Applied Mathematics, 2011, 2, 452-460
doi:10.4236/am.2011.24057 Published Online April 2011 (http://www.SciRP.org/journal/am)
Efficiency and Duality in Nondiffer entiable Multiobjective
Programming Involving Directional Derivative
Izhar Ahmad
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals,
Saudi Arabia, India
Received December 24, 2010; revised February 24, 2011; accepted February 26, 2011
Abstract
In this paper, we introduce a new class of generalized dI-univexity in which each component of the objective
and constraint functions is directionally differentiable in its own direction di for a nondifferentiable multiob-
jective programming problem. Based upon these generalized functions, sufficient optimality conditions are
established for a feasible point to be efficient and properly efficient under the generalised dI-univexity re-
quirements. Moreover, weak, strong and strict converse duality theorems are also derived for Mond-Weir
type dual programs.
Keywords: Multiobjective Programming, Nondifferentiable Programming, Generalized dI-Univexity,
Sufficiency, Duality
1. Introduction
The field of multiobjective programming, also known as
vector programming, has grown remarkably in different
directions in the setting of optimality conditions and dual-
ity theory. It has been enriched by the applications of var-
ious types of generalizations of convexity theory, with and
without differentiability assumptions, and in the frame-
work of continuous time programming, fractional prog-
ramming, inverse vector optimization, saddle point theory,
symmetric duality and vector variational inequalities etc.
Hanson [1] introduced a class of functions by genera-
lizing the difference vector

xx in the definition of a
convex function to any vector function

,  xx . These
functions were named invex by Craven [2] and

-con-
vex by Kaul and Kaur [3]. Hanson and Mond [4] defined
two new classes of functions called Type I and Type II
functions, which were further generalized to pseudo
Type I and quasi Type I functions by Rueda and Hanson
[5]. Zhao [6] established optimality conditions and dual-
ity in nonsmooth scalar programming problems assum-
ing Clarke [7] generalized subgradients under Type I
functions.
Kaul et al. [8] extended the concept of type I and its
generalizations for a multiobjective programming prob-
lem. They investigated optimality conditions and derived
Wolfe type and Mond-Weir type duality results. Suneja
and Srivastava [9] introduced generalized d-type I func-
tions in terms of directional derivative for a multiobjec-
tive programming problem and discussed Wolfe type and
Mond-Weir type duality results. In [10], Kuk and Tanino
derived optimality conditions and duality theorems for
non-smooth multiobjective programming problems in-
volving generalized Type I vector valued functions. Gu-
lati and Agarwal [11] discussed sufficiency and duality
results for nonsmooth multiobjective problems under
(,,,
F
d

 -type I functions. Agarwal et al. [12] estab-
lished sufficient conditions and duality theorems for
nonsmooth multiobjective problems under V-type I func-
tions. Recently, Jayswal et al. [13] obtained some opti-
mality conditions and duality results fo r nonsmooth mul-
tiobjective problems involving generalized

,,,F


dV

-univexity.
Antczak [14] studied d-invexity is one of the genera-
lization of invex function, which is introduced by [15]. In
[14], Antczak established, under weaker assumptions
than Ye, the Fritz John type and Karush-Kuhn-Tucker
type necessary optimality conditions for weak Pareto
optimality and duality results which have been stated in
terms of the right differentials of functions involved in
the considered multiobjective programming problem.
Some authors [16-18] proved that the Karush-Kuhn-
Tucker type necessary conditions [14] are sufficient un-
der various generalized d-invex functions. Antczak [19]

I. AHMAD 453
corrrected the Karush-Kuhn-Tucker necessary conditions
in [14] and discussed the sufficiency and duality under
drtype I functions. Recently, Silmani and Radjef
[20] introduced generalzed dI-invexity in which each
component of the objective and constraint functions is
directionally differentiable in its own direction and es-
tablished the necessary and sufficien t conditions for effi-
cient and properly efficient solutions. The duality results
for a Mond-Weir type dual are also derived in [20]. They
also observed that the Karush-Kuhn-Tucker sufficient
conditions discussed in [16-18] are not applicable. More
recently, Agarwal et al. [21] introduced a new class of
generalized

