American Journal of Operations Research, 2013, 3, 413-420
http://dx.doi.org/10.4236/ajor.2013.34039 Published Online July 2013 (http://www.scirp.org/journal/ajor)
 -Optimality in Multivalued Optimization
Surjeet Kaur Suneja, Megha Sharma
Department of Mathematics, University of Delhi, Delhi, India
Received March 2, 2013; revised April 3, 2013; accepted April 11, 2013
Copy ri ght © 2 013 Surjeet Kaur Suneja, Megha Sharm. This is an open access article distributed under the Creative Commons Attribu-
tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper we apply the directional derivative technique to characterize D-multifunction, quasi D-multifunction and
use them to obtain ε-optimality for set valued vector optimization problem with multivalued maps. We introduce the
notions of local and partial-ε-minimum (weak) point and study ε-optimality, ε-Lagrangian multiplier theorem and
ε-duality results.
Keywords: D-Multifunction; Partial-ε-Minimum Point; ε-Optimality; ε-Duality
1. Introduction
The theory of efficiency plays an important role in vari-
ous knowledge fields. It is proposed as a new frontier in
mathematical physics and engineering in context of pri-
orities concerning the alternative energies, the climate
exchange and education. Pareto efficiency or Pareto op-
timality is a central theory in economics with broad ap-
plications in game theory, social sciences, management
sciences, various industries etc. In set valued vector op-
timization problems, it is important to know when the set
of efficient points is nonempty to establish its main
properties (existence, connectedness and compactness)
and to extend the concepts to set valued vector optimiza-
tion in infinite dimensional ordered vector spaces. The
notion of proper efficiency was first introduced by Kuhn
and Tucker [1] in their well known paper on nonlinear
programming and many other notions have been pro-
posed since then. Some of the well known notions are
Geoffrion proper efficiency [2], Borwein proper effi-
ciency [3], Benson proper efficiency [4] and super effi-
ciency [5]. Chinaie and Zafarani [6] introduced the con-
cepts of feeble multifunction minimum (weak) point,
multifunction minimum (weak) point and obtained
optimality conditions for set valued vector optimiza-
tion problem having multivalued objective and con-
straints.
While it is theoretically possible to identify the com-
plete set of solutions, finding an exact descrip tion of this
set often turns out to be practically impossible or com-
putationally too expensive. In practical situations we
often stop the calculations at values that are sufficiently
close to the optimal solutions, that is, we use algorithms
that find approximate of the Pareto optimal set. Stability
aspect in set valued vector optimization deals with the
study of behaviour of the solution set under perturbations
of the data. One of the approaches in this regard is the
convergence of sequence of ε-solutions to a solution of
the original problem. These facts justify the need of
study of approximate efficiency which is equivalent to
ε-optimality for set valued vector optimization problems.
Some of the researchers who contributed in this area are
Hamel [7], Rong and Wu [8].
Chinaie and Zafarani [9] introduced the concepts of
ε-feeble multifunction minimum (weak) point and ob-
tained optimality conditions for set valued vector opti-
mization problem having multivalued objective and con-
str aints. In this paper, we have given the notio n s o f (l o ca l )
partial-ε-minimum point and (local) partial-ε-weak
minimum point, for set valued vector optimization prob-
lem and used them to study ε-optimality, ε-Lagrangian
multiplier theorem and ε-duality results.
This paper is organized as follows: In Section 2 we
have given the preliminaries and results related to quasi
D-multifunction. In Section 3 we apply the directional
derivative technique used by Yang [10] to characterize
ε-optimality conditions for set valued vector optimization
problem in terms of ε-feeble multifunction minimum
point given by Chinaie and Zafarani [9]. In Section 4, we
introduce (local) partial-ε-minimum point and (local)
partial-ε-weak minimum point, and show that it is dif-
ferent from ε-feeble mutifunction minimum point. Also,
we prove that every local partial-ε-minimum (weak)
point is a partial-ε-minimum (weak) point if the objective
C
opyright © 2013 SciRes. AJOR

