Vol 3, No 5 (2012)

Available online through

ISSN 2229 – 5046

NONDIFFERENTIABLE

MULTIOBJECTIVE

MIXED

SYMMETRIC

DUALITY

FOR

NONLINEAR

PROGRAMMING

INVOLVING

( , )

φ ρ

-UNIVEX

FUNCTION.

A. K. Tripathy

1*

& G. Devi

2

1

Department of Mathematics, Trident Academy of Technology, F2/A, Chandaka Industrial Estate, In front of Infocity, Patia, Bhubaneswar-751024, Odisha, India

2

Department of Computer Science, Ajayabinayak Institute of Technology, Cuttack, Odisha, India

(Received on: 07-05-12; Accepted on: 31-05-12)

________________________________________________________________________________________________

ABSTRACT

I

n this paper we have presented a pair of mixed symmetric dual for a class of nondifferentiable multiobjective programming involving square root term like (xTAx)1/2 .We established weak duality , strong duality and converse duality theorems with their proofs under (Φ, ρ)-univexity and (Φ, ρ)-pseudounivexity assumption. Also the self duality theorem with example is established. Discussion on some particular cases shows that our results generalize earlier results in related domain.

AMS 2002 Subject Classification: 90C29, 90C30, 90C46.

Keywords- (Φ, ρ)-univexity; (Φ, ρ)-pseudo-univexity; mixed symmetric dual, Schwartz inequality, quadratic term, efficient solution; self duality.

_______________________________________________________________________________________________

1. INTRODUCTION

Duality theory has played an important role in the development of optimization theory. For nonlinear programming problems a number of duals have been suggested, among which the Wolfe dual proposed by Dorn [12] is well known. Subsequently Dantzing et al [13] and Bazarra et al [5] established symmetric duality results for convex/concave functions. Devi [11], Weir and Mond [34], Mond and Schechter [25] studied non differentiable symmetric duality for a class of optimization problem in which the objective function consist of support function. Husain et al [15] have formulated a pair of Mond Weir type second order symmetric dual and establish the duality results under pseudo convexity –pseudo concavity assumption.

In recent years, several extension and generalization have been considered for classical convexity. A significant generalization of convex function is that of in-vex function introduced by Hanson [14]. Bector et al [6] have introduced the concept of pre-univex function, univex functions and pseudo-univex function as a generalization of in-vex function. Further development on the application of univex function and generalized univex function can be found in Rueda et al [28], Mishra [22] and Mishra et al [23], [24]. Ojha[27] has established symmetric duality results for (Φ, ρ)-univex function and Thakur et al [32] have established second order symmetric duality results for second order (Φ,ρ)-univex function .Earlier Chandra et al [8] had formulated mixed symmetric duality for a class of nonlinear programming problems. Yang et al [33] have discussed a mixed symmetric duality for a class of nondifferentiable nonlinear programming problems. Later on Ahmad [4] has formulated mixed type symmetric dual in multiobjective programming problem ignoring nonnegative restriction of Bector et al [7].Very recently Mishra et al [20] and Mishra [21] have presented a mixed symmetric first and second order duality in nondifferentiable mathematical programming problem under F-convexity. Li et al [18] and Agarwal et al [1] introduced a model of mixed symmetric duality for a class of non differentiable multiobjective programming problem with multiple arguments.

In motivation of Bector et al.[7], Ahmed [4], Ahmed et al[3], Mishra[21], Ojha[27], Thakur et al [32], Agarwal et al [1] and Li et al [18], we formulated a pair of multiobjective mixed symmetric dual program using square root term and established weak and strong duality and the converse duality theorems under (Φ,ρ)-univexity and (Φ,ρ) -pseudo-univexity condition. These results generalized the known works of Thakur et al [32], Mishra et al [20], Mishra [21], Agarwal et al [1], Ojha [27], Suneja et al [27], Chandra et al [8], Ahmad et al [4] and Wier and Mond l [30].

Corresponding author:

A. K. Tripathy

1*

1

Page: 1940-1956

© 2012, IJMA. All Rights Reserved 1941

2.NOTATION AND PRELIMINARIES

Let

R

n be n-dimensional Euclidean space and

R

+n its nonnegative orthant .The following conventions for vectors

n

R

y

x

,

∈

will be followed throughout this paper:

x

<

y

⇔

x

i

<

y

i

,

i

=

1

,

2

,....

n

.

x

≤

y

⇔

x

i

≤

y

i_.For any

vector we denote

∑

=

=

n

i i i T

y

x

y

x

1

.

Let X and Y are open subset of

R

n and

R

m respectively. Let

f

i

(

x

,

y

)

be a real valued twice differentiable function defined onX×Y . Let

∇

₁

f

_i

(

x

,

y

)

and

∇

₂

f

_i

(

x

,

y

)

denote the gradient vectors of

f

_i

(

x

,

y

)

with respect to first variable

x

and second variable yrespectively. Also let

∇

₁2

f

_i

(

x

,

y

)

and

∇

2₂

f

_i

(

x

,

y

)

denote the Hessian matrix of

)

,

(

x

y

f

i with respect to the first variable

x

and second variable y respectively.

