,..., p , j  1 , 2 ,... m

such that

















a) (a)(x a)h (x, β - (a) h - (x) h

a) (x (a) g

(a) a) f (x, (a) α

g (a) f (x) g

(x) f X a (VI)x,

' j j j j

' i

' i i

i i i

i



Let



_i

( x , a )  1  

_j

( x , a ), for i  1 , 2 ,... p , j  1 , 2 ,..., m

and p=1, the scalar problem (FP) becomes the strongly pseudo convex programming problem.

Example 1:- Consider the function

Z : R

ⁿ

 R

^pdefined by

 





 



  (φφ(x)

g .., f (φφ(x)),..

g Z(x) f

p p 1

1 where

p 1,2,..., i

R, R

g :

f

n

i

 

are strongly pseudo-convex functions with real positive functions

α

_i

(x, a), φ : R

ⁿ

 R

ⁿ^is

subjective with



^'

( a )

onto for each

a  R

ⁿ. Then, the function Z is V-convex. To see this, Let

x , a  R

ⁿ

),

( ),

( x v a

u    

Then, by strong pseudo convexity,

) v (v) (u g

(u) v) f (u, (v) α

g (v) f (u) g

(u) f (φφ(a) g

(φφ(a) f

(φφ(x) g

(φφ(x) f

' i

' i i

i i i

i i

i

    

Since



^'

( a )

is onto

( u  v)  φ

^'

(a)(x  a )

is solvable for some

( x  a )  R

ⁿ

Hence

φ(a)(x a)

(v) g

(v) v) f (u, (φφ(a) α

g

(φφ(a) f

(φφ(x) g

(φφ(x) f

' i

' i i

i i i

i

  

=



_i

 x , a  ( f

_i

.  )

^'

( a )( x  a ),

Where



_i

 x , a   

_i

  ( x ),  ( a )   0 .

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Example 2:- Consider the composite vector function

,

) ( (

) ( ,..., (

) ( (

) ( ) (

(

1 1

 





 



 

x G g

x F f x G g

x F X f

Z

p p

p p i

i where for each i=1,2,…,p

.

X R

G F

i

:

_



is continuously differentiable and pseudo linear with the possible proportional function



_i(. , . ) and

R g R

f

i

: 

is convex. The Z is V- Convex with this follows from the following convex inequality and pseudo linear equality conditions.

 



 

 

 



 

 

 



 

 

 



 



(a) G

(a) F (x) G

(x) F (a) G

(a) f F

(a) G

(a) f F

(x) G

(x) f F

i i i

i i

' i i i

i i i

i i

   x a 

(a) G

(a) a F x, (a) α G

(a)

f F

_'

i ' i i

i ' i

i

 





 

 

for a simple numerical example of a composite vector function, consider

  , Wher e

x x

x , x e x , x Z

2 1

2 /x 1

x 2

1 1 2

 



 





  ^X   ^x ^, ^x  ^R ^/x

1

^1, ^x

2

¹  ^.

2 2

1

  





We now show that the V-Convexity is preserved under smooth convex transformation.

Necessary Theorem:-

Theorem:- Let

 : R  R

be differentiable and convex with positive derivative everywhere. Let

P : X

_

 R

^p be V-convex.

Then, the function

P

_

( x )    ( p , ( x )),...  ( h

_p

( x )  , x  X



is V-Convex

Proof:- Let

x , a  X

_, Then from the monotonicity of



and V-Convexity of P, we get

 ^P

_i

⁽ ^x ⁾  ^   ^P

_i

⁽ ^a ⁾  ^ 

_i

⁽ ^x ^, ^a ⁾ ^P

_i^'

⁽ ^a ⁾⁽ ^x ^ ^a ⁾⁾ ^   ^P

_i

⁽ ^a ⁾  ^ 

^'

 ^P

_i

⁽ ^a ⁾  

_i

⁽ ^x ^, ^a ⁾ ^P

^'

⁽ ^a ⁾⁽ ^x ^ ^a ⁾



 P

_i

( a )  

_i

( x , a )  . P

_i



^'

( a )( x  a )

   

Thus,

P .  ( x )

is V-Convex the notations of pseudo-convexity and quasi-convexity for a scalar function can now be extended to a vector function. A vector function ^p

i

X R

g

f :

_



is said to be V-pseudo convex if there exists functions

R

p

X X

_

*

_



 :

and



_i

: X

_

* X

_such that for each

x , a  X

_,

  











p

1

i i

p

1

i i

p

1

i

g (a)

(a) a) f (x, (x) β

g (x) a) f (x, β 0

a) (a) (x gi'

(a) fi'

The vector function

i i

g

f

is said to be V-quasi-convex if there exists functions

 : X

_

* X

_

 R

^pand

} 0 {

\

*

: X X  R

_

r

_i _ _ such that for each

x , a  X

_,







p

1

i i

i i p

1 i

i

g (a)

(a) a) f (x, (x) r

g (x) a) f (x,

r

is inconsistent. Assume that

0 ) (

) (

1 ' '

 

 p

i i

i

g a

a

 f

, for

some

0    R

^p

, 

_i

 0

. Suppose that

a

is not a weak minimum for

i i

g

f

. Then there exists

x 

_

R

ⁿsuch

that

( )

) ( ) (

) (

a g

a f x g

x f

i i i

i





 , i=1,2,…,p. Since

i i

g

f

is V-convex, there exist



_i

( x

_

, a ), i  1 , 2 ,.., p

) ( ) (

) ( ) , (

1

' '

a x a g

a f a g

a f x g

x f a

x

_i

i i



 







 



 

_





 ^.

