An International Journal of Optimization
and Control: Theories & Applications
Vol.4, No.1, pp.21-33 (2014) © IJOCTA
ISSN: 2146-0957eISSN: 2146-5703
DOI: 10.11121/ijocta.01.2014.00175
http://www.ijocta.com
Mond-Weir type second order multiobjective mixed symmetric duality with
square root term under generalized univex function
Arun Kumar Tripathy
Department of Mathematics, Trident Academy of Technology
F2/A, Chandaka Industrial Estate, In front of Infocity, Patia, Bhubaneswar-751024, India
Email: [email protected]
(Received July 10, 2013; in final form December 03, 2013)
Abstract. In this paper, a new class of second order ( ,  ) -univex and second order ( ,  )
pseudo univex function are introduced with example. A pair Mond-Weir type second order mixed
symmetric duality for multiobjective nondifferentiable programming is formulated and the duality
results are established under the mild assumption of second order ( ,  ) univexity and second order
pseudo univexity. Special cases are discussed to show that this study extends some of the known
results in related domain.
Keywords: Second order ( ,  ) univex, mixed symmetric duality, efficient solution, square root
term, Schwartz inequality.
AMS Classification: 90C29, 90C30, 90C46
Xu [19] formulated two mixed type duals in
multiobjective programming and also proved
duality theorems. Earlier, Chandra et al. [9] and
Bector et al. [7,8] formulated mixed symmetric
duality for a class of nonlinear programming
problems. Yang et al. [20] discussed a mixed
symmetric duality for a class of nondifferentiable
nonlinear programming problems. Aghezzaf [2]
formulated a second order multiobjective mixed
type dual and obtained various duality results
involving a new class of generalized second order
(F,ρ)-convex function. Mishra et al. [14, 15] and
Mishra [13] presented mixed symmetric first and
second order duality in nondifferentiable
mathematical programming problem under Fconvexity. Recently, Ahmad and Husain [3]
discussed a pair of multiobjective mixed
symmetric dual programs over arbitrary cones
and established duality results under K-preinvex
/K-pseudo invexity assumption. Also, Kailey et
al. [10] established mixed second order
multiobjective symmetric duality with cone
1. Introduction
Mond [16] initiated second order symmetric
duality type in nonlinear programming and
proved second order symmetric duality theorems
under second order convexity. Mangasarian [12]
discussed second order duality in nonlinear
programming under inclusion condition. Mond
[16] and Mangasarian [12] also indicated
possible computational advantages of the second
order dual over the first order dual. Later, Bector
and Chandra [6] presented a pair of Mond-Weir
type second order dual programs and proved
weak, strong and self duality theorems under
pseudo bonvexity and pseudo convexity
assumption. Ahmad and Sharma [5] established
second order duality for non differentiable
multiobjective programming under generalized
F-convexity. The concept of mixed duality is
interesting and useful both from theoretical as
well as from algorithmic point of view.
Corresponding Author. Email: [email protected]
21
A. K. Tripathy / Vol.4, No.1, pp.21-33 (2014) © IJOCTA
22
constrain under -bonvexity. Li and Gao [11] 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 this paper, a pair of second order
multiobjective mixed symmetric dual programs
using square root term is formulated and duality
theorems are proved for these programs under
generalized ( ,  ) univexity. Special cases are
discussed to show that this study extends some of
the known results in related domain.
2. Notations and definitions
Let R n and R m are n-dimensional and mdimensional Euclidean space respectively. R n
m
and R  their respective nonnegative orthant. The
following conventions for vectors x , u  R n will
be
followed
throughout
this
paper:
x  u  xi  u i ,
For
any
vector
x  u  x i  u i ; i  1, 2 ,.... n .
x, u  R
n
we
denote
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
f ( x )  f ( x ), f ( x )  f ( x ).
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
j  {1, 2, ..., r } 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 :( Schwartz Inequality) Let
n
n
n
be a positive semi
x , y  R and A  R  R
n
x u 
T
 xu
i
1
i
.
i 1
Let X and Y are open subset sets of R n and R m
respectively. Let f i ( x , y ) be a real valued twice
differentiable function defined on X  Y . Let
 x f i ( x , y ) and  y f i ( x , y ) denote the gradient
vectors of f i ( x , y ) with respect to first variable
x and second variable y respectively. Also
 xx f i ( x , y ) and  yy f i ( x , y ) denote the Hessian
matrix of f i ( x , y ) with respect to the first
variable x and second variable y respectively.
For p  R ,  x (  yy f i ( x , y ) p ) and
m
 y (  yy f i ( x , y ) p )
are n  m and
matrix obtained by differentiating the elements of
 yy f i ( x , y ) p with respect to x and
y
respectively.
Consider
the
following
programming problem (MP):
MP :( Primal) Minimize
multiobjective
X0
i
i
| K i | 1
, for i  1, 2; respectively such
i
i
i
that  ( x , u ,  ) and 1 ( v , y ,  ) , i  1, 2. are
convex on R n 1 , R m 1 respectively
with
R
|K i |
R
|K i |
i
0
i
R
i
 0 ( x , u , (0, r ))  0,
i
i
i
and 1i ( v i , y i , (0, r ))  0,
i  1, 2 and r  R  . b 0 and b 1 are non
i
for
i
negative function defined on R | J |  R | J | and
|K |
|K |
R  R , i  1, 2. respectively .
i
i
i
Throughout this chapter, we assume that
i
i
i
 0 , 1 : R  R satisfying  0 ( u )  0  u  0,
 1 (u )  0  u  0
i
i
h ( x )  0, x  X  R ,
n
where f : X  R r , h : X  R m .
Let
i
and
 0 (  a )    0 ( a ),
i
i
 1 (  a )    1 ( a ) for i  1, 2.
f ( x )  ( f1 ( x ), f 2 ( x ), ....., f r ( x ))
Subject to
i
i
Let r ,   R . Suppose  0 and  1 are a real valued
|J |
|J |
| J | 1
function defined on
R R R
and
i
mm
1
definite matrix, then x T A y  ( x T A x ) 2 ( y T A y ) 2 ,
equality holds if for some   0, Ax   Ay .
be the set of all feasible solutions of
problem (P); that i.e. X 0  { x  X | h ( x )  0} .
i
Now we define a new class of second order
second order ( ,  ) -pseudo
univex and second order ( ,  ) -quasi-univex as
follows.
Definition 2.5 A real-valued twice differentiable
function
f i (, y ) : X  X  R is said to be
( ,  ) -univex,
Mond-Weir type second order multiobjective mixed symmetric duality …
second order
respect
to
( ,  ) -univex
qR
n
for
at
u  X with
b : X  X  R ,
n 1
 R,
 : R  R , if there exist  : X  X  R
  R such that

  u f i (u , y )
   x, u; 

   uu f i ( u , y ) q , 

f : R  R  R defined as
Let
x
2
e
y
2
.
2
 xx f ( x , y )  (4 x  2 ) e
2

 

2
,
f ( x , y )  f (u , y )  ( x  u )  x f (u , y )
T
with respect to q  R n for b : X  X  R  and
 : R  R , if there exist  : X  X  R
  R such that
x
Now f ( x , y ) is not convex at u=0, as
second order ( ,  ) -pseudo univex at u  X
n 1
 R,
 ( x , u ; (  u f i ( u , y )   uu f i ( u , y ) q ,  ))  0
 f i ( x , y )  f i (u , y ) 
  0.
 b ( x , u )  1 T
  q  f ( u , y ) q 
uu i
 2

