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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
The investigation of the optimality conditions is one of the most attractive topics of
optimization theory. For vector optimization, the optimality solutions can be characterized
with the help of different geometrical concepts. Miettinen and M
¨
akel
¨
a 1 and Huang and
Liu 2 derived several optimality conditions for efficient, weakly efficient, and properly
efficient solutions of vector optimization pro blems with the help of several kinds of cones.
Engau and Wiecek 3 derived the cone characterizations for a pproximate solutions of vector
optimization problems by using translated cones. In 4, Aghezzaf and Hachimi obtained
second-order optimality conditions by means of a second-order tangent set which can be
considered an extension of the tangent cone; Cambini et al. 5 and Penot 6 introduced
a new second-order tangent set called asymptotic second-order cone. Later, second-order
optimality conditions for vector optimization problems have been widely studied by using
these second-order tangent sets; see 7–9.
2 Journal of Inequalities and Applications
During the past decades, researchers and practitioners in optimization had a keen
interest in approximate solutions of optimization problems. There are several important
reasons for considering this kind of solutions. One of them is that an approximate solution of
an optimization problem can be computed by using iterative algorithms or heuristic methods.
In vector optimization, the notion of approximate solution has been defined in several ways.
The first concept was introduced by Kutateladze 10 and has been used to establish vector
variational principle, approximate Kuhn-Tucker-type conditions and approximate duality
theorems, and so forth, see 11–20. Later, several authors have proposed other -efficiency
concepts see, e.g., White 21;Helbig22 and Tanaka 23.
In this paper, we derive di fferent characterizations for approximate solutions by
treating convex case and nonconvex cases. Giving up convexity naturally means that
we need local instead of global analysis. Some definitions and notations are given in
Section 2.InSection 3, we derive some characterizations for global approximate solutions of
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
and D
s
, respectively.
Consider the following multiobjective optimization problem:
min
f
x
: x ∈ S
, 2.1
where S ⊂ R
n
is an arbitrary nonempty set, f : S → R
m
. As usual, the preference relation ≤
defined in R
m
by a closed c onvex pointed cone D ⊂ R
m
is used, which models the preferences
used by the decision-maker:
y, z ∈ Y, y ≤ z ⇐⇒ y − z ∈−D. 2.2
We recall that x
0
∈ S is an efficient solution of 2.1 with respect to D if fx
0
− 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.
ii x ∈ S is said to be a efficient q-solution of problem 2.1 if fS−fxq∩−D \
{0}∅.
iii
x ∈ S is said to be a properly q-efficient solution of problem 2.1,ifclconefS
q D − fx ∩ −D{0}.
The sets of q-efficient solutions, weakly q-efficient solutions, a nd properly q-
efficient solutions of problem 2.1 are denoted by AEf, S, q,WAEf, S, q,and
PAEf, S, q, respectively.
Remark 2.2. If 0, then q-efficient solution, weakly q-efficient solution, and properly q-
efficient solution reduce to efficient solution, weakly efficient solution and properly efficient
solution of problem 2.1.
Definition 2.3. Let Z ⊂ R
m
be a nonempty convex set.
The contingent cone of Z at z ∈ Z is defined as
T
z, Z
d ∈ R
m
:thereexistst
j
↓ 0andd
j
−→ d such that z t
j
d
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
≤ , ∀x ∈ Z
. 2.5
Lemma 2.4 see 26. Let N, K ⊂ R
m
be closed convex cones such that N ∩ K {0}. Suppose that
K is pointed and locally compact, or int K
/
∅,then,−N
∩ K
s
/
∅.
3. Cone Characterizations of Approximate Solutions: Convex Case
In this section, we assume that fS is a convex set.
Theorem 3.1. Let
x ∈ S and ≥ 0.If
F
f
x
,f
S
∩
−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
/
∈ PAEf, S, then, there exists q ∈−D \{0} such
that q ∈ cl conefS − f
xD. Hence, there exist λ
n
∈ R
, x
n
∈ S and q
n
∈ D, n ∈ N such
that λ
n
fx
n
− fxq
n
→ q.Sinceq
/
0, there exists n ∈ N such that λ
n
> 0.