,d

  type I for a non-smooth
multiobjective programming problem and discussed op-
timality conditions and duality results.
In this paper, we introduce I dV-univexity and ge-
neralized I dV-univexity in which each component of
the objective an d constraint functions of a multiobj ective
programming problem is semidirectionally differentiable
in its own direction d i. Various Karush-Kuhn-Tucker suf-
ficient optimality cond itions for efficient and properly ef-
ficient solutions to the problem are established involv ing
new classes of semidirectionally differentiable generali-
zed type I functions. Moreover, usual duality theorems
are discussed for a Mond-Weir type dual involving afo-
resaid assumptions. The results in this paper exten d many
earlier work appeared in the literature [9,10,12,14-16,
19].
2. Preliminaries and Definitions
The following conventio ns for equalities and inequalities
will be used. If

11
=,,, =,,n
nn
xx yyxy , then
==,=1,,
ii
x
yi nxy ; <<,=1,,
ii
x
yi nxy ;
,=1, ,;
ii
x
yi n xy and xy xyxy,
We also note q
 (resp. q

 or q

) the set of vectors
q
y with 0y (resp. 0y or >0y).
Definition 1 [22]. Let D be a nonempty subset of n
,
:n DD   and let 0
x
be an arbitrary point of
D.The set D is said to be invex at 0
x
with respect to

,
if for each
x
D,
 

00
,,0,1.xxxD   
D is said to be an invex set with respect to

, if D is
invex at each 0
x
D with respect to the same

.
Definition 2 [23]. Let n D be an invex set with
respect to :n
DD  .A function :fD is
called pre-invex on D with respect to

, if for all
0
,
x
xD,
 
 



00 0
1,,0,1.fxfx fxxx    
Definition 3 [14]. Let n D be an invex set with
respect to :n
DD  . A m-dimensional vector
valued function :m D is pre-invex with respect
to

, if each of its components is pre-invex on D with
respect to the same function

.
Definition 4 [7]. Let D be a nonempty open set in n
.
A function :fD is said to be locally Lipschitz at
0
x
D

, if there exist a neighborhood

0
x
 of 0
x
and a constant >0K such that



 
0
, ,,
f
yfx Kyxxyx  
where . denotes the Euclidean norm. We say that
f
is locally Lipschitz on D if its locally Lipschitz at any
point of D.
Definition 5 [7]. If :n
f
DR is locally Lip-
schitz at 0
x
D

, the Clarke generalized directional de-
rivative of
f
at 0
x
in the direction n d, denoted
by


000
0
;= sup.
lim
yx
t
f
ytd fy
fxd t





And the usual one-sided directional derivative of f at
0
x
in the direction d is defined by
 
00
00
;= ,
lim
f
xdfx
fxd







whenever this limit exists. Obviou sly,


000
;;
f
xdf xd

.
We say that f is directionally differentiable at 0
x
if
its directional derivative

0;
f
xd
 exists finite for all
n
d.
Definition 6 [15]. Let :
N
fD be a function de-
fined on a nonempty open set n
D and directional-
ly differentiable at 0.
x
D

f is called d-invex at 0
x
on
D with respect to

, if there exists a vector function
:,
n
DD   such that for any
x
D,




    
0000
,
,;,,
for all=1,,,
ii i
fy fxKyx
xyxf xf xfxxx
iN



 



where




00
;,
i
f
xxx 
 denotes the directional derivative
of i
f
at 0
x
in the direction






  
000
000
0
,: ;,
,
=.
lim
i
ii
xxf xxx
f
xxxfx








If Inequalities (1) are satisfied at any point 0
x
D

,
then
f
is said to be d-i nvex on D with respect to

.
Definition 7 [20]. Let D be a nonempty set in n 
and :n DD   a function.
 We say that :fD is a semi-directionally

I. AHMAD 454
differentiable at 0
x
D,if there exist a nonempty
subset n
S such that

0;
f
xd
 exists finite
for all dS

 We say that f is a semi-directionally differentiable
at 0
x
D in the direction

0
,
x
x  , if its direc-
tional derivative



00
;,fx xx 
 exists finite for
all
x
D.
Definition 8 [20]. Let :
N
fD be a function de-
fined on a nonempty open set n
D and for all
=1,,, i iNf is semi-directionally differentiable at
0
x
D in the direction :n
iDD
 . f is called
dI-invex at 0
x
on D with respect to