Definition 2.7 A real-valued twice differentiable
function f i (, y ) : X  X  R is said to be
second order ( ,  ) -quasiunivex at u  X with
for
Example 2.1
So we have  x f ( x , y )   2 xe  x ,
Definition2.6 A real-valued twice differentiable
function f i (, y ) : X  X  R is said to be
respect to q  R n
then the above definitions reduce to second
order F-convexity and second order F-pseudo
convexity as introduced by Mishra [13].
f ( x, y )  e
 f i ( x , y )  f i (u , y ) 

b ( x , u )  1 T
  q  f ( u , y ) q 
uu i
 2

23
b : X  X  R  and
n 1
 R,
 : R  R , if there exist  : X  X  R
  R such that
e
e
x
x
2
2
e
with F is sub linear in third argument and ρ=0,
)
1
2
q  xx f ( u , y ) q
T
 ( x  u ) [  x f ( u , y )   xx f ( u , y ) q ]
T
e
x
2
e
u
2

 ( x  u )[  2 ue
e
x
2
1
2
u
(4 u  2 ) e
2
2
u
2
 (4 u  2) e
2
u
2
]
 2 x  2  0, for u  0, x  12 .
So f ( x , y ) is not second order convex/bonvex at
u=0,
Let  : R  R  R 2  R defined as

 xu
 ( x , u ; ( a ,  ))  (1  2 ) e  u a ;
 : R  R defined as  ( x )  x ;
T
b : R  R  R  defined as b ( x , u )  1 .
b ( x , u ) [ f ( x , y )  f ( u , y ) 
1
2
q  xx f ( u , y ) q ]
T
 ( x , u , (  x f ( u , y )   xx f ( u , y ) q ,  ))
x
2
e
[  2 ue
2
2
 1  0 for u  0, x  0.
f ( x , y )  f (u , y ) 
e
 F ( x , u ;  u f ( u )   uu f ( u ) q )   d ( x , u )
u
0 
  ( x , u ; (  u f i ( u , y )   uu f i ( u , y ) q ,  ))  0.
 ( x , u ; (  u f ( u )   uu f ( u ) q ,  ))
 ( x  u )(  2 ue
1 
Now
Remark 2.1 If we consider the case b  1,
2
Let q    . Then
 f i ( x , y )  f i (u , y ) 
0
b ( x , u )  1 T
  q  f ( u , y ) q 
uu i
 2

Definition 2.8 A real valued twice differentiable
function f is second order ( ,  ) - unicave and
second order ( ,  ) -pseudounicave if  f is
second order ( ,  ) -univex and second order
( ,  ) pseudounivex respectively.
u
u
u
2
2

1
2
(4 u  2) e
2
 (4 u  2) e
2
u
u
2

 (1  2 ) e
 xu
2
]

e
x
2
 1  1  (1  2 )  2 at u  0,
e
x
2
 2  1  0 for   0,  x  R .

So f ( x , y ) is second order ( ,  ) -univex
function at u=0 .
Hence from the above example it is clear that
second order ( ,  ) -univex function is more
generalized than convex and second order convex
function.
A. K. Tripathy / Vol.4, No.1, pp.21-33 (2014) © IJOCTA
24
Example 2.2
Let g : R   R   R defined as
3. Mond-Weir type second order mixed
symmetric dual program
g ( x , y )  x ln x  y ln y ,  x g ( x , y )  1  ln x ,
For N={1,2,…,n} and M={1,2,…,m}, let J 1  N
and J 2  N \ J 1 . Similarly K 1  M and K 2  M \ K 1 .
 xx g ( x , y ) 
1
x
.
Now g ( x , y ) is not convex at u=1, since
Let J 1 denote the number of elements in J 1 .
g ( x , y )  g (u , y )  ( x  u )  x g (u , y )
The
 x ln x  u ln u  x  x ln u  u  u ln u
similarly. Notice that if | J 1 | 0 , then | J 2 | n . It
T
 x ln x  x  x ln u  x ln x  x  0
for
x  (0, e ).
Again g ( x , y )  g ( u , y )  q  xx g ( u , y ) q
1
2
T
 ( x  u ) [  x g ( u , y )   xx g ( u , y ) q ]
T
 x ln x  u 
1
2u
 1  x  x ln u 
x
u
 x ln x   2 x for u  1.
5
2
 x ln x 
5
2
 2 x  0, for x  [2,  )
So g ( x , y ) is not second order convex/bonvex at
u=1,
Let  : R  R  R 2  R defined as

 xu
 ( x , u ; ( a ,  ))  (1  2 ) e  u a ;
 : R  R defined as  ( x )  x ;
T
b : R  R  R  defined as b ( x , u )  1 .
b ( x , u ) [ g ( x , y )  g ( u , y ) 
 x ln x  u ln u 
1
2
q  xx g ( u , y ) q ]
T
1
2u


 (1  2 ) e
 xu
 xu
 (  u g   uu gq )
 (1  ln u  u1 )  0

x
 2  0, at u  1,

x
 2  0  x  0, for   1.
 (1  2 ) e
 (1  2 ) e
1 
Let q    .
x  ( x , x ), x  R
1
1
2
defined
J1
J2
and x 2  R
. Similarly,
any y  Y  R , can be written as y  ( y 1 , y 2 ) ,
m
y R
1
K1
,y R
K2
R
|K 1 |
2
|J1|
Let f i : R
gi : R
R
|J 2 |
.
 R and
 R
|K 2 |
are thrice continuously
differentiable functions.
Let  i  R , p i1  R | K | , w i1  R | K
1
q R
1
i
| J1 |
,q  R
2
i
1|
, a R
|J 2 |
1
i
2
, pi  R
| J1 |
|K 2 |
2
, wi  R
, a R
2
i
|J 2 |
|K 2 |
,
,
p  ( p1 , p 2 , ..., p r ), p  ( p1 , p 2 , ..., p r ),
1
1
1
1
2
2
2
2
q  ( q 1 , q 2 , ..., q r ), q  ( q 1 , q 2 , ..., q r ),
1
1
1
1
2
2
2
2
a  ( a1 , a 2 , ..., a r ), a  ( a1 , a 2 , ..., a r ),
1
1
1
1
2
2
2
2
w  ( w 1 , w 2 , ..., w r ), w  ( w 1 , w 2 , ..., w r ), and
1
1
1
2
1
1
2
2
2
2
2
E i , E i , C i , C i are positive semi definite
matrices of order | J 1 |, | J 2 |, | K 1 | and | K 2 |
respectively for i  1, 2, ..., r .
Now we formulate the following pair of
multiobjective mixed symmetric dual programs
and prove duality theorems:
(SMSP): Second order mixed symmetric
primal:
0 
Now
b ( x , y ) [ g ( x , y )  g ( u , y ) 
 x ln x  u ln u 
are
is clear that any x  X  R n can be written as
1
Let  ( x , u ; (  u g   uu gq ),  ))  0
 (1  2 ) e
numbers J 2 , K 1 , K 2
1
2u
1
2
q  xx g ( u , y ) q ]
T
Minimize
 0 , for x  0.
 H 1 ( x , y , w1 , p1 ),



