DUALITY IN MULTIOBJECTIVE
FRACTIONAL PROGRAMING UNDER
GENERALIZED UNIVEX FUNCTION
*ARUN KUMAR TRIPATHY
Department of Mathematics, Trident Academy of Technology, F2/A, Chandaka Industrial Estate, Infocity Area, Patia, Bhubaneswar-751024, Odisha, India.†
Email address: [email protected]‡
GAYATRI DEVI
Department of Computer Science and Engineering, A.B.I.T., Cuttack, Odisha, India. Email address: gayatridevi13@yahoo,com.
AMERANDRA BARAL
Department of Mathematics, Trident Academy of Technology, F2/A, Chandaka Industrial Estate, Infocity Area, Patia, Bhubaneswar-751024, Odisha, India
Abstract:
In this paper we introduced three approaches of nondifferentiable multiobjective fractional primal and dual programming problem involving square roots of positive semi definite quadratic forms. Necessary and sufficient optimality conditions for this type of problem are established. The weak duality and strong duality theorems are proved under ∝-univexity assumption.
Keywords: Fractional programming; ∝-univexity function; Quadratic form; Schwartz Inequality; Efficient solution.
1. Introduction
A fractional programming problem arises in many types of optimization problem such as portfolio selection, production, information theory and numerous decision making problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical function or both; such as cost/time, cost/volume, cost/benefit, etc. in order to measure the efficiency or productivity of the system. In these types of problem, the objective function is usually given as a rational of function in fractional programming form. Many economic, noneconomic and indirect application of fractional programming problem have also been given by Craven [6] and Schaible and Ibaraki [22].
Duality in fractional programming is an important class of duality theory and several contribution have been made in the past[4,5,9,10,11,12,17,20 ,21,22,23]for its development .Recently , Liang et al [13] obtained some duality results for a nonlinear fractional programming problem by defining a new class of generalized convexity called (F,∝,ρ,d) convexity.
In this paper we introduced three approach given by Dinkelbaih[7] and Jagannathan [9] for both primal and dual of a non differentiable multiobjective fractional programming problem in which the numerator of objective function contains square root of positive semi definite quadratic form. Also we established the necessary and sufficient optimality condition and used a parameterizations technique to established duality results under generalized ∝-univexity assumption..
2. Notation and Preliminaries
Let
R
nbe the n-dimensional Euclidean space andR
nbe its non-negative orthant .The following conventions for inequality will be used in this paper. For anyx
( ,
x x
1 2,...,
x
n),
y
( ,
y y
1 2,..,
y
n),
we denote (i)x
y
x
i
y
i, for alli
1, 2,.., .
n
(ii)
x
y
x
iy
i andx
y
.
Throughout the paper let X be a non empty open subset ofR
n.:
pf X
R
,g X
:
R
p, :
h X
R
m.
Consider the following problem;
Multiobjective fractional primal problem MFP0:
1 2
1 2
( ) (
)
min
( ),
( ),...,
( )
( )
T
p
f x
x Bx
K x K x
K
x
g x
where
1 2
( ) (
)
( )
( )
T
i i
i
i
f x
x B x
K x
g x
,i=1,2,….p
MFP1:
1 2
min
F x
( )
F x F x
( ),
( ),...,
F x
p( )
where
1 2
( )
( ) (
T)
( ),
1, 2,..,
i i i i i
F x
f x
x B x
g x i
p
,
i are fixedparameter.MFP
: Min
F x
( )
,
is p-dimensional strictly positive vector.
All with subject to same constraint
h x
( )
0,
x
X
R
n (2.1) LetX
0
{
x
X h x
: ( )
0}
be the feasible solution of MFP0, MFP1 and MFP.
In the following definition let
f X
:
R
p,
:
X
X
R
n ,
:
X
X
R
\ {0}.
