Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 620928, 17 pages
doi:10.1155/2010/620928
Research Article
Optimality Conditions for Approximate Solutions
in Multiobjective Optimization Problems
Ying Gao,
1
Xinmin Yang,
1
and Heung Wing Joseph Lee
2
1
Department of Mathematics, Chongqing Normal University, Chongqing 400047, China
2
Department of Applied Mathematics, The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong
Correspondence should be addressed to Ying Gao,
Received 18 July 2010; Accepted 25 October 2010
Academic Editor: Mohamed El-Gebeily
Copyright q 2010 Ying Gao et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study first- and second-order necessary and sufficient optimality conditions for approximate
weakly, properly efficient solutions of multiobjective optimization problems. Here, tangent cone,
-normal cone, cones of feasible directions, second-order tangent set, asymptotic second-order
cone, and Hadamard upper lower directional derivatives are used in the characterizations. The
results are first presented in convex cases a nd then generalized to nonconvex cases by employing
local concepts.
1. Introduction
multiobjective optimization problems by using tangent cone, the cone of feasible directions
and -normal cone. Finally, in Section 3, we introduce some local approximate concepts
and present some properties of these notions, and then, first and second-order sufficient
conditions for local properly approximate efficient solutions of vector optimization
problems are derived. These conditions are expressed by means of tangent cone, second-order
tangent set and asymptotic second-order set. Finally, some sufficient conditions are given for
local weakly approximate efficient solutions by using Hadamard upper lower directional
derivatives.
2. Preliminaries
Let R
n
be the n-dimensional Euclidean space and let R
n
be its nonnegative orthant. Let C
be a subset of R
n
, then, the cone generated by the set C is defined as coneC∪
α≥0
αC,
and int C and cl C referred to as the interior and the closure of the set C, respectively. A set
D ⊂ R
n
is said to be a cone if cone D D. We say that the cone D is solid if int D
/
∅,and
pointed if D ∩ −D ⊂{0}. The cone D is said to have a base B if B is convex, 0
/
∈ cl B and
D cone B. The positive polar cone and strict positive polar cone of D are denoted by D
− D ∩
fS{fx
0
}. x
0
∈ S is a weakly efficient solution of 2.1 with respect to D if fx
0
−
int D ∩ fS∅ in this case, it is assumed that D is solid. x
0
∈ S is a Benson properly
efficient solution see 24 of 2.1 with respect to D if cl conefSD −fx
0
∩ −D{0}.
x
0
∈ S is a Henig’ properly efficient solution see 24 of 2.1 with respect to D if x
0
∈
Ef, D
, for some convex cone D
with D \{0}⊂int D
.
Journal of Inequalities and Applications 3
Definition 2.1 see 18, 25.Letq ∈ D \{0} be a fixed element, and ≥ 0.
i x ∈ S is said to be a weakly q-efficient solution of problem 2.1 if fS − fx
q ∩ − int D∅ in this case it is assumed that D is solid.
j
∈ Z
. 2.3
The cone of feasible directions of Z at z ∈ Z is defined as
F
z, Z
{
d ∈ R
m
:thereexistst>0suchthatz td ∈ Z
}
. 2.4
Let ≥ 0, the -normal set of Z at z ∈ Z is defined as
N
z, Z
y ∈ R
m
: y
T
x − z
∩
−q − D \
{
0
}
∅, 3.1
then
x ∈ AEf, S, q.
Proof. Suppose, on the contrary, that
x
/
∈ AEf, S, q, then, there exist x ∈ S and p ∈ D \{0}
such that fx − f
xq −p.Thatis,fxfx−q − p. T herefore, −q − p ∈
Ff
x,fS, which is a contradiction to Ffx,fS ∩ −q − D \{0}∅. This completes
the proof.
4 Journal of Inequalities and Applications
Theorem 3.2. Let
x ∈ S.
i If Tf
x,fS ∩ −D \{0}∅,thenx ∈ PA E f, S.
ii Let >0,andD is solid set and q ∈ int D.IfTf
x,fS ∩ −q − D \{0}∅,then
x ∈ PAEf, S, q.
Proof. i Suppose, on the contrary, that
x
x.
On the other hand, from u ∈ D
s
,wehaveu, q < 0. Therefore, there exists n
1
∈
N such that u, fx
n
1
− fxq
n
1
< 0, and so u, fx
n
1
− fx < 0, which deduces a
contradiction, and the proof is completed.
ii Now, we let >0. From Tf
x,fS ∩ −q − D \{0}∅,wehave
T
f
x
,f
S
− int D
∅. 3.3
By using the convex separation theorem, there exists u ∈ R
m
\{0} such that u, y≥0, for all
y ∈−int D and u, y≤0, for all y ∈ cl conefS − f
x. It is easy to get that u, y≥0, for
all y ∈−D.Hence,u, y > 0, for all y ∈−int D.
