SOME OPTIMALITY AND DUALITY RESULTS FOR AN EFFICIENT SOLUTION OF MULTIOBJECTIVE NONLINEAR FRACTIONAL PROGRAMMING PROBLEM

SOME OPTIMALITY AND DUALITY

RESULTS FOR AN EFFICIENT

SOLUTION OF MULTIOBJECTIVE

NONLINEAR FRACTIONAL

PROGRAMMING PROBLEM

Paras Bhatnagar1

Department of Mathematics, Ideal Institute of Technology, Ghaziabad, U. P., India

Email: [email protected]

Sanjeev Rajan2

Department of Mathematics, Hindu College, Moradabad, U.P., India

Ridhi Garg3

Department of Mathematics, Moradabad Institute of Technology Moradabad, UP India

Abstract

Kaul and Kaur [7] obtained necessary optimality conditions for a non-linear programming problem by taking the objective and constraint functions to be semilocally convex and their right differentials at a point to be lower semi-continuous. Suneja and Gupta [12] established the necessary optimality conditions without assuming the semilocal convexity of the objective and constraint functions but their right differentials at the optimal point to be convex.

Suneja and Gupta [13] established necessary optimality conditions for an efficient solution of a multiobjective non-linear programming problem by taking the right differentials of the objective functions and constraint functions at the efficient point to be convex. In this paper we obtain some results for a properly efficient solution of a multiobjective non-linear fractional programming problem involving semilocally convex and related functions by assuming generalized Slater type constraint qualification.

Key words: Nonlinear programming, Multi-objective fractional programming, Semilocally convex functions,

Pseudoconvex functions.

1.Introduction

Consider the following Multi-objective fractional programming problem:

(MFP)

Minimize 1 2

1 2

( )

( )

( )

,

,...

( )

( )

( )

k k

f x

f x

f x

g x

g x

g x













Subject to

h

_j

(

x

)



0

,

j



1

,

2

,...

.,

m

,

x



S

Where

S



R

nis a locally star shaped set and

f

_i

,

g

_i

:

S



R

,

i



1

,

2

,...

.,

k

and

h

_j

:

S



R

,

j



1

,

2

,...,

m

, are semidifferentiable functions.

,

0

)

(

x



f

i

g x

i

( )



0,

i



1

,

2

,...

.,

k

for all

x



S

. Let

X

0





x



S h x

_j

( )



0,

j



1, 2,...,

m



and

( )

,

( )

i i

i

f x

t

g x



i



1

,

2

,...

.,

k

t

(MFP )

Minimize



(

f

1



t

1

g

1

)(

x

),

(

f

2



t

2

g

2

)(

x

),...

.,

(

f

k



t

k

g

k

)(

x

)



Subject to

h

j

(

x

)



0

,

j



1

,

2

,...

.,

m

,

x



S

We now have the following lemma connecting the properly efficient solutions of

(MFP)

and

(MFP )

_t .

Lemma:1 Let

x

*be a properly efficient solution of

(MFP)

, then there exists

t

*



R

_ksuch that

x

*is properly

efficient solution of

(MFP )

_t* .Conversely if

*

x

is a properly efficient solution

of

(MFP )

_t* where

* *

*

(

)

t

(

)

i i

i

f x

g x



,

i



1, 2,...,

k

then

x

* is a is properly efficient solution of

(MFP)

.

2. Optimality Conditions

Definition: The problem

(MFP )

_t* is said to satisfy generalized Slater type constraint qualification at

*

x



S

if the functions

f

_i



t g

_i* _i,

i



1, 2,...,

k

and

h

_j,

j



I

are semilocally pseudoconvex at

x

*and for each

1, 2,...,

r



k

there exists

x

_r

 

S

such that

*

(

f

_i



t g

_{i i}

)( )

x

_r





0

,

i



1, 2,...,

k

,

i



r

and

h x

_I

( )

_r

 

0

,where

I



I x

(

*

)





i h x

_i

(

*

)



0



and

J



J x

(

*

)





j h x

_j

(

*

)



0



Now we obtain necessary optimality conditions corresponding to

(MFP )

_t* .

Theorem: 1 Let

x

* be an efficient solution of

(MFP)

and S be convex. Let

h

_jbe continuous

at

x

*for

j



I

and

d f

(

i

t g

*i i

) ( ,

x x

*

x

*

)







,

i



1, 2,...,

k

,

(

dh

_j

) ( ,



x x

*



x

*

)

be convex functions of

x

.

If

(MFP )

_t* satisfies generalized Slater type constraint qualification at

*

x

for

* *

*

(

)

t

(

)

i i

i

f x

g x



,

i



1, 2,...,

k

then there exists



_i*



0

,

i



1, 2,...,

k

and



*_j



0

,

j



I

such that

*

* * * * * * * * *

1 ( )

(

) ( ,

)

(

) ( ,

)

(

) ( ,

)

0

k

i i i i j j

i j I x

df

x x

x

t dg

x x

x

dh

x x

x



 





 

































(1)

for all

x



S

.