Assume that:
R
R
satisfyingu
0
( )
u
0
and
(
u
)
( )
u
,K X
:
X
R
. For,x x
,
X
Write0
( , )
lim ( , , )
0
K x x
b x x
.Definition 2.1(Mishra et al [18, 19]: f is ∝-univex with respect to ∝,Ψ,K at
x
,if for allx
X
,
K x x
( , ) [ ( )
f x
f x
( )]
( , )
x x
f x
( ), ( , )
x x
.Definition 2.2(Mishra et al [18, 19]: f is ∝-pseudounivex with respect to ∝Ψ,K at
x
,if for allx
X
,
( , )
x x
f x
( ), ( , )
x x
0
K x x
( , ) [ ( )
f x
f x
( )]
0
.Definition 2.3(Mishra et al [18, 19]: f is ∝-quasiunivex with respect to ∝,Ψ,K at
x
,if for allx
X
,
K x x
( , ) [ ( )
f x
f x
( )]
0
( , )
x x
f x
( ), ( , )
x x
0
.Definition 2.5(Geoffrion [8]) A feasible point
x
is said to be properly efficient for (MFP1), if it is efficient and there existM
0
such that for eachi
{1, 2,.. }
p
and for all feasible point x in (MFP1) satisfyingF x
i( )
F x
i( )
, we haveF x
i( )
F x
i( )
M F x
(
r( )
F x
r( ))
for somer
such that( )
( ).
r r
F x
F x
We assume that
1 2
( ) (
T)
0,
( )
0,
1, 2,..
i i i
f x
x B x
g x
i
p
for allx
X
,
Lemma 2.1
(Generalized Schwartz inequality)
Let B be a positive semi definite matrix of
order n. Then, for all ,
x w
R
n,
x Bw
T
(
x Bx
T) (
21w Bw
T) .
21The equality holds if
Bx
Bw
for some
0.
Let
J x
( )
{
j
M
{1, 2,.. }:
m
h x
j( )
0}
and(
)
{
n:
T j(
)
0,
(
)}
W x
w
R
w
h x
j
J x
satisfying any one of the following condition;(a)
x Bx
T
0
12 0
(
)
(
)
0
(
)
T
T
Bx
w
f x
v
g x
x Bx
(b)
x Bx
T
0
.
120
(
)
(
)
(
)
0
T T
w
f x
v
g x
w Bw
Lemma 2.2
(a) If
x
0 is an optimal solution of MFP, thenx
0 is properly efficient for MFP0 . (b) If( ) (
T)
12i i
f x
x B x
andg x
i( )
are ∝-univex (i=1, 2, . . . p) andx
0 is properly efficient for MFP1 ,thenit is optimal for MFP .
Theorem 2.1.
The
x
0
X
0 is an efficient solution for MFP0 iff it is an efficient solution of MFP1 with F (x
0) =0.Proof: Suppose
x
0
X
0 is an efficient solution of MFP0 .If we choose1 2
0 0 0
0
(
) (
)
(
)
T
i i
i
i
f x
x B x
v
g x
,then
(
0)
(
0) (
0T)
12(
0)
0
i i i i i
F x
f x
x B
v g x
. Since forx
X
,( ) (
T)
120,
0
i i i
f x
x B x
g
for each i and eachv
iis assumed to be non negative .So1 2
( )
( ) (
T)
( )
0,
i i i i i
F x
f x
x B x
v g x
for allx
X
,i
1, 2,... .
p
Therefore min F(x) cannot be smaller (component wise ) than zero vector (0,0,…,0).But (0,0,..,0) is attend at
0
x
if1 2
0 0 0
0
(
) (
)
(
)
T
i i
i
i
f x
x B x
v
g x
,i
1, 2,... .
p
Sox
0
X
0 is an efficient solution of MFP1.Conversely, suppose
x
0is an efficient solution of MFP1 withF x
(
0)
0.
Ifx
0 is not an efficient solution of MFP0, then there existx
X
such that1 1
2 2
0 0 0
0
(
) (
)
( ) (
)
(
)
( )
T T
i i i i
i
i i
f x
x B x
f x
x B x
v
g x
g x
Therefore
( )
( ) (
T)
12( )
0,
i i i i i
F x
f x
x B x
v g x
this contradicts toF x
(
0)
0
andx
0is efficient value of MFP1.Hencex
0is efficient for MFP0,Theorem2.2(Necessary optimal Condition)
If
x
X
is an optimal solution of (MFP1) such thatW x
( )
, then there existv
0
R w
,
R
nandm
( )
T( )
F x
u
h x
1 1
[
( )
(
( ))]
( )
0,
p m
T
i i i i i i i
i i
f x
B w v
g x
y
h x
(2.3.1)1 2
0
( )
( ) (
T)
( ( )
0,
i i i i
F x
f x
x B x
v g x
(2.3.2)( )
0,
T
y h x
(2.3.3)1,
T
w Bw
(2.3.4)1 2
(
x Bx
T)
x Bw
T,
(2.3.5)0,
y
(2.3.6)0.
i
v
i
1, 2,.., .
p
(2.3.7)Theorem 2.3 (Sufficient Optimality condition) Let
x
X
0be a feasible solution of MFP1 and there exist0
,
n,
i
R w
R v
R
andy
R
msatisfying the condition in theorem 2.2 atx
.