Let

φ

be a real valued function defined on

X

×

X

×

R

n+1 such that

φ

(x,u,∗) is convex on

R

n+1 and

0 )) , 0 ( , ,

(x u r ≥

φ

for every (x,u)∈X ×X and r∈R . Also let b:X ×X →R₊ and

ψ

:R→ R

satisfying

ψ

(

u

)

≤

0

⇒

u

≤

0

and

ψ

(

−

a

)

=

−

ψ

(

a

)

.

Let C⊂ Rn be a compact convex set .The support function of C is defined by

s

(

x

|

C

)

=

max{

x

T

y

|

y

∈

C

}.

Consider the following multiobjective programming problem (MP):

P :(Primal) Minimize

f x

( )

=

( ( ),

f x

₁

f x

₂

( ),...,

f x

_r

( ))

Subject to

h

(

x

)

≤

0

,

x

∈

X

⊆

R

n , where

f X

:

→

R g X

r

, :

→

R

r

Let

X

₀ be the set of all feasible solutions of problem (P); that i.e.

X

₀

=

{

x

∈

X

|

h

(

x

)

≤

0

}

.

Definition 2.1.A vector

x

∈

X

₀ is said to be an efficient solution of problem (P) if there exists no

x

∈

X

₀ such that

).

(

)

(

),

(

)

(

x

f

x

f

x

f

x

f

≤

≠

Definition 2.2.A vector

x

∈

X

₀ is said to be a weakly efficient solution of problem (P) if there exists no

x

∈

X

₀

such that

f

(

x

)

<

f

(

x

).

Definition 2.3.A vector

x

∈

X

₀ is said to be a properly efficient of problem (P) if it is efficient and there exists a positive constant M such that whenever

f

_i

(

x

)

<

f

_i

(

x

)

for

x

∈

X

₀ and for

i

∈

{1, 2,... }

r

, there exist at least one

}

,...

2

,

1

{

p

j

∈

such that

f

j

(

x

)

<

f

j

(

x

)

and

f

i

(

x

)

−

f

i

(

x

)

≤

M

(

f

j

(

x

)

−

f

j

(

x

))

.

Definition 2.4: A real-valued twice differentiable function

f

_i

(

∗

,

y

)

:

X

×

Y

→

R

is said to be (Φ, ρ)-univex at

X

u

∈

with respect to

q

∈

R

n if for all

b

:

X

×

X

→

R

₊,

φ

:

X

×

X

×

R

n+1

→

R

, and

ψ

:

R

→

R

with

ρ

_i is a real number, we have

b x u

( , )[ { ( , )

ψ

f x y

_i

−

f u y

_i

( , )}]

≥

ϕ

( , ; (

x u

∇

₁

f u y

_i

( , )))

(2.1)

Definition 2.5: A real-valued twice differentiable function

f

_i

(

x

,

∗

)

:

X

×

Y

→

R

is said to be (Φ, ρ)-univex at

Y

y

∈

with respect to

p

∈

R

m if for all

b

:

Y

×

Y

→

R

₊,

φ

:

Y

×

Y

×

R

m+1

→

R

and

ψ

:

R

→

R

with

ρ

_i is a real number, we have

b v y

( , )[ { ( , )

ψ

f x v

_i

−

f x y

_i

( , )}]

≥

ϕ

( , ; (

v y

∇

₂

f x y

_i

( , ),

ρ

_i

))

(2.2)

Definition 2.6: A real-valued twice differentiable function

f

_i

(

∗

,

y

)

:

X

×

Y

→

R

is said to be (Φ- ρ)-pseudounivex at

X

u

∈

with respect to

q

∈

R

n if for all

b

:

X

×

X

→

R

₊,

φ

:

X

×

X

×

R

n+1

→

R

and

ψ

:

R

→

R

with

ρ

_iis a real number ,we have

ϕ

( , ; (

x u

∇

1

f u y

i

( , ),

ρ

i

))

≥

0

⇒

b x u

( , )[ { ( , )

ψ

f x y

i

−

f u y

i

( , )}]

≥

0

(2.3)

Page: 1940-1956

© 2012, IJMA. All Rights Reserved 1942

Definition 2.7: A real-valued twice differentiable function

f

_i

(

x

,

∗

)

:

X

×

Y

→

R

is said to be (Φ,ρ)-pseudounivex at

X

y

∈

with respect to

p

∈

R

m if for all

b

:

Y

×

Y

→

R

₊,

φ

:

Y

×

Y

×

R

m+1

→

R

,and

ψ

:

R

→

R

with

ρ

_iis a real number we have

ϕ

( , ; (

v y

∇

₂

f x y

_i

( , ),

ρ

_i

))

≥ ⇒

0

b v y

( , )[ { ( , )

ψ

f x v

_i

−

f x y

_i

( , )}]

≥

0

(2.4)

Definition 2.8: A real valued twice differentiable function

f

is

(

φ

,

ρ

)

- unicave and

(

φ

,

ρ

)

-pseudounicave if

−

f

is

(

φ

,

ρ

)

-univex and

(

φ

,

ρ

)

pseudounivex respectively.