Proof: We claim that the system

d f

(

i

t g

i* i

) ( ,

x x

*

x

*

)

0









,

i



1, 2,...,

k

(2)

d h

( ) ( ,

_j 

x x

*



x

*

)



0,

j



I x

(

*

)

(3) Has no solution

x



S

.

If possible let

x

*be the solution of the system. By the relation (2) we have

* * * * *

0

(

)(

(

)) (

)(

)

lim

f

i

t g

i i

x

x

x

f

i

t g

i i

x

0





















_

(4)

Which implies that there exists



₁



0

such that

(

f

i



t g

i* i

)(

x

*





(

x



x

*

)) (



f

i



t g

i* i

)(

x

*

)



0

for







0,



1



(5)

by (5) ,

(

f

_i



t g

_i* _i

)(

x

*





(

x



x

*

))



(

f

_i



t g

_i* _i

)(

x

*

)

,

i



1, 2,...,

k

similarly by the relation (3), there exists



₂



0

, such that

Let



*



min



  

₁

,

₂

,

_j





, then for





 

0,



* ,we have

* * * * *

(

f

_i



t g

_{i i}

)(

x





(

x



x

))



(

f

_i



t g

_{i i}

)(

x

)

,

i



1, 2,...,

k

,

* *

(

(

))

0

j

h x





x



x



,

j



I x

(

*

)

and

(

x

*





(

x



x

*

))



X

.

This implies that

x

*is not an efficient solution of

(MFP )

_t* .Hence,

*

x

is not an efficient solution of

(MFP)

, contrary to the given hypothesis. Thus, the system (2), (3) has no solution x in S. Also,

d f

(

_i



t g

_i* _i

) ( ,



x x

*



x

*

)

,

i



1, 2,...,

k

,

(

dh

_j

) ( ,



x x

*



x

*

)

,

j



I

are convex functions of x. By Mangasarian [10], there exists



_i*



0

,

i



1, 2,...,

k

,



*_j



0

,

j



I

not all zero such that (1) holds. We shall now show that



_i*



0

, for each

i



1, 2,...,

k

. If possible let



_r*



0

, for some r,

1

 

r

k

. By generalized Slater type constraint qualification, there exists

x

_r

 

S

, such that

* * *

(

f

_i



t g

_{i i}

)( )

x

_r





(

f

_i



t g

_{i i}

)(

x

)

,

i



1, 2,...,

k

i



r

( )

0

j r

h x

 

,

and

(

f

_i



t g

_i* _i

)

,

i



1, 2,...,

k

and

h

_j,

j



I

are semilocally pseudoconvex at

x

*. Therefore it follows that

d f

(

_i



t g

_i* _i

) ( ,



x x

* _r





x

*

)



0

,

i



1, 2,...,

k

and

(

dh

_j

)



(

x

*

,

x

_r





x

*

)



0

,

j



I x

(

*

)

Since at least one of the coefficients



_i*,

i



1, 2,...,

k

,

i



r

and



*_j,

j



I

is non zero, therefore we obtain

*

* * * * * * * * *

1 ( )

(

) ( ,

)

(

) ( ,

)

(

) ( ,

)

0

k

i i r i i j j r

i j I x

df

x x

x

t dg

x x

x

dh

x x

x



 





 





































Which contradicts (1) as



_i*



0

.

Thus



_i*



0

, for all

i



1, 2,...,

k

.

Following Geoffrion [5] we consider the scalar problem corresponding to

(

MFP

_t*

)

in order to prove sufficient

optimality conditions.



)

MFP

(

_t* Minimize







k i

i i i

i

f

x

t

g

x

1

*

))

(

)

(

(



Subject to

h

_j

(

x

)



0

,

j



1

,

2

,...,

m

We have the following results on the lines of Geoffrion in the light of Lemma1.

Lemma: 2 If

x

*is an optimal solution of

(

MFP

_t*

)

 for some

k

R





, with strictly positive components

where

)

(

)

(

* * *

x

g

x

f

t

i i

i



,then

*

x

is a properly efficient solution of

(MFP)

.

Theorem: 2 Suppose there exists a feasible

x

*for

(MFP)

and scalars



i*



0

,

i



1

,

2

,...,

k

,

0

*



j



,

I

j



such that for all

x



X

0,where

)

(

)

(

* * *

x

g

x

f

t

i i

i



,

i



1

,

2

,...,

k

. Let the following conditions be satisfied:

(i)

f

_i,



g

_i,

i



1

,

2

,...,

k

and

h

_j,

j



I

are semilocally convex at

x

*, (ii)







k i

i i i i

f

t

g

1

* *

)

(



is semilocally pseudoconvex.