Further more suppose that any one of the condition (a) or (b) holds.(a) 1 2 1 1
( )
[ ( ) (
)
( )]
( )
p m Ti i i i j j
i j
P x
f x
x Bx
v g x
y h x
is ∝-univex with respect to ∝, ,Ψ and K at0
x
X
(b)1 2 1
( )
[ ( ) (
)
( )]
p Ti i i i
i
Q x
f x
x Bx
v g x
and1
( )
( )
m j j j
H x
y h x
are ∝-univex withrespect to ∝, ,Ψ and K at
x
X
0. Thenx
is an efficient solution of MFP1. \Proof: If the hypothesis (a) holds, then from the ∝-univexity of
P x
( )
with respect to
, ,
and K at0
x
X
, we haveK x x
( , ) ( ( )
P x
P x
( ))
( , )
x x
P x
( ), ( , )
x x
But (2.3.1)
P x
( )
0
So (2.1) gives
K x x
( , ) ( ( )
P x
P x
( ))
0
and using the properties ofK
,
, we getP x
( )
P x
( )
1 1
2 2
1 1 1 1
[ ( ) (
)
( )]
( )
[ ( ) (
)
( )]
( )
p m p m
T T
i i i i i j j i i i i i j j
i j i j
f x
x B x
v g x
y h x
f x
x B x
v g x
y h x
(2.4.1) Suppose
x
is not efficient an efficient solution of MFP1, then there existx
X
0such that1 1
2 2
( ) (
T)
( )
( ) (
T)
( )
i i i i i i i i
f x
x B x
v g x
f x
x B x
v g x
,i
1, 2,.., .
p
and
1 1
2 2
( ) (
T)
( )
( ) (
T)
( )
t t t t t t t t
f x
x B x
v g x
f x
x B x
v g x
,for somet
{1, 2,.., }
p
The above relation together with the relation
i
0
, imply that1 1 2 2 1 1
[ ( ) (
)
( )]
[ ( ) (
)
( )]
p p T Ti i i i i i i i i i
i i
f x
x B x
v g x
f x
x B x
v g x
(2.4.2)From the relation (2.1), (2.3.3) and (2.3.6), we get
1 1
( )
( )
m m
j j j j
j j
y h x
y h x
(2.4.3) Consequently (2.4.2) and (2.4.3) yields1 1
2 2
1 1 1 1
[ ( ) (
)
( )]
( )
[ ( ) (
)
( )]
( )
p m p m
T T
i i i i i j j i i i i i j j
i j i j
f x
x B x
v g x
y h x
f x
x B x
v g x
y h x
This contradicts (2.4.1).
Hence
x
is an efficient solution for MFP1. If hypothesis (b) holds, the ∝-univexity of1
( )
m j j j
y h x
( , ) (
( )
( ))
( , )
( ), ( , )
K x x
H x
H x
x x
H x
x x
(2.4.4) From (2.4.3), we have( )
( )
(
( )
( ))
0.
H x
H x
H x
H x
So (2.4.4) imply that
1
( , )
( ), ( , )
0
( )
0
( )
0
m j j j
x x
H x
x x
H x
y h x
(2.4.5)From (2.3.1) and (2.4.5) , we get
1
[
( )
( )]
0.
p
i i i i i
i
f x
B w v g x
So for
( , )
x x
0
1[
( )
( )] ( , )
0.
p
i i i i i
i
f x
B w v g x
x x
Since
Q x
( )
is ∝-univex w.r.to
, ,
and K, we obtained( , ) [ ( )
( )]
( , )
( ), ( ,
0
K x x
Q x
Q x
x x
Q x
x x
Using the property of
K
,
and
,we get( )
( )
0
( )
( )
Q x
Q x
Q x
Q x
(2.4.6) Ifx
were not an efficient solution to MFP1 , then from (2.4.2) ,we have1 1
2 2
1 1
[ ( ) (
)
( )]
[ ( ) (
)
( )]
p p
T T
i i i i i i i i i i
i i
f x
x B x
v g x
f x
x B x
v g x
.This contradicts (2.4.6).Therefore
x
is an efficient solution for MFP1.3. Multiobjective Fractional Dual
MFD0:
1 2
( )
max
( ),
( ),..,
( )
( )
T
p
f u
u Bw
K u K u
K u
g u
Where
( )
( )
,
1, 2,.,
( )
T
i i
i
i
f u
u B w
K u
i
p
g u
MFD1:
max
F u
( )
F u F u
1( ),
2( ),..,
F u
p( )
Where
F u
i( )
f u
i( )
u B w v g u
T i
i i( )
MFD:max
F u
( )
All with subject to same constraint
[
( )
T( )]
0,
F u
y h u
(3.1)( )
T( )
0,
i i i i
f u
u B w v g u
fori
1, 2,..,
p
(3.2)( )
0,
,
T m
y h u
y
R
(3.3)1,
T i
w B w
i
1, 2,..,
p
(3.4)0,
i0.