S. K. SUNEJA, M. SHARM 414
function of set valued vector optimization problem is
strict quasi D-multifunction and constraint function is
quasi D-multifunction and show that this result is not true
in the case of local ε-feeble multifunction minimum point.
In Section 5, we obtain ε-Lagrangian multiplier theorem
in terms of partial-ε-weak minimum point and in Section
6, we establish ε-weak duality and ε-strong duality for
dual problem of set valued vector optimization prob-
lem.
2. Preliminaries and Definitions
Let X be locally convex topological vector space, Y, Z be
real locally convex Hausdorff topological vector spaces;
let be pointed closed convex cones with ,DYEZ
andint.DEint

 

for all yD
D
i
D


0 .yD\
Let Y be the dual space of Y, the positive dual cone D+
of D is given by


: 0,DfYfy

 .
The set of strictly positive functions in is denoted
by , that is

:0, for all
i
DfYfy


For a set
A
Y, we write


cone: 0,
A
aaA  
ti DD


int D
.
If D is convex cone in Y, then in and
equality holds if

 [1].
A partial order
D
12D yy in Y is defined by

iff,
, for all .
21
yyD12
Through out this paper, we denote and
,yy:int
o
DD
Y


:0.
o
DD
:
Let
F
UY
:
, be a multifunction defined on a non
empty subset U of X with values in Y, which is partially
ordered by cone D.
Now, for a multifunction
F
UY, denote by domF
and imF the domain and the image of F, respectively. In
other words



: ,XFxdomFx






im .
xX
F
FXFx


 

,
The set

 
gr :,:dom
xX
F
xy x
xF




FyF x
x



:
is called the graph of F.
Definition 2.1: [10,11] Let U be convex subset of X.
Let
F
UY be a multifunction:
1) F is said to be a D-multifunction on U if, for all
12
,
x
xU and


0,1 ,t we have









1212
11;tF xtF xFtxtxD


12
,
2) F is said to be a quasi D-multifunction on U if, for
all


x
xU 0,1 ,t we have:

and












1212
1;Fx DFxD FtxtxD

 
12
,,
3) F is said to be a strictly quasi D-multifunction on U
iff, for all


x
xU 12 and0,1 ,xx twe have:












1212
1.
o
Fx DFxD FtxtxD
:
Yang [10] gave the following definitions:
Definition 2.2: A function
f
XY is said to be a
continuous selection of F if f is continuous and




,
f
xFx. for all
x
X

Denote by CS (F) the set
of all continuous selections of F.
Definition 2.3: Let




00
,::the re e xists 0,,0SxVvXxtvVt




0
be the cone of feasible directions. Then the limit set of F
at
x
in the direction






 
,
00
00
0
0,
,is ,:
:lim, ,,
o
nu
n
y
F
nn
n
tv
n
vSxFY xv
fx tuy
zzu SxV
t




 


 







CS F

with
f
f
where 00 xy
0
.
x
in all directions The union of all limit sets of F at


0,vSxV

is denoted by



00
,,.
o
y
F
YxSxV
,
We need the following assumption:
Assumpti on 2.1: Let
x
yX. If




1,zFtx ty




for all 0,1 ,,tyFx then there exists a continu-
ous selection


f
CS F such that




1,zftx ty

for all

0, 1t
:
.
Theorem 2.1: Let U be convex subset of X and
F
UY

. If assumption 2.1 holds and F is D-multi-
function, then for any

, and ,
x
xUy Fx




,.
y
F
xy YxxxD


F


Proof: Since F is D-multifunction therefore, for


0,1 ,,,txxU

 we have









11;tFxt FxF txtxD



which gives that








11tFxty FtxtxD



.


