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RESEARC H Open Access
On ε-optimality conditions for multiobjective
fractional optimization problems
Moon Hee Kim
1
, Gwi Soo Kim
2
and Gue Myung Lee
2*
* Correspondence:
kr
2
Department of Applied
Mathematics, Pukyong National
University, Busan 608-737, Korea
Full list of author information is
available at the end of the article
Abstract
A multiobjective fractional optimization problem (MFP), which consists of more than
two fractional objective functions with convex numerator functions and convex
denominator functions, finitely many convex constraint functions, and a geometric
constraint set, is considered. Using parametric approach, we transform the problem
(MFP) into the non-fractional multiobjective convex optimization problem (NMCP)
v
with parametric v Î ℝ
p
, and then give the equivalent relation between (weakly) ε-
efficient solution of (MFP) and (weakly)
¯
ε
-efficient solution of


(
NMCP
)
¯
v
. Using the
equivalent relations, we obtain ε-optimality conditions for (weakly) ε-efficient solution
for (MFP). Furthermore, we present examples illustrating the main results of this
study.
2000 Mathematics Subject Classification: 90C30, 90C46.
Keywords: Weakly ε-efficient solution, ε-optimality condition, Multiobjective fractional
optimization problem
1 Introduction
We need constraint qualifications (for example, the Slater condition) on convex opti-
mization problems to obtain optimality co nditions or ε-optimality conditions for the
problem.
To get optimality conditions for an efficient solution of a multiobjective optimization
problem, we often formulate a corresponding scalar problem. However, it is so difficult
that such scalar program satisfies a constraint qualification which we need to derive an
optimality condition. Thus, it is very impo rtant to investigate an optimality condition
for an efficient solution o f a multiobjective optimization problem which holds without
any constraint qualification.
Jeyakumar et al. [1,2], Kim et al. [3], and Lee et al. [4], gave optimality conditions for
convex (scalar) optimization problems, which hold without any constraint qualification.
Very recently, Kim et al. [5] obtained ε-optimality theorems for a convex multiobjective
optimization problem. The purpose of this article is to extend the ε-optimality theo-
rems of Kim et al. [5] to a multiobjective fractional optimization problem (MFP).
Recently, many authors [5-15] have paid their attention to investigate properties of
(weakly) ε-efficient solutions, ε-optimality conditions, and ε-duality theorems for multi-
objective optimization problems, which consist of more than two objective functions

and a constrained set.
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>© 2011 Kim et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License ( 2.0), which permits unrestricted use, distribution, and reprodu ction in any medium,
provided the original work is properly cited.
In this article, an MFP, which consists of more than fractional objective functions
with convex numerator functions, and convex denominator functions and finitely
many convex constraint functions and a geometric constraint set, is considered. We
discuss ε-efficient solutions and weakly ε-efficient solutions for (MFP) and obtain ε-
optimality theorems for such solutions of (MFP) under weakened constraint qual ifica-
tions. Furthermore, we prove ε-optimality theorems for the solutions of (MFP) which
hold without any constraint qualifications and are expressed by sequences, and present
examples illustrating the main results obtained.
2 Preliminaries
Now, we give some definitions and preliminary results. The definitions can be found in
[16-18]. Let g : ℝ
n
® ℝ ∪ {+∞} be a convex function. The subdif ferent ial of g at a is
given by
∂g
(
a
)
:= {v ∈ R
n
| g
(
x
)
 g

(
a
)
+ v, x − a, ∀x ∈ domg}
,
where domg:={x Î ℝ
n
| g(x)<∞}and〈·, ·〉 is the scalar product on ℝ
n
.Letε ≧ 0.
The ε-subdifferential of g at a Î domg is defined by

ε
g
(
a
)
:= {v ∈ R
n
| g
(
x
)
 g
(
a
)
+ v, x − a−ε, ∀x ∈ domg}
.
The conjugate function of g : ℝ

n
® ℝ ∪ {+∞} is defined by
g

(
v
)
=sup{v, x−g
(
x
)
| x ∈ R
n
}
.
The epigraph of g, epig, is defined by
epig = {
(
x, r
)
∈ R
n
× R | g
(
x
)
 r}
.
For a nonempty closed convex set C ⊂ ℝ
n

, δ
C
: ℝ
n
® ℝ ∪ {+∞} is called the indicator
of C if
δ
C
(x)=

0ifx ∈ C,
+∞ otherwis
e
.
Lemma 2.1 [19]If h : ℝ
n
® ℝ ∪ {+∞} is a proper lower semicontinuous convex func-
tion and if a Î domh, then
epih

=

ε

0
{(v, v, a  + ε − h(a))|v ∈ ∂
ε
h(a)}
.
Lemma 2.2 [20]Let h : ℝ

n
® ℝ be a continuous convex function and u : ℝ
n
® ℝ ∪
{+∞} be a proper lower semicontinuous convex function. Then
epi
(
h + u
)

=epih

+epiu

.
Now, we give the following Farkas lemma which was proved in [2,5], but for the
completeness, we prove it as follows:
Lemma 2.3 Let h
i
: ℝ
n
® ℝ, i = 0, 1, , l be convex functions. Suppose that {x Î ℝ
n
|
h
i
(x) ≦ 0, i = 1, , l} ≠ ∅. Then the following statements are equivalent:
(i) {x Î ℝ
n
| h

i
(x) ≦ 0, i = 1, , l} ⊆ {x Î ℝ
n
| h
0
(x) ≧ 0}
(ii)
0 ∈ epih

0
+cl

λ
i

0
epi(

l
i=1
λ
i
h
i
)

.
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 2 of 13
Proof. Let Q ={x Î ℝ

n
| h
i
(x) ≦ 0, i = 1, , l}. Then Q ≠ ∅ and by Lemma 2.1 in [2],
epiδ

Q
=cl

λ
i

0
epi(

l
i=1
λ
i
h
i
)

. Hence, by Lemma 2.2, we can verify that (i) if and only
if (ii).
Lemma 2.4 [16]Let h
i
: ℝ
n
® ℝ ∪ {+∞}, i =, 1, , m be proper lower semi-continuous

convex funct ions. Let ε ≧ 0. if

m
i
=1
ri domh
i
=
0
, where ri domh
i
is the relative interior
of domh
i
, then for all
x ∈

m
i
=1
domh
i
,

ε
(
m

i
=1

h
i
)(x)=

{
m

i
=1

ε
i
h
i
(x) | ε
i
 0, i =1,··· , m,
m

i
=1
ε
i
= ε}
.
3 ε-optimality theorems
Consider the following MFP:
(MFP) Minimize
f (x)
g(x)