Since fS is convex set, cl conefS − f
x Tfx,fS.Hence,clconefS −
f
x ∩ −D \{0}∅.FromLemma 2.4,thereexistsu ∈ D
s
such that u, y≥0, for all
y ∈ cl conefS − f
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
∅. 3.2
In fact, if there exists p ∈ R
m
such that p ∈ Tfx,fS ∩ − int D, then, from q ∈ int D
and >0, there exists λ>0suchthatp
1
−λp − q ∈ D \{0}.Hence,−q − p
1
λp ∈
Tf
x,fS ∩ −q − D \{0}, which is a contradiction to the assumption.
Since fS is a convex set, cl conefS − f
x Tfx,fS.Hence,
cl cone
f
S
− f
x
∩
− 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
}
, 3.4
and there exist y
n
∈ conefSq D − fx,foralln ∈ N such that y
n
→ y.Thatis,there
exist λ
n
≥ 0,x
n
∈ S and p
n
∈ D,foralln ∈ N such that y
n
λ
n
fx
n
q p
n
− fx,forall
n ∈ N.Since
y
/
0, there exists n
1
∈ N such that λ
n
> 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
<λ
n
u, f
x
n
− f
x
≤ 0, ∀n ≥ n
1
. 3.5
Which implies u,
y < 0. On the other hand, from y ∈−D \{0},wehaveu, y≥0, which
yields a contradiction. This completes the proof.
Remark 3.3. If 0, then the conditions of Theorems 3.1 and 3.2 are a lso necessarysee 2.
But for >0, these are not necessary conditions, see the following example.
Journal of Inequalities and Applications 5
Example 3.4. 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 ∈ 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.
Proof. Assume that, there exists u ∈−D
\{0} such that −u, q≥1andu ∈ N
fx,fS.
Suppose, on the contrary, that
x
/
∈ WAEf, S,q, then, there exist p ∈−int D and x ∈ S such
that p fx−f
xq.Fromu ∈−D
\{0} and −u, q≥1, we have u, fx−fxq > 0.
Hence,
u, f
x
− f
x
> −
u, q
≥ . 3.6
On the other hand, from u ∈ N
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
s
such that −u, q≥1andu ∈ N
fx,fS.
Suppose, on the contrary, that
x
/
∈ PAEf, S,q, then, there exists p ∈ R
m
such that
p ∈ cl cone
f
S
q D − f
x
∩
−D \
{
0
}
, 3.7
and there exists p
n
∈ 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,
u, fx
n
qq
n
−fx > 0, for all n ≥ n
1
, which combing with q
n
∈ D and −u, q≥1 yields
u, fx
n
− fx > −u, q≥,foralln ≥ n
1
, which is a contradiction to u ∈ N
fx,fS.
Hence,
x ∈ PAEf, S, q.
Conversely, let
x ∈ PAEf, S, q, then,
cl cone
f
S
q D − f
x
∩
−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
.
6 Journal of Inequalities and Applications
Now, we prove that u ∈ N
fx,fS.Thatis,u, fx − fx≤,forallx ∈ S.
From u ∈−cl conefSq D − f
x
,wehave
u, f
x
− f
x
q p
≤ 0, ∀x ∈ S, p ∈ D. 3.9
Since 0 ∈ D and −u, q 1, we have
u, f
x
− 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
,
then u, p 1andu ∈ N
fx,fS {x ∈ R
2
: x
1
x
2
≥−1,x
1
≤ 0,x
0
≤ 0}.
Remark 3.8. i If 0andD R
m
, then Theorems 3.1 and 3.5 reduce to the corresponding
results in 1.
ii In 1, the cone characterizations of Henig’ properly efficient solution were
derived. We know that Henig’ properly efficient solution equivalent to Benson properly
efficient solution, when D is a closed convex pointed conesee 24. Therefore, if 0and
D R
m
,Theorems3.2 and 3.6 reduce to the corresponding results in 1.
4. Cone Characterizations of Approximate Solutions: Nonconvex Case
In this section, fS is no longer assumed to be convex. In nonconvex case, the corresponding
local concepts are defined as follows.