H ( x , y , w , p )   H 2 ( x , y , w 2 , p 2 ), 
 ..., H ( x , y , w , p ) 
r
r
r 

Subject to
So g ( x , y ) is second order ( ,  ) -pseudounivex function at u=1.
Hence from the above example it is clear that
second order ( ,  ) -pseudo-univex function is
more generalize than convex and second order
convex function.
r
  [
i
1
y
1
1
1
f i ( x , y )  C i wi  
1
1 1
y y
1
1
1
(3.1)
2
2
2
(3.2)
f i ( x , y ) p i ]  0,
i 1
r
  [
i
2
y
2
2
2
2
g i ( x , y )  C i w i   2 2 g i ( x , y ) p i ]  0,
y y
i 1
r
1 T
(y )
  [
i
i 1
1
y
1
1
1 1
f i ( x , y )  C i wi  
1
1 1
y y
1
1
f i ( x , y ) p i ]  0,
(3.3)
Mond-Weir type second order multiobjective mixed symmetric duality …
25
r
2 T
(y )

2 2
2 2
2 1 2
i [  y 2 g i ( x , y )  C i w i   y 2 y 2 f i ( x , y ) p i ]  0,
(3.4)
i 1
Remark 3.1
Since the objective function of (MSP) and
(MSD) contain the quadratic term like (xTAx)
these
problems
are
nondifferentiable
multiobjective programming problems.
( x , x )  0,
(3.5)
( w i ) C i w i  1, i  1, 2, ...r .
(3.6)
( w i ) C i w i  1, i  1, 2, ...r .
(3.7)
Theorem 3.1(Weak duality) Let
1
2
1
2
1
2
1
2
( x , x , y , y ,  , w , w , p , p ) be feasible solution
(3.8)
of (SMSP) and ( u 1 , u 2 , v 1 , v 2 ,  , a 1 , a 2 , z 1 , z 2 )
feasible solution (MSD) and
1
1 T
2
1
T
2
1
2
2
r
  0,   i  1
i 1
(SMSD): Second order Mixed Symmetric
Dual:
Maximize
 G1 ( u , v , a1 , q1 ),



G ( u , v , a , q )   G 2 ( u , v , a 2 , q 2 ), 
 ..., G ( u , v , a , q ) 
r
r
r 

r
1.
i
u
1
fi (u , v ) 
2
2
1 1
Ei a i

1
fi (u , v
1 1
u u
1
1
) qi ]
 0,
2
g i (u , v ) 
2 2
Ei a i

2
2
u u
2
2
g ( u , v )]  0,
(3.9)
1
  [ f ( x ,  )  ( )
1
i
T
i
1
1
is second order
C i wi ]
1
  [ g (, v
i
1
2
T
2
2
)  ( ) E i a i ]
i
is second order
i 1
(3.10)
( o ,  )
2
-pseudo univex at
u
2
for fixed
v
2
r

i [
1
u
1
1 1
fi (u , v )  E i a i  
1
1
1 1
u u
1
1
f ( u , v ) q i ]  0,
(3.11)
4.
  [g (x
i
2
i
T
2
2
,  )  ( ) C i w i ]
is
second
i 1
i 1
r
2 T
(u )
-univex at u 1 for fixed v 1 .
( 0 ,  )
r
r
1 T
is second
(1 ,  ) -unicave at y for fixed x .
i 1
(u )
1 1
1
1
r
u
T
)  () E i a i ]
i 1
3.
i
1
i
r
2.
i 1
  [
i
order
r
  [
  [ f (, v
i 1
Subject to
1
be

i [
2 2
2 2
2 2 2
g ( u , v )  E a i   2 2 g i ( u , v ) q ]  0,
i
i
u i
u u
2
1
2
 1; i  1, 2, .....r ,
1 T
( ai )
1 1
E i ai
2 T
( ai )
2 2
E i ai
 1; , i  1, 2, .....r ,
(3.13)
(3.14)
(3.15)
2
fixed x 2 .
i 1
( v , v )  0,
-pseudo unicave at y 2 for
(1 ,  )
order
(3.12)
5.  1 ( x 1 , u 1 ; ( 1 ,  ))  ( u 1 ) T  1
0
 0; where
r
 
1
  [
i
1
u
1
1
1 1
1
1
1
f ( u , v )  E i a i   u 1u 1 f ( u , v ) q i ].
i 1
r
  0,

i
 1.
(3.16)
i 1
6.  2 ( x 2 , u 2 ; ( 2 ,  ))  ( u 2 ) T  2
0
where
r


 f ( x 1 , y 1 )  g ( x 2 , y 2 )  (( x 1 ) T E 1 x 1 ) 2 
i
i
i


1


2 T
2 2
1 T 1 1


 (( x ) E i x ) 2  ( y ) C i w i
H i ( x, y, w, p )  
,
  ( y 2 )T C 2 w 2  1 ( p 1 )T  1 1 f ( x1 , y 1 ) p 1 
i i
i
i
y y


2


1
2 T
2
2
2


 ( pi )  2 2 g ( x , y ) pi
y y


2

1
1 

 f ( u 1 , v 1 )  g ( u 2 , v 2 )  (( v 1 ) T C 1v 1 ) 2 
i
i
i


1


2 T
2 2
1 T 1 1
 (( v ) C i v ) 2  ( u ) E i a i


G i (u , v , a , q )  
,
1
  ( u 2 )T E 2 a 2  ( q 1 )T  1 1 f ( u 1 , v1 ) q 1 
i i
i
i
u u


2


1 2 T
2
2
2


 ( qi )  2 2 g (u , u ) qi
u u


2
for i  1, 2, ..., r .
 0; where
2
  [

2
i
u
2
2
2
2
2
2
2
g ( u , v )  E i a i   u 2 u 2 g ( u , v ) q i ].
i 1
7.  1 ( v 1 , y 1 ; ( 1 ,  ))  ( y 1 ) T  1
1
 0; where
r
 
1
  [
i
1
y
1
1
1
1
1
1
1
f ( x , y )  C i w i   y 1 y 1 f ( x , y ) p i ].
i 1
8.
1 ( v , y ; ( ,  ))  ( y ) 
2
2
2
2
2 T
2
 0, where
r

2
  [

i
2
y
2
2
2
2
2
i 1
Then Inf ( SM SP )  Sup ( SM SD ).
r
Proof: Since
  [ f (, v
i
i
1
T
1 1
)  () E i a i ]
is second
i 1
order
( 0 ,  )
1
2
2
g ( x , y )  C i w i   y 2 y 2 g ( x , y ) p i ].
-univex at u 1 for fixed v 1 and   0,
A. K. Tripathy / Vol.4, No.1, pp.21-33 (2014) © IJOCTA
26
with
respect
0 : R
1
| J1 |
| J1 |
R



1
1 1
1 
b ( x , u ) 0 
0



R
r

i 1
1
to
| J1 | 1
b0 : R
 R
| J1 |
R
| J1 |
 R ,
1
1
  [ f (x ,v
1
i
, we have
r



 [ f ( x , y )  ( y )
1
1
1 T
i
1
1
1
C i wi ] 
1
u
1
1 1
f i (u , v )  E i ai  
1
1
1 1
u u
1
1
f i ( u , v ) q i ],  )).
1
  [( x )
1 T
we get
1
1 T
1
1
1 T
1 1
1 T
1
1
 [ f ( u 1 , v1 )  ( u 1 ) T E 1 a 1  1 ( q 1 ) T  1 1 f ( u 1 , v1 ) q 1 ]
i i
i
i
u u i
 i
2