=1, ,
iiN
 if for
any
x
D,
  


000
;,,for all ,2,,,
ii ii
fx fxfxxxiN 


where
 
00
;,
ii
f
xxx 
 denotes the directional deriva-
tive of i f at 0
x
in the direction
 


  
00 0
000
0
,:; ,
,.
lim
iii
ii i
xxf xxx
fxxx fx









If Inequalities (2) are satisfied at any point 0
x
D

,
then f is said to be
I
d-invex on D with respect to

=1, ,
iiN

Consider the following multiobjective programming
problem
 



 
12
Minimize=,,,N
M
Pfxfxfxfx


Subject to0,gx
,
x
D
where :, :,
N
k
fD gD D is a nonempty open
subset of n . Let


=: 0XxDgx be the set of
feasible solutions of (MP). For 0
x
D, we denote by

0
J
x the set
 




00
1,2,,:= 0,=
j
jkgxJJx
and by
  
00
resp.
J
xJx
 the set
  
0
1, 2,,:0
j
jkgx (resp.



0>0 j
gx . we
have
  
000
1, 2,,
J
xJxJx k
 and if

00
, =xXJx.
We recall some optimality concepts, the most often
studied in the literature, for the prob lem (MP).
Definition 9. A point 0
x
X

is said to be a local
weakly efficient solution of the problem (MP), if there
exists a neighborhood

0
Nx around 0
x
such that
  
00
for all
f
xfxxNxX
Definition 10. A Point 0
x
X

is said to be a weakly
efficient (an efficient) solution of the problem (MP), if
there exists no
x
X such that



  


00
<.fxfx fxfx
Definition 11. An efficient solution 0
x
X of (MP)
is said to be properly efficient, if there exists a positive
real number
M
such that inequality



 
00ii jj
fxfxMfx fx




is verified for all


1, ,iN and
x
X such that




0
<
ii
f
xfx, and for a certain

1, ,jN such
that




0
>.
jj
f
xfx
Following Jeyakumar and Mond [24], Kaul et al. [8]
and Slimani and Radjef [20], we give the following defi-
nitions.
Definition 12.


,
f
g is I dV-univex type I at
0
x
D

if there exist positive real valued functions
and
ij

defined on
X
D, nonnegative functions
01
andbb, also defined on 01 ,: ,:XDR RR


;:,and :
nn
ij
RXDR XDR   such that







 
00000 00
,,;,
iiiii
bxxfxfxxx fxxx 
 



(3)
and





1101 00 00
,,;,
jjjj
bxxg xxxg xxx 
  
 (4)
for every
x
X

and for all =1,2, ,iN and
=1,2, ,jk.
If the inequality in (3) is strict (wheneve r 0
x
x

), we
say that (MP) is of semistrictly I dV-univex type I
at 0
x
with respect to

 
=1, =1,
and
ij
iN
j
k
 .
Definition 13.


,
f
g is quasi-I dV-univex type I
at 0
x
D

if there exist positive real valued functions

and
j

, defined on
X
D, nonnegative functions
01
andbb, also defined on 0 , :,
X
DRR


1:RR

 and


Nk

dimensional vector functions
:,=1,
n
i
X
DRi N   and :,=1,
n
j
X
DRj k  
such that for some vectors and
N
k
RR  

:
 



000 00
=1
00
=1
,,
0(;(,))0
N
iii i
i
N
ii i
i
bxxxx fx fx
fxxxx X














(5)
and
 


1010 0
=1
00
=1
,,0
;, 0.
k
jj j
j
k
jj j
j
bxxxxg x
g
xxx xX











(6)
If the second inequality in (5) is strict

0
x
x, we
say that (MP) is of semi-strictly quasi
I
d-V-univex type
I at X with respect to



=1, =1,
and
ij
iN
j
k
 .

I. AHMAD 455
Definition 14.

,
f
g is pseudo-I dV-univex type
I at 0
x
D if there exist positive real valued functions
i

and
j

, defined on
X
D, nonnegative functions
0
b and 1 b, also defined on
X
D, 0:RR

,
1:RR

 and

Nk dimensions vector functions
:,=1,
n
i
X
DRi N   and :,=1,
n
j
X
DRj k  
such that for some vectors
N
R 

 and k R 

:
 
   
00
=1
000 00
=1
;,0
,,()0
N
iii
i
N
iii i
i
fx xx
bxxxxfxfxx X











(7)
and
 
 
00
=1
1010 0
=1
;, 0
,,0 .
k
jj j
j
k
jj j
j
gx xx
bxxxxg xxX











(8)
Definition 15.

