Applied Mathematics, 2011, 2, 1236-1242
doi:10.4236/am.2011.210172 Published Online October 2011 (http://www.SciRP.org/journal/am)
Multiobjective Nonlinear Symmetric
Duality Involving Generalized
Pseudoconvexity
Mohamed Abd El-Hady Kassem
Mathematical Department, Faculty of Science, Tanta Universit y, Tanta, Egypt
Received April 8, 2011; revised June 9, 2011; accepted June 16, 20 1 1
Abstract
The purpose of this paper is to introdu ce second order (K, F)-pseudoconvex and second order strong ly (K, F)-
pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex
functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using
the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are es-
tablished. Finally, a self duality theorem is given.
Keywords: Multiobjective Programming, Second-Order Symmetric Dual Models, Duality Theorems,
Pseudoconvex Functions, Cones
1. Introduction
Duality is an important con cept in the study of nonlinear
programming. Symmetric duality in nonlinear program-
ming in which the dual of the dual is the primal was first
introduced by Dorn [1]. Subsequently Dantzig et al. [2]
established symmetric du ality results for convex/concave
functions with nonneg ative orthan t as th e cone. Th e sym-
metric duality result was generalized by Bazaraa and
Goode [3] to arbitrary con es. Kim et al. [4] formulated a
pair of multiobjective symmetr ic dual programs for pseu-
doinvex functions and arbitrary cones. The weak, strong,
converse and self duality theorems were established for
that pair of dual models.
The study of second order duality is sign ificant due to
the computational ad vantage over first order duality as it
provides tighter bounds for the value of the objective
function when approximations are used (see Hou and
Yang [5], Yang et al. [6,7], Yang et al. [8]).
Hou and Yang [5] introduced a pair of second order
symmetric dual non-differentiable programs and second
order F-pseudoconvex and proved the weak and strong
duality theorems for these second order symmetric dual
programs under the F-pseudoconvex assumption. Suneja
et al. [9] formulated a pair of multiobjective symmetric
dual programs over arbitrary cones for cone-convex func-
tions. The weak, strong, converse and self-duality theo-
rems were proved for these programs. Yang et al. [6]
formulated a pair of Wolf type non-dif ferentiable second
order symmetric primal and dual problems in mathemati-
cal programming. The weak and strong duality theorems
were established under second order F-convexity assump-
tions. Symmetric minimax mixed integer primal and dual
problems were also investigated. Khurana [10] intro-
duced cone-pseudoinvex and strongly cone-pseudoinvex
functions, and formulated a pair of Mond-Weir type
symmetric dual multiobjective programs over arbitrary
cones. The duality theorems and the self-dual theorem
were established under these functions. Yang et al. [8]
proved the weak, strong and converse duality theorems
under F-convexity conditions for a pair of second order
symmetric dual programs. Yang et al. [7] established
various duality results for nonlinear programming with
cone constraints and its four dual models introduced by
Chandra and Abha [11].
In this paper, we present new definitions dealing with
second order (K, F)-pseudoconvex and second order
strongly (K, F)-pseudoconvex functions which are a gen-
eralized of cone-pseudoconvex and strongly cone-pseu-
doconvex functions. We suggest a pair of multiobjective
nonlinear second order symmetric dual programs. More-
over, we establish the duality theorems using the above
generalization of cone-pseudoconvex functions. Finally,
a self-duality theorem is given by assuming the skew-

M. ABD EL-H. KASSEM
1237
,
symmetric of the functions.
2. Notations and Definitions
The following conventions for vectors in Rn will be used:
,1,2,,
ii
x
yxyi n 
,1,2,, ,
ii
x
yxyi≦≦ n
1, 2,,
ii
x
yxyi n ≦, but
x
y.
A general multiobjective nonlinear programming
problem can be expressed in the form:
(P):
  
12
min,,, m

f
xfxfxfx


0,1,2, ,
nj

s
ubject toxXxRgxjk ≦
where :and:
nm nk
.
f
RR gRR
Definition 1. A point
x
X is said to be an efficient
(or a Pareto optimal) solution of problem (P) if there
exists no other
x
X such that
 
