SOME OPTIMALITY AND DUALITY
RESULTS FOR AN EFFICIENT
SOLUTION OF MULTIOBJECTIVE
NONLINEAR FRACTIONAL
PROGRAMMING PROBLEM
Paras Bhatnagar1
Department of Mathematics, Ideal Institute of Technology, Ghaziabad, U. P., India
Email: bhatnagarparas@gmail.com
Sanjeev Rajan2
Department of Mathematics, Hindu College, Moradabad, U.P., India
Ridhi Garg3
Department of Mathematics, Moradabad Institute of Technology Moradabad, UP India
Abstract
Kaul and Kaur [7] obtained necessary optimality conditions for a non-linear programming problem by taking the objective and constraint functions to be semilocally convex and their right differentials at a point to be lower semi-continuous. Suneja and Gupta [12] established the necessary optimality conditions without assuming the semilocal convexity of the objective and constraint functions but their right differentials at the optimal point to be convex.
Suneja and Gupta [13] established necessary optimality conditions for an efficient solution of a multiobjective non-linear programming problem by taking the right differentials of the objective functions and constraint functions at the efficient point to be convex. In this paper we obtain some results for a properly efficient solution of a multiobjective non-linear fractional programming problem involving semilocally convex and related functions by assuming generalized Slater type constraint qualification.
Key words: Nonlinear programming, Multi-objective fractional programming, Semilocally convex functions, Pseudoconvex functions.
1.Introduction
Consider the following Multi-objective fractional programming problem:
(MFP)
Minimize 1 21 2
( )
( )
( )
,
,...
( )
( )
( )
k
k
f x
f x
f x
g x
g x
g x
Subject to
h
j(
x
)
0
,
j
1
,
2
,...
.,
m
,x
S
Where
S
R
nis a locally star shaped set andf
i,
g
i:
S
R
,
i
1
,
2
,...
.,
k
and
h
j:
S
R
,
j
1
,
2
,...,
m
, are semidifferentiable functions.,
0
)
(
x
f
ig x
i( )
0,
i
1
,
2
,...
.,
k
for allx
S
.Let
X
0
x
S h x
j( )
0,
j
1, 2,...,
m
and( )
,
( )
i i
i
f x
t
g x
i
1
,
2
,...
.,
k
t
(MFP )
Minimize
(
f
1
t
1g
1)(
x
),
(
f
2
t
2g
2)(
x
),...
.,
(
f
k
t
kg
k)(
x
)
Subject toh
j(
x
)
0
,
j
1
,
2
,...
.,
m
,x
S
We now have the following lemma connecting the properly efficient solutions of
(MFP)
and(MFP )
t .Lemma:1 Let
x
*be a properly efficient solution of(MFP)
, then there existst
*
R
ksuch thatx
*is properlyefficient solution of
(MFP )
t* .Conversely if*
x
is a properly efficient solutionof
(MFP )
t* where* *
*
( )
t
( )
i ii
f x
g x
,i
1, 2,...,
k
thenx
* is a is properly efficient solution of(MFP)
.2. Optimality Conditions
Definition: The problem
(MFP )
t* is said to satisfy generalized Slater type constraint qualification at*
x
S
ifthe functions
f
i
t g
i* i,i
1, 2,...,
k
andh
j,j
I
are semilocally pseudoconvex atx