:=

f
1
(x)
g
1
(x)
, ··· ,
f
p
(x)
g
p
(x)

subject to x ∈ Q := {x ∈
R
n
|h
j
(x)  0, j =1, , m}
.
Let f
i
: ℝ
n
® ℝ, i = 1, , p be convex functions, g
i
: ℝ

n
® ℝ, i =1, ,p,concave
functions such that for any x Î Q, f
i
(x) ≧ 0 and g
i
(x) >0, i = 1, , p, and h
j
: ℝ
n
® ℝ, j
= 1, , m, convex functions. Let ε =(ε
1
, , ε
p
), where ε
i
≧ 0, i = 1, , p.
Now, we give the definition of ε-efficient solution of (MFP) which can be found in
[11].
Definition 3.1 The point
¯
x ∈
Q
is said to be an ε-efficient solution of (MFP) if there
does not exist x Î Q such that
f
i
(x)
g

i
(x)

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i
,foralli =1, , p,
f
j
(x)
g
j
(x)
<
f
j
(
¯
x)
g
j

(
¯
x)
− ε
j
,forsomej ∈{1, , p}
.
When ε = 0, t hen the ε-efficiency becomes the efficiency for (MFP) (see the defini-
tion of efficient solution of a multiobjective optimization problem in [21]).
Now, we give the definition of weakly ε-efficient solution of (MFP) which is weaker
than ε-efficient solution of (MFP).
Definition 3.2 Apoint
¯
x ∈
Q
is said to be a weakly ε-efficient solution of (MFP) if
there does not exist x Î Q such that
f
i
(x)
g
i
(
x
)
<
f
i
(
¯

x)
g
i
(
¯
x
)
− ε
i
,foralli =1, , p
.
When ε = 0, then the weak ε-efficiency becomes the weak efficiency for (MFP) (see
the definition of efficient solution of a multiobjective optimization problem in [21]).
Using parametric approach, we transf orm the problem (MFP) into the nonfr actional
multiobjective convex optimization problem (NMCP)
v
with parametric v Î ℝ
p
:
(NMCP)
v
Minimize (f (x) − vg(x)) := (f
1
(x) − v
1
g
1
(x), , f
p
(x) − v

p
g
p
(x)
)
sub
j
ect to x ∈ Q.
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 3 of 13
Adapting Lemma 4.1 in [22] and modifying Proposition 3.1 in [12], we can obtain
the following proposition:
Proposition 3.1 Let
¯
x ∈
Q
. Then the following are equivalent:
(i)
¯
x
is an ε-efficient solution of (MFP).
(ii)
¯
x
is an
¯
ε
-efficient solution of
(
NMCP

)
¯
v
,where
¯
v :=

f
1
(
¯
x)
g
1
(
¯
x)
− ε
1
, ,
f
p
(
¯
x)
g
p
(
¯
x)

− ε
p

and
¯ε =(ε
1
g
1
(
¯
x), , ε
p
g
p
(
¯
x)
)
.
(iii)
Q ∩ S
(
¯
x
)
=

or
p


i=1

f
i
(x) −

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(x)

 0=
p

i=1

f
i

(
¯
x) −

f
i
(
¯
x)
g
i
(
¯
x
)
− ε
i

g
i
(
¯
x)


p

i=1
ε
i

g
i
(
¯
x) for any x ∈ Q ∩ S(
¯
x)
,
where
S(
¯
x)={x ∈ R
n
| f
i
(x)−

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i


g
i
(x)  0=f
i
(
¯
x)−

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(
¯
x)−¯ε
i
, i =1, , p
}
.

Proof. (i) ⇔ (ii): It follows from Lemma 4.1 in [22].
(ii) ⇒ (iii): Let
¯
x
be an
¯
ε
-efficient solution of
(
NMCP
)
¯
v
,where
¯
v :=

f
1
(
¯
x)
g
1
(
¯
x)
− ε
1
, ,

f
p
(
¯
x)
g
p
(
¯
x)
− ε
p

and
¯ε =(ε
1
g
1
(
¯
x), , ε
p
g
p
(
¯
x)
)
.Then
Q ∩ S

(
¯
x
)
=

or
Q ∩ S
(
¯
x
)
=

. Suppose that
Q ∩ S
(
¯
x
)
=

. Then for any
x ∈ Q ∩ S
(
¯
x
)
and all i =1, p,
f

i
(x) −

f
i
(
¯
x)
g
i
(
¯
x
)
− ε
i

g
i
(x)  f
i
(
¯
x) −

f
i
(
¯
x)

g
i
(
¯
x
)
− ε
i

g
i
(
¯
x) −¯ε
i
.
Hence the
¯
ε
-efficiency of
¯
x
yields
f
i
(x) −

f
i
(

¯
x)
g
i
(
¯
x
)
− ε
i

g
i
(x)=f
i
(
¯
x) −

f
i
(
¯
x)
g
i
(
¯
x
)

− ε
i

g
i
(
¯
x) −¯ε
i
for any
x ∈ Q ∩ S
(
¯
x
)
and all i = 1, , p. Thus we have, for all
x ∈ Q ∩ S
(
¯
x
)
,
p

i
=1

f
i
(x) −


f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(x)

=
p

i
=1

f
i
(
¯
x) −


f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(
¯
x)


p

i
=1
¯ε
i
.
(iii) ⇒ (ii): Suppose that
Q ∩ S
(

¯
x
)
=

. Then there does not exist x Î Q such that
x ∈ S
(
¯
x
)
; that is, there does not exist x Î Q such that
f
i
(x) −

f
i
(
¯
x)
g
i
(
¯
x
)
− ε
i


g
i
(x)  f
i
(
¯
x) −

f
i
(
¯
x)
g
i
(
¯
x
)
− ε
i

g
i
(
¯
x) −¯ε
i
for all i = 1, , p. Hence, there does not exist x Î Q such that
f

i
(x) −

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(x)  f
i
(
¯
x) −

f
i
(
¯
x)
g

i
(
¯
x)
− ε
i

g
i
(
¯
x) −¯ε
i
, i =1, , p,
f
j
(x) −

f
j
(
¯
x)
g
j
(
¯
x)
− ε
j


g
j
(x) < f
j
(
¯
x) −

f
j
(
¯
x)
g
j
(
¯
x)
− ε
j

g
j
(
¯
x) −¯ε
j
,forsomej ∈{1, , p}
.