Suppose, on the contrary, that
x
/
∈ PAEf, S,q, then, there exists y ∈ R
m
such that
y ∈ cl cone
f
S
q D − f
x
∩
−D \
{
> 0, for all n ≥ n
1
.From>0, q ∈ int D
and p
n
∈ D,foralln ∈ N,wehaveq p
n
∈ int D,foralln ∈ N. Therefore,
u, y
n
λ
n
u, f
x
n
− f
x
u, q p
n
1
≥ 0,x
2
≥ 0},f : S → R
2
, fxx,
1/2and
x 1/2, 1/2
T
, then, x ∈ AEf, S, q and x ∈ PAEf, S, q.ButFfx,fS
R
2
Tfx,fS.Hence,Ffx,fS ∩ −q − D \{0}
/
∅ and Tfx,fS ∩ −q − D \
{0}
/
∅.
Theorem 3.5. Let
x ∈ S, ≥ 0, D be a solid set and q ∈ int D.Ifthereexistsu ∈−D
\{0}
such that −u, q≥1 and u ∈ N
fx,fS,thenx ∈ WAEf, S,q. Conversely , if x ∈
WAE f, S, q, then there exists u ∈−D
\{0} such that −u, q 1 and u ∈ N
fx,fS.
fx,fS,wehaveu, fx − fx≤,whichisa
contradiction to the above inequality. Hence,
x ∈ WAEf, S, q.
Conversely, let
x ∈ WAEf, S,q, then, fS − fxq ∩ − int D∅.SincefS is
convex and D is a convex cone, there exists
u ∈−D
\{0} such that u, fx − fxq≤
0, for all x ∈ S.Sinceq ∈ int D,thereexistsu ∈−D
\{0} such that −u, q 1and
u, fx − f
xq≤0, for all x ∈ S. Therefore, u, fx − fx≤−u, q ,forallx ∈ S,
which implies u ∈ N
fx,fS. This completes the proof.
Theorem 3.6. Let x ∈ S and ≥ 0.Ifthereexistsu ∈−D
s
such that −u, q≥1 and u ∈
N
fx,fS,thenx ∈ PAEf, S, q. Conversely, assume that D is a locally compact set, if x ∈
PA Ef, S, q, then there exists u ∈−D
s
such that −u, q 1 and u ∈ N
fx,fS.
Proof. Assume that, there exists u ∈−D
∈ conefSq D − fx,foralln ∈ N such that p
n
→ p.From
u ∈ D
s
and p ∈ −D \{0},wehaveu, p > 0. Hence, there exists n
1
∈ N such that
u, p
n
> 0, for all n ≥ n
1
.Fromp
n
∈ conefSq D − fx,foralln ∈ N,thereexist
λ
n
≥ 0, x
n
∈ S,andq
n
∈ D such that p
n
λ
n
fx
n
q q
n
− fx,foralln ∈ N. Therefore,
∩
−D
{
0
}
. 3.8
Since fS is a convex set, cl conefSq D − f
x is a closed convex cone. From
Lemma 2.4,thereexists
u ∈ −D
s
−D
s
such that u ∈−cl conefSq D − fx
.
Since q ∈ int D, D
s
and cl conefSq D − fx
are cone, there exists u ∈ −D
s
such
that −u, q 1andu ∈−cl conefSq D − f
x
≤−
u, q
, ∀x ∈ S. 3.10
Which implies u ∈ N
fx,fS. This completes the proof.
Example 3.7. Let D R
2
, q 1 , 1
T
, S {x ∈ R
2
: x
1
≥ 0,x
2
≥ 0},f : S → R
2
, fxx,
1/2and
x 1/2, 1/2
T
, then, x ∈ WAEf,S, and x ∈ PAEf, S, .Letu −1/2, 1/2
T
,
a neighborhood V of x such that fS ∩ V − fxq ∩ − int D∅ in this case,
it is assumed that D is solid.
ii x ∈ S is said to be a local q-efficient solution of problem 2.1,ifthereexistsa
neighborhood V of x such that fS ∩ V − fxq
∩ −D \{0}∅.
iii x ∈ S is said to be a local properly q-efficient solution of problem 2.1,ifthere
exists a neighborhood V of x such that cl conefS∩V qD−fx∩−D{0}.