Example 2.1: We present here a function which is

(

φ

,

ρ

)

−

univex

function .We can proceed similarly for the other classes of function introduced.

Let

X

=

(

0

,

∞

),

f

:

X

→

R

defined by

f

(

x

)

=

x

log

x

.Obviously f is non convex.

We have

∇

f

(

u

)

=

1

+

log

x

.

Let

φ

:

X

×

X

×

R

→

R

defined by

φ

(

x

,

y

;

(

a

,

ρ

))

=

a

ρ

,

Let









=

−

+

0

)

,

(

log log

log 1

u u x x

u

u

x

b

if

if

0

log

log

0

)

log

log

(

≤

−

>

−

u

u

x

x

u

u

x

x

, and

ψ

(

x

)

=

3

x

We have

φ

(

x

,

u

;

(

∇

f

(

u

),

ρ

))

=

ρ

(

1

+

log

u

).

So the definition of

(

φ

,

ρ

)

−

univex

function becomes

3

)

log

1

(

)

log

log

(

3

log

log

log

1

≤

⇒

+

≥

−

×

−

+

_ρ

_ρ

u

u

u

x

x

u

u

x

x

u

It now follows that for

ρ

≤

3

, the function

f

(

x

)

=

x

log

x

becomes

(

φ

,

ρ

)

−

univex

function with respect to

b

,

ψ

.

Remark 1:

(i) If we consider the case

b

≡

1,

ϕ

( , ; (

x u

∇

f u

( ), ))

ρ

=

F x u

( , ;

∇

f u

( ))

with F is sub linear in third argument, then the above definition reduce to F-convexity and F-pseudo convexity introduced by Mishra [21].

(ii) If

b

≡

1,

ϕ

( , ; (

x u

∇

f u

( ), ))

ρ

=

η

( , )

x u

T

∇

f u

( ))

, where

η

:

X

× →

X

R

n, the above definition reduces to

η

-bonvex function given by Devi [11].

Definition 2.9: (Schwartz Inequality)

Let

x y

,

∈

R

n and

A

∈

R

n

×

R

n be a positive semi definite matrix, then

1 1

2 2

(

) (

)

T T T

x Ay

≤

x Ax

y Ay

, equality holds if for some

λ

≥

0,

Ax

=

λ

Ay

.

3. MIXED SYMMETRIC DUAL PROGRAM

For N={1,2,…n} and M={1,2,….m},let

J

₁

⊆

N

and

J

₂

=

N

\

J

₁

,

K

₁

⊆

M

and

K

2

=

M

\

K

1 .Let J1 denote

the number of elements in

J

₁ . The numbers J₂ , K1 , K2 are defined similarly. Notice that, if

J

1

=

φ

, then

N

J

₂

=

that is

|

J

₁

|

=

0

, and

|

J

₂

|

=

n

. It is clear that any x∈X ⊆ Rn can be written as,

x

=

(

x

1

,

x

2

)

_,

1

1 J

R

x

∈

and

_x

2

∈

_R

J2

.Similarly for any

y

∈

Y

⊆

R

m ,

y

=

(

y

1

,

y

2

),

y1∈RK1 ,y2 ∈RK2 . Let

R

R

R

f

i J

×

K

→

| | | | 1 1

:

and

g

i

R

J

×

R

K

→

R

| | | | 2 2

:

are twice continuously differentiable functions.

Page: 1940-1956

© 2012, IJMA. All Rights Reserved 1943

(SMSP): Mixed Symmetric Primal:

{

1 2

}

( , , )

( , , ),

( , , ),...,

_r

( , , )

H x y w

=

Min imize H x y w H x y w

H x y w

Subject to ₂ 1 1 1 1

1

[

( ,

)

]

0

r

i i i i

i

f x y

C w

λ

=

∇

−

≤

∑

, (3.1)

2 2 2 2 2

1

[

(

,

)

]

0

r

i i i i

i

g x

y

C w

λ

=

∇

−

≤

∑

, (3.2)

{

}

1 1 1 1 1

2 1

(

)

( ,

)

i

0

r T

i i i

i

y

λ

f x y

C w

=

∇

−

≥

∑

, (3.3)

{

}

2 2 2 2 2

2 1

(

)

(

,

)

i

0

r T

i i i

i

y

λ

g x

y

C w

=

∇

−

≥

∑

(3.4)

0

)

,

(

x

1

x

2

≥

, (3.5)

1 1 1

(

w

_i

)

T

C w

_{i i}

≤

1,

i

=

1, 2,... .

r

(3.6)

2 2 2

(

w

_i

)

T

C w

_{i i}

≤

1,

i

=

1, 2,... .

r

(3.7)

1

,

0

1

=

>

∑

= r

i i

λ

λ

(3.8)

(SMSD): Mixed Symmetric Dual:

{

1 2

}

( , , , )

( , , ),

( , , ),....,

_r

( , , )

G u v a

=

Ma xmize G u v a G u v a

G u v a

Subject to ₁ 1 1 1 1

1

[

( , )

i

]

0,

r

i i i

i

f u v

E a

λ

=

∇

+

≥

∑

(3.9)

2 2 2 2 1

1

[

(

,

)

i

]

0

r

i i i

i

g u v

E a

λ

=

∇

+

≥

∑

, (3.10)

1 1 1 1 2

1 1

( )

[

( , )

i

]

0

r T

i i i

i

u

λ

f u v

E a

=

∇

+

≤

∑

, (3.11)

2 2 2 2 2

1 1

(

)

[

(

,

)

i

]

0

r T

i i i

i

u

λ

g u v

E a

=

∇

+

≤

∑

, (3.12)

0

)

,

(

v

1

v

2

≥

, (3.13)

1 1 1

( )

a

i T

E a

i i

≤

1

,

i

=

1

,

2

,...

r

(3.14)

2 2 2

(

a

_i

)

T

E a

_{i i}

≤

1

,

i

=

1

,

2

,...

r

(3.15)

1

,

0

1

=

>

∑

= r

i i

λ

λ

. (3.16)

Where

1 1

1 1 2 2 1 1 1 ₂ 2 2 2 ₂ 1 1 1 2 2 2

( , , )

( ,

)

(

,

) (( )

T

)

((

)

T

)

(

)

T

(

)

T

i i i i i i i i i

H x y w

=





f x y

+

g x

y

+

x

E x

+

x

E x

−

y

C w

−

y

C w









1 1

1 1 2 2 1 1 1 ₂ 2 2 2 ₂ 1 1 1 2 2 2

( , , )

( , )

(

,

) (( )

T

)

((

)

T

)

( )

T

(

)

T

i i i i i i i i i

G u v a

=



_

f u v

+

g u v

−

v

C v

−

v

C v

+

u

E a

+

u

E a



_





1 2

1 2

| | | |

1 2 1 2 1 1 1 1 2 2 2 2

1 2 1 2

| | | |

1 2 1 1 1 1 2 2 2 2 1 2

1 2 1 2

,

(

,

),

,

,

(

,

,...

),

(

,

,...

),

( ,

),

( ,

,... ),

(

,

,...

),

,

K K

i i i r r

J J

r r i i

R w

w w

w

R

w

R

w

w w

w

w

w w

w

a

a a

a

a a

a

a

a a

a

a

R

a

R

λ

∈

=

∈

∈

=

=

=

=

=

∈

∈

Page: 1940-1956

© 2012, IJMA. All Rights Reserved 1944

Remark 3.1

Since the objective function of (MSP) and (MSD) contains the square root terms like

1 2

(

x Ax

T

)

, these problems are nondifferentiable programming problems.

For the following theorems let assume

ρ

is a real number and

φ

0i

,

φ

1i are a real valued function defined on 1

| | | | |

|Ji

×

Ji

×

Ji+

R

R

R

and

R

|Ki|

×

R

|Ki|

×

R

|Ki|+1 respectively such that 0

(

x

,

u

,

(

0

,

r

))

≥

0

i i i

φ

,for every

| | | |

)

,

(

_x

i

_u

i

∈

_R

Ji

×

_R

Ji

.

i

=

1

,

2

and 1

(

y

,

v

,

(

0

,

r

))

≥

0

i i i

φ

for

(

_y

i

,

_v

i

)

∈

_R

|Ki|

×

_R

|Ki| _and

+

∈

R

r

. Let

b

i

b

i

1 0

,

be

non negative real valued function defined on

_R

|Ji|

×

_R

|Ji|

and

R

|Ki|

×

R

|Ki| respectable for

i

=

1

,

2

We assume that

ψ

0i

,

ψ

1i

:

R

→

R

satisfying 0

(

u

)

≤

0

⇒

u

≤

0

i

ψ

and

ψ

₁i

(

u

)

≤

0

⇒

u

≤

0

and assume that

i i

1 0

,

ψ

ψ

are odd function, for

i

=

1

,

2

_.

Theorem 3.1(Weak duality)

Let

( , , , , )

x y

λ

w p

be feasible solution of (SMSP) and

( , , , , )

u v

λ

a q

be feasible solution (MSD) and

(I) 1 1 1

1

[ ( , ) ( )

i

]

r

T

i i i

i

f

v

E a

λ

=

∗

+ ∗

∑

is

( , )

φ ρ

₀1 -univex at

u

1for fixed

v

1,

(II) 1 1 1

1

[ ( , ) ( )

i

]

r

T

i i i

i

f x

C w

λ

=

∗ − ∗

∑

is

( , )

φ ρ

₁1 -unicave at

y

1 for fixed

x

1,

(III) 2 2 2

1

[

( ,

) ( )

i

]

r

T

i i i

i

g

v

E a

λ

=

∗

+ ∗

∑

is

(

φ ρ

_o2

, )