Therefore,
¯
x
is an
¯
ε
-efficient solution of
(
NMCP
)
¯
v
,where
¯
v :=

f
1
(
¯
x)
g
1
(
¯
x)
− ε
1
, ,
f

p
(
¯
x)
g
p
(
¯
x)
− ε
p

.
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 4 of 13
Assume that
Q ∩ S
(
¯
x
)
=

. Then, from this assumption
p

i
=1

f

i
(x) −

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(x)


p

i
=1

f
i
(
¯

x) −

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(
¯
x)


p

i
=1
¯ε
i
,
(3:1)

for any
x ∈ Q ∩ S
(
¯
x
)
. Suppose to the contrary that
¯
x
is not an
¯
ε
-efficient solution of
(
NMCP
)
¯
v
. Then, there exist
ˆ
x ∈
Q
and an index j such that
f
i
(
ˆ
x) −
¯
v

i
g
i
(
ˆ
x)  f
i
(
¯
x) −
¯
vg
i
(
¯
x) −¯ε
i
, i =1, , p,
f
j
(
ˆ
x) −
¯
v
j
g
j
(
ˆ

x) < f
j
(
¯
x) −
¯
v
j
g
j
(
¯
x) −¯ε
j
,forsomej ∈{1, , p}
.
Therefore,
ˆ
x ∈ Q ∩ S
(
¯
x
)
and

p
i=1

f
i

(
ˆ
x) −

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(
ˆ
x)

<

p
i=1

f
i

(
¯
x) −

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(
¯
x)



p
i=1
¯ε
i
,

which contradicts the above inequality. Hence,
¯
x
is an
¯
ε
-efficient solution of
(
NMCP
)
¯
v
.
We can easily obtain the following proposition:
Proposition 3.2 Let
¯
x ∈
Q
and suppose that
f
i
(
¯
x
)
 ε
i
g
i
(

¯
x
)
, i =1, ,
p
. Then the fol-
lowing are equivalent:
(i)
¯
x
is a weakly ε-efficient solution of (MFP).
(ii)
¯
x
is a weakly
¯
ε
-efficient solution of
(
NMCP
)
¯
v
, where
¯ε =(ε
1
g
1
(
¯

x), , ε
p
g
p
(
¯
x)
)
and
¯ε =(ε
1
g
1
(
¯
x), , ε
p
g
p
(
¯
x)
)
.
(iii) there exists
¯
λ := (
¯
λ
1

, ,
¯
λ
p
) ∈ R
p
+
\{0
}
such that
p

i=1
¯
λ
i

f
i
(x) −

f
i
(
¯
x)
g
i
(
¯

x)
− ε
i

g
i
(x)

 0=
p

i
=1
¯
λ
i

f
i
(
¯
x) −

f
i
(
¯
x)
g
i

(
¯
x)
− ε
i

g
i
(
¯
x)


p

i
=1
¯
λ
i
ε
i
g
i
(
¯
x) for any x ∈ Q
.
Proof. (i) ⇔ (ii): The proof is also following the similar lines of Proposition 3.1.
(ii) ⇒ (iii): Let (x)=(

1
(x), , 
p
(x)), ∀x Î Q,where
ϕ
i
(x)=f
i
(
¯
x) −

f
i
(
¯
x)
g
i
(
¯
x)
− ε
i

g
i
(x), i =1,··· , p
.Then,
i

(x), i = 1, , p, are convex. Since
¯
x ∈
Q
is a weakly ε-efficient solution of
(
NMCP
)
¯
v
,where
(
ϕ
(
Q
)
+ R
p
+
)

(
−intR
p
+
)
=

,
(

ϕ
(
Q
)
+ R
p
+
)

(
−intR
p
+
)
=

, and hence, it follows from
separation theorem that there exist
¯
λ
i
 0
, i = 1, , p,
(
¯
λ
1
, ,
¯
λ

p
) =
0
such that
p

i
=1
¯
λ
i
ϕ
i
(x)  0 ∀x ∈ Q
.
Thus (iii) holds.
(iii) ⇒ (ii): If (ii) does not hold, that is,
¯
x
is not a weakly
¯
ε
-efficient solution of
(
NMCP
)
¯
v
, then (iii) does not hold. □
We present a necessary and sufficient ε-optimality theorem for ε-efficient solution of

(MFP) under a constraint qualification, which will be called the closedness assumption.
Theorem 3.1 Let
¯
x ∈
Q
and assume that
Q ∩ S
(
¯
x
)
=

and
f
i
(
¯
x
)
 ε
i
g
i
(
¯
x
)
, i =1, ,
p

i = 1, , p. Suppose that

λ
j
0
m

j=1
epi(λ
j
h
j
)

+

μ
i
0
p

i=1

epi(μ
i
f
i
)

+epi(−

¯
v
i
μ
i
g
i
)


is closed, where
¯
v
i
=
f
i
(
¯
x)
g
i
(
¯
x)
− ε
i
, i = 1, , p. Then the following are equivalent.
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 5 of 13

(i)
¯
x
is an ε-efficient solution of (MFP).
(ii)

0
0

T


p
i=1

epif

i
+epi(−
¯
v
i
g
i
)


+

λ

j
0

m
j=1
epi(λ
j
h
j
)

+

μ
i

0
p

i=1

epi(μ
i
f
i
)

+epi(−
¯
v

i
μ
i
g
i
)


.
(iii) there exist a
i
≧ 0,
u
i
∈ ∂
α
i
f
i
(
¯
x
)
, b
i
≧ 0,
y
i
∈ ∂
β

i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
, i = 1, , p, l
j
≧ 0, g
j

0,
w
j
∈ ∂
γ
j

j
h
j
)(
¯

x)
, j = 1, , m, μ
i
≧ 0, q
i
≧ 0,
s
i
∈ ∂
q
i

i
f
i
)(
¯
x
)
, z
i
≧ 0,
t
i
∈ ∂
z
i
(−
¯
v

i
μ
i
g
i
)(
¯
x
)
i = 1, , p such that
0=
p

i=1
(u
i
+ y
i
)+
m

j
=1
w
j
+
p

i=1
(s

i
+ t
i
)
and
p

i=1

i
+ β
i
+ q
i
+ z
i
)+
m

j
=1
γ
j
=
p

i=1
ε
i
(1 + μ

i
)g
i
(
¯
x)+
m

j
=1
λ
j
h
j
(
¯
x)
.
Proof. Let
h
0
(x)=
p

i
=1

f
i
(x) −

¯
v
i
g
i
(x)

.
(i) ⇔ (by Proposition 3.1) h
0
(x) ≧ 0,
∀x ∈ Q ∩ S
(
¯
x
)
.