1 1
1 T 1 1
1 T
1 1 1
1
  [ f i ( x , y )  ( y ) C i wi  ( p i )  1 1 f i ( x , y ) p i ] 
y y
2


r

i 
i 1
1
(3.22)
Now from (6), we get
1
 ( x , u ; ( ,  ))  ( u ) 
2
Using (3.11) in the above inequality, we get
1
1
1
1
1 T 1
 ( x , u ; ( ,  ))   ( u )   0
and with the
2
2
2
2 T
2
0
 ( x , u ; ( ,  ))   ( u ) 
2
2
2
property of
b

and
0
1
0
2 T
2
 0.
, (3.17) becomes
2
2
  [ f (x ,v
1
1
i
1 T
  [ g (, v
Since
i
( o ,  )
2
i 1
r
1
1
1 T
1 1
i [ f i (u , v )  (u ) E i ai 
i 1
1
2
1 T
(qi ) 
1
1 1
u u
1
1
f i ( u , v ) q i ].
2
using (3.23) with the properties of b and 
0
  [ f ( x , )  ( )
i
  [g (x
we get
T
i
1 1
Ci wi ]
is
second order

i
  [ g (u
i
1
|K1 |
R



1 1 1
1
b1 ( v , y ) 1 



1
 1 ( v , y ; (
R
| K1 | 1
 R


1 1
1 T 1 1
[ f i ( x , v )  ( v ) C i w i ]



1 1
1 T 1 1 
 i  [ f i ( x , y )  ( y ) C i wi ]  


1
  ( p 1i ) T  1 1 f ( x 1 , y 1 ) p 1i  
y
y
 2
 
r

i 1
  [
1
i
Hypothesis
2
y
1
1
2
2
2 T
2
2
, v )  (u ) E i ai 
1
2
1 T
( qi ) 
2
2
u u
2
2
2
g i (u , v ) qi ]
  [g (x
i
1 1
f i ( x , y )  C i wi  
1
1 1
y y
2
, v )  ( v ) C i wi ]
2
, y )  ( y ) C i wi  1 ( p i )  2 2 g i ( x , y ) p i ]
y y
2
i
2
2 T
2
2
i 1
r

1
  [g (x
i
i
2
2 T
2
2
1 T
2
2
2
i 1
1
f i ( x , y ) p i ],  )).
(3.25)
Subtracting (3.25) from (3.24), we get
(7)
  [
i
i
(3.24)
in
light
of
(3.3)
implies
r
  [x
1
y
1
1
1 1
f i ( x , y )  C i wi  
1
1 1
y y
1
i 1
So from (3.19), (3.20)
1
1
b1 and  1 , we get
(3.20)
and with the property of
E i ai  v
2
2
2T
2
2
C i wi ]
i 1
r
1
f i ( x , y ) p i ],  ))  0.
2T
i
r
1
2
r
(3.19)
1
2 T
Similarly from the hypothesis (6), (8) and (3.12)
2
2
with the property of b 1 ,  1 we get
, we have
i 1
1 ( v , y ; (
2
, v )  ( x ) E i ai ]
1
r
1
2
i
i 1
-unicave at y 1 for fixed x 1 , for   0
with respect to b11 : R | K |  R | K |  R  , 11 : R  R ,
1
1
,
r
(1 ,  )
|K1 |
2
0
i 1
1
i 1
1
is second order
-pseudo univex at u 2 for fixed v 2 and
r
1 : R
T
)  () E i a i ]
r
(3.18)
Now
2
i
i 1
1 1
E i ai ]
)  (x )
(3.23)
r
r
1
2
0
1
, which implies in
light of (3.12) as
0

1 1
1
f ( x , y ) pi 

E i a i  ( v ) C i wi ] 
i
0
i
1 1
y y
i 1
  0 ( x , u ; ( ,  ))   ( u )  .

1 T
( pi ) 
2
(3.21)
 0 ( x , u ; ( ,  ))  ( u )   0
1
1
r
( i [
From the hypothesis (5),
1
1
Subtracting (3.21) from (3.18), we get
(3.17)
1
1 T
)  ( v ) C i wi ]
i 1
i 1
1
1
i
i 1


1 1
1 T 1 1
[ f i ( x , v )  ( x ) E i a i ]  


1 1
1 T 1 1 
i  [ f i (u , v )  (u ) E i a i ]  


1
  ( q 1i ) T  1 1 f ( u 1 , v 1 ) q 1i  
u
u
 2
 
r
1
 0 ( x , u ;
r


i 1
 g (u 2 , v 2 )  u 2T E 2 a 2 
i
i i
i 
 g ( x 2 , y 2 )  y 2T C 2 w 2 
i i
 i
1
2
g i (u , v ) qi  

1T
2
2
2
pi  2 2 g i ( x , y ) pi 
y y

1T
qi 
1
2
2
2
u u
2
2
2
(3.26)
Adding (3.22) and (3.26), we obtain
r
  ( x )
1 T
i 1
1 1
1 T 1 1
2 T
2 2
2 T
2 2
E i ai  ( v ) C i wi  ( x ) E i a i  ( v ) C i wi 

Mond-Weir type second order multiobjective mixed symmetric duality …


i 1
r

Theorem 3.2 (Strong Duality) Suppose
 [ f i ( u 1 , v 1 )  ( u 1 ) T E i1 a i1  1 ( q i1 ) T  1 1 f i ( u 1 , v 1 ) q i1 ]

2
u u


  [ f i ( x 1 , y 1 )  ( y 1 ) T C i1 w i1  12 ( p i1 ) T  1 1 f i ( x 1 , y 1 ) p i1 ]

y y


i 

2
2
2 T
2 2
2 T
2
2
2
1

[
g
(
u
,
v
)

(
u
)
E
a

(
q
)

g
(
u
,
v
)
q
]
2 2


i
i
i
i
i
i
2
u u

2
2
2 T
2
2
2 T
2
2
2 
1

[
g
(
x
,
y
)

(
y
)
C
w

(
p
)

g
(
x
,
y
)
p
]
2
2
i
i
i
i
i
i 
2
y y


r

i
27
fi : R
| J1 |
R
thrice
|K1 |
and
 R
gi : R
differentiable
|J 2 |
|K 2 |
 R
be
and
let
function
1
2
1
2
1
2
1
2
( xˆ , xˆ , yˆ , yˆ , ˆ , wˆ i , wˆ i , pˆ i , pˆ i )
 ( x1 ) T E 1i a 1  ( v 1 ) T C 1 w1

i
i i

i 
  ( x 2 )T E 2 a 2  ( v 2 )T C 2 w 2 
i
i
i
i


R
be
a
weak
efficient solution for (SMSP). Let    be fixed
1 1 T


1 1
1 T 1 1
1 1 1
[ f i (u , v )  (u ) E i ai  ( qi )  1 1 f i (u , v ) qi ]


u u


2

i 

1
1
1
1
T
1
1
1
T
1
1
1
 [ f ( x , y )  ( y ) C w  ( p )  1 1 f ( x , y ) p ] 
i 1
i
i i
i
i
i
y
y