,wFx by assumption 2.1, there exist If

S. K. SUNEJA, M. SHARM 415


f
CS F such that





.t xD

 

11twtyftx 
That is,

1ftx t
wy t

 xy D



,.xxxD

 
Thus,
y
F
wy Y



Hence,



–, .
y
F
F
xyY


xxxD


:
Theorem 2.2: Let U be convex subset of X,
F
UY be quasi D-multifunction on U and assump-
tion 2.1 hold then, for any





x D,, ,xxU yFxyF


 

,.xx D
implies that,
y
F
Yx



 .
Proof: For


,, ,
x
xUy Fx
 
 let




yFx D


That is,


yF


xD and

.yFx D



Since F is quasi D-multifunction, therefore for

0,1 ,t


1,
 
 
F
xDFxD Ftx

tx D


which implies that



1.tx D

 yFtx
Then, by assumption 2.1 there is


f
CS F such
that





1 ,yftxtx D

for all 0,1,t
which gives that



r all 0,1
ftxtxyDt
t


 





,.xx D
1,fo .
That is, y
F
Yx



 

 
n
xU
DF
Gx E
x
3. ε-Optimality in Terms of Directional
Derivatives
In this section, we obtain ε-optimality conditions for set
valued vector optimization problem in terms of direc-
tional derivatives given by Yang [10] for ε-feeble multi-
function minimum point given by Chinaie and Zafarani
[9]. We consider the following set valued vector optimi-
zation proble m :

VP mi
s.t.



UX:

where is non empty set,
F
UY:GU Z , ,
are multifunctions with nonempty values. The set of fea-
sible solutions of (VP) is denoted by V, that is






:VxUGx E


.
Chinaie and Zafarani [9] gave the following defini-
tions.
Definition 3.1: Let
x
,o VD.



1)
x
is called a ε-feeble multifunction minimum
point (ε-f. m. m. p.) of problem (VP), if there exists,


y
Fx, such that






(3.1) ;FV xyD

  
2)
x
is called a ε-feeble multifunction weak mini-
mum point (ε-f. m. w. m. p.) of problem (VP), if there
exists,


,
y
Fx such that






–;
o
FV xyD

  (3.2)
The set of
x
V

which satisfies (3.1) or (3.2) is de-
noted by


ˆ,SFD
 

ˆ,WSF D
  and respectively.
When


Vx is replaced by



NxVx in
(3.1) and (3.2),


Nx being neighbourhood of
x
, then
we have local ε-f. m. m. p. and local ε-f. m. w. m. p. of
problem (VP).
We now give the necessary optimal conditions for lo-
cal ε-feeble we ak minimum point [9] of (VP).
Theorem 3.1: Let 0
x
V

and 00

y
Fx be local
ε-feeble multifunction weak minimum point of problem
(VP). Then,


 

00
,,, .
o
yo
F
o
YxSxV DD  
Proof Suppose 0
x
V

is local ε-feeble multifunction
weak minimum point of problem (VP) and f is any con-
tinuous selection of F such that

00
y
fx.
Then,








000,
o
FVxN xyD

 


0
Nx

is neighbourhood of x0. where
If


00
,,
o
y
F
zY xSxV

,


then there exists v, 0,
n
uSxVn uv0
n
t, , 
 

such that
,0,
lim nu
n
onn o
tv n
fx tu
t
zy



,


f
for someCS F
0nnn
.
x
xtu0 nN; then there exist such that Let





00 0
, for all
nVx Nxxnn;






00
, for all
n
o
F
yD nnx 

.
Since f is any continuous selection of F such that

S. K. SUNEJA, M. SHARM 416

00
y
fx
00
,for allDnn


, therefore: 00
,
o
y
F
kYxxx 
  

–
n
o
fx fx 

;





0
–
,f
o
n
n
fx fx
t


 

0 orallDnn
o
zD
.
It follows that

 
0
.
Now, we give the sufficient conditions for ε-feeble
multifunction minimum point of problem (VP).
Theorem 3.2: Let
x
V,

00
y
Fx, ,:
o
DFU Y


:GU Z
be a D-mulifunction and be quasi D-multi-
function and f be continuous selection of F such that