{x|f
i
(
x
)

¯
v
i
g
i
(

x
)

0
, i = 1, , p, h
j
(x) ≦ 0, j = 1, , m} ⊂ {x | h
0
(x) ≧ 0}.
⇔ (by lemma 2.3)

0
0

T

p

i=1

epif

i
+epi(−
¯
v
i
g
i
)



+cl




λ
j
0
m

j=1
epi(λ
j
h
j
)

+

μ
i
0
p

i=1

epi(μ
i

f
i
)

+epi(−
¯
v
i
μ
i
g
i
)





.
Thus by the closedness assumption, (i) is equivalent to (ii).
(ii) ⇔ (iii): (ii) ⇔ (by Lemma 2.1), there exist a
i
≧ 0,
u
i
∈ ∂
α
i

i

f
i
)(
¯
x
)
, i = 1, , p, b
i

0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
, i = 1, , p, l
j
≧ 0, g

j
≧ 0,
w
j
∈ ∂
γ
j

j
h
j
)(
¯
x)
, j =1, ,m, μ
i
≧ 0, q
i
≧ 0,
s
i
∈ ∂
q
i

i
f
i
)(
¯

x
)
, i = 1, , p, z
i
≧ 0,
t
i
∈ ∂
z
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
, i = 1, , p such that

0
0

T
=
p


i=1


u
i
u
i
,
¯
x + α
i
− f
i
(
¯
x)

T
+

y
i
y
i
,
¯
x + β
i
− (−

¯
v
i
g
i
)(
¯
x)

T

+
m

j=1

w
j
w
j
,
¯
x + γ
j
− (λ
j
h
j
)(
¯

x)

T
+
p

i=1


s
i
s
i
,
¯
x + q
i
− (μ
i
f
i
)(
¯
x)

T
+

t
i

t
i
,
¯
x + z
i
− (−
¯
v
i
μ
i
g
i
)(
¯
x)

T

.
⇔ there exist a
i
≧ 0,
u
i
∈ ∂
α
i


i
f
i
)(
¯
x
)
, b
i
≧ 0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
, i = 1, , p, l
j
≧ 0, g

j
≧ 0,
w
j
∈ ∂
γ
j

j
h
j
)(
¯
x
)
, j = 1, , m, μ
i
≧ 0, q
i
≧ 0,
s
i
∈ ∂
q
i

i
f
i
)(

¯
x
)
, z
i
≧ 0,
t
i
∈ ∂
z
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
i = 1, , p such that
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 6 of 13
0=
p

i=1

(u
i
+ y
i
)+
m

j
=1
w
j
+
p

i=1
(s
i
+ t
i
)
and
p

i=1

i
+ β
i
+ q
i

+ z
i
)+
m

j=1
γ
j
=
p

i=1


f
i
(
¯
x) −
¯
v
i
g
i
(
¯
x)+(μ
i
f
i

)(
¯
x) − (
¯
v
i
μ
i
g
i
)(
¯
x)+
m

j=1
λ
j
h
j
(
¯
x)


.
⇔ (iii) holds. □
Now we give a necessary and sufficient ε- optimality theorem for ε-efficient solution
of (MFP) which holds without any constraint qualification.
Theorem 3.2 Let

¯
x ∈
Q
. Suppose that
Q ∩ S
(
¯
x
)
=

and
f
i
(
¯
x
)
 ε
i
g
i
(
¯
x
)
, i =1, ,
p
, i
= 1, , p. Then

¯
x
is an ε-efficient solution of (MFP) if and only if there exist a
i
≧ 0,
u
i
∈ ∂
α
i

i
f
i
)(
¯
x
)
, i =1, ,p, b
i
≧ 0,
y
i
∈ ∂
β
i
(−
¯
v
i

μ
i
g
i
)(
¯
x
)
, i =1, ,p,
λ
n
j

0
,
γ
n
j
 0
,
w
n
j
∈ ∂
γ
n
j

n
j

h
j
)(
¯
x
)
, j = 1, , m,
μ
n
k

0
,
q
n
k

0
,
s
n
k
∈ ∂
q
n
k

n
k
f

k
)(
¯
x
)
,
z
n
k

0
,
t
n
k
∈ ∂
z
n
k
(−
¯
v
k
μ
n
k
g
k
)(
¯

x
)
, k = 1, , p such that
0=
p

i=1
(u
i
+ y
i
) + lim
n→∞


m

j=1
w
n
j
+
p

k=1
(s
n
k
+ t
n

k
)


and
p

i=1
ε
i
g
i
(
¯
x)=
p

i=1

i
+ β
i
) + lim
n→∞



m

j=1


γ
n
j
− (λ
n
j
h
j
)(
¯
x)

+
p

k
=1

q
n
k
+ z
n
k
− μ
n
k
ε
k

g
k
(
¯
x)


.
Proof.
¯
x
is an ε-efficient solution of (MFP)
⇔ (from the proof of Theorem 3.1)

0
0

T

p

i=1

epif

i
+epi(−
¯
v
i

g
i
)


+cl




λ
j
0
m

j=1
epi(λ
j
h
j
)

+

μ
i
0
p

i=1


epi(μ
i
f
i
)

+epi(−
¯
v
i
μ
i
g
i
)





.
⇔ (by Lemma 2.1) there exist a
i
≧ 0,
u
i
∈ ∂
α
i


i
f
i
)(
¯
x
)
, i =1, ,p , b
i
≧ 0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
, i =1, ,p,
λ

n
j

0
,
γ
n
j

0
,
w
n
j
∈ ∂
γ
n
j

n
j
h
j
)(
¯
x
)
, j =1, ,m,
μ
n

k
 0
,
s
n
k
∈ ∂
q
n
k

n
k
f
k
)(
¯
x
)
,
s
n
k
∈ ∂
q
n
k

n
k

f
k
)(
¯
x
)
,
z
n
k

0
,
t
n
k
∈ ∂
z
n
k
(−
¯
v
k
μ
n
k
g
k
)(

¯
x
)
, k = 1, , p, such that

0
0

T
=
p

i=1


u
i
u
i
,
¯
x + α
i
− f
i
(
¯
x)

T

+

y
i
y
i
,
¯
x + β
i
− (−
¯
v
i
g
i
)(
¯
x)

T

+ lim
n→∞



m

j=1


w
n
j
w
n
j
,
¯
x + γ
n
j
− (λ
n
j
h
j
)(
¯
x)

