Advances in Pure Mathematics, 2011, 1, 184-186
doi:10.4236/apm.2011.14032 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Well-Posedness for Tightly Proper Efficiency in Set-Valued
Optimization Problem*
Yangdong Xu#, Pingping Zhang
College of Mathematics and Statistics, Chongqing University, Chongqing, China
Received April 3, 2011; revised April 30, 2011; accepted May 10, 2011
Abstract
In this paper, a characterization of tightly properly efficient solutions of set-valued optimization problem is
obtained. The concept of the well-posedness for a special scalar problem is linked with the tightly properly
efficient solutions of set-valued optimization problem.
Keywords: Set-Valued Optimization Problem, Tightly Proper Efficiency, Well-Posedness
1. Introduction
One important problem in vector optimization is to find
the efficient points of a set. As observed by Kuhn,
Tucker and later by Geoffrion, some efficient points ex-
hibit certain abnormal properties. To eliminate such ab-
normal efficient points, various concept of proper effi-
ciency have been introduced. The original concept was
introduced by Kuhn and Tucker [1] and Geoffrion [2],
and later modified and formulated in a more generalized
framework by Borwein [3], Hartley [4], Benson [5],
Henig [6], Borwein and Zhuang [7]; also see the refer-
ences there in. Particularly, the concept of tightly proper
efficiency was introduced by Zaffaroni [8], and he used a
special scalar function to characterize the tightly proper
efficiency, and obtained some properties of tightly
proper efficiency.
In this paper, we study the characterization and well-
posedness for tightly proper efficiency in set-valued
vector optimization problem. The paper is organized as
follows. In Section 2, some concepts of tightly proper
efficiency and some preliminary results are given. In
Section 3, the characterization and well-posedness for
tightly proper efficiency in set-valued vector optimiza-
tion problem is discussed.
2. Preliminaries
Throughout this paper, let
X
be a linear space, and Y
Z
be two finite dimensional, with topological dual
spaces and
*
Y*
Z
. For a set
A
Y, , , clA intA
A

,
and c
A
denote the closure, the interior, the boundary
and the complement of
A
, respectively. Moreover, we
will denote with the closed unit ball of Y. A set
is said to be a cone if
B
CYcC

 for any cC

and λ ≥ 0. A cone C is said to be convex if CC C

,
and it is said to be pointed if . In the
sequel we suppose that is a convex, closed,
pointed cone with nonempty interior. We say that the set

CC

=0
YC
Y

 is a base for if is convex with 0CclC

and




,  \0=Cc=e: =Yy  , >0  ony

.
Definition 2.1: A point
y
SY is said to be
efficient with respect to (denoted C

,

y
ESC) if


=0
Y
C

Sy
Definition 2.2: [8] The point
y
SY is called
tightly proper efficient with respect to C (denoted


,
y
TPES C
K
) if there exists an open convex set
Y
 with 0Y
K

 satisfying

Sy=K and
there exists >0

such that


c
K
BC B 


It is easy to verify that


,,SS C
0
y

TP
clS
E CE
S Definition 2.3: [9] Let be a nonempty subset of
. The contingent cone (or the Bouligand tangent cone)
to at
Y
S

is the set
 



00
with= lim
nn
n
yy v yy 


0
,: ,y YR

 :,Y
nn
y
n
TS v 
*This research was partially supported by the National Natural Science
Foundation of China (Grant number: 10871216).

Y. D. XU ET AL.
185
Jahn [9] have gotten the following proposition on the
contingent cone to at S0 yclS

.
Proposition 2.1 a nonempty convex
subset of a real normed space. Then
: [9] Let S be




,=TS CclconeSy
3. Tightly Proper Efficiency and
Well-Posedness
onsider the following vector optimization problem with C
set-valued maps:
(VP) min

F
x,
 
s.t. Gx D,
x
X,
where :2
Y
FX and :2
Z
GX are set-valued
maps, respectively. D is ax, pointed cone
of
closed,conve
Z
.
Denuote the feasible soltion set of (VP) by
thimage of
 


:= :9AxXGx D
ande
A
under
F
by
 
=
xA
F
AFx


Definition 3.1: A point
x
is said to
proper efficient solution (VP)
be a tightly
of , if there exists

y
Fx such that

,yTPEFAC



, and the point

,
x
y
of
is said to be a tightly properly efficient minimizer
e define
(VP).
Definition 3.2: For a set SY. Let the function

R b
 
\
=
SSYS
:
SY d as
y
dy dy
where
 
=inf :
S
dy sys ith

=dy

S w

.
S was introduceThe function d inin
properties are ged together in the following pro-
position.
osition 3.1: [8] Letbee wi
,

ather
[8], its ma
Prop a convex conth
no
C
nempty interiorthen the function C  is convex,
positively homogenous and lipschitzian. Moreover, this
function is negative on the interior of C null on C,

and positive on int c C.
We consider the parameterized scalar problem:

y
P min

C
y
y


s.t.