2


in (MSD). Assume that
r

1 2 T


2 2
2 T 2 2
2 2 2
r
[ g i ( u , v )  ( u ) E i a i  ( q i )  u 2 u 2 g i ( u , v ) q i ]



2

i 
.
  [ g ( x 2 , y 2 )  ( y 2 )T C 2 w 2  1 ( p 2 )T  2 2 g ( x 2 , y 2 ) p 2 ] 
i
i
i 1
i i
i 
y y

2 i


1 T
1
1
2 T
2
2
2 T
2
1
1
1 1
1 T
2
1
1 1
1 T
1
1 1
1 T
 (( x )
1
1 1
1 T
2 T
2 2
1
2 T
2 2
 (( x ) E i x ) 2  (( v ) C i v ) 2  (( x ) E i x ) 2  (( v ) C i v ) 2
(3.28)
Using (3.28) in (3.27), we get
r

1
1
1
1
1 T 1 1 2
1 T 1 1 2
2 T 2 2 2
2 T 2 2 2
i [(( x ) E i x )  (( v ) C i v )  (( x ) E i x )  (( v ) C i v ) ]
i 1
1 1
1 T 1 1
1 T
1 1 1


[ f (u , v )  (u ) E a  1 ( q )  1 1 f i (u , v ) q ]
i i
i
 i

u u
2 i


r
  [ f i ( x1 , y1 )  ( y1 )T C 1 w1  1 ( p1 )T  1 1 f i ( x1 , y1 ) p1 ]

i i
i
y y


2 i

i 

  [ g i ( u 2 , v 2 )  ( u 2 )T E 2 a 2  1 ( q 2 ) T  2 2 g i ( u 2 , v 2 ) q 2 ] 
i
i
i
i
u
u
i 1
2



2 2
2 T 2 2
2 T
2 2 2 
1
  [ g i ( x , y )  ( y ) C i wi  2 ( p i )  y 2 y 2 g i ( x , y ) p i ] 



1




1 1
2 2
1 T 1 1 2
 f i ( x , y )  g i ( x , y )  (( x ) E i x )



1
r




2 T 2 2 2
1 T 1 1
2 T 2 2

 i   (( x ) E i x )  ( y ) C i wi  ( y ) C i w i



i 1
 1 1 T
1 1 1 1
2 T
2 2 2
  ( p i )  y1 y1 fi ( x , y ) p i  ( p i )  y 2 y 2 g i ( x , y ) p i 
2
 2






are nonsingular for all
i  1, 2,..., r ;

matrix
i
y
1
(
1
1 1
y y
and
f i pˆ i )
i
y
2
(
2
y y
2
2
g i pˆ i )
are positive definite or
i 1
negative definite and
(iii) the set
y
1
1 1
f1  C1 wˆ 1  
y y
2 2
g 1  C1 wˆ 1  
y y
1 1
1
f1 pˆ 1 , ..., 
y
1 1
f r  C r wˆ r  
1
1 1
y y
1
f r pˆ r }
y
2
2
2
2
g 1 pˆ 1 ,..., 
y
2
2 2
g r  C r wˆ r  
are linearly independent.
Then there exist a i1  R | J | , a i2  R | J
1
2|
2
2
y y
2
g r pˆ r }
such that
1
2
1
2
1
2
1
2
( xˆ , xˆ , yˆ , yˆ , ˆ , aˆ i , aˆ i , qˆ i  0, qˆ i  0)
is
feasible solution of (SMSD) and the two objective
values are equal. Also if the hypotheses of
theorem 3.1 are satisfied for all feasible solution
of
(SMSP)
and
(SMSD),
then
1
2
1
2
1
2
1
2
is
( xˆ , xˆ , yˆ , yˆ , ˆ , aˆ i , aˆ i , qˆ i  0, qˆ i  0)
efficient solution of (SMSD).
1
2
1
2
1
2
1
2
Proof. Since ( xˆ , xˆ , yˆ , yˆ , ˆ , wˆ i , wˆ i , pˆ i , pˆ i ) is
a weak efficient solution of (SMSP) by Fritz-John
condition [12] there exist
1
  R,   R
| K1 |
, R
|K 2 |
1
,  R
| J1 |
r
r
r
  R ,  R ,   R ,
2
,  R
|J 2 |
such that
r
  [
i
1 1
x
f i  E i aˆ i 
1
1
2
1
 x1 (  y 1 y 1 f i pˆ i )]
i 1
1




1 1
2 2
1 T 1 1 2
 f i ( u , v )  g i ( u , v )  (( u ) E i u )



1
r




2 T 2 2
1 T 1 1
2 T 2 2

 i   (( v ) E i v ) 2  ( y ) C i w i  ( y ) C i w i



i 1
 1 1 T
1 1 1 1
2 T
2 2 2

(
p
)

f
(
x
,
y
)
p

(
p
)

g
(
x
,
y
)
p
1
1
2
2

i
i
i
i 
y y i
y y i
2
 2

r

  [
i
1 1
y x
1
1
1
1
f i   x1 (  y 1 y 1 f i pˆ i )](    yˆ )    0,
i 1



and
r
{
1
1 1
1
f i ( xˆ , yˆ )
and
1
1
1
2 2
2 T 2 2
2 T 2 2
2 T 2 2
E i x ) 2 (( a i ) E i a i ) 2  (( v ) C i v ) 2 (( w i ) C i w i
1
1 T
the

{
1 1
 (( x ) E i x ) 2 (( a i ) E i a i ) 2  (( v ) C i v ) 2 (( w i ) ) C i w i ) 2
2 T
2
2
g i ( xˆ , yˆ )
2
1
1 1
y y
i 1
( x ) E i a i  ( v ) C i wi  ( x ) E i a i  ( v ) C i wi
1 T
2
y y
(ii)
(3.27)
From Schwartz inequality, (3.6), (3.7), (3.14)
and (3.15), we have
1 1


r

1 T
(i) the Hessian matrices
(3.29)
r
  [


i
x
2
2 2
g i  E i aˆ i 
1
2
2
 x 2 (  y 2 y 2 g i pˆ i )]
i 1
r
 Inf ( M SP )  Sup ( M SD ).