T
+
p

k
=1


s

n
k
s
n
k
,
¯
x + q
n
k
− (μ
n
k
f
k
)(
¯
x)

T
+

t
n
k
t
n
k
,
¯

x + z
n
k
− (−
¯
v
k
μ
n
k
g
i
)(
¯
x)

T

.
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 7 of 13
⇔ there exist a
i
≧ 0,
u
i
∈ ∂
α
i


i
f
i
)(
¯
x
)
, i = 1, , p, b
i
≧ 0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
, i = 1, ,
p,
λ

n
j
 0
,
γ
n
j
 0
,
w
n
j
∈ ∂
γ
n
j

n
j
h
j
)(
¯
x
)
, j =1, ,m,
μ
n
k


0
,
q
n
k

0
,
s
n
k
∈ ∂
q
n
k

n
k
f
k
)(
¯
x
)
,
t
n
k
∈ ∂
z

n
k
(−
¯
v
k
μ
n
k
g
k
)(
¯
x
)
,
t
n
k
∈ ∂
z
n
k
(−
¯
v
k
μ
n
k

g
k
)(
¯
x
)
, k = 1, , p, such that
0=
p

i=1
(u
i
+ y
i
) + lim
n→∞


m

j=1
w
n
j
+
p

k=1
(s

n
k
+ t
n
k
)


and
p

i=1
ε
i
g
i
(
¯
x)=
p

i=1

i
+ β
i
) + lim
n→∞




m

j=1

γ
n
j
− (λ
n
j
h
j
)(
¯
x)

+
p

k
=1

q
n
k
+ z
n
k
− μ

n
k
ε
k
g
k
(
¯
x)


.
We present a necessary and sufficient ε-optimality theorem for weakly ε-efficient
solution of (MFP) under a constraint qualification.
Theorem 3.3 Let
¯
x ∈
Q
and assume that
f
i
(
¯
x
)
 ε
i
g
i
(

¯
x
)
, i =1, ,
p
, i = 1, , p, and

λ
j
0

m
j=1
epi(λ
j
h
j
)

is closed. Then the following are equivalent.
(i)
¯
x
is a weakly ε-efficient solution of (MFP).
(ii) there exist μ
i
≧ 0, i = 1, , p,

p
i

=1
μ
i
=
1
such that

0
0

T

p

i=1

epi(μ
i
f
i
)

+epi(−
¯
v
i
μ
i
g
i

)


+

λ
j
0
m

j=1
epi(λ
j
h
j
)

,
where
¯
v
i
=
f
i
(
¯
x)
g
i

(
¯
x
)
− ε
i
, i = 1, , p.
(iii) there exist μ
i
≧ 0,

p
i
=1
μ
i
=
1
, a
i
≧ 0,
u
i
∈ ∂
α
i

i
f
i

)(
¯
x
)
, b
i
≧ 0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
,
i = 1, , p, l
j
≧ 0, g
j
≧ 0,

w
j
∈ ∂
γ
j

j
h
j
)(
¯
x
)
, j = 1, , m, such that
0=
p

i=1
(u
i
+ y
i
)+
m

j
=1
w
j
and

p

i=1
μ
i
ε
i
g
i
(
¯
x)=
p

i=1

i
+ β
i
)+
m

j
=1

γ
j
− (λ
j
h

j
)(
¯
x)

.
Proof. (i) ⇔ (ii):
¯
x
is a weakly ε-efficient solution of (MFP)
⇔ (by Proposition 3.2) there exist μ
i
≧ 0, i = 1, , p,

p
i
=1
μ
i
=
1
such that
p

i
=1
μ
i
[f
i

(x) −
¯
v
i
g
i
(x)]  0 ∀x ∈
Q
⇔ there exist μ
i
≧ 0, i = 1, , p,

p
i
=1
μ
i
=
1
such that
{x|h
j
(x)  0, j =1, , m}⊂{x|
p

i
=1
μ
i


f
i
(x) −
¯
v
i
g
i
(x)

 0
}
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 8 of 13
⇔ (by Lemma 2.3) there exist μ
i
≧ 0, i = 1, , p,

p
i
=1
μ
i
=
1
such that

0
0


T

p

i=1

epi(μ
i
f
i
)

+epi(−
¯
v
i
μ
i
g
i
)


+cl




λ
j

0
m

j=1
epi(λ
j
h
j
)




.
Thus, by the closedness assumption, (i) is equivalent to (ii).
(ii) ⇔ (iii): (ii) ⇔ (by Lemma 2.1) there exist μ
i
≧ 0,

p
i
=1
μ
i
=
1
, a
i
≧ 0,
u

i
∈ ∂
α
i

i
f
i
)(
¯
x
)
, b
i
≧ 0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯

x
)
, i = 1, , p, l
j
≧ 0, g
j
≧ 0,
w
j
∈ ∂
γ
j

j
h
j
)(
¯
x
)
, j
= 1, , m, such that

0
0

T
=
p


i=1


u
i

u
i
,
¯
x

+ α
i
− (μ
i
f
i
)(
¯
x)

T
+

y
i

y
i

,
¯
x

+ β
i
− (−
¯
v
i
μ
i
g
i
)(
¯
x)

T

+
m

j
=1

w
j

w

j
,
¯
x

+ γ
j
− (λ
j
h
j
)(
¯
x)

T
.
⇔ (iii) holds. □
Now, we propose a n ecessary and sufficient ε-optimality theorem for weakly ε-effi-
cient solution of (MFP) which holds without any constraint qualification.
Theorem 3.4 Let
¯
x ∈
Q
and assume that
f
i
(
¯
x

) 
ε
i
g
i
(
¯
x
)
, i =1, ,
p
. Then
¯
x
is a
weakly ε-efficient solution of (MFP) if and only if there exist μ
i
≧ 0, i =1, ,p,

p
i
=1
μ
i
=
1
, a
i
≧ 0,
u

i
∈ ∂
α
i

i
f
i
)(
¯
x
)
, i =1, ,p, b
i
≧ 0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯

x
)
, i = 1, , p,
γ
n
j

0
,
γ
n
j

0
,
w
n
j
∈ ∂
γ
n
j

n
j
h
j
)(
¯
x

)
, j = 1, , m, such that
0=
p

i=1
(u
i
+ y
i
) + lim
n→∞
m

j
=1
w
n
j
and
p

i=1
μ
i
ε
i
g
i
(

¯
x)=
p

i=1

i
+ β
i
) + lim
n→∞
m

j
=1

γ
n
j
− (λ
n
j
h
j
)(
¯
x)

.
Proof.