,
f
g is quasi pseudo-I dV-univex
type I at 0
x
D if there exist positive real valued func-
tions i

and
j

, defined on
X
D, nonnegative fun-
ctions 0 b and 1 b, also defined on
X
D,0:,RR


1:RR

 and ()Nk dimensions vector functions
:,
n
i
X
DR   =1,iN and :,
n
j
X
DR   =1,jk
such that the relation (5) and (8) are satisfied. If the
second inequality in (8) is strict 0 (
x
x, we say that
()VP is of quasi strictly-pseudo I d-Vtype I at 0
x
with respect to


=1, =1,
and .
ij
iN
j
k

Definition 16.


,
f
g is pseudoquasi -I d-V-univex
type I at 0
x
D if there exist positive real valued func-
tions i

and
j

, defined on
X
D, nonnegative
functions 0 b and 1 b, also defined on
X
D

,
01
:, :RR RR

 and ()Nk dimensions vector
functions :, =1,
n
i
X
DRiN   and
:, =1,
n
j
X
DRjk   , such that k R   the rela-
tions (7) and (6) are satisfied. If the second inequality in
(7) is strict

0
x
x, we say that

MP is of strictly-
pseudo quasi I dVtype I at 0
x
with respect to

=1,
iiN
 and

=1, .
j
j
k

3. Optimality Conditions
In this section, we discuss some sufficient conditions for
a point to be an efficient or properly efficient for (MP)
under generalized I dVunivex type I assumptio ns.
Theorem 3.1. Let 0
x
be a feasible solution for (MP)
and suppose that there exist


NJ vector functions
:,=1,,
n
i
X
XRiN  


0
:,
n
j
X
XRjJx   
and scalars =1
0, =1,, =1;
N
ii
i
iN 

 0,
j
 


0
jJx such that
   
000 0
=1( )
0
;,;, 0,
,
N
ij
iij j
ijJx
f xxxgxxx
xX
 

 


(9)
Further, assume that one of the following conditions is
satisfied:
a) i)


,
f
g is quasi strictly-pseudo I dV

-univex
type I at 0
x
with respect to



=1, 0
,,,
ij
iN jJx



and for some positive functions ,=1, ,
iiN
 ,
j



0
jJx,
ii) for any uR

,

0
00;uu  


1<0u

<0; u


00
,>0,bxx

10
,>0;bxx
b) i)


,
f
g is strictly-pseudo I dV-univex type I at
0
x
with respect to


=1,
iiN
 ,


0, ,
jjJx


 and
for some positive functions , =1,, ,
ij
iN



0,jJx
ii) for any uR

, 0( )>0>0;uu


1
0()0,uu


00
(, )>0,bxx 10 (, )0.bxx 
Then 0
x
is an efficient solution for

MP .
Proof: Condition a). Suppose that 0
x
is not an effi-
cient solution of


MP . Then there exists an
x
X

such that

 
0,
f
xfx
which implies that
  
00
=1
,0.
N
iiii
i
xxfx fx 
 

 (10)
Since


00
,>0bxx ;

0
00uu  , the above
inequality gives
   
000 00
=1
,, 0.
N
iiii
i
bxxxxfx fx 

 



From the above inequality and Hypothesis i) of a), we
have


00
=1 ;, 0.
N
iii
i
fx xx 


By using the Inequality (9) we deduce that



00
()
0
;, 0,
jjj
jJx
gx xx 




which implies from the condition part ii) of a) that



0
1010 0
,,<0.
jj j
jJx
bxxxxg x 






Since




10 1
,>0; <0<0,bxxuu   we get

I. AHMAD 456


00
0
,<0.
jj j
jJx
xx gx 

 (11)
As 0   and
 
00
=0;
j
g
xjJx, it follows that
 
00
=0, ,
jj
g
xjJx   which implies that


00
0
,=0.
jj j
jJx
xx gx 


The above equation contradicts Inequality (11) and
hence the conclusion of the theorem follows:
Condition b): Since