,
f
xfx
 
,

ii
f
xf≦x1, but 2, ,im


 
.
f
xfx
Recall the following three definitions aiming to give
the desired definition (i.e., Definition 5).
Definition 2. [5,7,8] A functional

:n

n
F
XXRRXR 

is sublinear in its third
component if, for all ,,
x
uX

,;,; X

1)

1212 12
,;, n
F
xua aFxua≦FxuaaaR ;
and
2)

,;,; ,
n
F
xuaF xuaaRR   0,

≧.
For notational convenience, we write



,,; xu
F
aFxua.
Let K be a closed convex pointed cone in with
m
R
int K

 and :nm
f
RR be a differentiable func-
tion.
Definition 3. [4,10,12] The polar cone *
K
of K is
defined as


*0. mT
K
zRxz xK ≧
Definition 4. [5] The function f is said to be second-
order F-pseudoconvex at if uX

,n
x
pXR ,
 
  
,0
1.
2
xu uuu
Tuu
Ffu fup
fx fupfup



≧
f is second-order F-pseudoconcave if is second-order
F-pseudoconvex. f
Now, we are in position to give our definitions of sec-
ond-order (K, F)-pseudoconvex functions and second-
order strongly (K, F)-pseudoconvex functions.
Definition 5. The function f is said to be second-order
(K, F)-pseudoconvex at if uX

,,
n
x
pXR


 
,int
1int ;
2
xu uuu
Tuu
Ffu fupK
f
xfup fupK
  





and the function
f
is said to be second-order strongly
(K, F)-pseudoconvex at uX

if

,,
n
x
pXR


 
,int
1.
2
xu uuu
Tuu
F
fufupK
f
xfup fupK
 

 
f is second-order (K, F)-pseudoconcave if
f
 is second-
order (K, F)-pseudoconvex and f is second-order strongly
(K, F)-pseudoconcave if
f

is second-order strongly
(K, F)-pseudoconvex.
Remark 1. If p = 0 and
  


,, xu
F
fuxu fu   
where


,,
nn
X R:XX R   the second-order
strongly (K, F)-pseudoconvex functions and second-order
(K, F)-pseudoconvex functions reduce to strongly K-
pseudoinvex functions and K-pseudoinvex functions de-
fined by Khurana [10 ] .
Remark 2. Every second-order strongly (K, F)-pseu-
doconvex function is second-order (K, F)-pseudoconvex
but converse is not necessarily true as can be seen from
the following example.
Example 1. Let



2
,4 ,0
2
,,1
x
x
Kxyxy x
fxx xep

,


 



 
≦≦
and



3
,. xu
F
AAxu
It can be seen that


f
x is second-order (K, F)-pseu-
doconvex at u
0 but


f
x is not second-order strongly
(K, F)-pseudoconvex at u
0 because for x 1


,int xu uuu
F
fufup K 


and
 
1.
2
Tuu
f
xfup fupK

 
The following example show that a function which is
second-order strongly (K, F)-pseudoconvex but not sec-
ond-order F-pseudoconvex where K is a closed convex
cone.
Example 2. Let






22
,,,
3, ,1
Kxyyxyxx
fxxxxp

 
≧≧ ≧0,
and



3
,. xu
F
AAxu

1238 M. ABD EL-H. KASSEM
Then


f
x is second-order strongly (K, F)-pseudoco-
nvex at u
0. However,


f
x is not F-pseudoconvex
at u
0, because for x 3,
 
,0 xu uuu
Ffu fup  

but
 
11.
2
Tuu
f
xfupfup
We formulate the following multiobjective nonlinear
symmetric dual problems:
(SP):



1
min ,,
2
TT
yy
f
xypfxy p  




*
2
,,
TT
yyy ,
s
ubjecttofx yfx ypC 

 

(1)





,,
TTT
yyy
yfxy fxyp 


≧0, 
(2)
*1
,
K
xC  .
(SD):



1
max ,,
2
TT
uu
f
uvrf uvr  




*
1
,,
TT
uuu ,
s
ubjecttofu vfu vrC   
0,
(3)




(, ,
TT T
uuu
ufuv fuvr 


≦


(4)
*2
,
K
vC  
m
,
where :nl
f
RR R is a thrice differentiable func-
tion of x and y. C1 and C2 are closed convex cones- with
nonempty interiors in Rn and Rl respectively. For exam-
ple, the nonnegative orthant


0
n
xRx≧ is a convex
cone). and are positive polar cones of C1 and
C2, respectively. K is a closed convex pointed cone in Rm
such that
*
1
C*
2
C
int K

 and *
K
is its positive polar cone.