-pseudo univex at

u

2 for fixed

v

2,

(IV) 2 2 2

1

[ (

, ) ( )

i

]

r

T

i i i

i

g x

C w

λ

=

∗ − ∗

∑

is

(

φ ρ

₁2

, )

-pseudo unicave at

y

2 for fixed

x

2,

(V)

0

1 1 1 1 1 1

( ,

x u

; ( , )) ( )

u

T

0

φ

ξ ρ

+

ξ

≥

, where 1 ₁ 1 1 1 1

1

(

( , )

)

r

i i i i

i

f u v

E a

ξ

λ

=

=

∑

∇

+

,

(VI)

0

2 2 2 2 2 2

(

x u

,

; (

, )) (

u

)

T

0

φ

ξ ρ

+

ξ

≥

, where 2 ₁ 2 2 2 2

1

(

(

,

)

)

r

i i i i

i

g u v

E a

ξ

λ

=

=

∑

∇

+

.

(VII)

1

1 1 1 1 1 1

( ,

v y

; ( , )) (

y

)

T

0

φ

ς ρ

+

ς

≤

, where 1 ₂ 1 1 1 1

1

(

( ,

)

)

r

i i i i

i

f x y

C w

ς

λ

=

=

∑

∇

−

,

(VIII) ₁2

(

2

,

2

; (

2

, )) (

2

)

T 2

0

v

y

y

φ

ς ρ

+

ς

≤

, where 2 ₂ 2 2 2 2

1

(

(

,

)

)

r

i i i i

i

g x

y

C w

ς

λ

=

=

∑

∇

−

.

Then

H x y

( , , , )

λ

a

≥

G u v

( , , , )

λ

w

Proof: Since 1 1 1

1

[ ( , ) ( )

i

]

r

T

i i i

i

f

v

E a

λ

=

∗

+ ∗

∑

is

(

φ

₀

,

ρ

)

-univex at

u

1 for fixed

v

1and

λ

>

0

_, Then for 1 |1| | 1| 1 |1| | 1| |1| 1

0

:

,

0

:

J J J J J

b

R

×

R

→

R

₊

φ

R

×

R

×

R

+

→

R

, o

:

R

→

R

1

ψ

and

ρ

∈

R

we have

0

1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1

( ,

)

{

[ ( , ) ( )

i

( , ) ( )

]}

r

T T

i i i i i i

i

b x u

ψ

λ

f x v

x

E a

f u v

u

E a

=

+

−

−

∑

₀1 1 1 ₁ 1 1 1 1

1

( ,

; (

( [

( , )

], ))

r

i i i i

i

x u

f u v

E a

φ

λ

ρ

=

≥

∑

∇

+

, for

i

=

1

,

2

,...

r

. (3.17)

Using (3.11) in hypothesis (V) we get

1 1 1 1 1 1 1

0

( ,

; (

₁

[

1

( ,

)

], )

0

r

i i i i

i

x u

f u v

E a

φ

∑

₌

λ

∇

+

ρ

≥

Page: 1940-1956

© 2012, IJMA. All Rights Reserved 1945

So from (3.17) and (3.18) with the property of 1

0

b

and

ψ

₀1 we obtain

∑

∑

= =

+

≥

+

r

i

i i T i

i r

i

i T i

i

f

x

v

x

E

i

a

f

u

v

u

E

a

1

1 1 1 1 1 1

1 1 1 1 1

]

)

(

)

,

(

[

]

)

(

)

,

(

[

λ

λ

. (3.19)

Now it follows from the

(

φ

₁1

,

ρ

)

-unicavity of

∑

=

∗

−

∗

r

i

i T i

i

f

x

C

w

i

1

1 1 1

]

)

(

)

,

(

[

λ

at

y

2 for fixed

x

2, for

λ

>

0

_,

1 1 1 1 1

| | | | | | | | | |

1 1 1

1

:

,

1

:

K K K K K

b

R

×

R

→

R

₊

φ

R

×

R

×

R

+

→

R

,

:

R

→

R

1 1

ψ

and

ρ

∈

R

that

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 2

1 1

( , ) {

[ ( , ) ( )

i

( ,

) ( )

]}

( , ;

( [

( ,

)

], ))

r r

T T

i i i i i i i i i i

i i

b y v

ψ

λ

f x v

v

C w

f x y

y

C w

φ

v y

λ

f x y

C w

ρ

= =

−

−

+

≤

∇

−

∑

∑

(3.20) Hypothesis (VII) in light of (3.3) implies

1 1 1 1 1 1 1

1 2

1

( ,

; (

[

( ,

)

], ))

0

r

i i i i

i

v y

f x y

C w

φ

λ

ρ

=

∇

−

≤

∑

. (3.21)

So from (3.20), (3.21) and with the property of

b

₁1 and

ψ

₁1 we get

1 1 1 1 1 1 1 1 1 1

1 1

[ ( ,

) ( )

]

[ ( ,

) (

)