u
v
(3.5)Theorem 3.1 (Weak Duality): Let
x
be a feasible solution for the primal and( , , , )
u y v w
be feasible for dual. LetF
i(.)
f
i(.) (.)
TB w vg
i
i(.)
is ∝-pseudounivex with respect to , Ψ, K and fory
R
m, the functiony h
T is ∝-quasiunivex with respect to , Ψ, K, then
Inf
(
F x
( ))
Sup
(
F u
( )).
Proof: Now from the primal and dual constraint, we have
h x
( )
0
andy h u
T( )
0.
So( )
( )
0
T T
( , ) [
x u
y h u
T( )], ( , )
x u
0.
( , ) [
x u
y h u
T( )]
T( , )
x u
0
. (3.1.2) Again from the dual constraint (3.1), we have
[
F u
( )
y h u
T( )]
0.
Since
( , )
x u
R
n,
0
we have
( , ) [
x u
F u
( )
y h u
T( )]
T( , )
x u
0
( , ) (
x u
F u
( ))
T( , )
x u
( , ) (
x u
y h u
T( ))
T( , )
x u
0
Using (3.1.2) ,we get
( , ) (
x u
F u
( ))
T( , )
x u
( , ) (
x u
y h u
T( ))
T( , )
x u
0
1
( , ) [
{ ( )
( )}]
( , )
0.
p
T T
i i i i i
i
x u
f u
u B w v g u
x u
(3.1.3)Since
F u
i( )
is ∝-pseudounivex with respect to , Ψ, K ,by definition and (3.1.3) ,we get1
( ( )
( ))
( , ) {
}
( , )
0.
( ( )
( ))
T p
i i i i T
i T
i i i i i
f x
x B w v g x
K x u
x u
f u
u B w v g u
Using the property of Ψ, K ,we get
1
[ ( )
( )]
p
T
i i i i i
i
f x
x B w v g x
1
[ ( )
( )]
p
T
i i i i i
i
f u
u B w v g u
(3.1.4)Now by Schwartz Inequality and (3.4), we have
1 1 1 2 2 2
(
) (
)
(
)
T T T T
x Bw
x Bx
w Bw
x Bx
(3.1.5)So both (3.1.4) and (3.1.5) implies 12
1 1
[ ( ) (
)
( )]
[ ( )
( )]
p p
T T
i i i i i i i i i i
i i
f x
x B x
v g x
f u
u B w v g u
Inf
(
F x
( ))
Sup
(
F u
( )).
Theorem 3.2 Let
x
be properly efficient solution of MFP0 and a constraint qualification (Mangasarian [15]) is satisfied .Then there exists a feasible solution( , , , )
u y v w
for dual and corresponding objective values are equal to zero. Further if( , , , )
u y v w
is feasible for dual,F u
i( )
is ∝-pseudounivex andy h
T is ∝-quasiunivex then(
x
u y v w
, , , )
is properly efficient for MFD0.Proof: Since
x
is a properly efficient solution of MFP0, it is optimal for of MFP by lemma 2.2 and theorem2.1 . Then by theorem 2.2, we have
( )
T( )
0,
i
F x
y
h x
( )
0,
i
F x
( )
0,
T
y h x
1,
T
w Bw
1 2
(
x Bx
T)
x Bw
T,
0,
u
0
0.
v
which are nothing but the dual constraints . So( , , , )
x y v w
is feasible for dual.So the objective values of MFP and MFD are equal. It follows from theorem 3.1 and for any feasible solution
( , , , )
u y v w
of dual
F u
( )
F x
( )
.So( , , , )
x y v w
is optimal solution of MFD .Then applying lemma2.2 and theorem 2.1 , we conclude that( , , , )
x y v w
is properly efficient for MFD0 .References
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