00
=0, 0, ,
jj
g
xjJx  
and
 
00
,>0, ,
j
x
xjJx   we obtain




00
0
,=0,.
jj j
jJx
x
xgx xX 



By Hypothesis ii) of b), we get



101 0
0
,,0.
jj
jJx
bxxxx 






From the above inequality and the Hypothesis i) of
b)( in view of reverse implication in (8), if follows that




00 0
0
;,<0, \.
jj j
jJx
g
xxx xXx 



By using Inequality (9), we deduce that



00 0
=1 ;,>0, \,
N
iii
i
f
xxx xXx 

 (12)
which by virtue of relation (7) implies that
   


000 00
=1
0
,, >0,
\.
N
iii i
i
bxxxx fx fx
xX x







The above inequality along with Hypothesis ii) of b)
gives
    
00 0
=1 ,>0,\.
N
iii i
i
x
xfxfx xXx  
 (13)
Since (10) and (13) contradicts each other, and hence
the conclusion follows:
Theorem 3.2. Let 0
x
be a feasible solution for (MP)
and suppose that there exist


NJ vector functions


0
:, =1,, :,
nn
ij
X
XRi NXXRjJx   
and scalars

0
=1
0, =1,, =1, 0,
N
iij
i
iN jJx

 


such that Inequality (9) of Theorem 3.1 is satisfied.
Moreover, assume that one of the following conditions
is satisfied.
a) i)

,
f
g is pseudo quasi I dVunivex type
I
at x0 with respect to
 

=1, 0
,,,
ij
iN jJx


 and for
some positive functions

0
,=1, and,,
ij
iN jJx

 
ii) for any uR

,


10
00, 00,uuuu  


00 10
,>0, ,0;bxx bxx
b) i)


,
f
g is strictly pseudo I dVunivex type
I at x0 with respect to




=1, 0
,,,
ij
iN jJx


 and
for positive functions =1,
iN
 and


0
,,
jjJx
 
ii) for any uR



 
01
00 10
00; 00;
,>0, ,0.
uuuu
bxx bxx
 

Then 0
x
is an efficient solution for

MP . Further
Suppose that these exist positive real numbers ,
ii
nm
such that


0
<,<, =1,
ii i
nxxmiN  for all feasible
x.Then 0 x is a properly efficient solution for


MP
Proof: Condition a). Suppose that 0
x
is not an effici-
ent solution of


MP . Then there exists an
x
X

such that




0
f
xfx which implies that
  

00
=1 ,<0.
N
iii i
i
xxfxfx  
 (14)
Since


0=0, 0 jj
gx   and


00
,>0,
j
x
xjJx  
we obtain



00
0
,=0.
jj j
jJx
xx gx 


From the above inequality and Hypothesis ii) of a), we
have
 
10100
()
0
,,0.
jj j
jJx
bxxxxg x 






Using Hypothesis i) of a), we deduce that



00 0
()
0
,;,0.
jjj j
jJx
xx gxxx  



 (15)
The Inequalities (9) and (14) yield that



00
=1 ;, 0,
N
iii
ifx xx



which by Hypothesis i) of a), we obtain
   

000 00
=1
,, 0,
N
iii i
i
bxxxx fxfx 




 (16)
The Inequality (16) and Hypothesis ii) of a) give
  

00
=1
,0.
N
iii i
i
xxfxf x  
 (17)
Since (14) and (17) contradict each other, we conclude

I. AHMAD 457
that 0
x
is not an efficient solution of

MP . The pro-
perly efficient solution follows as in Hanson et al. [25].
For the proof of part b), we proceed as in part b) of
Theorem 3.1, we get Inequality (17). Thus complete the
proof.
4. Mond-Weir Type Duality
Consider the following multiobjective dual to problem

MP

MD Maximize





12
=,,,
N
f
yfyfy fy
subject to




=1 =1
;,;, 0,
Nk
iiij jj
ij
f
yxygyxy xX  




0, =1,2,,, , ,
N
k
jj
g
yj kyDRR    
:, =1,2,,,
:,=1,2,,.
n
i
n
j
X
DR iN
XD Rjk


 



Let Y be the set of feasible solutions of problem

MD ; that is,








2
=1 =1
=,,, ,:
;,;, 0,
ij
ij
Nk
iij jj
ij
Yy
f yxygyxy
 
 