*and for each1, 2,...,
r
k
there existsx
r
S
such that *(
f
i
t g
i i)( )
x
r
0
,i
1, 2,...,
k
,i
r
and
h x
I( )
r
0
,whereI
I x
( )
*
i h x
i( )
*
0
andJ
J x
( )
*
j h x
j( )
*
0
Now we obtain necessary optimality conditions corresponding to
(MFP )
t* .Theorem: 1 Let
x
* be an efficient solution of(MFP)
and S be convex. Leth
jbe continuousat
x
*forj
I
andd f
(
i
t g
i* i) ( ,
x x
*
x
*)
,i
1, 2,...,
k
,(
dh
j) ( ,
x x
*
x
*)
be convex functions ofx
. If(MFP )
t* satisfies generalized Slater type constraint qualification at*
x
for* *
*
(
)
t
(
)
i ii
f x
g x
,i
1, 2,...,
k
then there exists
i*
0
,i
1, 2,...,
k
and
*j
0
,j
I
such that*
* * * * * * * * *
1 ( )
(
) ( ,
)
(
) ( ,
)
(
) ( ,
)
0
k
i i i i j j
i j I x
df
x x
x
t dg
x x
x
dh
x x
x
(1)for all
x
S
.Proof: We claim that the system
d f
(
it g
i* i) ( ,
x x
*x
*)
0
,i
1, 2,...,
k
(2)d h
( ) ( ,
j x x
*
x
*)
0,
j
I x
(
*)
(3) Has no solutionx
S
.If possible let
x
*be the solution of the system. By the relation (2) we have
* * * * *
0
(
)(
(
)) (
)( )
lim
f
it g
i ix
x
x
f
it g
i ix
0
(4)
Which implies that there exists
1
0
such that
(
f
i
t g
i* i)(
x
*
(
x
x
*)) (
f
i
t g
i* i)( )
x
*
0
for
0,
1
(5) by (5) ,(
f
i
t g
i* i)(
x
*
(
x
x
*))
(
f
i
t g
i* i)( )
x
* ,i
1, 2,...,
k
similarly by the relation (3), there exists
2
0
, such thatLet
*
min
1,
2,
j
, then for
*0,
,we have* * * * *
(
f
i
t g
i i)(
x
(
x
x
))
(
f
i
t g
i i)( )
x
,i
1, 2,...,
k
,* *
(
(
))
0
j
h x
x
x
,j
I x
( )
* and(
x
*
(
x
x
*))
X
.This implies that
x
*is not an efficient solution of(MFP )
t* .Hence,*
x
is not an efficient solution of(MFP)
, contrary to the given hypothesis. Thus, the system (2), (3) has no solution x in S. Also,d f
(
i
t g
i* i) ( ,
x x
*
x
*)
,i
1, 2,...,
k
,(
dh
j) ( ,
x x
*
x
*)
,j
I
are convex functions of x. By Mangasarian [10], there exists
i*
0
,i
1, 2,...,
k
,
*j
0
,j
I
not all zero such that (1) holds.We shall now show that
i*
0
, for eachi
1, 2,...,
k
. If possible let
r*
0
, for some r,1
r
k
. By generalized Slater type constraint qualification, there existsx
r
S
, such that* * *
(
f
i
t g
i i)( )
x
r
(
f
i
t g
i i)( )
x
,i
1, 2,...,
k
i
r
( )
0
j r
h x
,and
(
f
i
t g
i* i)
,i
1, 2,...,
k
andh
j,j
I
are semilocally pseudoconvex atx
*. Therefore it follows that
d f
(
it g
i* i) ( ,
x x
* rx
*)
0
,i
1, 2,...,
k
and
(
dh
j)
(
x
*,
x
r
x
*)
0
,j
I x
( )
*Since at least one of the coefficients
i*,i
1, 2,...,
k
,i
r
and
*j,j
I
is non zero, therefore we obtain*
* * * * * * * * *
1 ( )
(
) ( ,
)
(
) ( ,
)
(
) ( ,
)
0
k
i i r i i j j r
i j I x
df
x x
x
t dg
x x
x
dh
x x
x
Which contradicts (1) as
i*
0
.Thus
i*
0
, for alli
1, 2,...,
k
.Following Geoffrion [5] we consider the scalar problem corresponding to
(
MFP
t*)
in order to prove sufficientoptimality conditions.
