American Journal of Oper ations Research, 2011, 1, 243-248
doi:10.4236/ajor.2011.14028 Published Online December 2011 (http://www.SciRP.org/journal/ajor)
243
Optimality for Multi-Objective Programming Inv olvi ng
Arcwise Connected d-Type-I Functions*
Guolin Yu, Min Wang
Research Institute of Information and System Computation Science, The North University fo r Ethn ics,
Yinchuan, China
Received April 26, 2011; revised May 27, 2011; accepted June 11, 2011
Abstract
This paper deals with the optimality conditions and dual theory of multi-objective programming problems
involving generalized convexity. New classes of generalized type-I functions are introduced for arcwise
connected functions, and examples are given to show the existence of these functions. By utilizing the new
concepts, several sufficient optimality conditions and Mond-Weir type duality results are proposed for
non-differentiable multi-objective programming problem.
Keywords: Multi-Objective Programming, Pareto Efficient Solution, Arcwise Connected d-Type-I
Functions, Optimality Conditions, Duality
1. Introduction
Investigation on sufficiency and duality has been one of
the most attraction topics in the theory of multi-objectiv e
problems. It is well known that the concept of convexity
and its various generalizations play an important role in
deriving sufficient optimality conditions and duality re-
sults for multi-objective programming problems. The
concept of type-I functions was first introduced by Han-
son and Mond [1] as a generalization of co nvexity. With
and without differentiability, the type-I functions were
extended to several classes of generalized type-I func-
tions by many researchers, and sufficient optimality cri-
teria and duality results are established for multi-objec-
tive programming (vector optimization) problems in-
volving these functions (see [1-12]). Another meaningful
generalization of convex functions is the introduction of
arcwise connected functions, which was given by Avriel
and Zang [13]. Singh [14] and Mukherjee and Yadav [15]
discussed some properties of arcwise connected sets and
functions. Bhatia and Mehara [16] investigated some
properties of arcwise connected functions in terms of
their directional derivatives and established optimality
conditions for scalar-valued nonlinear programming
prblems involving these functions. Mehar and Bhatia [17]
and Davar and Mehra [18] studied optimality conditions
and duality results for minmax problems and fractional
programming problems involving arcwise connected and
generalized arcwise connected functions, respectively.
In this paper, we first introduce new classes of gener-
alized convex type-I functions by relaxing definitions of
arcwise connected function and type-I function. We pre-
sent some sufficient optimality cond itions and dual theo-
rems for non-differential multi-objective programming
problem under various generalized convex type-I func-
tions assumptions. This paper is divided into four sec-
tions. Section 2 recalls some definitions and related re-
sults which will be used in later sections, and introduces
new classes generalized convex type-I functions. In Sec-
tion 3 and Section 4, the sufficient optimality conditions
and Mond-Weir type duality results are established for
non-differential multi-objective programming problem
involving these generalized convex functions, respec-
tively.
2. Preliminaries
In this section, we first recall some concepts and results
related arcwise connected functions. Let be the n-
dimensional Euclidean space and be the set of all
real numbers. Throughout this paper, the following con-
vention for vectors in will be followed:
n
R
1
R
n
R
*This research is supported by Zizhu Science Foundation of Beifang
University of Nationalities; Natural Science Foundation for the Youth
(No. 10901004).
x
y

if and only if ii
x
y, , 1, 2,,in

G. L. YU ET AL.
244
x
y if and only if ii
x
y

，, 1, 2,,in
x
y if and only if ii
x
y1, , , but 2, ,in
x
y

,
x
y≮ is the negation of
x
y
n
.
Definition 2.1. (See [15]) A subset
X
R is said
to be an arcwise connected (AC) set, if for every
x
X