0, ; ,, ;
:=1,2,, ;
N
k
jj
n
i
g
yxXyDRR
XD RiN


 
 



:,=1,2,,.
n
j
X
DRj k   
We denote by rD PY, the projection of set Y on D.
Theorem 4.1. (Weak Duality). Let
x
and



=1, =1,
,,, ,
ij
iN
j
k
y  
be feasib le solution for (MP)
and (MD) respectively. Moreover, assume that one of the
following conditions is satisfied:
a) i)

,
f
g is pseudo quasi
I
d-V-univex type I at
y with respect to

=1,
>0, , ,
iiN


=1,
j
j
k
 and
for some positive functions i

,
j

for =1,2, ,iN
and=1,2, ,jk,
ii) for any uR
 
 
01
01
00; 00;
,>0, ,0
uuu u
bxy bxy
  

b) i)

,
f
g is strictly-pseudo quasi
I
d-V-univex
type I at y with respect to

=1,
, , iiN   ,

=1,
j
j
k
 and
for some positive function i

,
j

for =1,2, ,iN
and=1,2, ,jk,
ii) for any uR,


 
01
10
0>0; 00;
,0, ,>0;
uuu u
bxy bxy
 

c) i)


,
f
g is quasi strictly-pseudo I dV

-univex
type I at y with respect to



=1, =1,
,,,
ij
iN
j
k
 
and for some positive functions ,for=
ij i

1,2,, N and =1,2, ,jk,
ii) for any uR

,


 
01
01
>0>0; >0>0;
,>0, ,>0.
uuu u
bxy bxy
 
Then




f
xfy.
Proof: Since


 

11
0,=1,2,,,
00,,>0
and,>0,=1,2,,
jj
j
gy jk
uubxy
x
yjk







 ,
we have
 
11
=1
,,0.
k
jj j
j
bxyxyg y 



By Condition a) (in view of definition 16), it follows
that
  
=1 ,;,0.
k
jjjj
j
xyg yxy  
 (18)
Since




=1, =1,
,,, ,
ij
iN
j
k
y   is a feasible solu-
tion for (MD), the first dual constraint with (18) implies
that
 
=1
;, 0.
N
iii
i
fy xy 

 (19)
From (19) and Hypothesis i) of a), we obtain
 

00
=1
,, 0.
N
iii i
i
bxyxy fxfy 




 (20)
Condition ii) of a) and Inequality (20) give


=1
,0.
N
iii i
i
xyf xf y  
 (21)
Assume that




f
xfy. Since
>0, =1,2,,and>0
iiN

 , we obtain
  
=1 ,<0,
N
iii i
i
xyf xf y  
 (22)
which contradicts (21), Therefore, the conclusion follows:
The proof of part b ) and c) are ve ry similar to proof of
part a), except that: for part b), the Inequality (21) beco-
mes strict


> and Inequality (22) becomes non strict


. For part c), the Inequality (18) becomes strict


<,

I. AHMAD 458
it follows that the Inequalities (20) and (21) become strict

>. Since 0

, then the Inequality (22) becomes non
strict

. In this cases, the Inequalities (21) and (22)
contradicts each other always.
Remark 1: If we omit the assumption >0

in the
condition i) of a) or the word “strictly” in the condition
b),we obtain, for this part of theorem,



f
xfy.
Theorem 4.2. (Weak Duality). Let
x
and



=1, =1,
,,,,
ij
iN
j
K
y  
be feasible solutions for


MP and


MD respectively, Assume that
1)


,
f
g is semi-strictly I dV-univex type I at y
with respect to >0

,

=1,
, iiN 
,

=1,
j
j
k
 and for
some positive functions
 
=1, =1,
,
ij
iN jk


,
2) for any uR,
 
 
01
01
>0>0, 00,
,>0, ,0.
uauu
bxy bxy
 

Then
 
f
xfy.
Proof: Since

0, =1,2,,,
jj
g
yjk   which imp-
lies that

=1 ,0.
k
jj j
j
xyg y 

 (23)
By (23) and Hypothesis i) (with



1,, j
bxy xy 
) in
Definition 12 replaced by

,
j
x
y 
 it follows that


=1 ,, 0.
k
jj j
j
gy xy 

 (24)
The first dual constraint and (24) give
 
=1
,, 0.
N
iii
i
fy xy 

 (25)
Dividing both sides of (3) by

,
i
x
y  and taking
x
y, by Hypothesis i), we get

  