The sets of local q-efficient solutions, local weakly q-efficient solutions and local
properly q-efficient solutions of problem 2.1 are denoted by LAEf, S,q,LWAEf, S, q
and LPAEf, S, q, respectively.
If 0, then, i, ii,andiii reduce to the definitions of local weakly effi
cient
solution, local efficient solution and local properly efficient solution, respectively, and
the sets of local weakly, properly efficient solutions of problem 2.1 are denoted by
LEf, SLWEf, S,LPEf, S, respectively.
Journal of Inequalities and Applications 7
Definition 4.2 see 4, 5.LetZ ⊂ R
m
and y, v ∈ R
m
.
i The second-order tangent set to Z at y, v is defined as
T
2
Z, y, v
d ∈ R
m
: ∃
t
n
,r
n
↓
0, 0
, ∃d
n
−→ d
such that
t
n
r
n
−→ 0,y
n
x t
n
v
1
2
t
n
r
Z, y, v
/
∅.Ify ∈ int Z,thenT
2
Z, y, vT
Z, y, vR
m
,andT
2
Z, y, 0
T
Z, y, 0Ty, Z.
iii Let Z is convex. If v ∈ Ty, Z and T
Z, y, v
/
∅,thenT
2
Z, y, v ⊂ T
Z, y, v
cl cone coneZ − y − vTv, TZ, y.
Definition 4 .4 see 27.LetK ⊂ R
n
and φ : K → R be a nonsmooth function. The Hadamard
upper directional derivative and the Hadamard lower directional derivative derivative of φ
at x ∈ K in the direction d ∈ R
inf
h → d
φ
x th
− φ
x
t
.
4.3
Lemma 4.5 see 7. Let Y be a finite-dimensional space and y
0
∈ E ⊂ Y . If the sequence y
n
∈
E \{y
0
} converges to y
0
, then there exists a subsequence (denoted the same) y
n
such that y
n
−
y
0
/t
that t
n
/r
n
→ 0 and y
n
− y
0
− t
n
u/1/2t
n
r
n
converges to some vector z ∈ T
E, y
0
,u ∩ u
⊥
\{0},
where u
⊥
denotes the orthogonal subspace to u.
In the following theorem, we derive several properties of local weakly, properly
approximate efficient solutions.
8 Journal of Inequalities and Applications
Theorem 4.6. i Let int D
/
∅, then, for any fixed q ∈ D \{0},
V − fxq D is a closed set, and cl conefS ∩ V − fxq D ∩ −D{0},then
x ∈ LPEf, S.
Proof. i Let
x ∈ LWEf, S, then, there exists a neighborhood V
1
of x such that fS ∩ V
1
−
f
x ∩ − int D∅.Fromq ∈ D \{0 },wehave
f
S ∩ V
1
− f
x
∩
−q − int D
∅, ∀>0 . 4.5
Which implies
x ∈
>0
LWAEf,S, q.
Conversely, we assume that there exists a neighborhood
x−p −q − p/2 ∈−q − D \{0}, which is a contradiction to the
assumption. This completes the proof.
iii It is easy to see that LPEf, S ⊂
>0
LPAEf,S, q.
Conversely, we assume that there exists a neighborhood
V of x such that for any fixed
q ∈ D \{0} and >0, conefS ∩
V − fxq D is a closed set, and cl cone fS ∩
V − fxq D ∩ −D{0}. Suppose, on the contrary, that x
/
∈ LPEf,S, then, for any
neighborhood V of
x,wehaveclconefS ∩ V − fxD ∩ −D \{0}
/
∅.TakeV V , then,
there exist λ>0,
p
1
∈ D \{0}, p
2
∈ D and x ∈ S ∩ V such that λfx − fxp
2
−p
1
.Take
q p
1
/2λ and 1, similar to the proof of ii we can complete the proof.
∅,
T
f
S
,f
x
,v
∩ v
⊥
∩
−cl cone
D q v
{
0
}
,
4.6
then
x ∈ LAEf, S, q.
x
f
x
n
− f
x
∈−
1
f
x
n
− f
x
q
.
4.7
Since fx
n
→ fx and q ∈ D \{0},thereexistsn
1
∈ N such that
1
f
x
n
− f
x
− 1
q ∈ D, ∀n ≥ n
1
. 4.8
−→ d ∈ T
f
x
,f
S
∩
−q − D
.
