Applied Mathematics, 2011, 2, 922-925
doi:10.4236/am.2011.27126 Published Online July 2011 (http://www.SciRP.org/journal/am)
Optimality for Henig Proper Efficiency in Vector
Optimization Involving Dini Set-Valued Directional
Derivatives
Guolin Yu, Huaipeng Bai
Research Institute of Information and System Computation Science, Beifan g Un iversity
of Nationalities, Yinchuan, China
Received March 21, 2011; revised June 8, 2011; accepted Ju ne 11, 2011
Abstract
This note studies the optimality conditions of vector optimization problems involving generalized convexity
in locally convex spaces. Based upon the concept of Dini set-valued directional derivatives, the necessary
and sufficient optimality conditions are established for Henig proper and strong minimal solutions respec-
tively in generalized preinvex vector optimization problems.
Keywords: Vector Optimization, Dini Set-Valued Directional Derivative, Generalized Preinvex Function,
Henig Proper Efficiency
1. Introduction
The study on the optimality conditions for non-smooth
and generalized convex vector optimization problem in
abstract spaces is a lively subject. Recently, there is a
growing interest on this topic by using Dini set-valued
directional derivatives. For example: Yang [1] intro-
duced Dini set-valued directional derivatives for a vector
valued function in infinite dimensional vector spaces and
used this concept to establish the optimality conditions
for weakly efficient solution in vector optimization
problem under cone-convexity assumption; Ginchev [2]
obtained first-order necessary and sufficient optimality
conditions in terms of Dini set-valued directional deriva-
tives in finite dimensional linear spaces for locally
Lipschitz vector optimization.
It is well known that the concept of convexity and its
various generalizations play an important role in opera-
tions research and applied mathematics. A meaningful
generalized convex function was called the preinvex
functions, which was introduced by Weir and Mond [3]
and by Weir and Jeyakumer [4] in -dimensional
Euclidean space. Nowadays, this class of functions has
been extended to the abstract spaces and applied to es-
tablish optimality criteria and duality in vector optimiza-
tion [5-8]. Recently, Qiu [9] in normed linear spaces
considered a class of functions called generalized prein-
vex and established the unified optimality conditions for
set-valued vector optimization problems.
n
On the other hand, the (weakly) efficient solution is a
kind of extremely efficient solutions in vector optimiza-
tion. Since the range of the set of (weak) efficient solu-
tions is often too large, contracting the solution range is a
basic topic in vector optimization. For this purpose,
many kinds of proper efficiency have been presented.
Among them, an important proper efficiency is called
Henig proper efficiency, which was introduced by Henig
[10]. It is worthy to notice that the super efficiency, in-
troduced by Borwein [11], equals to the Henig efficiency
when the convex cone has a bounded base.
The aim of this paper is to deal with the optimality of
Henig proper efficient solutions for vector optimization
problems in terms of Dini set-valued directional deriva-
tives under the generalized preinvex assumptions.
2. Preliminaries
In this note, it is assumed that
X
and Y are two lo-
cally convex spaces with topological duals
X
and Y,
respectively. The partially order of Yis defined by a
closed convex cone C with Yint C

. On the
other hand, we assume that Y is a complete vector lattice,
i.e., sup


12
,
y
y exists for all 12 and every
bounded nonempty subset has an infimum and a supre-
,yy Y

G. L. YU ET AL.923


mum. The sets of minimal elements and maximal ele-
ments of are defined respectively by
KY
min
max
VK
VK

  

  

:
: ;
yKyCCy
yKyCCy
 
 
For a set
A
Y
cone
, we write

A
 
:0,aa A  .
The closure and interior of a set
A
are denoted by
and . Let and C be the dual cone
and strictly dual cone of convex cone , defined by

cl AAint
i
Cf
Cf


Ci
C

 
 

0 for all ,
:0 for all \0
fy yC
fy yC
 
 

*:Y
C

A nonempty convex subset of the convex cone
is called a base of , if and
B
Cc CC oneB 0clB

.
In this paper, it is always assumed that
B
is a base of
. Set
C

CB

{f


i
C

: There exists such that 0t
f
bt
0clB
, for all . }bB
inf
Since there exists such that

*\0Y
 

() :0rbbB  .
Let


:2
B
{UV
U
Vy
:U

U
C B
Ybr  
Define the neighborhood family of in as fol-
lows:
0YY
B is an open convex circled neighbor-
hood of zero in}.

For each , Let
 
cone BU
It has been pointed out in [5] that for each U

,
is a pointed convex cone in Y with
.