  [
i
2
y x
2
2
2
2
2
g i   x 2 (  y 2 y 2 g i pˆ i )](    yˆ )    0,
i 1
(3.30)
A. K. Tripathy / Vol.4, No.1, pp.21-33 (2014) © IJOCTA
28
r
 (
r
 ˆi )[ 
i
y
 ˆ ( 
1 1
f i  C i wˆ i ] 
1
i
i 1
1 1
y y
1
1
1
f )(    yˆ   pˆ i )

[
y
(
1
1
1
1
f i pˆ i )][(    yˆ ) ˆi 
1 1
y y
1
2
1
 i pˆ i ]  0,
ˆ  0,

2T
ˆ  0,
(3.47)
r
 (
 ˆi )[  y 2 g i 
i
2
2
C i wˆ i
1 1
xˆ   0,
(3.49)
2 2
xˆ   0,
(3.50)
1T
1 1
aˆ i E i aˆ i  1, i  1, 2,..., r ,
(3.51)
2T
2 2
aˆ i E i aˆ i  1, i  1, 2,..., r ,
(3.52)
(3.32)
(  ,  ,  ,  ,  ,  )  0,
(3.53)
(3.33)
(  ,  ,  ,  ,  ,  )  0.
(3.54)
]
i 1
r
 ˆ ( 
i
2
y y
g i )( 
2
2
(3.48)
(3.31)
i 1

1T
i 1
r


2
2
  yˆ   pˆ i )
i 1
r

 [
(  y 2 y 2 g i pˆ i )][( 
2
y
2
2
2
  yˆ ) ˆi 
1
2
 i pˆ i ]  0,
2
i 1
(    yˆ ) [ 
1
1 T
y
fi 
1
1 1
C i wˆ i

ˆ 1i ] 
1 1 fi p
y y
1
i
 0,
i  1, 2, ..., r ,
Since ˆ  0,
(
  yˆ ) [ 
2
2 T
y
gi 
2
2 2
C i wˆ i

ˆ i2
2 2 gi p
y y
]
2
i
 0,
i  1, 2, ..., r ,
(3.34)
implies that  1  0 ,  2  0 .
Consequently (3.34) and (3.35) implies that
1
1 T
(    yˆ ) [ 
1 T
[(    yˆ ) ˆi   i pˆ i ] 
1
[( 
2
1
1
y y
2
2 T
  yˆ ) ˆi   i pˆ i ] 
f i  0, i  1, 2, ...r ;
1
2
y y
2
g i  0, i  1, 2, ..., r ; (3.36)
1 1
1
1 T
1
1 1
 i C i yˆ  (    yˆ ) ˆi C i  2 i C i wˆ i , i  1, 2,..., r ,
2
2
 i C i yˆ  ( 
2
2 T
2
2 2
  yˆ ) ˆi C i  2 i C i wˆ i , i  1, 2,..., r ,
1
2
1 T
1 1
1 T
1 1
( xˆ ) E i aˆ i  (( xˆ ) E i xˆ ) , i  1, 2,..., r ,
2 T
( xˆ )
2 2
E i aˆ i
2 T
 (( xˆ )
2 2
E i xˆ
(3.35)
(3.37)
(3.38)
(3.39)
1
y
2
1
1 1
y y
f i pˆ i ]  0.
2 2
g i  C i wˆ i  
and  y
2 2 g i
y
i  1, 2 ,..... r .
(3.55)
2
2
y y
2
g i pˆ i ]  0.
(3.56)
are nonsingular
(3.35) and (3.36)
(3.57)
and (  2   yˆ 2 ) ˆi   i pˆ i2 .
Using (3.57) in (3.31) we get
 (
 ˆi )[  y 1 f i  C i wˆ i  (  y 1 y 1 f i pˆ i )]
1
i
(3.58)
1
1
i 1
r

1
1
1
 i [  y 1 f i  C i wˆ i   y 1 y 1 f i pˆ i ]  0, i  1, 2,..., r ,
(3.41)
r
( )
1
y y
1
f i  C i wˆ i  
1
1
1
(    yˆ ) ˆi   i pˆ i

1
2
  [
i
y
2
gi 
2 2
C i wi

ˆ i2 ]
2 2 gi p
y y
 ˆ [
i
y
1
1
1
1
(  y 1 y 1 f i pˆ i )](    yˆ )  0.
(3.59)
i 1
i 1
2 T
1
fi
matrix for
implies
(3.40)
) , i  1, 2,..., r ,
1
y
2 T
  yˆ ) [ 
2
Since 
r
1 T
( )
(
and
r
1
2
inequalities (3.47) and (3.48)
Similarly by using (3.58) in (3.32), we get
 0, i  1, 2,..., r ,
r
i 1
(3.42)
 (
i
1 1
2
 ˆi )[  y 2 g i  C i wˆ i   y 2 y 2 g i pˆ i )]
i 1
r
(  yˆ )
1 T
  [
i
y
1
fi 
1 1
C i wˆ i
  y1 y1
1
f i pˆ i ]
 0,
(3.43)
i 1
r

1
2
 ˆ [
i
(  y 2 y 2 g i pˆ i )]( 
2
y
2
2
2
  yˆ )  0.
(3.60)
i 1
r
2 T
(  yˆ )
  [
i
y
2
2 2
2
g i  C i wˆ i   y 2 y 2 g i pˆ i ]  0,
(3.44)
i 1
 i [( wˆ i ) C i wi  1]  0, i  1, 2,..., r ,
1 T
1
1
(3.45)
Multiplying (3.59) by (  1   yˆ 1 ) T
(3.55) the result reduces to
and using
r
 i [( wˆ i ) C i w i  1]  0, i  1, 2, ...r ,
2
T
2
2
(3.46)
1
1 T
(    yˆ )

i
i 1
y
1
1
1
1
(  y 1 y 1 f i pˆ i )(    yˆ )  0. (3.61)
Mond-Weir type second order multiobjective mixed symmetric duality …
And multiplying (3.60) by (  2  
2 T
yˆ )
and using
(
2 T
  yˆ )

i
y
2
(
2
y y
ˆ i2 )( 
2 gi p
2
2
  yˆ )  0.
(3.62)
Using the hypothesis (2) in (3.61) and (3.62), we
   ŷ
1

i 1
i 1
get

r
r
2
r
i 1
(3.56), we get
1
i
2 2
2
[  x 2 g i  E i aˆ i ]   , which by (3.67) gives
ˆi [  x f i  E i aˆ i ] 
1
and
1

1
r
(3.63)

ˆ [ 
1
 0.


2
aˆ i ] 
2 gi  E
x
2
i
i
i 1

2
Now ( xˆ 1 ) T   i [  x
2
  yˆ .
(3.64)
Therefore (3.59) and (3.60) reduced to
i
 ˆi )[ 
y
1
f i  C i wˆ i  ( 
y y
i
 ˆi )[ 
y
2
2 2
g i  C i wˆ i  
y y
1
1
1
1 1
(3.65)
2
(3.66)
i 1
r
 (
2
2
g i pˆ i )]  0 .
i 1
Using hypothesis (iii) in (3.65) and (3.66) we get
 i  ˆi , i  1, 2 ,..... r .
(3.67)
If   0, then  i  0; i  1, 2 ,..... r . and (3.63)
and (3.64) implies  1   2  0.
Therefore (3.29) and (3.30) implies  1   2  0
and (3.37), (3.38) implies  i  0, i  1, 2 ,..... r .
Thus ( ,  ,  ,  ,  ,  )  0 .
  0.
(3.68)
Since  i  0, i  1, 2 ,.... r , (3.67) implies
 i  0,
i  1, 2, ....r .
(3.69)
Using (3.63) in (3.57) and (3.64) in (3.58), we get
 i pˆ i  0, for i  1, 2 ,.... r and j  1, 2 . (3.70)
j
(3.71)
Again using (3.63) and (3.71) in (3.29) and (3.64)
and (3.71) in (3.30), it gives

i 1
2
2T 2
xˆ 
2 2
g i  E i aˆ i ] 