,
T
y
f
xy   and yy


,

T
f
xy


,

T
 are the gradient
and the Hessian matrix of

f
xy 
with respect to y,
respectively.
Similarly,



,
T
u
f
xy   and


,
T
uu

f
xy   are
the gradient and the Hessian matrix of



,
T
f
xy 
with respect to u, respectively.
Observe that if then (SP) and (SD) be-
comes (P) and (D) given by Khurana [8], respectively.
0,pr
3. Symmetric Duality
Now, we establish the symmetric duality theorems for
the problems (SP) and (SD) as follows.
Theorem 1. (Weak duality). Let (,, ,)
x
yp

be feasible
solution for the problem (SP) and (,, ,)uvr

be feasible
solution for the problem (SD). Suppose there exist sub-
linear functionals :n
F
XXRR

n
X
R and
:l GY YRR 


l
YR satisfying:


,
T1
,
xu
F
aau 1 CaC (5)


,
T
by
2
.CbC
vy
Gb
(., )
2
(6)
Furthermore, assume that either
1)
f
v
(., ) is second-order (K, F)-pseudoconvex at u and
f
v
(., )
is second-order (K, G)-pseudoconcave at y;
or 2)
f
v(., is second-order strongly (K, F)-pseudoco-
nvex at u and )
f
v is second-order strongly (K, G)-
pseudoconcave at y.
Then






1
,,
21
2
TT
uu
TT
yy
r f
,, int
fuv uvr
f
xyfxy pK p




Proof: Suppose th e contrary, i.e.,






1
,,
21
,,int
2
fuv uvr
TT
uu
TT
yy
r f
f
xyfxy pK p




(7)
Since (,, ,)
x
yp

is a feasible solution for the prob-
lem (SP) and (,,uv ,)r

is a feasible solution for the
problem (SD), we h av e:
By the dual constraint (3), the vector






,
TT
uuu
afovfu  ,vr


 

 belongs to ,
and so by (5) we get from (4)
*
1
C









,,,
,,
TT
xu uuu
TT T
uuu
Ffuvfuvr
ufuv fuvr






 

≧≧
0.
This gives





,,,
TT
xu uuuint.
F
fuvfuvrK 

 

(*)
In a similar fashion,





,,,
TT
vy yyy
Gfxyfxyp 

 

≧0
for the vector





,,
TT
yyy
bfxyfx  yp


 


in and so
*
2,C





,,,
TT
vy yyy
Gfxyfxyp 

 

int K. (**)
(1) Since the function (., )
f
v is second-order (K, F)-
pseudoconvex at u, relation (*) implies to
 


1
,, ,in
2
TT
uu t
f
xvfuvrfuvrK 




. (8)
Similarly from (1) and (6), where

M. ABD EL-H. KASSEM
1239
C






*
2
,,
TT
yyy
bfxyfxyp 

 
,
we get






,,,
TT
vy yyy
Gfxyfxyp 

 
int.K


(**)
interiors in Rm and Rk, respectively, we will make use the
following proposition which gives generalized form of
Fritz-John optimality conditions established by suneja et
al. [9] for a point to be a weak minimum point of the
following multiobjective nonlinear programming prob-
lem:
(MONLP):



 
12
min,,, m
K
fxfxf xfx 
Also, since the function is second-order (K, G)-
pseudo conc ave at y (i.e., is second-order (K, G)-
pseudoconvex at y), we have
(,.)fx
(,.)fx


  

12
, ,...,
n
k
subject toxXxRGx
g
xgxgx Q
 

 


1
,, ,in
2
TT
yy t
f
xvfxypfxy pK 

 

. (9)
Definition 6. [6,8] A point
x
X is said to be a
weak minimum point of (MONLP) if for every
x
X

,




int
f
xfx K.
Adding (8) and (9 ), we get






1
,,
21
,),int,
2
TT
uu
TT
yy
fuvrf uvr
fxypfxypK



 
Proposition. [9]. If
x
X

is a weak minimum point
of (MONLP), then there exist
K


, Q 


 not
both zero such that
 
0,
TT
f
xGxxxx 


C



≧
this contradicts (7). Hence, the result follows for (1).