U
CB

\0

int U CC

B
Definition 2.1. (See [12]) Let
K
be a nonempty sub-
set of and let be a base of . 0 is said
to be a Henig proper efficient point of
YBC yK
K
with respect
to if there exists U such that B

0
Ky

int U CB

 
Now, let us recall the concepts of upper and lower
Dini set-valued directional derivatives given by Yang [1].
Definition2.2. (See [1]) Let :
f
XY be a vector
valued function and ,
x
dX
. The limiting set of
at
f
x
in the direction is defined as follows d
 
0
;:lim
ft
f
xtd fx
t

 


Yx d zz 
(2.1)
For our approach in this note, the following assump-
tion will be needed.
Assumption 2.1. (See [1]) The subset has a
minimal element and a maximal element.

;
f
Yxd

Definition 2.3. Let :
f
XY be a vector valued
function. Let ,
x
dX

be two points. The upper and
lower Dini-directional derivatives of at f
x
in the
direction are defined respectively by
d


 
max
min
;;
;;
f
f

f
xdVYxd
f
xdVY xd









(2.2)
Remark 2.1. It is obviously that




;;
f
xdfxd

.
In addition, it has been pointed out in Ref. [1] that if
Assumption 2.1 holds, then


;fxd;,fxd




,
and


;
f
xd
,


;
f
xd
 as functions of are posi-
tively homogeneous.
d
Definition 2.4. (See [9]) Let :
X
XX

 be a
map and




:0,10,   be a function such that


0
lim 0k




.
The set is called a generalized invex set with
respect to
SX

and

if for any ,
x
yS and any
[0,1]


,

 
,
y
xy S 

.
Suppose that is a generalized invex set with
respect to
SX

and

. A vector valued function
:
f
SY is called generalized-preinvex on with
respect to
CS

and

if for any ,
x
yS and any
[0,1]


,







1,
f
xfyfyxy   C

 
. (2.3)
Remark 2.2. It is clear that the concepts of general-
ized invex sets and generalized preinvex functions are
the generalizations of the invex sets and preinvex func-
tions which introduced by Weir [3,4]. In addition, the
function




:0,10,   in Definition 2.4 has prop-
erty

0
lim 0




.
In fact,
 
00
limlim0 0k



 





We need the next assumptions.
Assumption 2.2. (See [1]) Let be defined
as in (2.1). The domination property is said to hold for if

;
f
Yxd






  
min max
;; ;
ff f
YxdV Yxd CVYxd C

(2.4)

G. L. YU ET AL. 924
The following important property of Generalized
cone-preinvex functions will be used in the sequel.
Proposition 2.1. Let be a generalized invex
set with respect to
SX

and

and the vector valued
function :
f
SY
be Generalized C preinvex on
with respect to
S

and

. Then for any Syx

,,
 


;,
f
fyfxkY xxyC   . (2.5)
If, in addition, Assumption 2.2 holds, then
 


;,fyfxkf xxyC 

 . (2.6)
Proof: For any ,
x
yS and any


0, 1  , it fol-
lows from Definition 2.4 that
 


1,
f
yfxfxxy    C
.
This means that
  


,fxxyfx
fy fxC



 
.
Thus,
   




,fxxyfx
f
yfxC




 
.
Then, it yields from Remark 2.2 that
 


;,
f
f
yfxkYxxyC   
.
Furthermore, if follows from Assumption 2.2 and
positive homogeneous property of Dini set-valued direc-
tional derivatives that
   
min
;, ;,
ff
Yx xyVYx xyC 

.
Thus, we get
 


;,
f
yfxkfxxy C 

 .
3. Optimality Criteria
In this section, we apply the Dini set-valued directional
derivatives defined in the last section to characterize op-
timality conditions for a vector optimization problem
involving the generalized preinvex functions. We begin
by presenting the following vector optimization problem
,

VOP
 
min xS VOPCf x



where is a nonempty open subset of
S
X
and
:
f
XY.
Definition3.1. a) The point is said to be a Henig
proper efficient solution of


VOP with respect to if
there exists such that U

 
 
int .
U
fSfxC B

 
b) The pointis said to be a strong efficient solution of


VOP , if



,
f
xfxC xS

.
Theorem 3.1. Consider the vector optimization prob-
lem


VOP .
1) If
x
S

is a Henig proper efficient of


VOP
with respect to , then there exists such that
BU



 
;, int,
fU
Yx xxCBxS 


. (3.1)
In particular,





;, int,
U
f
xxxCB xS 



. (3.2)
2) Assume that is a generalized invex with respect
to
S

and

and
f
is generalized -preinvex on
with respect to
C
S

and

. If (3.1) holds, then
x
is
a Henig proper efficient solution of .