,
, there exists a continuous vector-valued functions
uX


,xu :0,1
H
X
,xu
, called an arc, such that

0
H
x,

1
,xu
H
u.
Definition 2.2. (See [15]) Let
f
be a real-valued
function defined on an AC set n
X
R. Then
f
is
said to be an arcwise connected function (CN) if, for
every
x
X, , there exists an arc uX,
x
u
H
such
that
 
)1

,xu
f
Hfxfu   , for 01



Definition 2.3. (See [13,14]) Let n
X
R be an AC
set, and Let f be a real-valued function defined on .
For any X
x
X
,
, , the directional derivative of f
with respect to uX
x
u
H
at 0



is defined as





,
0
mli xu
f
Hf 

x
 ,
provided the limit exists and is denoted by
. If
 
,0 xu
fH



,xu
0
mli
H
x





,0 xu
H


exists and it is denoted by , then vector
is called directional derivative of

0
,xu
H,
x
u
H
at
0

.
Consider the following multiobjective programming
problem:


0,
fx(MP) min
s.t. ,
g
xxX
where :m
f
XR , :p
g
XR, X is a nonempty
open AC set of . Let F denote the feasible solutions
of (MP) assumed to be nonempty, that is
n
R


ngx

:0xRF.
Definition 2.4. A point
x
X

is said to be a Pareto
efficient solution of problem (MP), if



f
xfx for
all
x
X.
Definition 2.5. A point
x
X

is said to be a weak
Pareto efficient solution of problem (MP), if
 
f
xfx for all
x
X.
Along the lines of [1,5], we now define the following
classes of functions.
Definition 2. 6.

,
ij
f
g, 1, 2,, im

 and
1,2, ,jp, is said to be CN-d-type-I with respect
to *,
x
x
H, at *
x
X

if there exist an arc


*,:0,1
xx
H
X such that for all
x
X,





*
*
,0
ii i
xx
fxfxf H


and




*
*
,0
jj
xx
gx gH


Definition 2. 7.


,
ij
f
g, and 1,2, ,im
1, 2,,jp

, is said to be quasi-CN-d-type-I with re-
spect to *,
x
x
H at *
x
X

if there exist an arc


*,:0,1
xx
H
X such that for all
x
X,





*
*
,00
ii i
xx
fxfxfH


,
and




*
*
,
00
jj
xx
gx gH
0

 .
Definition 2. 8.


,
ij
f
g, and 1,2, ,im
pj ,,2,1 

, is said to be pseudo-CN-d-type-I with re-
spect to at if there exist an arc
xx
H,
*Xx 
*


*,:0,1
xx
H
X such that for all , Xx





**
,00
ii
xx i
f
Hfx 
fx
,
and



 **
,00
jj
xx
gH gx
 0.
Definition 2. 9.


,fg
ij
, and 1, 2,,im
pj ,,2,1 

, is said to be quasipseudo-CN-d-type-I
with respect to at if there exist an arc
xx
H,
*Xx 
*


*,:0,1
xx
H
X such that for all , Xx




 *
*
,00
ii i
xx
fxfxfH


,
and



 **
,00
jj
xx
gH gx
 0.
Definition 2.10.


,fg
ij
, and 1, 2,,im
1, 2,,jp

, is said to be pseudoquasi-CN-d-type-I
with respect to at
xx
H,
*
*
x
X if there exist an arc


*,:0,1
xx
H
X such that for all , Xx




 **
,00
ii
xx
fHfx fx
 
i
,
and




*
*
,
00
jj
xx
gx gH
0

 .
To show the existence of the CN-d-type-I functions

245 G. L. YU ET AL.
we give the following exam ple:
Example 2.1. Define a set 2
X
R as


22
12 1212
,:1, 0,0Xxxxxx x
.
Then x is an AC set with respect to


,:0,1 xy
H
X
given by
 




12 12
22 22
,112
1,1
xy
Hxyx     
2
y

12
,
x
xx X,



12
,0yyy X  ,1.
Define , :fX R:
g
XR as

22
12 12
,if 1 and 1
0, otherwise,
xx xx
fx  


2
21 2
,if 1 and 1
0, otherwise,
xx x
gx  

*

f and g are not differentiable at

1,1
x
X

. How-
ever, and


*,0
xx
fH



*,0
xx
gH
 existed, and
they are given by


*
*
,
22
12 ,
0
,if both components of 1
0, otherwise,
xx
xx
fH
xx H








*
*
2
2,
,
,if both components of
00, otherwise,
xx
xx
x
gH 





H

1

where

12
,
x
xx. It is obviously that

,

f
g is CN-d-
type-I at .