]

r r

T T

i i i i i i i

i i

f x v

v

C w

f x y

y

C w

λ

= =

−

≤

−

∑

∑

. (3.22)

Subtracting (3.22) from (3.19) we get

{

}

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1

[( )

( )

]

[ ( , ) ( )

] [ ( ,

) (

)

]

r r

T T T T

i i i i i i i i i i i i

i i

x

E a

v

C w

f u v

u

E a

f x y

y

C w

λ

λ

= =

+

≥

+

−

−

∑

∑

. (3.23)

Similarly hypothesis (VI) in light of (3.12) implies

0

2 2 2 2 2 2 2

1 1

(

,

; (

[

(

,

)

], ))

0

r

i i i i

i

x u

g u v

E a

φ

λ

ρ

=

∇

+

≥

∑

. (3.24)

So the

(

φ ρ

o2

, )

-pseudo univexity of

2 2 2

1

[

( ,

) ( )

i

]

r

T

i i i

i

g

v

E a

λ

=

∗

+ ∗

∑

at

u

2 for fixed

v

2 _{and for}

0

>

λ

_, 2 | 2| | 2| 2 | 2| | 2| | 2| 1

0

:

,

0

:

J J J J J

b

R

×

R

→

R

₊

φ

R

×

R

×

R

+

→

R

,

ψ

_o2

:

R

→

R

and

ρ

∈

R

, implies

(

)

0

2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 ₁

(

,

)

(

,

) (

)

i

(

,

) (

)

0

r _{T T}

i i i i i i

i

b x u

ψ

∑

₌

λ



_

g x v

+

x

E a

−

g u v

−

u

E a



_

≥

and by the properties of

b

20 and

2 0

ψ

, we get

2 2 2 1 1 2 2 2 2 2

1 1

[

(

,

) (

)

]

[ (

,

) (

)

]

r r

T T

i i i i i i i i

i i

g x v

x

E a

g u v

u

E a

λ

λ

= =

+

≥

+

∑

∑

. (3.25)

Similarly from (3.4) and from the hypothesis (iv), (viii) with the property of

b

₁2and

ψ

₁2 , we get

2 2 2 2 2 2 2 2 2 2

1 1

[

(

,

) (

)

]

[

(

,

) (

)

]

r r

T T

i i i i i i i i

i i

g x v

v

C w

g x

y

y

C w

λ

λ

= =

−

≤

−

∑

∑

. (3.26)

Subtracting (3.26) from (3.25) we get

{

}

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1

[(

)

(

)

]

[ (

,

) (

)

] [ (

,

) (

)

]

r r

T T T T

i i i i i i i i i i i i

i i

x

E a

v

C w

g u v

u

E a

g x

y

y

C w

λ

λ

= =

+

≥

+

−

−

∑

∑

(3.27) Adding (3.23) and (3.27), we obtain

1 1 1 1 1 1 2 2 2 2 2 2 1

[( )

( )

(

)

(

)

]

r

T T T T

i i i i i i i i i

i

x

E a

v

C w

x

E a

v

C w

λ

=

+

+

+

∑

1 1 1 1 1 1 1 1 1 1 2 2

1

{[ ( , ) ( )

[ ( ,

) (

)

] [ (

,

)

r _{T T}

i i i i i i i i

i=

λ

f u v

u

E a

f x y

y

C w

g u v

≥

∑

+

−

−

+

2 2 2 2 2 2 2 2

(

u

)

T

E a

i i

] [

g x

i

(

,

y

) (

y

)

T

C w

i i

]}

+

−

−

Page: 1940-1956

© 2012, IJMA. All Rights Reserved 1946

From the Definition 2.9 (Schwartz inequality), we have

1 1

2 2

1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 1

( )

x

T

E a

_{i i}

+

( )

v

T

C w

_{i i}

+

(

x

)

T

E a

_{i i}

+

( )

v

T

C w

_{i i}

≤

(( )

x

T

E x

_i

) (( )

a

_i T

E a

_{i i}

)

1 1 1 1 1 1

2 2 2 2 2 2

1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2

(( )

v

T

C v

_i

) ((

w

_i

)

T

C w

_{i i}

)

((

x

)

T

E x

_i

) ((

a

_i

)

T

E a

_{i i}

)

(( )

v

T

C v

_i

) ((

w

_i

)

T

C w

_{i i}

)

+

+

+

(3.29)

Using (3.6), (3.7), (3.14) and (3.15) in (3.29) we obtain

1 1 1 1 1 1 2 2 2 2 2 2

( )

x

T

E a

i i

+

( )

v

T

C w

i i

+

(

x

)

T

E a

i i

+

(

v

)

T

C w

i i

1 1 1 1

2 2 2 2

1 1 1 1 1 1 2 2 2 2 2 2

(( )

x

T

E x

_i

)

(( )

v

T

C v

_i

)

((

x

)

T

E x

_i

)

(( )

v

T

C v

_i

)

≤

+

+

+

. (3.30)

Using (3.30) in (3.28), we get

1 1 1 1

2 2 2 2

1 1 1 1 1 1 2 2 2 2 2 2

(( )

x

T

E x

_i

)