4.9
Let t
n
fx
n
− fx and z
n
2/t
n
fx
n
− fx/t
n
− d,foralln ∈ N. Similar to
the proof of Lemma 4.3,wehavethereexistsz ∈ R
x
,v
∩ v
⊥
∩
−cl cone
D v
∅,
T
f
S
,f
x
,v
∩ v
⊥
∩
x
,v
,T
f
S
,f
x
,v
∩ v
⊥
∩
−cl cone
D q v
{
0
}
,
4.11
∩ d
⊥
∩
−cl cone
D q βU d
∅,
T
f
S
,f
x
,d
∩ d
⊥
∩
−cl cone
D q βU d
−λb − B V
λ
∅. 4.13
Suppose, on the contrary, that
x
/
∈ LPAEf, S, q, then, for any neighborhood V of 0, we have
cl cone
f
S ∩
x
V
n
− f
x
q D
∩
−D \
{
0
}
k
n
∈ S ∩ x V/n and p
k
n
∈ D such that z
k
n
λ
k
n
fx
k
n
− fxq p
k
n
and z
k
n
→ z
n
.Sincez
k
n
→ z
n
,thereexistsk
1
− f
x
∈ z
n
V − λ
k
n
q p
k
n
, ∀k ≥ k
1
. 4.16
Let p
k
n
β
k
n
θ
k
n
for β
k
n
1 λ
k
n
β
k
n
λ
k
n
β
k
n
θ
k
n
1 λ
k
n
β
k
n
−
αλ
k
n
b
1 λ
n
θ
k
n
/1 λ
k
n
β
k
n
, then, γ
k
n
∈ B,sinceB is a convex set, and so,
λ
k
n
1 λ
k
n
β
k
n
f
x
k
n
4.18
On the other hand, from x
k
n
∈ S ∩ x V/n,wehavex
k
n
→ x when n →∞and k →∞.
Since f is a continuous function, fx
k
n
→ fx when n →∞and k →∞, which combining
with the assumption fS−f
x ∩δU ⊂ Tfx,fS yields there exist n
1
∈ N and k
n
1
∈ N
such that
f
x
k
n
1
− f
x
/
0, there exists
k
n
1
∈ N such that λ
k
n
1
> 0, for all k ≥ k
n
1
.Takek
2
max{k
n
1
, k
n
1
},
and let λ αλ
k
2
n
1
/1 λ
k
2
n
− f
x
∈
−B − λb V
λ
. 4.20
Which is a contradiction to 4.13. This completes the proof.
ii Suppose, on the contrary, that
x
/
∈ LPAEf, S, , then, for any γ>0andn ∈ N,we
have
cl cone
f
S ∩
x
γU
n
− f
fx
k
n
− fxq p
k
n
and z
k
n
→ z
n
.Itis
12 Journal of Inequalities and Applications
obvious that fx
k
n
/
fx.Otherwise,z
n
∈ q D ∩ −D \{0}, which is a contradiction to
the assumption that D is a pointed cone. Since z
n
/
0andz
k
n
→ z
n
,thereexistsk
n
− fx→d ∈ Tfx,fS.Fromz
k
n
λ
k
n
fx
k
n
− fxq p
k
n
,
we have for sufficiently large n, k ∈ N
f
x
k
n
− f
x
f
x
k
n
− f
x
.
4.22
On the other hand, we have
z
k
n
− λ
k
n
q p
k
n
λ
k
n
k
n
f
x
k
n
− f
x
∈
z
n
V − λ
k
n
q D
λ
k
n
λ
k
n
f
x
k
n
− f
x
∈−q − D
V
λ
k
n
f
x
n
− f
x
z
k
n
− λ
k
n
q p
k
n
λ
k
n
f
x
fx
k
n
− fx and z
k
n
2/t
k
n
fx
k
n
− fx/t
k
n
− d. Similar to the proof of
Lemma 4.3,wehavethereexistsz ∈ R
m
such that z ∈ T
2
fS,fx,d∩d
⊥
∩−cl coneqD
d βU or z ∈ T
fS,fx,d∩d
⊥
\{0}∩−cl cone qD d βU, which is a contradiction
to the assumptions. This completes the proof.
Journal of Inequalities and Applications 13
,T
f
S
,f
x
,d
∩ d
⊥
∩
−cl cone
D q βU d
{
0
}
.
4.27
Remark 4.12. The conditions of Theorem 4.7, Corollary 4.8 and Theorem 4.10 are not
necessary conditions, see Examples 4.14 and 4.15.