00
.
y
fx

0
forall,,xSxV
If
 
00
,,
o
y
F
Yxxx D  
then x0 is ε-feeble multifunction minimum point of problem
(VP).
Proof: Let x0 be not ε-feeble multifunction minimum
point of problem (VP), then there exists


00
y
Fx
such that






0o
FV xyD



,
which implies that, there exists


0,
x
xVyFx
y

.y
D
,
such that 0


Since G is quasi D-multifunction, therefore feasible set
V is convex,


00
xtxx

for 01.Vt
Thus,

0,
x
SxV which implies that



D
00
,
o
y
F
Yxxx



0
,,
(3.3)
Since F is D-multifunction therefor e,

00
o
y
F
F
xyY xxxD 

00
,.xxxD 

which gives that
0
o
y
F
yy Y
Thus, there exists

00
,
o
y
Fxxx
0
yykD
0.ky yD
0
yyD
kY
such that
,
that is
Also,


.kD
which implies that
Hence,



00
,,
o
y
F
Yxxx D


 Also,


which is contradiction to given condition (3.3).
4. Partial-ε-Minimum (Weak) Point
In this section we introduce the notion of partial-ε-mini-
mum point, and partial-ε-weak min imum point .
Definition 4.1: Let ,
x
VD .



1)
x
is called a partial-ε-minimum point (p.-ε-m. p.)
of problem (VP), if there exists,


y
Fx , such that






;
o
FV xyD

   (4.1)
2)
x
is called a partial-ε-weak minimum point (p.-ε-
w. m. p.) of problem (VP), if there exists,


y
Fx,
such that





;
o
FV xyD

   (4.2)
The set of
x
V

which satisfies (4.1) or (4.2) is de-
noted by


ˆ,PFD
 

ˆ,WPFD
  and respectively.
When


Vx is replaced by



NxVx in
(4.1) and (4.2),


Nx being neighbourhood of
x
, then
we have local p.-ε-.m. p. and local p.-ε-w. m. p. of prob-
lem (VP). If


x
,gryF satisfies (4.1) then it is called
partial-ε-minimizer of (VP) and if satisfies (4.2) then it is
called partial-ε-weak min imizer of (VP ) .
Now we show that partial-ε-minimum point is differ-
ent from ε-feeble multifunction minimum point.
The following example illustrates that


ˆˆ
,,.SFDPFD  
UXR 2 YR2 , , Example 4.1: Let
Z
R

,

11
,
22




2
DR

2 ER


:GU Z

, ,
and be defined by





0,0, if0
,0if 0
xxx
Gx xx







: and UY
 
be defined by
F



0,0 ,,0if0
0,0,if 0
xx
Fx
xx x









:0.VxRx then,
Let
 
0, 0,0
x
Vy Fx  .
Then,






.FV xyD

  

S. K. SUNEJA, M. SHARM 417

Thus, ˆ,
x

SFD   ...
But








0,0 FVx .
o
y D  
Thus,


ˆ,.
x
PFD

ˆ
,,.SFD  
R2 YR
 
The following example illustrates that

ˆ
PFD  
Example 4.2: Let , , UX
2
Z
R, 5,3
2





, 0xyy

0,0yy x
,

,: ,Dxyxy ,

,:Ex
:GU Z
,
and be defined by






0,
,0
Gx 




:
0, if0
if 0
xxx
xx


and
F
UY be defined by
  



0,0,2.5,3.1 ,(
1, 1,
Fx
xx






:0.VxRx 
,0) if0
if 0
xx
x


then,
Let
0
x
V ,




0,0 .
y
Fx
Then,




o
DFVxy



.
Thus,


ˆ,.
x
PFD  
But


  
2.5, 3.1
F
Vx y D  .
Thus,


ˆ,.
x
SFD  
:
The following lemma can be proved as in [9].
Lemma 4.1: Let
F
UY
:Z
be a strictly quasi
D-multifunction and GU be a quasi D-multi-
function. Then,