)
MFP
(
t* Minimize
ki
i i i
i
f
x
t
g
x
1*
))
(
)
(
(
Subject to
h
j(
x
)
0
,j
1
,
2
,...,
m
We have the following results on the lines of Geoffrion in the light of Lemma1. Lemma: 2 If
x
*is an optimal solution of(
MFP
t*)
for somek
R
, with strictly positive componentswhere
)
(
)
(
* * *
x
g
x
f
t
i i
i
,then *x
is a properly efficient solution of(MFP)
.Theorem: 2 Suppose there exists a feasible
x
*for(MFP)
and scalars
i*
0
,i
1
,
2
,...,
k
,
*j
0
,I
j
such that for allx
X
0,where)
(
)
(
* * *
x
g
x
f
t
i i
i
,i
1
,
2
,...,
k
. Let the following conditions besatisfied:
(i)
f
i,
g
i,i
1
,
2
,...,
k
andh
j,j
I
are semilocally convex at *x
,(ii)
ki
i i i i
f
t
g
1* *
)
(
is semilocally pseudoconvex.Then
x
* is a properly efficient solution of(MFP)
.Then
) ( * * ) ( * * *)
(
)
(
x I j j j x I j jj
h
x
h
x
(6)(i) Since each
h
j,j
I
is semi locally convex at *x
, therefore it follows that
( ) * * *
)
(
x I j j jh
x
is semilocallyconvex. From (6) we obtain
) ( * * * *0
)
,
(
)
(
x I j jj
dh
x
x
x
(7)Using (1) and (7), we get
0
]
)
,
(
)
(
)
,
(
)
[(
1 * * * * * *
k i i i ii
df
x
x
x
t
dg
x
x
x
, for allx
X
0 (8)Now ,
f
i,
g
i,i
1
,
2
,...,
k
are semilocally convex at *x
, therefore
k i i i i if
t
g
1* *
)
(
is alsosemilocally convex at
x
*, and (8) gives)
)
(
)
(
(
))
(
)
(
(
1 * * * * 1 * *
k i i i i i k i i i ii
f
x
t
g
x
f
x
t
g
x
for allx
X
0 (9)Which implies that
x
*is an optimal solution of(
MFP
*)
*
t , where *
has strictly positive components. Bylemma 2, it follows that
x
*is properly efficient for(MFP)
.(ii) Using the semilocal convexity of
( ) * * x I j j j
h
atx
* in (6) we get (7) which when combined with (1) gives(8). Now
k i i i i if
t
g
1* *
)
(
is semilocally pseudoconvex, therefore, we get (9), which implies thatx
*isproperly efficient for
(MFP)
.In relation to
(MFP)
,we now associate the following multiple objective Schaible type dual program:(MFD)
Maximize
(t)
t
(
t
1,
t
2,...
t
k)
Subject to 1
[(
) ( ,
)
(
) ( ,
)
k
i i i i
i
df
u x u
t dg
u x u
1
[(
) ( ,
)]
0
m
j j j
dh
u x u
(10)for all
x
X
0where
(
(
)
(
))
0
1
k i i i ii
f
u
t
g
u
, (11)1
0
m j j jh (u)
, (12) andu
S
,
i
0
,
t
i
0
,
i
1
,
2
,...,
k
,u
j
0
,j
1
,
2
,...
m
. (13) We now prove some duality results.3. Weak Duality
Theorem: 3 Let x be feasible for
(MFP)
and
u
,
,
,
t
be feasible for(MFD)
. If the following holds: (i)f
i,
g
i,i
1
,
2
,...,
k
andh
j,j
I
are semilocally convex at u,(ii)
k i i i i if
t
g
1)
(
is semilocally pseudoconvex and 1 m j j jη
h
is semilocally convex at u, Thenf
(
x
)
is
not
t
Proof: If possible let
f
(
x
)
t
This gives i i i
t
x
g
x
f
)
(
)
(
and r r
r
t
x
g
x
f
)
(
)
(
, for some r.