y
FA
where
y
Y.
Definition 3.3: Let
y
Y , the parameterized scalar
optimization problem

y
P
is Tikhonov well-posed if
1)




=>0
CC
yy dy

 for all y


yFA
with
y
y;
2) for all

n

y
FA with

0
Cn
dyy



implies that n yy.
denote by We



=y WPVP Fyov we
Lemma 3.1onsid llowing
APis Tikhonllposed.
: [8] Cer the fo statements:
(a) the point
y
is a tightly properly efficient point in
(b) there exists an open convex set such that
S;
KY


\0
Y
CK

 and


=Sy K

;
(c)




,=0
Y
TS CC.
If Y
y
n it holds that
finite dim
all stateents are equivalent.
eorem ccterize the relation
between
parameterized scro
is a any normed space, the
). If is ensional, we have
() and
()) (abc
also that()ca
followi
( Y
h
m
The harang t
tightly properly efficient points of(VP) and the
alar pblem

y
P.
Theore et m 3.1: L
x
A

,

y
Fx. The


,
x
y is
a tightly properly efficient minimizer of (VP), then
y
is
a solution of


y
P.
Proof. We show that
y
is a solution of the scalar
problem


y
P. Indeed3.1 and Lemma
3.1(b), we have
, by Proposition




=0,
CC
y
yd y yFA


Noting
y
that


=0
C
dyy
, thus we have that
y
is
just the solution of

y
P. the problem

otRemark 3.1: The converse of Theorem 3.1 mabe n
valid, the following example can illustrate the case.
1: Let
y
Example 3.=
X
R, 2 =YR and =
Z
R.
Given 2 =CR

, =DR

.
  


 
[0,1]
,,
F
2
,111,xyR R yxifx
=x
 

x
yRR otherwise









=, 1,forany
g
xxx xX


Thus, the feasible set of (VP)

 


=| =[0,AxXGx R
)

 


F
A can see Figure 1. The set of

F
A. Figure 1. The set of

Y. D. XU ET AL.
186
, but

 
1, 1,0
(VP). Th

1,0
y
Y is a solution of
y
P is
not a tightly properly efficient minimizer ofus,
the converse of Theorem 3.1 is not valid.
Theorem 3.2: Let ,

y
Fx
x
A. Then, the

,
x
y is a tightly properly efficient minimizer of


y
P
if and only if the scalar problem

y
P is Tikhonnov
well-posed.
Proof. If

,
x
y is a tightly properly efficient
minimizer of

y
P, let us show that the scalar problem

y
P is Tik well-posed. We argue by contr-
diction: if the conclusion is false, then there exists a
sequence
honnov

y
a

n

FA such that n yyB

 for some
>0

and

0
n
yy, which means that there
exists a sequence

n
cC with
C
d

0
nn
ycy and
2B
. Since we ay=nnn c

 can alws writen
c

 with
n

, it follows that n

does not coverge to zero,
i.e., there exists a subsequence (we agall it n
n
ain c

) with
>
n

for som> e
0

. Now take
1
2
=
nnn
ycy   ,
since n

is bounded away from zero and 0
n

, thus
we have <
nn

 . Set

=
nnnn
c


 to obtain
=0
nn n
ycy 

 
nnn
ycy  and

1
nnn
ycy 

  And by1, we 0  
. the Lemma 3.
get contrathat diction, therefore it shows


y
P
is
posed. Tikhonov well-
Con
en
real norme and L), there exist
th
versely, we argue by contradiction: if the
conclusion is false, then by Y is a finite dimsional
al spacemma 3.1(cs

\0 suchat
Y
vC

vTSC e

,y. Thus ther
exist

n

y
FA,

n
cC with n y and
n
c y

nR
 that
 such


=lim nn ny
n
vyc 
  (1)
It follows that






=0
CnnCnn n
yy ycy 

. Since

y
P is Tikhonov well-posed, we have

0
nn
yy  
By (1), we have nn cv

, but since C is pointed
and

0
Y. Therefore it is a contradiction. \vC

Wewi ple to ile Theorem lu
3.2.
example
strat
Example 3.2: In this , we shall continue to
consider Example 3.1. By Definition 3.1 and Definition
3.3, we can get





give the follong exam
 

2
=, ,|=111, 0,1xxy yxx 
and
,TPE FAC
 


2
=,|=111,0,1WPVPx yyxx 
Therefore, Theorem 3.2 is valid.
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] A. M. Geoffrion, “Proper Efficiency and the Theory of
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July-12 Augus
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