+

(( )

v

T

C v

_i

)

+

((

x

)

T

E x

_i

)

+

(( )

v

T

C v

_i

)

1 1 1 1 1 1 1 1 1 1 2 2

1

{[ ( , ) ( )

] [ ( ,

) (

)

] [ (

,

)

r _{T T}

i i i i i i i i

i=

λ

f u v

u

E a

f x y

y

C w

g u v

≥

∑

+

−

−

+

2 2 2 2 2 2 2 2

(

u

)

T

E a

_{i i}

[

g x

_i

(

,

y

) (

y

)

T

C w

_{i i}

]

+

−

−

1 1

2 2

1 1 2 2 1 1 1 2 2 2 1 1 1 2 2 2

1

{ ( ,

)

(

,

) (( )

)

((

)

)

(

)

(

)

r _{T T}

i i i i i i i i i

i=

λ

f x y

g x y

x E x

x E x

y

C w

y

C w

⇒

∑

+

+

+

−

−

1 2

1 1 2 2 1 1 1

1

{ ( ,

)

(

,

) (( )

)

r

T

i i i i

i

f u v

g u v

v

C v

λ

=

≥

∑

+

−

1

2

2 2 2 1 1 1 2 2 2

(( )

v C v

_i

)

( )

u

T

E a

_{i i}

(

u

)

T

E a

_{i i}

}

−

+

−

∑

=

≥

∑

=

⇒

r

i

r i i i i

i

H

x

y

a

G

u

v

w

1

λ

(

,

,

)

1

λ

(

,

,

)

That is

H

(

x

,

y

,

λ

,

a

)

≥

G

(

u

,

v

,

λ

,

w

)



Theorem 3.2 (Strong Duality):

f

i

R

J

×

R

K

→

R

| | | | 1 1

:

and

g

i

R

J

×

R

K

→

R

| | | | 2 2

:

be thrice differentiable function

and let

(

x

ˆ

1

,

x

ˆ

2

,

y

ˆ

1

,

y

ˆ

2

,

λ

ˆ

,

w

ˆ

1i

,

w

ˆ

i2

)

be a weak efficient solution for (SMSP) .Let

λ

ˆ

=

λ

be fixed in (MSD).Assume

that (i) the matrix

(

2₂

)

1

i r

i

i

∇

f

∑

=

λ

and

(

2₂

)

1

i r

i

i

∇

g

∑

=

λ

are positive definite or negative definite and (ii) the set

{

1 1

}

2 1 1 1 1 1

2

f

−

C

w

ˆ

,...

..

∇

f

r

−

C

r

w

ˆ

r

∇

and

{

ˆ

,...

..

2

ˆ

2

.

}

2 2 1 2 1 1

2

g

−

C

w

∇

g

r

−

C

r

w

r

∇

are linearly independent. Then there exist

_ˆ

1 | 1|

_,

_ˆ

2 |J2|

i J

i

R

a

R

a

∈

∈

such that

,

ˆ

,

ˆ

)

ˆ

,

ˆ

,

ˆ

,

ˆ

,

ˆ

(

x

1

x

2

y

1

y

2

λ

a

i1

a

i2 is feasible solution of (SMSD) and the

two objective values are equal. Also if the hypothesis of Theorem 3.1 are satisfied for all feasible solution of (SMSP)

and (SMSD), then

(

ˆ

1

,

ˆ

2

,

ˆ

1

,

ˆ

2

,

ˆ

,

ˆ

1

,

ˆ

2

)

i i

a

a

y

y

x

x

λ

is properly efficient solution of (SMSD).

Proof: Since

(

x

ˆ

1

,

x

ˆ

2

,

y

ˆ

1

,

y

ˆ

2

,

λ

ˆ

,

w

ˆ

i1

,

w

ˆ

i2

)

is a weak efficient solution of (SMSP) by Fritz-John constraint condition (Mangasarian, (1969)) there exist

| | 2 | | 1 | | 2 | |

1 1

_,

2

_,

1

_,

2

,

,

,

,

r r K K J J

r

R

R

R

R

R

R

R

R

∈

∈

∈

∈

∈

∈

∈

∈

τ

δ

γ

β

β

ξ

ξ

α

Such that

∑

∑

= =

=

−

∇

∇

+

+

∇

r i i i r i i i i

i

f

E

a

f

y

1 1 1 1 2 1 1 1 1

1

ˆ

]

[

(

)(

ˆ

)]

[

λ

β

γ

ξ

α

(3.31)

∑

∑

= =

−

∇

∇

+

+

∇

r i r i i i i i i

i

g

E

a

g

y

1 1 2 2 2 1 2 2

1

ˆ

]

[

(

)(

ˆ

)]

[

λ

β

γ

α

, (3.32)

,

0

]

ˆ

)

ˆ

)][(

[(

])

ˆ

)[

ˆ

(

1 1 1 2 2 1 1 1 2

∑

∑

= =

=

−

∇

+

−

∇

−

r i i i r i i i i i

i

γ

λ

f

C

w

f

β

γ

y

λ

α

(3.33)