Then

x

* is a properly efficient solution of

(MFP)

.

Then





 



) ( * * ) ( * * *

)

(

)

(

x I j j j x I j j

j

h

x



h

x



(6)

(i) Since each

h

j,

j



I

is semi locally convex at

*

x

, therefore it follows that



 ( ) * * *

)

(

x I j j j

h

x



is semilocally

convex. From (6) we obtain



 

_

_

) ( * * * *

0

)

,

(

)

(

x I j j

j

dh

x

x

x



(7)

Using (1) and (7), we get

0

]

)

,

(

)

(

)

,

(

)

[(

1 * * * * * *











   k i i i i

i

df

x

x

x

t

dg

x

x

x



, for all

x



X

0 (8)

Now ,

f

i,



g

i,

i



1

,

2

,...,

k

are semilocally convex at

*

x

, therefore







k i i i i i

f

t

g

1

* *

)

(



is also

semilocally convex at

x

*, and (8) gives

)

)

(

)

(

(

))

(

)

(

(

1 * * * * 1 * *





 







k i i i i i k i i i i

i

f

x

t

g

x



f

x

t

g

x



for all

x



X

0 (9)

Which implies that

x

*is an optimal solution of

(

MFP

*

)

*



t , where

*



has strictly positive components. By

lemma 2, it follows that

x

*is properly efficient for

(MFP)

. (ii) Using the semilocal convexity of



 ( ) * * x I j j j

h



at

x

* in (6) we get (7) which when combined with (1) gives

(8). Now







k i i i i i

f

t

g

1

* *

)

(



is semilocally pseudoconvex, therefore, we get (9), which implies that

x

*is

properly efficient for

(MFP)

.

In relation to

(MFP)

,we now associate the following multiple objective Schaible type dual program:

(MFD)

Maximize



(t)



t

(

t

₁

,

t

₂

,...

t

_k

)

Subject to

1

[(

) ( ,

)

(

) ( ,

)

k

i i i i

i

df

u x u

t dg

u x u



 



 

 



1

[(

) ( ,

)]

0

m

j j

j

dh

u x u













(10)

for all

x



X

0

where

(

(

)

(

))

0

1







 k i i i i

i

f

u

t

g

u



, (11)

1

0

m j j j

h (u)









, (12) and

u



S

,



_i



0

,

t

_i



0

,

i



1

,

2

,...,

k

,

u

_j



0

,

j



1

,

2

,...

m

. (13) We now prove some duality results.

3. Weak Duality

Theorem: 3 Let x be feasible for

(MFP)

and



u

,



,



,

t



be feasible for

(MFD)

. If the following holds:

(i)

f

i,



g

i,

i



1

,

2

,...,

k

and

h

j,

j



I

are semilocally convex at u, (ii)







k i i i i i

f

t

g

1

)

(



is semilocally pseudoconvex and

1 m j j j

η

h





is semilocally convex at u, Then

f

(

x

)

is

not



t

Proof: If possible let

f

(

x

)



t

This gives _i

i i

t

x

g

x

f

_

)

(

)

(

and _r r

r

t

x

g

x

f

_

)

(

)

(

, for some r.

Multiplying these inequalities by



i



0

,

i



1

,

2

,...,

k

and adding we get

(

(

)

(

))

0

1







 k i

i i i

i

f

x

t

g

x



(14)

Now x is feasible for

(MFP)

and



u

,



,



,

t



is feasible for

(MFD)

therefore





 



m j

j j m

j j

j

h

x

h

u

1 1

)

(

)

(





Since

h

j,

j



1

,

2

,...,

m

are semilocally convex and consequently

1

m j j j

η

h





is semilocally convex at u, therefore

0

)

,

(

)

(

1







_

_

m j

j

j

dh

u

x

u



Using (10), we obtain

0

)]

,

(

)

(

)

,

(

)

[(

1









_

_

_

_

k i

i i i

i

df

u

x

u

t

dg

u

x

u



Now,

f

_i,



g

_i,

i



1

,

2

,...,

k

are semilocally convex therefore







k i

i i i i

f

t

g

1

)

(



is semilocally convex so we

get







 k i

i i i

i

f

x

t

g

x

1

))

(

)

(

(



(

(

)

(

))

0

1







 k i

i i i

i

f

u

t

g

u



This contradicts (14). Hence

f

(

x

)

is

not



t

.

4. Strong Duality

Theorem: 4 Let

x

*be feasible for

(MFP)

and



u

*

,



*

,



*

,

t

*



be feasible for

(MFD)

such that

* * *

(

)

(

)

i

i i

f x

t

g x



,

i



1

,

2

,...,

k

. let

f

i,



g

i,

i



1

,

2

,...,

k

be strictly semilocally convex and

j

h

,

j



1

,

2

,...,

m

be semilocally convex at

u

*,then

x

*



u

*.