Definition 4.1. Let q ∈ D \{0} be a fixed element and ≥ 0.
i x ∈ S is said to be a local weakly q-efficient solution of problem 2.1,ifthereexists
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
↓ 0, ∃d
n
−→ d such that y
n
y t
n
v
1
2
t
2
n
d
n
∈ Z, ∀n ∈ N
.
4.1
ii The asymptotic second-order tangent cone to Z at y, v is defined as
T
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
n
d
n
∈ Z, ∀n ∈ N
.
4.2
In 4–9, some properties of second-order tangent sets have been derived, see the
following Lemma.
Lemma 4.3. Let y ∈ cl Z and v ∈ R
m
,then,
i T
2
Z, y, v and T
Z, y, v are closed sets contained in cl cone cone Z − y − v,and
T
Z, y, v is a cone.
ii If v
/
∈ Ty, Z,thenT
2
Z, y, vT
Z, y, v∅.Ifv ∈ Ty, Z,thenT
2
Z, y, v ∪
T
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
n
are given by
φ
x, d
lim
t↓0
sup
h → d
φ
x th
− φ
x
t
,
φ
−
x, d
lim
t↓0
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
n
converges to some nonnull vector u ∈ Ty
0
,E,wheret
n
y
n
− y
0
, and either y
n
− y
0
−
t
n
u/1/2t
2
n
converges to some vector z ∈ T
2
E, y
0
,u∩ u
⊥
or there exists a sequence r
n
→ 0
such
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},
LWE
f, S
⊂
>0
LWAE
f, S, q
.
4.4
Conversely, if
x ∈ S, and there exists a neighborhood V of x such that fS ∩ V − fx ∩ −q −
int D∅, for all >0,thatis,
x ∈ WAEf, S ∩ V,q, for all >0,thenx ∈ LWEf, S.
ii For any fixed q ∈ D \{0},LEf, S ⊂
>0
LAEf, S, q. Conversely, if x ∈ S and there
exists a neighborhood V of x such that for any fixed q ∈ D \{0} and >0, fS ∩ V − fx ∩
−q − D \{0}∅,then
x ∈ LEf, S.
iii For any fixed q ∈ D \{0},LPEf, S ⊂
>0
LPAEf, S,q. Conversely, if x ∈ S and
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 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
V of x such that x ∈
WAEf,S ∩
V,q,forall>0. Suppose, on the contrary, that x
/
∈ LWEf, S, then, for any
neighborhood V of
xfS ∩ V − fx ∩ − int D
/
∅.TakeV V , then, there exist p ∈ int D
and x ∈ S ∩
V such that fx − fx−p. Therefore, if >0issufficiently small, we
have fx − f
x −p −q − p − q ∈−q − int D, which is a contradiction to
x ∈ WAEf, S ∩ V,q,forall>0. This completes the proof.
ii It is easy to see that LEf, S ⊂
>0
LAEf, S, q.
Conversely, we assume that there exists a neighborhood
V of x such that for any fixed
q ∈ D \{0} and >0, fS ∩
V − fx ∩ −q − D \{0}∅.Suppose,onthecontrary,that
x
/
∈ LEf, S, then, for any neighborhood V of x,wehavefS ∩V −fx ∩−D \{0}
/
∅.Take
V
V , then, there exist p ∈ D \{0} and x ∈ S ∩ V such that fx − fx−p.Takeq p/2
and 1, then, fx − f
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.
Journal of Inequalities and Applications 9
Theorem 4.7. Let f be a continuous function on S,
x ∈ S,and>0.
i If Tf
x,fS ∩ −q − D∅,thenx ∈ LAEf, S, q.
ii If Tf
x,fS ∩ −q − D
/
∅,andforeachv ∈ Tfx,fS ∩ −q − D
T
2
f
S
,f
x
,v
∩ v
⊥
∩
−cl cone
D q v
∅,
T
f
S
,f
x
,v
∩ v
⊥
∩
−cl cone
D q v
{
0
}
,
4.6
then
x ∈ LAEf, S, q.