¯
x
is a weakly ε-efficient solution of (MFP)
⇔ ((from the proof of Theorem 3.3) there e xist μ
i
≧ 0, i =1, ,p,

p
i
=1
μ
i
=
1
such
that

0
0

T

p

i=1

epi(μ
i
f
i

)

+epi(−
¯
v
i
μ
i
g
i
)


+cl




λ
j
0
m

j=1
epi(λ
j
h
j
)





.
⇔ (by Lemma 2.1) there exist μ
i
≧ 0, i = 1, , p,

p
i
=1
μ
i
=
1
, a
i
≧ 0,
u
i
∈ ∂
α
i

i
f
i
)(
¯
x

)
,
i = 1, , p, b
i
≧ 0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
, i =1, ,p,
λ
n
j

0
,
γ

n
j

0
,
w
n
j
∈ ∂
γ
n
j

n
j
h
j
)(
¯
x
)
, j
= 1, , m, such that

0
0

T
=
p


i=1


u
i

u
i
,
¯
x

+ α
i
− (μ
i
f
i
)(
¯
x)

T
+

y
i

y

i
,
¯
x

+ β
i
− (−
¯
v
i
μ
i
g
i
)(
¯
x)

T

+ lim
n→∞



m

j=1


w
n
j

w
n
j
,
¯
x

+ γ
n
j
− (λ
n
j
h
j
)(
¯
x)

T



.
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 9 of 13

⇔ there exist μ
i
≧ 0, i =1, ,p,

p
i
=1
μ
i
=
1
, a
i
≧ 0,
u
i
∈ ∂
α
i

i
f
i
)(
¯
x
)
, i = 1, , p, b
i


0,
y
i
∈ ∂
β
i
(−
¯
v
i
μ
i
g
i
)(
¯
x
)
, i = 1, , p,
λ
n
j

0
,
γ
n
j

0

,
w
n
j
∈ ∂
γ
n
j

n
j
h
n
j
)(
¯
x
)
, j = 1, , m,such
that
0=
p

i=1
(u
i
+ y
i
) + lim
n→∞

m

j
=1
w
n
j
and
p

i=1
μ
i
ε
i
g
i
(
¯
x)=
p

i=1

i
+ β
i
) + lim
n→∞
m


j
=1

γ
n
j
− (γ
n
j
h
j
)(
¯
x)

.

Now, we give examples illustrating Theorems 3.1, 3.2, 3.3, and 3.4.
Example 3.1 Consider the following MFP:
(MFP)
1
Minimize

x
1
,
x
2
x

1

subject to
(
x
1
, x
2
)
∈ Q := {
(
x
1
, x
2
)

R
2
|−x
1
+1 0, −x
2
+1 0}
.
Let
ε =(ε
1
, ε
2

)=(
1
2
,
1
2
)
, and f
1
(x
1
, x
2
) =x
1
, g
1
(x
1
, x
2
) = 1, f
2
(x
1
, x
2
) =x
2
, g

2
(x
1
, x
2
) =
x
1
, h
1
(x
1
, x
2
) =-x
1
+ 1 and h
2
(x
1
, x
2
)=-x
2
+1.
(1)Let
(
¯
x
1

,
¯
x
2
)=(
3
2
,
9
4
)
. Then
(
¯
x
1
,
¯
x
2
)
is an ε-efficient solution of (MFP)
1
.
Let
¯
v
1
=
f

1
(
¯
x
1
,
¯
x
2
)
g
1
(
¯
x
1
,
¯
x
2
)
− ε
1
and
¯
v
2
=
f
2

(
¯
x
1
,
¯
x
2
)
g
2
(
¯
x
1
,
¯
x
2
)
− ε
2
. Then
¯
v
1
=
¯
v
2

=
1
, and
Q ∩ S
(
¯
x
1
,
¯
x
2
)
= Q ∩{(
¯
x
1
,
¯
x
2
) ∈ R
2
|f
1
(
¯
x
1
,

¯
x
2
) −
¯
v
1
g
1
(
¯
x
1
,
¯
x
2
)  0, f
2
(
¯
x
1
,
¯
x
2
) −
¯
v

2
g
2
(
¯
x
1
,
¯
x
2
)  0
}
= {
(
1, 1
)
}.
Thus,
Q ∩ S
(
¯
x
1
,
¯
x
2
)
=


. It is clear that
f
1
(
¯
x
1
,
¯
x
2
)
 ε
1
g
1
(
¯
x
1
,
¯
x
2
)
and
f
2
(

¯
x
1
,
¯
x
2
)
 ε
2
g
2
(
¯
x
1
,
¯
x
2
)
. Let
A =

λ
1
≥0,
λ
2
≥0


2
j=1
epi(λ
j
h
j
)

+

μ
1
≥0,
μ
2
≥0

2
j=1
[epi(μ
j
f
j
)

+epi(−
¯
v
i

μ
i
g
i
)

]
.
Then
A =

λ
1
≥0, λ
2
≥0
μ
1
≥0, μ
2
≥0
epi


2

j=1
λ
j
h

j
+
2

i=1
μ
i
(f
i

¯
v
i
g
i
)



=coneco{
(
−1, 0, −1
)
,
(
0, −1, −1
)
,
(
1, 0, 1

)
,
(
−1, 1, 0
)
,
(
0, 0, 1
)
}
,
wherecoDistheconvexhullofasetDand cone coD is the cone generated by coD.
Thus A is closed. Let
B =

2
i=1
[epif

i
+epi(−
¯
v
i
g
i
)

]+
A

. Then
B = {(1, 0)} × [0, ∞)+{(0, 0)} × [1, ∞)+{(0, 1)} × [0, ∞)+{(-1, 0)} × [0, ∞)+A. Since (0,-
1,-1) Î A, (0, 0, 0) Î B. Thus (ii) of Theorem 3.1 holds. Let a
1
= b
1
= g
1
= q
1
= z
1
= a
2
= b
2
= g
2
= q
2
= z
2
=0,and let μ
1
= μ
2
=1,and l
1
=0and l
1