*1,1x
3. Sufficient Optimality Conditions
In this section, we establish sufficient optimality condi-
tions for a feasible solution
x
to be a weak Pareto effi-
cient solution for (MP) under the CN-d-type-I and pseu-
doquasi-CN-d-type- I assumptions.
Theorem 3.1. Suppose that there exist a feasible solu-
tion xF and

1,,
P

 , 0

 such that
 
,00
T
xx
fHg x 

X (3.1)

0
jj
gx  , 1, 2,,jp

 (3.2)
If

,
ij

f
g is CN-d-type-I with respect to ,
x
x
H
at
x
, then
x
is a weak Pareto efficient solution for (MP).
Proof Suppose that
x
is not a weak Pareto efficient
solution of (MP). Then th ere is a feasible solution
x
of
(MP) such that
() ()
ii
f
xfx for any . 1, 2,,im
By CN-d-type-I assumption on, we get
i
f






,
00
ii ixx
fxfxf H

 for any . 1, 2,, im
Thus,




,0 ixx
fH
0

for any . 1,2,,im
So, we have


,
1
0
m
ixx
i
fH



0. (3.3)
It yields from (3.1) that


,
10
p
jj xx
j
gH 



0. (3.4)
from CN-d-type-I assumption on
j
g
, we get






,0 jjxx
gx gH

 , for any . 1, 2,,jp
Since

0 and Equation (3.2) holds, we can get




,
00
jjjj xx
gx gH 

 , for any . 1, 2,,jp
Therefore,


,
100
p
jj xx
j
gH 



,
which contradicts to (3.4).
Theorem 3.2. Suppose that there exist a feasible solu-
tion x

F and


1,,
P

 , 0

 such that (3.1)
and (3.2) hold. If


,
ijj
f
g  is pseudoquasi-CN-d-type-
I with respect to ,
x
x
H
at
x
, then
x
is a weak Pareto
efficient solution for (MP).
Proof Since (3.2) holds and xF, by pseudoquasi-
CN-d-type-I hypothesis on
j
j
g

at
x
, for all
x
X

we have



*,00
jj xx
gH 
.
Thus



*,
10
p
jj xx
j
gH 




x
X. (3.5) 0 for all
Let
x
not be a weak Pareto efficient solution for
(MP). Then there is a feasible solution ˆ
x
for (MP) such
that




ˆ
ii
f
xfx for any . 1, 2,,im
from pseudoquasi-CN-d-type-I hypothesis on i
f
at
x
,
it yields




ˆ ,0 ixx
fH
0

for any . 1,2,,im
so,
 
ˆ
,
1
0
m
ixx
i
fH



0. (3.6)

G. L. YU ET AL.
246
Combing (3.5) and (3.6), we get




ˆˆ
,,
11
00
p
m
ixx jjxx
ij
fH gH 



 0.
But this is a contradiction to (3.1). The proof is com-
pleted.
Remark 3.1. For the functions ()
f
x and ()
g
x in
Example 2.1, we consider the programming problem
(MP). Let 1

, we are easy to get that
   
ˆˆ ,,
000
T
xx xx
f
HgH x 

X
,
which implies that

1,1x is an optimal solution (is
also weak Pareto efficient) of (MP).
4. Duality Results
Now, in relation to (MP) we consider the following
Mond and Weir type dual under the CN-d-type-I and
generalized CN-d-type-I assumptions.
  