Proof. i Let Tf
x,fS ∩ −q − D∅. Suppose, on the contrary, that x
/
∈ LAEf, S, q,
then, there exists x
n
∈ S and x
n
→ x such that fx
n
− fxq ∈−D \{0},foralln ∈ N.
Since f is a continuous function and D is a pointed cone, fx
n
/
fx,foralln ∈ N and
fx
n
→ fx. Therefore, fx
n
− fx/fx
n
− fx→d ∈ Tfx,fS.
On the other hand, for any n ∈ N,wehave
f
x
n
− f
x
f
x
n
− f
x
∈−
1
f
x
n
− f
x
q D \
{
0
}
⊂−
q D \
{
0
}
1
f
x
n
− f
x
− 1
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
Hence, d ∈−q D, which is a contradiction to the assumption. This completes the proof.
ii Suppose, on the contrary, that
x
/
∈ LAEf, S, q. Similar to the proof of i,wehave
there exists x
n
→ x such that
f
x
n
− f
x
f
x
n
− f
x
−→ 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
m
such that z ∈ T
2
fS,fx,d ∩ d
⊥
∩
−cl coneq D d or z ∈ T
fS,fx,d ∩ d
⊥
\{0}∩−cl cone q D d,whichisa
contradiction to the assumptions. This completes the proof.
10 Journal of Inequalities and Applications
Corollary 4.8. Let f be a continuous function on S,
x ∈ S and 0.
i If Tf
x,fS ∩ −D{0},thenx is a local efficient solution of problem 2.1.
ii If Tf
x,fS ∩ −D \{0}
/
∅,andforeachv ∈ Tfx,fS ∩ −D \{0}
T
2
f
S
,f
x
,v
∩ v
⊥
∩
−cl cone
D v
∅,
T
f
S
,f
x
,v
∩ v
⊥
∩
−cl cone
D v
{
0
}
,
4.10
then
x is a local efficient solution of problem 2.1.
Proof. The proof is similar to Theorem 4 .7.
Remark 4.9. If fS is convex, then the condition ii of Theorem 4.7 is equivalent to the
following condition
ii
Tfx,fS ∩ −q − D
/
∅,andforeachv ∈ Tfx,fS ∩ −q − D
0
/
∈ T
2
f
S
,f
x
,v
,T
f
S
,f
x
,v
∩ v
⊥
∩
−cl cone
D q v
{
0
}
,
4.11
since T
2
fS,fx,v ⊂ T
fS,fx,v by Lemma 4.3iii.
Theorem 4.10. Let f be continuous on S,
x ∈ S,and ≥ 0.
i Assume that D has a compact base B, p αb for b ∈ B and α>0, and there exists δ>0
such that fS − f
x ∩ δU ⊂ Tfx,fS.IfTfx,fS ∩ −q − D \{0}∅,
then
x ∈ LPAEf, S, q.
ii Assume that Tf
x,fS ∩ −q − D \{0}
/
∅, and there exists β>0 such that for each
d ∈ Tf
x,fS \{0} ∩ −q − D βU the following conditions hold
T
2
f
S
,f
x
,d
∩ d
⊥
∩
−cl cone
D q βU d
∅,
T
f
S
,f
x
,d
∩ d
⊥
∩
−cl cone
D q βU d
{
0
}
,
4.12
then
x ∈ LPAEf, S, q,where,U denotes the closed unit ball of R
m
.