=2.Moreover,
∂f
2
(
¯
x
1
,
¯
x
2
)
= {
(
0, 1
)}
,
∂f
2
(
¯
x
1
,
¯
x
2
)
= {
(

0, 1
)}
,

(

¯
v
1
g
1
)(
¯
x
1
,
¯
x
2
)
= {
(
0, 0
)}
,

(

¯
v

2
g
2
)(
¯
x
1
,
¯
x
2
)
= {
(
−1, 0
)}
,

(
λ
2
h
2
)(
¯
x
1
,
¯
x

2
)
= {
(
0, −2
)
}
,
,

(
λ
2
h
2
)(
¯
x
1
,
¯
x
2
)
= {
(
0, −2
)
}
,


(
μ
1
f
1
)(
¯
x
1
,
¯
x
2
)
= {
(
1, 0
)}
,

(

¯
v
1
μ
1
g
1

)(
¯
x
1
,
¯
x
2
)
= {
(
0, 0
)}
,

(

¯
v
1
μ
1
g
1
)(
¯
x
1
,
¯

x
2
)
= {
(
0, 0
)}
,

(

¯
v
2
μ
2
g
2
)(
¯
x
1
,
¯
x
2
)
= {
(
−1, 0

)}
.
Thus,

2
i
=1
∂(f
i

¯
v
i
g
i
)(
¯
x
1
,
¯
x
2
)+

2
i
=1
∂(λ
i

h
i
)(
¯
x
1
,
¯
x
2
)+

2
i
=1
∂(μ
i
f
i

¯
v
i
μ
i
g
i
)(
¯
x

1
,
¯
x
2
)={(0, 0)
}
and

2
i=1

i
+ β
i
+ q
i
+ z
i
)+

2
j
=1
γ
j
=0=

2
i=1

ε
i
(1 + μ
i
)g
i
(
¯
x
1
,
¯
x
2
)+

2
i=1
λ
j
h
j
(
¯
x
1
,
¯
x
2

)
.
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 10 of 13
Thus, (iii) of Theorem 3.1 holds.
(2) Let
(
˜
x
1
,
˜
x
2
)=(
3
2
,
15
4
)
. Then
(
˜
x
1
,
˜
x
2

)
is not an ε-efficient solution of (MFP)
1
, but
(
˜
x
1
,
˜
x
2
)
is a weakly ε-efficient solution of (MFP)
1
.
Let
C =

λ
1
≥0,
λ
2
≥0

2
i=1
epi(λ
i

h
i
)

. Then
C =coneco{
(
−1, 0, −1
)
,
(
0, −1, −1
)
,
(
0, 0, 1
)
}
.
Hence, C is closed. Moreover,
f
1
(
˜
x
1
,
˜
x
2

)
− ε
1
g
1
(
˜
x
1
,
˜
x
2
)
=1
0
, and
f
2
(
˜
x
1
,
˜
x
2
)
− ε
2

g
2
(
˜
x
1
,
˜
x
2
)
=3
0
. Let
¯
v
1
=
f
1
(
¯
x
1
,
¯
x
2
)
g

1
(
¯
x
1
,
¯
x
2
)
− ε
1
and
¯
v
2
=
f
2
(
¯
x
1
,
¯
x
2
)
g
2

(
¯
x
1
,
¯
x
2
)
− ε
2
. Then,
˜
v
2
=2
,
˜
v
2
=
2
. Let μ
1
=1and μ
2
=1.Then,
2

i=1

[epi(μ
i
f
i
)

+epi(−
˜
v
i
μ
i
g
i
)

]
= {
(
1, 0
)
}×R
+
+ {
(
0, 0
)
}×[1, ∞
)
+ {

(
0, 0
)
}×R
+
.
Since (-1, 0,-1) Î C,
(0,0,0) ∈

2
i
=1
[epi(μ
i
f
i
)

+epi(−
˜
v
i
μ
i
g
i
)

]+
C

. So, (ii) of Theo-
rem 3.3 holds. Let a
1
= b
1
= g
1
= a
2
= b
2
= g
2
=0,l
1
=1and l
2
=0.Then,
2

i=1
∂(μ
i
f
i
)(
˜
x
1
,

˜
x
2
)+
2

i=1
∂(−
˜
v
i
μ
i
g
i
)(
˜
x
1
,
˜
x
2
)+
2

j
=1
∂(λ
j

h
j
)(
˜
x
1
,
˜
x
2
)={(0, 0)
}
and
2

i=1
μ
i
ε
i
g
i
(
˜
x
1
,
˜
x
2

)=
1
2
=
2

i=1

i
+ β
i
)+
2

j
=1

j
− (λ
j
h
j
)(
˜
x
1
,
˜
x
2

)]
.
Thus, (iii) of Theorem 3.3 holds.
Example 3.2 Consider the following MFP:
(MFP)
2
Minimize

−x
1
+1,
x
2
−x
1
+1

sub
j
ect to [max{0, x
1
}]
2
 0, −x
2
+1 0
.
Let
ε =(ε
1

, ε
2
)=(
1
2
,
1
2
)
, and f
1
(x
1
, x
2
)=-x
1
+1,g
1
(x
1
, x
2
)=1,f
2
(x
1
, x
2
)=x

2
, g
2
(x
1
,
x
2
)=-x
1
+1,h
1
(x
1
, x
2
) = [max{0, x
1
}]
2
and h
2
(x
1
, x
2
)=-x
2
+1.
(1) Let

(
¯
x
1
,
¯
x
2
)=(−
1
2
,
9
4
)
. Then,
(
¯
x
1
,
¯
x
2
)
is an ε-efficient solution of (MFP)
2
. Let
A =


λ
1
0,
λ
2

0

2
j=1
epi(λ
j
h
j
)

+

μ
1
0,
μ
2

0

2
i=1
[epi(μ
i

f
i
)

+epi(−
¯
v
i
μ
i
g
i
)

]
. Then,clA = cone co{(0, -1,
-1), (1, 0, 0), (-1, 0, 0), (1, 1, 1), (0, 0, 1)}. Here, (1, 0, 0) Î clA, but (1, 0, 0) Î A, where
clA is the closure of the set A. Thus, A is not closed. Let Q ={(x
1
, x
2
) Î ℝ
n
| h
1
(x
1
, x
2
)

≦ 0, h
2
( x
1
, x
2
) ≦ 0}. Then,
Q ∩ S
(
¯
x
1
,
¯
x
2
)
= {
(
1, 1
)}
. Let
v
i
=
f
i
(
¯
x

1
,
¯
x
2
)
g
i
(
¯
x
1
,
¯
x
2
)
− ε
i
, i =1,2.Then,
¯
v
1
=
¯
v
2
=
1
. Let a

1
= b
1
= a
2
= b
2
=0,
λ
n
1
=
0
,
λ
n
2
=
1
,
γ
n
1
= γ
n
2
=
0
,
w

n
1
=(0,0
)
,
w
n
2
=(0,−1
)
. Let u
1
=(-1,0)u
2
= (0, 1), y
1
= (0, 0) and y
2
= (1, 0). Let
q
n
1
= q
n
2
= z
n
1
= z
n