   
12
,,
(DMP) max,,,
s.t. 000 for all
m
TT
yx yx
fyfy f yfy
fH gHy 




F 
(4.1)

10
p
jj
j
gy 


, (4.2)
0

, , (4.3)

12
,,, T
m
 

0

, . (4.4)

12
,,, T
p
  

Theorem 4.1. (Weak Duality) Let
x
and


,,y


be feasible soltuions for (MP) and (DMP), respectively.
Moreover, we assume that any one of the following con-
ditions holds:
a)

,
ij

f
g is CN-d-type-I with respect to ,
y
x
H
at
;
yb)

,
ijj

f
g 
,
is pseudoquasi-CN-d-type-I with re-
spect to
y
x
H
at y.
Then
 
f
xfy. (4.5)
Proof Since

,,y


 is feasible solution for (DMP),
we have




,,
000
TT
yx yx
fH gHxX 



0
(4.6)
and (4.2) holds. We proceed by contradiction. Suppose
that



f
xfy.
Then, there exists an index such that k


kk

f
xfy,




ii
f
xfy for all i. k
Since 0
i

, `1,2 ,,im

, we get




kk kk
f
xf  y,


ii ii

f
xf  y for all ik

.
Thus,
 
11
mm
ii ii
ii
f
xf 



y. (4.7)
by condition (a), we get





,0 iiiyx
fyfxf H



and



,0 jjyx
gy gH





0
.
Therefore, we can get
  
,
11 10
mm m
iiiiiiyx
ii i
fxfyf H  

 

 (4.8)
and
 

,
11
0
pp
jjjjyx
jj
gy gH 



 . (4.9)
Combing (4.2), (4.7) , (4.8) and (4.9 ), we get


,
100
m
ii yx
i
fH 



,
and


,
100
p
jj yx
j
gH 



.
so




,,
11
00
p m
i iyxijyx
ij
fH gH 




 (4.10)
which contradicts (4.6).
By condition (b), noticing that (4.2) holds, with the
similar argument as that of Theorem 3.2, we can get


,
1
00
m
ii yx
i
fH 



,
and
 
,
100
p
jj yx
j
gH 



.
The above two inequalities imply (4.10), again a con-
tradiction to (4.6). This completes the proof.
Theorem 4.2. Suppose that there exist feasible solu-
tions
x
and


,,y

for (MP) and (DMP), respec-
tively, such that

247 G. L. YU ET AL.
 
ii
f
xfy, . (4.11) 1, 2,,im
Moreover, we assume that the hypotheses of Theorem
4.1 hold at
y
, then
x
is a Pareto efficient solution for
(MP).
Proof For any feasible solution
x
for (MP), we get
from Theorem 4.1 that



f
xfy.
Suppose that
x
is not a Pareto efficient solution for
(MP). Then, there exist a feasible solution ˆ
x
for (MP)
and an index k such that
 
ˆ
kk
f
xfx,
 
ˆ
kk
f
xfx for all ik

.
Using condition (4.11), we get
 
ˆ
kk
f
xfy,
 
ˆ
kk
f
xfy for all ik

.
This contradicts to Theorem 4.1.
Theorem 4.3. (Converse Duality) Let


,,y


y
be a
Pareto efficient solution for (DMP). Moreover, we as-
sume that the hypotheses of Theorem 4.1 hold at, then
is a Pareto efficient solution for (MP). yProof We proceed by contradiction. Suppose that
is not a Pareto efficient solution for (MP), that is, there
exist and an index k such that
y
xF
 
kk
f
xfy,
 
ii
f
xfy for all ik

.
If any one of the hypotheses of Theorem 4.1 holds, it
yields in light of Theorem 4.1 that (4.5) is satisfied. This
leads to the similar contradiction as in the proof of
Theorem 4.1.
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