ˆ
PFD  
ˆ
,, WPFD.
Now, we show that every local partial-ε-minimum
(weak) point is a partial-(weak) point if F is srictly quasi
D-multifunction and G is quasi D-multifunction and
prove local ε-feeble multifunction minimum point is
not ε-feeble multifunction minimum point of problem
(VP) in above conditions.
Theorem 4.1: Let F be strictly quasi-D-multifunction
and G be a quasi D-multifunction. Then, any local par-
tial-ε-minimum point of problem (VP) is a partial-ε-mi-
nimum point o f pr obl em (VP).
Proof: Let
x
be local partial-ε-minimum point of
problem (VP), then there exists a neighbourhood


Nx
of
x
and


y
Fx such that






.
o
FVN xxyD

  (4.3)
Let if possible,
x
be not partial-ε-minimum point of
problem (VP). Then, there exist


y
Fx such that






.
o
FV xyD

  
Thus, there exists



,
x
Vx
y
and Fx such
that o yy D, which gives that



.
o
yyDyD




That is,








.yFxDFxD   
Since F is a strict quasi D-multifunction, therefore for
each


0, 1t. We have











1,
o
Fx DFxDFtxtxD
which implies that




–.
o
yFxtxx D  
Let


–.
txtxx


x
0,1 ,t Then, for each

to
y
FxD 



tt
y
Fx such that, and consequently there exists
to
yyD




.

On the other hand for 0,t

0
with

 small
enough,




–.
x
tx xNx Since G is quasi D-multi-
function, therefore, feasible set is convex and we have










–,for 0,.
t
xxtxxV Nxxt 

Thus, we deduce that






–,
to
yFV NxxyD  
UXR
which contradicts (4.3).
The following example illustrates that above result is
not true for ε-feeble multifunction minimum point of
(VP).
2
YR2 , ,
Example 4.3: Let
Z
R

,



2
DR

, 3.5, 3.5  2 ER

:GU Z

, and de-
fined b y





0,0,if0
,0if 0
xx x
Gx xx








S. K. SUNEJA, M. SHARM 418
and :
F
UY defined by

 






22
22
11
1,1 ,,,,
22
1,1 ,,
,0 ,,
xx
Fxxx
xxx














if 1
if 10
if 0
x
x
x



Here G is quasi D-multifunction and F is strict quasi
D-multifunction.
Then, Let 0
x
V ,




1,1
y
Fx .
Then








.yDFVxNx

 
Thus,
x
is local ε-feeble multifunction minimum
point.

52, 52
y
DD    ,
 


.D3, 3FV xy

 
Thus,
x
is not ε-feeble multifunction minimum poin t
of problem (VP).
5. ε-Lagrangian Multiplier Theorem
In this section, let L(Z, Y) be the set of co ntinuous lin-
ear operators from Z to Y, and let







:,,
L
ZYTL ZY
 TED
Denote by (F, G) the multivalued map from X to Y  Z
defined by



,
F
GxFx Gx,
for all x  X.
If , hY




,TLZ:hF XRY , we define and
:
F
TG XY as








hF

xhFx

and



F
TGxFxT Gx
:
,
respectively.
Lemma 5.1: [14]. Let
F
XY be D-multifunc-
tion on X. Then, one and only one of the following
statements is true:
1) there exists
x
X such that



.
o
DFx


2) there exists


0D  

such that

0y   for
all

.
y
FX
Theorem 5.1: Let o D


,



o
EGV


,


y
Fx and let

–,
F
yG   be D-multifunction
on V. If
x
is partial-ε-weak minimum point of problem
(VP), then there exists


,ZTL
 such that Y
x
is
partial-ε-weak minimum point of following problem:
  



T
VP min
xV
F
xTGx

and









0
o
TGxE D
Proof: Since
x
is partial-ε-weak minimum point of
problem (VP), therefore there exists,