i 1
[  x1 f i  E aˆ ]  
1
 0.
(3.75)
1 1
1 1
2 2
2 2
C i yˆ  tC i wˆ i , C i yˆ  tC i wˆ i .
(3.77)
This is a condition of Schwartz Inequality.
Therefore ( yˆ 1 ) T C i1 wˆ 1i  (( yˆ 1 ) T C i1 yˆ 1 ) ( wˆ 1i T C i1 wˆ 1i )
(3.78)
2 T
2 2
2 T
2 2
2T
2 2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
and ( y ) C i w i  (( y ) C i y ) ( w i C i w i ) .
(3.79)
In case  i  0 , from (3.45) and (3.46), we get
1
2
1
2
1T
1 1
2T
2 2
wˆ i C i wˆ i  1, wˆ i C i wˆ i  1
so we get
1
2
and
1
1 T
1 1
1 T
1 1
( yˆ ) C i wˆ i  (( yˆ ) C i yˆ ) 2
2 T
2 2
2 T
2 2
( yˆ ) C i wˆ i  (( yˆ ) C i yˆ )
1
2
and
.
2 i
i
 t  0.
implies C i1 yˆ 1
2 2
 C i yˆ  0.
Hence
1 T
1 1
1 T
1 1
( yˆ ) C i wˆ i  (( yˆ ) C i yˆ ) 2  0
2 T
2 2
2 T
2 2
( yˆ ) C i wˆ i  (( yˆ ) C i yˆ )
1
1
2
So (3.77)
and
 0.
Thus in either case
1
i
1
i
(3.74)
(by using (3.50))
Also from (3.63), (3.64) and (3.53), we have
1
2


1
2
yˆ 
 0, yˆ 
 0.
(3.76)


Hence from (3.51), (3.52) and from (3.71) to
(3.76), we get
1
2
1
2
1
2
1
2
is a
( xˆ , xˆ , yˆ , yˆ , ˆ , aˆ i , aˆ i , qˆ i  0, qˆ i  0)
feasible solution of (SMSD).
2 i
 t , then t  0 . From (3.37), (3.38),
Let
i
(3.63) and (3.64), we get
r
1
i
 0,

(by using (3.49) )
In case  i  0 we get
Using (3.69) in (3.70), we obtain
j
pˆ i  0, for i  1, 2 ,.... r and j  1, 2 .
(3.73)
1
2
This is a contradiction to (3.54).
Hence
1T 1
xˆ 
1 1
f i  E i aˆ i ] 
r
f i pˆ i )]  0 .
 0.

i 1
and ( xˆ 2 ) T   i [  x
r
 (
1
(3.72)
2
r
and
29
1 T
1 1
1 T
1 1
( yˆ ) C i wˆ i  (( yˆ ) C i yˆ ) 2
and
and
(3.80)
1
2 T
2 2
2 T
2 2
( yˆ ) C i wˆ i  (( yˆ ) C i yˆ ) 2
.
(3.81)
A. K. Tripathy / Vol.4, No.1, pp.21-33 (2014) © IJOCTA
30
Therefore using (3.39), (3.40), (3.78) and (3.79),
we get
1
1
1
1
2
2
1T 1 1
2T
2 2
f i ( xˆ , yˆ )  g i ( xˆ , yˆ )  ( xˆ E i xˆ ) 2  ( xˆ E i xˆ ) 2
1
2
1
2
1
2
1
2
( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , ˆ , qˆ  0 , qˆ  0 )
of (SMSD) and an index i such that

1
2
1
2
1
2
{G i ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0)
1T 1 1
2T
2 2
 yˆ C i wˆ i  yˆ C i wˆ i

1
2
1
2
1
2
 G i ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0)}
1
1
2
2
1T 1 1
2T
2 2
 f i ( xˆ , yˆ )  g i ( xˆ , yˆ )  ( xˆ E i aˆ i )  ( xˆ E i aˆ i )
 G  ( xˆ 1 , xˆ 2 , yˆ 1 , yˆ 2 , aˆ 1 , aˆ 2 , 0, 0) 
 j

 M 


1
2
1
2
1
2
  G j ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0) 
 ( yˆ
1T
1 1
C i yˆ )
 ( yˆ
1
2
2T
2 2
C i yˆ ) .
(3.82)
i  1, 2, 3, ..., r ;
for each
or
1
2
(3.84)
and for all j satisfying

j
1
2
1
2
1
2
G ( xˆ , xˆ , yˆ , yˆ , wˆ , wˆ , 0, 0)

1
2
1
2
1
2
1
2
H i ( xˆ , xˆ , yˆ , yˆ , wˆ , wˆ , pˆ  0, pˆ  0) 
 G j ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0),
1
2
1
2
1
2
1
2
G i ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , qˆ  0, qˆ  0)

1
2
1
2
1
2
whenever G i ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0)
1
for each i  1, 2 ,3,...... r ,
or H ( xˆ1 , xˆ 2 , yˆ 1 , yˆ 2 , wˆ 1 , wˆ 2 , 0, 0)  G ( xˆ1 , xˆ 2 , yˆ 1 , yˆ 2 , aˆ1 , aˆ 2 , 0, 0).
(3.83)
2
1
2
1
2
(3.85)

1
2
1
2
1
2
 G i ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0)
(3.86)
This implies that
So the objective values of both problems are
equal.
Now we claim that
1
2
1
2
1
2
1
2
( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ ,  , qˆ  0, qˆ  0)
is properly
efficient solution for (SMSD).
First we have to show it is an efficient solution of
(SMSD). If this would not be the case, then there
would exist a feasible solution
1
2
1
2
1
2
1
2
of (SMSD)
( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , ˆ , qˆ  0, qˆ  0)
such that
1
2
1
2
1
2
1
2
G ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , qˆ  0, qˆ  0)
1
2
1
2
1
2
1
 G ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , qˆ  0, qˆ
2
1
2
1
2
1
2
1
2
 G ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , qˆ  0 , qˆ  0 ) .
This is a contradiction to Theorem 3.1.
Hence ( xˆ 1 , xˆ 2 , yˆ 1 , yˆ 2 , aˆ 1 , aˆ 2 ,  , qˆ 1  0 , qˆ 2  0 )
is an efficient solution of (SMSD).
Now we have to claim
1
2
1
2
1
2
 0 ) is
properly efficient for (SMSD).
For that rewriting the objective function of
(SMSD) into minimization form we get
m in

1
G i ( xˆ ,
2
1
2
1
2
1
xˆ , yˆ , yˆ , aˆ , aˆ , qˆ  0, qˆ
2
 0)
If ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ ,  , qˆ  0 , qˆ  0 ) were
not properly efficient for (SMSD), then for every
scalar M  0 , there exist a feasible solution
1
2
1
2
1
2
1
2
(3.87)
for all j satisfying
1
2
1
2
1
2
1
2
1
2
1
2
G j ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0)  G j ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0),
(3.88)
whenever G i ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0)
1
2
1
2
1
2
1
2
1
2
1
2
 G i ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0)
(3.89)
Since M  0 and using (3.88) in (3.87), we get
G i ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0)
1
2
1
2
1
2
1
2
1
2
1
2
 G i ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0)  0.
Using
(3.83)
in
(3.90),
we
G i ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0)
1
2
1
2
1
2
1
2
1
2
1
2
 H i ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0)  0
 G i ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0) 
1
( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ ,  , qˆ  0 , qˆ
2
 G j ( xˆ 1 , xˆ 2 , yˆ 1 , yˆ 2 , aˆ 1 , aˆ 2 , 0, 0) 
,
M 
  G ( uˆ 1 , uˆ 2 , vˆ 1 , vˆ 2 , aˆ 1 , aˆ 2 , 0, 0) 
j