0.
TGx


(2) From (*) and since the function is sec-
ond-order strongly (K, F)-ps eudoconvex at u, we get
(., )fv
 


1
,,,
2
TT
uu
fxvfuvrf uvrK 

 





. (10)
Theorem 2. (Strong duality). Let

,,,
x
yp  be a
weak minimum point for the problem (SP): fix



and rr

in the problem (SD). Assume that
1) the matrix




,
T
yy
f
xy   is nonsingular,
Also, from (**) and since the function is sec-
ond-order strongly (K, G)-pseudoconcave at y (i.e.,
is second-order strongly (K, G)-pseudoconvex
at y), we get
(,.)fx
(,.)fx2) the set




,,1,2,,
yi
fxyi m is linearly in-
dependent,
3)






,,
TT
yyy
fxy fxyp 

0

 

,



1
,, ,
2
TT
yy
f
xyfxvpfxy pK 

 

. (11) then


,,, 0xyp r 

 is feasible solution for the
problem (SD) and the objective values of the problems
(SP) and (SD) are equal.
Adding (10) and (11), we get






1
,,
2
1
,,
2
TT
uu
TT
yy
fuvrf uvr
fxypfxyp K



 
,
Furthermore, under the assumptions of Theorem 1,


,,, 0xyp r 

 is a weak maximum point of the
problem (SD).
Proof: Since


,,,
x
yp  is a weak minimum point
for the problem (SP), by the Fritz-John conditions of the
above proposition , there exist
K
 , ,

22
CC 



0

,


,, 0 

, such that for each 1
x
C,
K


,
, 0p≧
this contradicts (7). Hence, the result follows for (2).
Therefore, the proof is completed.
For the closed convex cones K and Q with nonempty















  



1
,, ,
2
1
,,)
2
1
,,
2
,
T
T
TT T
xxy xyy
T
TTTT
yyyyyy
T
T
yyy
TT
yy
fxyyf xyypfxypxx
fxyypf xyypf xypyy
yfxyyp fxyp
yp fxy

 
 
 
 

 
 


 
 
 

 
 


 
 


 





 
,

0pp



≧
(12)





,,
TT T
yyy
fxy fxyp  


0



(13)

M. ABD EL-H. KASSEM
1240





,,
TT T
yyy
yfxy fxyp  


0


(14) Substituting 1
x
xC

, 2 yyC and pp

in
the inequality (12), we have
 

 
*
1
,,
2
1
,,
2
T
T
yyy
T
T
yyy
yfxyyp fxyp
yfxyyp fxyp
 
 


 







 





≧0
0,
K
this can be written in the following form







1
,, ,0
2
TTT TT
yyy yy
yfxyfxyppfxyp  

 

. (15)
Subtract (14) from (13), we have







,,
TTT
yyy
yfxy fxyp  

 

0,
then Equation (15) becom e s



,
TT
yy
pfxyp  0 (16)
Similarly, if
x
x,
y
y, and

  in Equation
(12), we ge t




,0
TT
yy
yp fxy    
using condition (1), we get
y
p



. (17)
We claim that 0


. Indeed, if 0

, then (17)
implies
y



. (18)
Therefore, equality (12) becomes












,,0
,,0
TT T
yyy
TT T
yyy ,
p
f
xyfxyPy yyR
fxy fxyP
 
 

 


 

≦
and from the condition (3), we have
0

 (19)
Therefore, (18) becomes
0

 (20)
Hence, , which contradicts the assump-
tion . Therefore,

,, 0  

,, 0  

 0

 and Equation (16)
take the form


,0
TT
pfxyp  
yy and since


,
T
yy
f
xy   is nonsingular (condition (1)) we get
0p (21)
So, Equation (17) becomes
y