,
y
Fx such
that






o
FV xyD

   (5.1)
Hence,








,,
oo
FV xyGVDE



Since


–,
F
yG   is D-multifunction on V, there-
fore by Lemma 5.1, there exists







,,0,0hpD E


such that


 
for all–0,
,,.
hy yps
x
VxyFxsGx
 


0.h
(5.2)
We claim that

In fact, if h = 0, then 0p

and


0,ps

for all

s
Gx (5.3). Since



,
o
GV E


there exists x1V and




11 o
s
Gx E.
Hence,


10ps

, which c ontradicts ( 5.3).
Therefore, 0.h

Fix with
o
dD


1hd

and
define as T(z) = p(z)d, f or all (5.4). :TZYzZ
Clearly,


,.TLZY
Using (5.2 ) and (5.4), we ge t




–0hy yTs 


. (5.5)
Since
x
V

, therefore
 
Gx E

.
Let




,
s
Gx E then


s
Gx and
s
E

.
This gives that


0ps (5.6).
Therefore we get that,







0.
o
Tspsd D 
Thus, we ha ve










0.
o
TGxE D
Suppose that
x
is not partial-ε-weak minimum point
of problem (VP)T, which gives that


 





o
FV xTGV xyD


 
Then there exist




000
,
x
Vxy Fx\ and


00
s
Gx such that




00 .
o
yyTs D   
Since


0,hD

 we get:





00 0hyy Ts




S. K. SUNEJA, M. SHARM 419
From (5.4), we get




00
–0,pshy y 
which contradicts (5.2).
Hence
x
is partial-ε-weak minimum point of prob-
lem (VP)T.
6. ε-Duality
Let us define a multivalued mapping 
by

:,LZY Y



T 
max 
= {y: there exists x  V, y  F(x) such that x
is partial-ε-weak minimum point of problem (VP)T}.
Consider the following maximum problem: (VD)
subject to

T


,ZY



TL ..
Definition 6.1: A point

,TLZY

.T
 is said to be a
feasible point of problem (VD) if

 

,Ty

We say
that 00
is partial-ε-weak maximizer of (VD) if
there exists no feasible point


,ZY

TL such that:



0
Ty

.
o
D

  
We now establish the following ε-duality results.
Theorem 6.1 (ε-Weak duality): If
x
V

and

,TLZY

 is a feasible point of problem (VD), then






–.
o
DFx T





T

Proof: Since ,

  for any


,
y
T   there
exists


,
x
Vy Fx such that
x
is partial-ε-weak
minimum point of (VD) corresponding to T.
It follows that,






––FTG Vxy

.
o
D

  


 (6.1)
Now, we show that



.
o
DFx y

  
On contrary, suppose that



.
o
DFx y

  
Then there exists


,
y
Fxsuch that

–,
o
yy D  
which implies that


–.
o
D  yy
Since
x
is a feasible point of problem (VP)T, there
exist
 
.x E

,TLZ


zG
It is given that therefore

,Y,Tz D


which implies that
–.
ooo
y
TzyTz DD   D D
Thus,






\FTG Vxy

,
o
D







which contradicts (6.1).
Therefore, we have






–.
o
DFx T





0–,

Theorem 6.2: (ε-Strong dual ity): Let
F
yG  
be D-multifunction on V. If


,,
00 00
x
VyFxx is
partial-ε-weak minimum point of problem (VP) and




o
GV E




,TLZY, then there exists 0 such
that


00
,Ty is partial-ε-weak maximizer of problem
(VD).
Proof: Suppose x0 is partial-ε-weak minimum point of
problem (VP) and




o
GV E

.

Then, by Theorem 5.1 there exists

,ZY
0
TL

such that x0 is partial-ε-weak minimum point of problem


0
T
It follows that,
, corresponding to T0. VP


T



0.


00
y
T  . Thus, T0 is feasible point of (VD) and
By ε-weak duality, we obtain


 
00 .
o
yT D



 

Thus,

00
,Ty is partial-ε-weak maximizer of prob-
lem (VD).
7. Acknowledgements
The first author is grateful to the University Grants
Commission (UGC ), In dia fo r offering financial support.
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