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Online version available at: www.crdeep.com So,





 





 

p



i 1 i

i i

i

(a) g

(a) f ) (x g

) (x f a) , (x α

τ



<0,

and hence



 p

i 1

. 0 ) ( ) (

) (

' '



 a x a g

a f

i i

i 



This is a contradiction.

Sufficiency theorem:-

Consider the multiobjective fractional problem (VFP). Suppose that the Lagrange multiplier conditions that

0 λ , R τ 0, τ 0, τ , R τ 0, (a) h λ , 0 (a) h (a) λ

g (a)

τ f

_j _j ^p ^m

m

1 j

' j ' j

i ' i p

1 i

i

        





, hold at a feasible

point

a  X

_. If

 





 





p p p 2 2 2 1 1

1

g

τ f ,..., g τ f g ,

τ f

is V-pseudo convex with respect to



and

 λ

₁

h

₁

,..., λ

_m

h

_m



is V-quasi

convex with respect to



then

a

is a global weak minimum point for (VFP).

Proof:- Suppose that

a

is not a global minimum point. Then there exists feasible

x 

_

X

_such that

(a) g

(a) f ) (x g

) (x f

i i i

i





 ,

i=1,2,..,p. So,

 





p

1

i i

i i i

i i i p

1 i

i

g (a)

(a) a)τ f , (x ) β

(x g

) (x a)τ f , (x

β

_



 . Now by the V-pseudo convexity condition, we

get

( , ) 0

) (

1 ' '

 

 p

i i

i

x a

a g

a f







. Since the Lagrangian condition holds at

a

,

( ) ( , ) 0

1

'







a x a h

m

j j

j







From the V-quasi-

convexity condition, we get

 

 



m

j

m

j

j j

j

x a h x r x a

r

1 1

) , ( )

( ) ,

(

_



_ _



this is a contradiction,

since



_j

h

_j

( x

_

)  0 , 

_j

h

_j

( a )  0 , and r

_j

( x

_

, a )  0 , forj  1 , 2 ,..., m

. Duality:-

Dual Formulation:-

V-maximize

    _ ^





 



  

p p 1

1

g ,..., f g

f

,

(VFD) Subject to

( ) λ h ( ) 0, λ 0, τ 0, τe 1, λ h ( ) 0, j 1,2,.., m .

g

τ f

_j _i _j _j

m

1 j

' j ' j

i ' i p

1 i

i

       







Where e= (1, 1… 1). Note that the Lagrangian conditions in sufficiency theorem hold for (VFP) at a weak minimum point

a

under a constraint qualification and they can equivalently be written as

m.

1,2,.., j

0, (a) h λ 1, τe 0, τ 0, λ 0, (a) h (a) λ

g (a)

τ f

_j _j

m

1 j

' j ' j

i ' i p

1 i

i

       





Weak Duality Theorem:- Consider the multiobjective fractional problem (VFP) and (VFD). Let

x

feasible for (VFP) and

  , ,   

be feasible for (VFD). If

 





 





p p p 2 2 2 1 1

1

g

τ f ,..., g τ f g ,

τ f

is V-pseudo convex and

 

₁

^h

₁

,..., 

_m

^h

_m



is V-quasi convex

with respect to the same



, then

   







 



 

 





 





^T _p

p p 1

1 T

p p 1

1

in R

(ξξ g

(ξξ ,...., f (ξξ g

(ξξ f (x)

g (x) ,...., f (x) g

(x)

f

.

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37

Online version available at: www.crdeep.com Proof:- from the feasibility conditions,

λ

_j

h

_j

(x)  h

_j

h

_j

(ξξ  0

, for each j=1,2,..,m. Since

r

_j

( x ,  )

is

positive

_  







m

1 j

j j

j

(x, ξ) h (x) h (ξξ 0

r

. Hence,

λ h (ξξ μ(x, ξ) 0 ,

m

1 j

' j

j







and so,

μ(x, ξ) 0,

(ξξ g

(ξξ τ f

p

1

i '

i ' i

i







the conclusion now follows from the V-pseudo convexity condition since

τe  1

^and

β

_i

(x, ξ)  0

.

Strong Duality Theorem:-

Assume that

a

is a weak minimum of (VFP) and that a suitable constraint qualification is satisfied at

a

. Then there exist

  ,  

such that

 a ,  ,  

is a feasible for (FVD).

Proof:- since

a

is a weak minimum for (FVP) and a constraint qualification is satisfied at

a

, from the Lagrangian conditions, there exists

  ,  

such that

 a ,  ,  

is a feasible for (VFD). Clearly the values of (VFP) and (VFD) are equal at

a

, since the objective function for both problems are the same. By the generalized V-convexity hypothesis, weak duality holds, hence if

 a ,  ,  

is not a weak optimum for (VFD), there must exist

  ^,  ^,  ^ 

feasible for (VFD),

  a

such that

 

  ^ ^ _ ^ ^

^



 



 

 





 





^T _p

p p 1

1 T

p p 1

1

in R

(a) g

(a) ,...., f (a) g

(a) f g

,...., f g

f



contradicting weak duality.

Conclusion

Necessary and sufficient theorems are contradicting the V-convex function and also contradicting weak and strong duality theorems.

References

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Egudo, R.R.and Hanson M.A, On sufficiency of Kuhn-Tucker conditions in nonsmooth multiobjective programming, FSU Technical Report M-888,(1993).

Hanson,M.A., On sufficiency of the Kuhn-Tucker conditions ,Journal of Mathematical Analysis and Applications,Vol.80,pp.544- 550,1981.

Jeyakumar,V., Equivalence of saddle-points and optima, and duality for a class of non smooth non convex problems, Journal of MathematicalAnalysis and applications , Vol.130,pp.334-343, 1988.

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