  (22)
Substituting from Equations (21), (22) and
x
x

in
the inequality (12), we get




,0
,0.
T
p
y
T
y
f
xyy yyR
fxy






 
≧
And since

,
y

  (23)
Using (21), (22) and (23) in (12), we have



1
,0,
,0
Tx
x
f
xyx xK
f
xyx xx C



 
≧
≧
(24)
As 1 is closed convex cone, C11 ,
x
xCxC 
hence from (24) and
K


, we get




1

f
xy is linearly independent (con-
dition (2)), we get
,0
TT
x
x
fxy C  x

≧ and by using (21), we
get






1
,,0
TT T
xxx,
x
fxyfxyp xC 

 

≧
this implies that






1
,,
TT
xxx.
f
xyfxyp C 

 
Similarly, by letting 0x

in (24) we have





,,0
TT T
xxx
xfxyfxyp 



≦.
Thus


,,, 0xy p 

is feasible solution for the prob-
lem (SD) and the values of the objective function for the
problems (SP) and (SD) are same at

,,, 0xy p 

.

M. ABD EL-H. KASSEM
1241
We will now show that


,,, 0uv r   is a weak
maximum point for the problem (SD).
Suppose not, then there exists a feasible solution

,,, 0uv r   such that






1
,
2
1
,,
2
TT
uu
TT
yy
fuvrf uvr,
int,
f
xypf xypK



 
which contradicts the weak duality theorem.
Theorem 3. (Converse duality). Let

,,,uv r  be a
wea k maximum point f or the prob lem (SD). Fix



,
pp in the problem (SP). Assume that
1) the matrix


,
T
uu

f
uv   is nonsingular,
2) the set


,,1,2,,
ui
f
uv im is linearly in-
dependent,
3)




,,
TT
uuu
fuvfuvr 



0 ,
then

,,, 0uv r  

is feasible solution for the problem
(SP) and the objective values of the problems (SP) and
(SD) are equal.
Furthermore, under the assumptions of Theorem 1,

,,, 0xy p  


is a weak minimum point for the prob-
lem (SP).
The proof follows on the same lines of Theorem 2.
4. Self Duality
A nonlinear programming problem is said to be self-dual
if, when the dual is recast in the form of the primal, the
new problem so obtained is the same as the primal prob-
lem.
Now we establish the self-duality of the problem (SP).
So, we assume that , nl

,,
f
xyf yx
CC (i.e., f is skew-symmetric) and 12

, . pr
The dual problem (SD) may be rewritten as a miniza-
tion form:
(SD)’:



1
min ,,
2
TT
uu
f
uvrf uvr  




1
,,
TT
uuu ,
s
ubjecttofuvfuvrC 

 





,,
TT T
uu
ufuvfuvr 



≦0,
2
,.
K
vC 


Since
 
,,,
f
uvf vu



,,
uv
f
uvf vu

, and uu

,
vv
f
uvf vu, the above dual problem
(SD)’ reduces to
(SD)”:



1
min ,,
2
TT
vv
f
vurfvu r  




1
,,
TT
vvv





,,
TT T
vvv
ufvu fvur 



≧0,
2
,.
K
vC 


Therefore, this dual problem (SD)’ is formally identi-
cal to the primal problem (SP), that is, the objective and
constraint functions of the problems (SP) and (SD)” are
identical. Hence, this problem is self dual. Consequently,
the feasibility point


,,, 0xyp r   for the primal
problem (SP) implies the feasibility point

,, ,yx


0pr

 for the dual problem (SD) and vice versa.
Theorem 4. (Self duality). Under the assumptions of
the weak duality theorem and the point

,,, 0xy p 

is a weak minimum point for the problem (SP), we as-
sume that
1) the primal problem (SP) is self dual,
2) the matrix




,
T
yy
f
xy   is nonsingular,
3) the set




,,1,2,,
yi
f
xy im is linearly in-
dependent,
4)





,,
TT
yyy
fxy fxyp 

0

 

,
then


,,, 0xy p 

is a weak minimum point and a
weak maximum point, respectively for both the problems
(SP) and (SD) and the common optimal value is zero.
Proof: From the strong duality theorem

,,, 0xy p 

is a weak maximum point for the problem (SD) and the
optimal values of the problems (SP) and (SD) are id enti-
cal. By using the self duality, we have

,,, 0xy p 

is
feasible for both problems (SP) and (SD) and using the
theorems 1-3, we get that it is optimal for both the prob-
lems (SP) and (SD).
To show that the common optimal value is zero, since
f is skew symmetric, we have


 
,,
,,
yy xx
fxy fxy,
.
f
xyf xy


Hence,









1
,,
21
,,
21
,,
2
TT
yy
TT
xx
TT
yy
fxypf xyp
,
f
yxpfyx p
f
xypf xyp





 
and so






1
,,
21
,,
2
TT
yy
TT
xx
fxypf xyp
fyxpf yxp



0.