Proof. i Let Tf
x,fS ∩ −q − D \{0}∅, then, Tfx,fS ∩ −λb − B∅,forall
λ>0. The assumptions and the separation result 28,page9 implies that for any λ>0there
exists a neighborhood V
λ
of 0 such that
T
f
x
,f
S
∩
−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
}
/
∅. 4.14
Journal of Inequalities and Applications 11
Therefore,
cl cone
f
S ∩
x
V
n
− f
x
q D
∩−B
/
∅. 4.15
That is, for any n ∈ N there exist z
n
∈ cl conefS ∩ x V/n − fxq D ∩ −B,
and so, for any n ∈ N there exists λ
k
n
≥ 0, x
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
∈ N such that
z
k
n
∈ z
n
V ,forallk ≥ k
1
.Byz
k
n
λ
k
n
fx
k
n
− fxq p
k
n
,wehave
λ
k
n
f
x
k
n
− 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
≥ 0andθ
k
n
∈ B, then,
λ
k
n
1 λ
k
n
β
k
n
f
x
k
n
− f
x
∈−
−
z
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
k
n
β
k
n
V
1 λ
k
n
β
k
n
. 4.17
Let γ
k
n
−z
n
/1 λ
k
n
β
k
n
λ
k
n
β
k
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
− f
x
∈−γ
k
n
−
αλ
k
n
b
1 λ
k
n
β
k
n
V
1 λ
k
n
β
k
n
, ∀k ≥ k
1
.
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
∈
f
S
− f
x
∩ δU ⊂ T
f
x
,f
S
, ∀k ≥ k
n
1
. 4.19
From z
n
1
/
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
1
β
k
2
n
1
> 0. Since V is an arbitrary set, it follows that
λ
k
2
n
1
1 λ
k
2
n
1
β
k
2
n
1
f
x
k
2
n
1
− 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
x
q D
∩
−D \
{
0
}
/
∅.
4.21
Let V γU. Similar to the proof of i, we have for any n ∈ N there exist λ
k
n
≥ 0, x
k
n
∈
S ∩
x V/n,andp
k
n
∈ D such that z
k
n
λ
k
n
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
1
∈ N such
that λ
k
n
> 0andz
k
n
∈ z
n
V ,forallk ≥ k
1
.Fromx
k
n
∈ S ∩ x V/n,wehavex
k
n
→ x,when
n →∞and k →∞.Sincef is a continuous function and fx
k
n
/
fx, it is easy to see that
fx
k
n
− fx/fx
k
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
z
k
n
− λ
k
n
q p
k
n
λ
k
n
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
f
x
k
n
− f
x
∈−q − D V,
4.23
for sufficiently large k, n ∈ N.Infact,forsufficiently large k,n ∈ N
z
k
n
− λ
k
n
q p
k
n
λ
k
n
f
x
k
n
− f
x
∈
z
n
V − λ
k
n
q D
λ
k
n
f
x
k
n
− f
x
.
4.24
Hence,
z
k
n
− λ
k
n
q p
k
n
λ
k
n
f
x
k
n
− f
x
∈−q − D
V
λ
k
n
f
x
k
n
− f
x
,
4.25
when k and n sufficiently large enough. Since γ>0 is arbitrary,
f
x
k
n
− f
x
f
x
k
n
− f
x
z
k
n
− λ
k
n
q p
k
n
λ
k
n
f
x
k
n
− f
x
−→ d ∈
−q − D βU
∩ T
f
x
,f
S
.
4.26
Let t
k
n
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
Remark 4.11. If fS is convex, then the conditions i and ii of Theorem 4.10 are equivalent
to i
and ii
, respectively.
i
D has a compact base B, p αb for some b ∈ B, α>0, and Tfx,fS ∩ −q −
D \{0}∅.
ii Tf
x,fS ∩ −q − D \{0}
/
∅,andthereexistsβ>0suchthatforeachd ∈
Tf
x,fS \{0} ∩ −q − D βU
0
/
∈ T
2
f
S
,f
x
,d
,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.
Now, we give some examples to verify the results of Theorem 4.7, Theorem 4.10 and
Corollary 4.8.
Example 4.13. Let D R
2
, S {x
1
,x
2
∈ R
2
: x
2
≥|x
1
|
3/2
},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
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
T
2
fS,fx,dcl coneD d βUR
2
.Infact,foranyβ>0, take d β/2,β/2
T
∈
Tf
x,fS \{0} ∩ −D βU, then, T
fS,fx,dT
2
fS,fx,dcl coneD d
βUR
2
and d
⊥
{y y
1
,y
2
T
∈ R
2
: y
1
y
2
0}. Hence, the condition ii of Theorem 4.10
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
∈ R
2
: x
2
≥ 1},fx : S → R
2
and fxx.Takex 0, 0
T
, then, it is easy to
see that there exists δ>0suchthatfS − f
x ∩ δU ⊂ Tfx.fS and Tfx,fS ∩
−q − D \{0}∅,forall ≥ 0. Hence,
x ∈ LPAEf, , p,forall ≥ 0. But x is not a global
properly efficient solution, where, U is closed unit ball of R
2
.