,

0

]

ˆ

)

ˆ

)][(

[(

]

ˆ

)[

ˆ

(

1 2 2 2 2 1 2 2 2

∑

∑

= =

=

−

∇

+

−

∇

−

r i i i r i i i i i

i

γ

λ

g

C

w

g

β

γ

y

λ

α

Page: 1940-1956

© 2012, IJMA. All Rights Reserved 1947

0

]

ˆ

[

)

ˆ

(

β

1

−

γ

y

1 T

∇

₂

f

_i

−

C

_i1

w

1_i

−

δ

_i1

=

, (3.35)

0

]

ˆ

[

)

ˆ

(

2 2 2 2

2

2

−

∇

−

−

=

i i i i T

w

C

g

y

δ

γ

β

, (3.36)

r

i

f

y

ˆ

)

ˆ

_i

]

T _i

0

,

1

,

2

,...

[(

β

1

−

γ

1

λ

∇

2₂

=

=

, (3.37)

r

i

g

y

ˆ

)

ˆ

_i

]

T _i

0

,

1

,

2

,...

[(

β

2

−

γ

2

λ

∇

2₂

=

=

(3.38) 1 1 1 1 1 1 1

ˆ

2

)

ˆ

(

ˆ

_{i i i i i}

i

i

C

y

β

γ

y

λ

C

τ

C

w

α

+

−

=

, (3.39)

2 2 2 2 2 2 2

ˆ

2

)

(

ˆ

T _{i i i i i}

i

i

C

y

β

γ

y

λ

C

τ

C

w

α

+

−

=

, (3.40)

2 1 1 1 1 1 1 1

)

ˆ

)

ˆ

((

ˆ

)

ˆ

(

x

E

w

x

E

i

x

T i

i

T

=

, (3.41)

2 1 2 2 2 2 2 2

)

ˆ

)

ˆ

((

ˆ

)

ˆ

(

x

E

w

x

E

i

x

T i

i

T

=

, (3.42)

0

]

ˆ

[

)

(

1 1 1 2

1

∑

∇

−

=

= r i i i i i T

w

C

f

λ

β

, (3.43)

0

]

ˆ

[

)

(

1 2 2 2

2

∑

_∇

₋

₌

= r i i i i i T

w

C

g

λ

β

, (3.44)

0

]

ˆ

[

)

ˆ

(

1 1 1 2

1

∑

∇

−

=

= r i i i i i T

w

C

f

y

λ

γ

, (3.45)

0

]

ˆ

[

)

)

ˆ

((

1 2 2 2

2

∑

∇

−

=

= r i i i i i T

w

C

g

y

λ

γ

, (3.46)

r

i

w

C

w

_i T _{i i}

i

[(

ˆ

)

ˆ

1

]

0

,

1

,

2

,...

1 1

1

−

=

=

τ

, (3.47)

r

i

w

C

w

_i T _{i i}

i

[(

ˆ

)

ˆ

1

]

0

,

1

,

2

,...

2 2

2

−

=

=

τ

_{, (3.48)}

,

0

1

λ

=

δ

T

(3.49)

0

ˆ

2

λ

₌

δ

T

, (3.50)

0

ˆ

1

ξ

1

=

x

, (3.51)

0

ˆ

2

ξ

2

=

x

, (3.52)

1

ˆ

ˆ

1 1 1

≤

i i T i

E

w

w

_, (3.53)

,

1

ˆ

ˆ

2 2 2

≤

i i T i

E

w

w

(3.54)

0

)

,

,

,

,

,

(

α

β

γ

τ

δ

ξ

≥

, (3.55) and

(

α

,

β

,

γ

,

τ

,

δ

,

ξ

)

≠

0

_. (3.56)

Since

λ

ˆ

>

0

, from (3.49) and (3.50), we get

δ

1

=

0

,

δ

2

=

0

_{. Consequently (3.35) and (3.36) implies}

0

]

ˆ

[

)

ˆ

(

2 1 1

1

1

−

∇

−

=

i i i T

w

C

f

y

γ

β

(3.57) and

(

β

2

−

γ

y

ˆ

2

)

T

[

∇

₂

g

_i

−

C

_i2

w

ˆ

_i2

]

=

0

. (3.58)

Pre-multiplying (3.33) by

(

β

1

−

γ

y

ˆ

1

)

T and then using (3.57) we get

.

0

)

ˆ

)(

(

ˆ

)

ˆ

(

1 1

1 2 2 1

1

₋

∑

_∇

₋

₌

=

y

f

y

r i i i

T

λ

β

γ

γ

β

(3.59)

Again pre- multiplying (3.34) by

(

β

2

−

γ

y

ˆ

2

)

Tand then using (3.58) we get

.

0

)

ˆ

)(

(

ˆ

)

ˆ

(

2 2

1 2 2 2

2

₋

∑

_∇

₋

₌

=

y

g

y

r i i i

T

λ

β

γ

γ

β