 
5. Conclusions
,
s
ubjecttofv ufv urC 


 
 
A pair of symmetric dual programs has been formulated

M. ABD EL-H. KASSEM
1242
by considering the optimization with respect to an arbi-
trary cone under th e assumptions of second order ( K, F)-
pseudoconvex and second order strongly (K, F)-pseu-
doconvex functions. The results may be further general-
ized by relaxing the condition of cone-pseudoconvex
functions to cone-pseudobonvex functions.
6. References
[1] W. S. Dorn, “A Symmetric Dual Theorem for Quadratic
Programs,” Journal of the Operations Research Society of
Japan, Vol. 2, 1960, pp. 93-97.
[2] G. B. Dantzig, E. Eisenberg and R. W. Cottle, “Symmet-
ric Dual Nonlinear Programs,” Pacific Journal of Mathe-
matics, Vol. 15, No. 3, 1965, pp. 809-812.
[3] M. S. Bazaraa and J. J. Goode, “On Symmetric Duality in
Nonlinear Programming,” Operation Research, Vol. 21,
No. 1, 1973, pp. 1-9. doi:10.1287/opre.21.1.1
[4] D. Sang Kim, Y. B. Yun and W. J. Lee, “Multiobjective
Symmetric Duality with Cone Constraints,” European
Journal of Operational Research, Vol. 107, No. 3, 1998,
pp. 686-691. doi:10.1016/S0377-2217(97)00322-6
[5] S. H. Hou and X. M. Yang, “On Second-Order Symmet-
ric Duality in Non-Differentiable Programming,” Journal
of Mathematical Analysis and Applications, Vol. 255,
2001, pp. 491-498. doi:10.1006/jmaa.2000.7242
[6] X M. Yang, X. Q. Yang and K. L. Teo, “Non-Differen-
tiable Second Order Symmetric Duality in Mathematical
Programming with F-Convexity,” European Journal of
Operational Research, Vol. 144, 2003, pp. 554-559.
doi:10.1016/S0377-2217(02)00156-X
[7] X. M. Yang, X. Q. Yang and K. L. Teo, “Converse Dual-
ity in Nonlinear Programming with Cone Constraints,”
European Journal of Operational Research, Vol. 170,
2006, pp. 350-354. doi:10.1016/j.ejor.2004.05.028
[8] X. M. Yang, X. Q. Yang, K. L. Teo and S. H. Hou, “Mul-
tiobjective Second Order Symmetric with F-Convexity,”
European Journal of Operational Research, Vol. 165, No.
3, 2005, pp. 585-591. doi:10.1016/j.ejor.2004.01.028
[9] K. Suneja, S. Aggarwal and S. Davar, “Multiobjective
Symmetric Duality involving Cones,” European Journal
of Operational Research, Vol. 141, No. 3, 2002, pp. 471-
479. doi:10.1016/S0377-2217(01)00258-2
[10] S. Khurana, “Symmetric Duality in Multiobjective Pro-
gramming involving Generalized Cone-Invex Functions,”
European Journal of Operational Research, Vol. 165, No.
3, 2005, pp. 592-597. doi:10.1016/j.ejor.2003.03.004
[11] S. Chandra and A. Abha, “A Note on Pseudo-Invex and
Duality in Nonlinear Programming,” European Journal
of Operational Research, Vol. 122, No. 1, 2000, pp. 161-
165. doi:10.1016/S0377-2217(99)00076-4
[12] M. Kassem, “Higher-Order Symmetric Duality in Vector
Optimization Problem involving Generalized Cone-Invex
Functions,” Applied Mathematics and Computation, Vol.
209, No. 2, 2009, pp. 405-409.
doi:10.1016/j.amc.2008.12.063