PVO
Proof: 1) If (3.1) does not hold, by the Definition 2.2,
then there exists ˆ
x
S

and small enough
ˆ0t
such that


ˆ
ˆˆ
x
txxS 

 and




 

ˆ
ˆˆ int U
f
xtxxfxC B  .
which contradicts to the assumption that
x
S

VOP is a He- nig proper efficient solution of problem . On the
other hand, It is obviously from (2.2) that the inequality
(3.2) holds.

2) Suppose that there exists such that U



 
;, int,
fU
Yx xxCBxS 

.
Then





;, int,
fU
kYxx xCBxS 

.
By Proposition 2.1, we get



 
;,
f
f
xfxkYxxx C 


Noticing that

 
\0int U CC
 
B, we get





 
;,
fU
f
xfxkYxxx CB   .
Hence



 
int ,
U
f
xfxCB xS 

 
This means that
x
is a Henig proper efficient solu-
tion of the problem


VOP .
Theorem 3.2. Consider the vector optimization Prob-
lem


VOP .
1) If
x
S

is a strong efficient solution of


VOP ,
then




;, ,
f
Yx xxCxS  . (3.3)
In particular,




;, ,
f
xxx CxS 
. (3.4)

G. L. YU ET AL. 925
[3] T. Weir and V. Jeyakumar, “A Class of Nonconvex Func-
tions and Mathematical Programming,” Bulletin of Aus-
tralian Mathematical Society, Vol. 38, No. 2, 1988, pp.
177-189. doi:10.1017/S0004972700027441
2) Assume that is generalized invex with respect to S

and

and
f
is generalized -preinvex on
with respect to
C S

and

. If (3.3) holds, then
x
is a
strong efficient solution of .

VOP [4] T. Weir and B. Mond, “Preinvex Functions in Multiple-
Objective Optimization,” Journal of Mathematical Analy-
sis and Applications, Vol. 136, No. 1, 1988, pp. 29-38.
doi:10.1016/0022-247X(88)90113-8
Proof: (1) We proceed by contradiction. Assuming
that the condition (3.3) is not satisfied, then there exists
ˆ
x
S such that


ˆ
;,
f
Yx xxC  . [5] L. Batista Dos Santos, R. Osuna-ómez, M. A.Rojas-Me-
dar and A. Rufián-Lizana, “Preinvex Functions and Weak
Efficient Solutions for Some Vectorial Optimization
Problem in Banach Spaces,” Computers and Mathematics
with Applications, Vol. 48, No. 5-6, 2004, pp. 885-895.
doi:10.1016/j.camwa.2003.05.013
Thus, there exists small enough such that
ˆ0t

ˆ
ˆˆ
x
txxS  .
and [6] A. J. V. Brandāo, M. A. Rojas-Medar and G. N. Silva,
“Optimality Conditions for Pareto Nonsmooth Noncon-
vex Programming in Banach Spaces,” Journal of Opti-
mization Theory and Applications, Vol. 103, No. 1, 1999,
pp. 65-73. doi:10.1023/A:1021769232224



ˆ
ˆˆ
f
xt xxfxC  ,
which contradicts to that
x
S

is a strong efficient
solution of . It is easy to obtain inequality (3.4)
from the definition of lower Dini directional derivative
for

VOP

f
.
[7] S. K. Mishra, G. Giorgi and S. Y. Wang, “Duality in
Vector Optimization in Banach Spaces with Generalized
Convexity,” Journal of Global Optimization, Vol. 29, pp.
2004, pp. 415-424. doi:10.1016/j.camwa.2007.05.002
2) Suppose that the condition (3.3) is satisfied, that is


;, ,
f
Yx xxCxS  . [8] G. L. Yu and S. Y. Liu, “Some Vector Optimization
Problems in Banach Spaces with Generalized Convex-
ity,” Computers and Mathematics with Applications, Vol.
54, No. 11-12, 2007, pp. 1403-1410.
doi:10.1016/j.camwa.2007.05.002
By Proposition 2.1,
 



;, ,
f
f
xfxkYxxx CCxS   
Hence [9] J. H. Qiu, “Cone-Directed Contingent Derivatives and
Generalized Preinvex Set-Valued Optimization,” Acta
Mathematica Scientia, Vol. 27, No. 1, 2007, pp. 211-218.
doi:10.1016/S0252-9602(07)60019-8
 
,
f
xfxCxS.
That is,
x
is a strong efficient solution of problem
.
)(VOP [10] M. I. Henig, “Proper Efficiency with Respect to Cones,”
Journal of Optimization T h eory and Application s, Vol. 36,
No. 3, 1982, pp. 387-407. doi:10.1007/BF00934353
4. References [11] J. M. Borwein and D. Zhuang, “Super Efficiency in Vec-
tor Optimization,” Transactions of the American Mathe-
matical Society, Vol. 338, No. 1, 1993, pp. 105-122.
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