We let 0 <<1and
x ,
T
∈ S, then, Tfx,fS ∩ −q − D \{0}
/
∅,forall
>0, and ii in Theorem 4.10 is false. In fact, for any β>0andd ∈ Tf
x,fS ∩ −q −
D βU ⊂−int R
2
, T
2
fx,fS,dT
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
,q
1, 1
T
and 1. We consider x q ∈ S. It is easy to see that x ∈ LAEf, S, qAEf, S, q.
But Tf
x,fS ∩ −q − D{y
1
,y
2
T
∈ R
2
: y
1
≤−1,y
2
≤−1,y
2
≥ y
1
}
/
∅,andthe
14 Journal of Inequalities and Applications
condition ii of Theorem 4.7 is false. In fact, if we take d −2 , −2
T
∈ Tfx,fS ∩
−q− D, then, d
⊥
{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
T
∩−cl coneDqv {y
1
,y
2
∈
R
2
: y
1
y
2
0,y
1
≤ 0}.
Example 4.16. Let D R
2
, S {x
1
,x
2
∈ R
2
: x
2
|x
1
|
3/2
},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
m
∅, for any unit vector d ∈ Tx, S,thenx ∈ LWAEf,S, q.
ii If f
−
x, d∩−q−R
m
\{0}∅, for any unit vector d ∈ Tx, S,thenx ∈ LAEf, S, q.
Where, f
−
x, df
−
1
x, d, ,f
−
m
x, d
T
.
Proof. i Suppose, on the contrary, that
x
/
∈ LW AEf, S, q, then, there exists x
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
− f
x
t
k
f
x t
k
d
k
− f
x
t
k
∈−q − int R
m
−
1
t
k
− 1
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
q
i
< 0, ∀i ∈
{
1, ,m
}
,k≥ k
1
.
4.29
Therefore,
f
−
i
x, d
q
i
lim
t↓0
inf
h → d
f
i
x th
− 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 ∈
{
1, ,m
}
.
4.30
Which is a contradictions to the assumption. This completes the proof.
ii Similar to the proof of i,wehavethereexistsx
k
∈ S\{x},k ∈ N and x
k
→ x such
that fx
k
− fx ∈−q − R
m
\{0}.Hence,thereexistsk
1
∈ N such that fx
k
− fx/t
k
∈
−q − R
m
\{0},forallk ≥ k
1
. It is easy to see that, if we take an appropriate subsequences x
k
n
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
n
q
i
≤ 0, ∀i ∈
{
1, ,m
}
, ∀k ≥ k
0
,n≥ n
0
,
f
i
0
x
k
n
− f
i
0
x
t
k
n
q
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
f, S, q
⇒ f
−
x, d
∩
−q − int R
m
∅, ∀d ∈ T
x, S
.
x ∈ LAE
f, S, q
⇒ f
−
x, d
∩
−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 sin
1
x
,x
/
0,
0,x 0,
4.33
f
2
xx, 2/π, q 1, 1
T
, S {x ∈ R : −2/π ≤ x ≤ 2/π}. Consider the following
problem:
min
x∈S
f
x
.
MP
It is easy to see that
x 0isanq-efficient solution of MP,but,{d ∈ R : f
−
x, dq ∈
− int R
2
}∩Tx, S
/
∅.Infact,
f
1
−
x
,d
lim
t↓0
inf
h↓d
f
1
th
− f
1
0
t
lim
t↓0
inf
h↓d
h sin
1
th
−
|
d
|
, ∀d ∈ R.
4.34
f
2
−
x, dd,foralld ∈ R. It is obvious that −1 ∈{d ∈ R : f
−
x, dq ∈−int R
2
}.Onthe
other hand, T
x, SR.Hence,{d ∈ R : f
−
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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