Proof: Let

x

*



u

*, now

x

*is feasible for

(MFP)

and



u

*

,



*

,



*

,

t

*



is feasible for

(MFD)

, therefore

* * * *

1 1

(

)

(

)

m m

j j j j

j j

h x

h u





 







Since

h

_j,

j



1

,

2

,...,

m

are semilocally convex and consequently *

1

m j j j

h







is semilocally convex at

*

u

,therefore

* * * *

1

(

) ( ,

)

0

m

j j j

dh

u x

u



 







Using the above inequality in the relation (10) for dual feasibility of



u

*

,



*

,



*

,

t

*



and for

x

*



X

0, we get

* * * * * * * *

1

[(

) ( ,

)

(

) ( ,

)] 0

k

i i i i

i

df

u x

u

t dg

u x

u



 













* * * * * *

1 1

(

)(

)

(

)(

)

k k

i i i i i i i i

i i

f

t g

x

f

t g

u





 











* * *

(

)

(

)

i

i i

f x

t

g x



gives that

* * * *

1

( (

)

( ))

0

k

i i i i

i

f x

t g x











,

therefore we get * * *

1

(

)(

)

0

k

i i i i i

f

t g

u











,

which contradicts the fact that



u

*

,



*

,



*

,

t

*



is dual feasible. Hence

x

*



u

*.

Theorem: 5 Let



u

*

,



*

,



*

,

t

*



be feasible for

(MFD)

,

f

_i,



g

_i,

i



1, 2,...,

k

and

h

_j,

j



1,2,....,

m

are semilocally convex at

u

*. Suppose there exists a feasible

x

*for

(MFP)

such that

* * *

(

)

(

)

i

i i

f x

t

g x



,

i



1

,

2

,...,

k

(15)

Then

x

*is properly efficient solution for

(MFP)

. Also, if for each feasible



u

,



,



,

t



of

(MFD)

, i

f

,



g

_i,

i



1

,

2

,...,

k

and

h

_j,

j



1

,

2

,...,

m

are semilocally convex at u then



u

*

,



*

,



*

,

t

*



is properly efficient solution for

(MFD)

.

Proof: Suppose that

x

*is not an efficient solution of

(MFP)

, then there exists a feasible x for

(MFP)

and an

index r such that

)

(

)

(

)

(

)

(

* *

x

g

x

f

x

g

x

f

i i i

i



,

i



1

,

2

,...,

k

,

i



r

And

)

(

)

(

)

(

)

(

* *

x

g

x

f

x

g

x

f

r r r

r



Using (15), we get

( )

*

( )

f x

t

g x



, which contradicts the weak duality. Therefore *

x

must be an efficient solution

of

(MFP)

.

If possible let

x

*be not properly efficient solution for

(MFP)

, then by lemma 2.2.1, it follows that

x

*is not a properly efficient solution of

(MFP

_t*

)

.

Therefore for every

M



0

, there exists a feasible

x

for

(MFP

_t*

)

and hence

(MFP)

and an index

i

such that

( ( )

f x

_i



t g x

_i* _i

( ))



( (

f x

_i *

)



t g x

_i* _i

(

*

))

And

M

x

g

t

x

f

x

g

t

x

f

x

g

t

x

f

x

g

t

x

f

j j j j

j j

i i i i

i

i















))

(

)

(

(

))

(

)

(

(

))

(

)

(

(

))

(

)

(

(

* *

* *

* *

* *

For all

j

satisfying

(

f

_j

(

x

)



t

*_j

g

_j

(

x

))



(

f

_j

(

x

*

)



t

*_j

g

_j

(

x

*

))

That is

t g x

_i* _i

( )



f x

_i

( )



M f x

(

_j

( )



t g x

*_{j j}

( ))

for all j satisfying

*

( )

( )

0

j j j

f x



t g x



Thus

t g x

_i* _i

( )



f x

_i

( )

can be made large and hence for



_i*



0

, we get the inequality

* * 1

(

( )

( ))

0

k

i i i i

i

t g x

f x









The above result can also be proved under the assumption that for each feasible



u

, , ,

 

t



,

1

( ( )

( ))

k

i i i i

i

f x

t g x









is semilocally pseudoconvex and *

1

m j j j

h







is semilocally quasiconvex.

5. Conclusion: In this paper the problem under consideration is multiobjective non-linear fractional

programming problem involving semilocally convex and related functions. We have discussed the interrelation between the solution sets involving properly efficient solutions of multiobjective fractional programming and corresponding scalar fractional programming problem. Necessary and sufficient optimality conditions are obtained for efficient and properly efficient solutions. Some duality results are established for multiobjective Shaible type dual.

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

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