 0).
This by (3.83) gives
1
2
1
2
1
2
1
2
H ( xˆ , xˆ , yˆ , yˆ , wˆ , wˆ , pˆ  0 , pˆ  0 )
1
1
2
1
2
1
2
1
2
1
2
1
2
G i ( uˆ , uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0)  G i ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0)
2
1
2
1
2
1
2
1
2
1
2
H i ( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0).
For each ˆi  0 , we have
r
 ˆ {G ( uˆ
i
i
1
, uˆ , vˆ , vˆ , aˆ , aˆ , 0, 0)} 
2
1
2
1
2
i 1
r
 ˆ H
i
i
1
2
1
2
1
2
( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ , 0, 0).
i 1
This again contradicts Theorem 3.1.
(3.90)
obtain
Mond-Weir type second order multiobjective mixed symmetric duality …
Hence
1
( xˆ , xˆ , yˆ , yˆ , aˆ , aˆ ,  , qˆ  0 , qˆ
1
31
2
1
2
1
2
1
(x
 0 ) is
2
jT
E i x ) 2  s ( x | C i ) and
j
j
j
1
j
C i y ) 2  s ( y | D i ) , where E i and
properly efficient solution of (SMSD).
(y
Theorem 3.3 (Converse Duality)
C i are positive semi definite,
jT
j
fi : R
be
| J1 |
thrice
R
|K1 |
and
 R
differentiable
gi : R
|J 2 |
functions
( uˆ , uˆ , vˆ , vˆ , ˆ , aˆ i , aˆ i , qˆ i , qˆ i )
1
2
1
2
1
2
1
2
R
|K 2 |
and
 x1 x1 f i ( xˆ , yˆ )
be a weak
 x 2 x 2 g i ( xˆ , yˆ )
2
and
2

1
1
i 1
and
1 1

i
i 1
(  x1 x1 g i qˆ i )
2
x
1
j
  0,
j
j
j
b  1,  I , C i  { E i x ; x
j
D i  {C i y ; y
j
j
j
jT
C i y
j
j
j
j
with
jT
j
Ei x
j
 1},
 1},
1
(x
r
 i  x (  x x f i qˆ i )
j
 ( x , u ; (  f ( u ),  ))  F ( x , u ;  f ( u ))
3. If
are
nonsingular for all i  1, 2 ,..... r . (ii) The matrix
r
j
then our problem (MSP) and (MSD) reduces
to the pair of dual and dual results given by Li
and Gao [11], Mishra et al. [14, 15].
let
in (SMSD). Assume that (i) the Hessian matrices
1
j
C i w i  z i , E i a i  w i , i  1, 2, ..., r ; j  1, 2,
 R
efficient solution for (SMSD). Let ˆ   be fixed
1
j
j
j
Let
j
jT
(y
jT
E i x ) 2  s ( x | C i ) and
j
j
j
1
j
C i y ) 2  s ( y | D i ), where E i and
j
j
j
j
j
C i are positive semi definite matrices,
j
C i w i  z i , E i a i  w i , i  1, 2, ..., r ; j  1, 2,
j
j
j
j
j
j
are positive definite or negative definite and
C i w i   y 1 f ( x , y ) E i a i   x1 f ( u , v )
(iii) The set
and then our problem (MSP) and (MSD)
reduce to the problem (MP) and (MD) given
by Agarwal et al. [1].
{
{
x
1
1 1
1
1 1
1
f1  E1 aˆ1   1 1 f1 qˆ1 , ...,  1 f r  E r aˆ r   1 1 f r qˆ r }
x x
x
x x
x
2
2 2
g1  E1 aˆ1  
2
2
x x
2
g1 qˆ1 , ..., 
x
2
1
and
4.
2 2
2
g r  E r aˆ r   2 2 g r qˆ r }
x x
If
1
1
1
1
1
1
b  1,   I , | J 2 | 0, | K 2 | 0,   0,
 ( x , u ; (  u f ( u )   uu f ( u ) q ,  ))  F ( x , u ;  u f ( u )   uu f (u ) q )
are linearly independent.
Then there exist
1
|K |
|K |
1
2
wi  R 1 , w i  R 2
1 2 1 2
1
2 1
2
( uˆ , uˆ , vˆ , vˆ , ˆ , wˆ i , wˆ i , qˆ i  0, qˆ i  0)
such that
is feasible solution
and, C i  E i  0, i  1, 2,..., r , then the problem
(SMSP) and (SMSD) reduces to a pair of
problems (MP) and (MD) and the results studied
by Suneja and Lalita [17].
of (SMSD) and the two objective values are equal.
5. If b  1,   I , | J 2 | 0, | K 2 | 0,   0,
Also if the hypotheses of theorem 3.1 are satisfied
 ( x , u ; (  u f (u )   uu f (u ) q ,  ))  F ( x , u ;  u f (u )   uu f (u ) q ),
for all feasible solution of (SMSP) and (SMSD),
efficient solution of (SMSP).
in (SMSP) and (SMSD), then we obtain a pair of
nondifferentiable second order symmetric dual in
multiobjective program considered by Ahmad and
Husain [4].
Proof: It follows on the lines of theorem 3.2.
5. Conclusion
4. Special Case
1. If | J 2 | 0,| K 2 | 0 , then our problem reduces
to a pair of (MP) and (MD) given by Thakur
et al. [18].
In this article, a new pair of nondifferentiable
multiobjective second order mixed symmetric
dual programs is presented and duality relations
between primal and dual problems are
established. The results developed in this paper
improve and generalize a number of existing
results in the literature. The results discussed in
this paper can be extended to higher order as well
as to other generalized convexity assumptions.
1
2 1 2
1
2
1
2
( uˆ , uˆ , vˆ , vˆ , ˆ , wˆ i , wˆ i , qˆ i  0, qˆ i  0)
then
2. If
 ( x , u ; (  f ( u ),  ))  F ( x , u ;  f ( u ))
b  1,  I , C i  { E i x ; x
j
D i  {C i y ; y
j
j
j
j
jT
C i y
j
j
j
jT
j
Ei x
 1},
j
 1},
is properly
with 
 0,
A. K. Tripathy / Vol.4, No.1, pp.21-33 (2014) © IJOCTA
32
These results can be extended to the case of
continuous – time problems as well.
Acknowledgments
The author is thankful to the referees for their
valuable suggestions for the improvement of the
paper.
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Arun Kumar Tripathy is an Assistant Professor
in Mathematics at Trident Academy of
Technology (under B.P.U.T), Bhubaneswar,
Odisha, India. He has received his MPhil in
Mathematics from Ravenshaw College (Utkal
University), Odisha. He has submitted his PhD
33
Thesis to Utkal University. He has 10 years
research experience. His research interests are in
non-linear mathematical programming and
convex analysis. He has qualified for CSIR
(NET), GATE. He has published many papers in
reputed international and national journals such
as Applied Mathematics And Computation
(Elsevier Science),
OPSEARCH (Springer),
International Journal of Mathematics and
Operational Research (Inderscience), Journal of
Applied Analysis and Computation, Journal of
Mathematical Modeling and Algorithm in
Operation Research (Springer) and many more.