,q 1, 1
T
,and>0. We consider x 0, 0
T
∈ S. It is easy to see that
Tf
x,fS ∩ −q − D∅ and fS − fx ⊂ Tfx,fS. That is, the condition i
of Theorem 4.10 is valid, and
x ∈ LPAEf, S,qPAEf, S, q,forall>0.
If we let 0 <<1and
x ,
3/2
T
∈ S, then, Tfx,fS ∩ −q − D
/
∅.Butthe
condition ii of Theorem 4.10 is valid. Hence,
x ∈ LPAEf, S, PEAf, S, .
Let 0, then, Tf
x,fS ∩ −D \{0}
/
∅.Butforalld ∈ Tfx,fS ∩ −D \{0},
the condition ii of Corollary 4.8 satisfies see Example 3.7 in 7,and
x is an efficient
solution of this problem, since fS is a convex set. But for any β>0, it is easy to
check that there exists d ∈ Tf
x,fS \{0} ∩ −D βU such that T
fS,fx,d
is false, and
x is not a properly efficient solution of this problem.
Example 4.14. Let D R
2
,q 1, 1
T
, S {x
1
,x
2
T
: x
1
x
2
≥ 0}∪{x
1
,x
2
∈ R
2
: x
1
≥
1}∪{x
1
,x
2
fx,fS,dR
2
,sincefx ∈ intfS.But
x ∈ LPAEf, S,. This implies that the conditions of Theorem 4.10 are not necessary.
Example 4.15. Let D R
2
, S {x
1
,x
2
∈ R
2
: x
2
≥|x
1
|}, f : S → R
2
, fx
1
,x
2
x
1
,x
2
T
{y
1
,y
2
T
: y
1
y
2
0}, −cl coneD q dR
2
and T
fS,fx,d
{y
1
,y
2
T
∈ R
2
: y
2
≥ y
1
}. Therefore, T
fS,fx,v∩v
},f : S → R
2
, fx
1
,x
2
x
1
,x
2
T
,
q 1, 1
T
,and>0. We consider x 0, 0
T
∈ S. It is easy to see that Tfx,fS ∩ −q −
D∅ and fS − f
x ⊂ Tfx,fS. That is, the condition i of Theorem 4.7 is valid, and
x ∈ LPAEf, S,qPAEf, S, q,forall>0.
Theorem 4.17. Let
x ∈ S, ≥ 0 and D R
m
.
i If f
−
x, d∩−q−int R
k
∈ S \{x},
k ∈ N and x
k
→ x such that fx
k
− fx ∈−q − int R
m
.Letd
k
x
k
− x/x
k
− x and
t
k
x
k
− x, then, t
k
→ 0, d
k
→ d ∈ Tx, S and d 1. Hence,
f
x
k
q.
4.28
Since t
k
↓ 0, there exists k
1
∈ N such that fx
k
− fx/t
k
∈−q − int R
m
,forallk ≥ k
1
.
Hence,
f
i
x
k
− f
i
x
t
k
− f
i
x
t
q
i
≤ lim inf
n →∞
f
i
x t
n
d
n
− f
i
x
t
n
q
i
< 0, ∀i ∈
{
Journal of Inequalities and Applications 15
and t
k
n
of x
k
and t
k
, respectively, then there exist an index i
0
∈{1, ,m}, n
0
∈ N and k
0
∈ N
such that
f
i
x
k
n
− f
i
x
t
k
i
0
< 0, ∀k ≥ k
0
,n≥ n
0
.
4.31
Therefore, f
−
i
x, dq
i
≤ 0, for all i ∈{1, ,m},andf
−
i
0
x, dq
i
0
< 0, which is a
contradiction to the assumption. This completes the proof.
Remark 4.18. The following necessary conditions for -local weakly efficient solutions may
not be true.
x ∈ LWAE
−q − R
m
\
{
0
}
∅, ∀d ∈ T
x, S
.
4.32
See the following example.
Example 4.19. Let fxf
1
x,f
2
x
T
: R → R
2
,
f
1
x
x, dq ∈
− int R
2
}∩Tx, S
/
∅.Infact,
f
1
−
x
,d
lim
t↓0
inf
h↓d
f
1
th
− f
1
0
x, dq ∈−int R
2
}∩Tx, S
/
∅.
16 Journal of Inequalities and Applications
Acknowledgments
This work was partially supported by the National Science Foundation of China no.
10771228 and 10831009, the Research Committee of The Hong Kong Polytechnic University,
the Doctoral Foundation of Chongqing Normal University no.10XLB015 and the Natural
Science Foundation project of CQ CSTC no. CSTC. 2010BB2090.
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