00
1
,>,,,
,
=1,2,, .
ii ii
i
bxyfxfyfyxy
xy
iN






On Multiplying by i

and taking

1
=,
ii
x
y

, we
get
 



00
,>,,,
=1,2, ,
iiiiii i
bxyfx fyfyxy
iN
 





Adding with respect to i, and applying (25) and Hy-
pothesis ii), we have
 

=1
,>0.
N
iii i
i
xyf xfy  
 (26)
Assume that




f
xfy. Since >0and >0
i

,
we have
 

=1 ,<0,
N
iii i
i
xyf xf y  

which contradicts (26).
Theorem 4.3. (Strong Duality ).Let x0 be a weakly effi-
cient solution for


MP . Assume that the function
g
satisfies the
I
d-constraint qualification at 0
x
with res-
pect to


=1,
j
j
k
 . Then there exist >> and
N
K
RR  
such that


0=1,
,,,iiN x  ,

=1,
jjk Y
  and objective
functions of


MP and


MD have the same values
at 0
x
and




0=1, =1,
,,, ,
ij
iN
j
k
x  
, respectively. If,
further, the weak duality between

MP and


MD in
theorem holds with the condition a) without >0

(resp. with the condition b) or c)), then



 0=1, =1,
,,, ,
ij
iN jk
x
Y    is a weakly efficient
(resp. an efficient) solutions of

MD .
Proof. By the Theorem 31 [20], there exists k 


and


0
J
x


 such that





0000
=1 =1
;,;, 0,
.
Nk
iiij jj
ij
f xxxgxxx
xX
 




It follows that
 


0=1, =1,
,,,,.
ij
iN jk
x
Y  

Tri-
vially, the objective function values of (MP) and (MD)
are equal.
Suppose that



 0=1, =1,
,,, ,
ij
iN jk
x
Y    is not a
weakly efficient solution of

MD . Then there exists
 


=1, =1,
,,, ,
ij
iN jk
y
Y   
  such that




0<
f
xfy
 which violates the weak duality
theorem. Hence




0=1, =1,
,,, ,
ij
iN jK
x
Y  

is in-
deed a weakly efficient solution of (MD).
Theorem 4.4. (Strict Converse Duality). Let 0
x
and




0=1, =1,
,,, ,
ij
iN jk
y  
be feasible solutions for (MP)
and (MD) respectively, such that
 
00
=1 =1
=.
NN
ii ii
ii
f
xfy 

(27)
Moreover, assume that


,
f
g is strictly pseudo quasi
I
dV

type I at o y with respect to



=1, =1,
,
ij
iN
j
k

and for

and

. Then 00 =
x
y.
Proof. Since


00 =1,2,, jj
g
yj k  , we have

I. AHMAD 459
 
100100 0
=1 ,0.
k
jj j
j
bxyxyg y 




Using the second part of the hypothesis, we get


000
=1 ;, 0.
k
jjj
j
gy xy 

 (28)
The Inequality (28) and feasibility of



0=1, =1 ,
,,, ,
ij
iN
j
k
y  
for (MD) give


000
=1
;, 0,
N
iii
i
fy xy 


which by the first part of Hypothesis ii), we obtain
 

000000 00
=1
,, >0,
.
N
iii i
i
bxyxyfxfy
xX







The above inequality along with Hypothesis iii) gives


00 00
=1 ,>0.
N
iii i
i
xy fxfy  
 (29)
By Hypothesis i), iii) and

00
,>0,
ixy

=1,2, , iN we have
 
00 00
=1
,=0.
N
iii i
i
xy fxfy  
 (30)
Now (29) and (30) contradict each other. Hence the
conclusion follows.
5. Conclusion and Future Developments
In this paper, generalized I dV

-univex functions have
been introduced. The sufficient optimality conditions are
discussed for a point to be an efficient or properly
efficient for (MP) under the introduced functions. App-
ropriate Mond-Weir type duality relations are established
under these assumptions. Sufficiency and duality with
generalized I dV-univex functions will be studied for
nonsmooth variational and nonsmooth control problems,
which will orient the future research of the author.
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