1
=
0
, and
μ
n
1
= μ
n
2
=
0
. Let
s
n
1
= s
n
2
=(0,0
)
and
t
n
1
= t
n
2
= {
(

0, 0
)
}
. Then,
u
i
∈ ∂f
i
(
¯
x
1
,
¯
x
2
)
, i =1,2,
y
i
∈ ∂
(

¯
v
i
g
i
)(
¯

x
1
,
¯
x
2
)
, i =1,2,
w
n
j
∈ ∂(λ
n
j
h
j
)(
¯
x
1
,
¯
x
2
)
, j =1,2,
s
n
k
∈ ∂(μ

n
k
f
k
)(
¯
x
1
,
¯
x
2
)
, k =1,2,and
t
n
k
∈ ∂(−
¯
v
k
μ
n
k
g
k
)(
¯
x
1

,
¯
x
2
)
, k =1,2.Moreover,
Kim et al. Fixed Point Theory and Applications 2011, 2011:6
/>Page 11 of 13
0=
2

i=1
(u
i
+ y
i
) + lim
n→∞


2

j=1
w
n
j
+
2

i=1

(s
n
k
+ t
n
k
)


and
2

i=1
ε
i
g
i
(
¯
x
1
,
¯
x
2
)
=
2

i=1


i
+ β
i
) + lim
n→∞
2

j
=1

n
j
− (λ
n
j
h
j
)(
¯
x
1
,
¯
x
2
)] +
2

k=1

[q
n
k
+ z
n
k
− μ
n
k
ε
k
g
k
(
¯
x
1
,
¯
x
2
)]
.
Thus, Theorem 3.2 holds.
(2) Let
(
˜
x
1
,

˜
x
2
)=(−
1
2
,
1
5
4
)
. Then,
(
˜
x
1
,
˜
x
2
)
is a weakly ε-effici ent solution of (MFP)
2
, but
not an ε-efficient solution of (MFP)
2
. Let
B =

λ

1
≥0,
λ
2
≥0
epi(

2
i=1
λ
i
h
i
)

. Then,clB = cone co
{(0, -1, -1), (1, 0, 0), (0, 0, 1)}. However, (1, 0, 0) ∉ B. Thus, B is not closed. Moreover,
f
2
(
˜
x
1
,
˜
x
2
)
− ε
2

g
2
(
˜
x
1
,
˜
x
2
)
=3
0
,
f
2
(
˜
x
1
,
˜
x
2
)
− ε
2
g
2
(

˜
x
1
,
˜
x
2
)
=3
0
. Let
˜
v
2
=
f
2
(
˜
x
1
,
˜
x
2
)
g
2
(
˜

x
1
,
˜
x
2
)
− ε
2
and
˜
v
2
=
f
2
(
˜
x
1
,
˜
x
2
)
g
2
(
˜
x

1
,
˜
x
2
)
− ε
2
. Then,
˜
v
1
=
1
and
˜
v
2
=
2
. Le t μ
1
=1,μ
2
=0,
a
1
= b
1
= a

2
= b
2
=0and
r
n
2
=0,λ
n
2
=
0
. Let
γ
n
1
=
1
2
+
1
4
n
,
λ
n
1
= n
,
γ

n
2
=
0
,
λ
n
2
=0
, n Î N.
Then,

(
μ
1
f
1
)(
˜
x
1
,
˜
x
2
)
= {
(
−1, 0
)

}, ∂
(
μ
2
f
2
)(
˜
x
1
,
˜
x
2
)
= {
(
0, 0
)}
,

(

˜
v
1
μ
1
g
1

)(
˜
x
1
,
˜
x
2
)
= {
(
0, 0
)}
,

γ
n
1

n
1
h
1
)(
˜
x
1
,
˜
x

2
)=

0, −n +

n
2
+4n(
1
2
+
1
4n
)

×{0} = [0, 1] ×{0
}
,

γ
n
2

n
2
h
2
)(
˜
x

1
,
˜
x
2
)={(0, 0)
}
,

γ
n
2

n
2
h
2
)(
˜
x
1
,
˜
x
2
)={(0, 0)
}
. Let u
1
=(-1,0)and u

2
= y
1
= y
2
= (0, 0). Then,
u
1
∈ ∂
(
μ
1
f
1
)(
˜
x
1
,
˜
x
2
)
,
u
2
∈ ∂
(
μ
2

f
2
)(
˜
x
1
,
˜
x
2
)
,
y
1
∈ ∂
(

˜
v
1
μ
1
g
1
)(
˜
x
1
,
˜

x
2
)
,
y
2
∈ ∂
(

˜
v
2
μ
2
g
2
)(
˜
x
1
,
˜
x
2
)
. Let
w
n
1
=(1,0

)
and
w
n
2
=(0,0
)
. Then,
w
n
1
∈ ∂
γ
n
1

n
1
h
1
)(
˜
x
1
,
˜
x
2
)
and

w
n
2
∈ ∂
γ
n
2

n
2
h
2
)(
˜
x
1
,
˜
x
2
)
. Thus,

2
i=1
(u
i
+ y
i
)+lim

n→∞

2
j=1
w
n
j
=(−1, 0) + (1, 0) = (0, 0
)
,
lim
n→∞

2
i=1

γ
n
j
− (λ
n
j
h
j
)(
˜
x
1
,
˜

x
2
)

= lim
n→∞

1
2
+
1
4n

=
1
2
and
lim
n→∞

2
i=1

γ
n
j
− (λ
n
j
h

j
)(
˜
x
1
,
˜
x
2
)

= lim
n→∞

1
2
+
1
4n

=
1
2
. Hence, Theorem 3.4 holds.
Acknowledgements
This study was supported by the Korea Science and Engineering Foundation (KOSEF) NRL program grant funded by
the Korea government(MEST)(No. ROA-2008-000-20010-0).
Author details
1
School of Free Major, Tongmyong University, Busan 608-711, Korea

2
Department of Applied Mathematics, Pukyong
National University, Busan 608-737, Korea
Authors’ contributions
The authors, together discussed and solved the problems in the manuscript. All Authors read and approved the final
manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 31 January 2011 Accepted: 21 June 2011 Published: 21 June 2011
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doi:10.1186/1687-1812-2011-6
Cite this article as: Kim et al.: On ε-optimality conditions for multiobjective fractional optimization problems.
Fixed Point Theory and Applications 2011 2011:6.
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