Multiplying these inequalities by
i
0
,i
1
,
2
,...,
k
and adding we get
(
(
)
(
))
0
1
k
i
i i i
i
f
x
t
g
x
(14)Now x is feasible for
(MFP)
and
u
,
,
,
t
is feasible for(MFD)
therefore
m jj j m
j j
j
h
x
h
u
1 1)
(
)
(
Since
h
j,j
1
,
2
,...,
m
are semilocally convex and consequently 1 mj j j
η
h
is semilocally convex at u, therefore0
)
,
(
)
(
1
m
j
j
j
dh
u
x
u
Using (10), we obtain
0
)]
,
(
)
(
)
,
(
)
[(
1
k
i
i i i
i
df
u
x
u
t
dg
u
x
u
Now,
f
i,
g
i,i
1
,
2
,...,
k
are semilocally convex therefore
ki
i i i i
f
t
g
1)
(
is semilocally convex so weget
k
i
i i i
i
f
x
t
g
x
1))
(
)
(
(
(
(
)
(
))
0
1
k
i
i i i
i
f
u
t
g
u
This contradicts (14). Hence
f
(
x
)
is
not
t
.4. Strong Duality
Theorem: 4 Let
x
*be feasible for(MFP)
and
u
*,
*,
*,
t
*
be feasible for(MFD)
such that ** *
( )
( )
ii i
f x
t
g x
,i
1
,
2
,...,
k
. letf
i,
g
i,i
1
,
2
,...,
k
be strictly semilocally convex andj
h
,j
1
,
2
,...,
m
be semilocally convex atu
*,thenx
*
u
*.Proof: Let
x
*
u
*, nowx
*is feasible for(MFP)
and
u
*,
*,
*,
t
*
is feasible for(MFD)
, therefore* * * *
1 1
(
)
(
)
m m
j j j j
j j
h x
h u
Since
h
j,j
1
,
2
,...,
m
are semilocally convex and consequently * 1 mj j j
h
is semilocally convex at *u
,therefore* * * *
1
(
) ( ,
)
0
m
j j j
dh
u x
u
Using the above inequality in the relation (10) for dual feasibility of
u
*,
*,
*,
t
*
and forx
*
X
0, we get* * * * * * * *
1
[(
) ( ,
)
(
) ( ,
)] 0
k
i i i i
i
df
u x
u
t dg
u x
u
By the strict semilocal convexity of
f
iand
g
i,i
1
,
2
,...,
k
at ** * * * * *
1 1
(
)( )
(
)( )
k k
i i i i i i i i
i i
f
t g
x
f
t g
u
* * *
( )
( )
ii i
f x
t
g x
gives that* * * *
1
( ( )
( ))
0
k
i i i i i
f x
t g x
,therefore we get * * * 1
(
)( )
0
k
i i i i i
f
t g
u
,which contradicts the fact that
u
*,
*,
*,
t
*
is dual feasible. Hencex
*
u
*.Theorem: 5 Let
u
*,
*,
*,
t
*
be feasible for(MFD)
,f
i,
g
i,i
1, 2,...,
k
andh
j,j
1,2,....,
m
are semilocally convex atu
*. Suppose there exists a feasiblex
*for(MFP)
such that* * *
( )
( )
ii i
f x
t
g x
,i
1
,
2
,...,
k
(15)Then
x
*is properly efficient solution for(MFP)
. Also, if for each feasible
u
,
,
,
t
of(MFD)
, if
,
g
i,i
1
,
2
,...,
k
andh
j,j
1
,
2
,...,
m
are semilocally convex at u then
u
*,
*,
*,
t
*
is properly efficient solution for(MFD)
.Proof: Suppose that
x
*is not an efficient solution of(MFP)
, then there exists a feasible x for(MFP)
and an index r such that)
(
)
(
)
(
)
(
* *
x
g
x
f
x
g
x
f
i i
i i
,
i
1
,
2
,...,
k
,i
r
And
)
(
)
(
)
(
)
(
* *
x
g
x
f
x
g
x
f
r r
r r
Using (15), we get
( )
*( )
f x
t
g x
, which contradicts the weak duality. Therefore *x
must be an efficient solutionof
(MFP)
.If possible let
x
*be not properly efficient solution for(MFP)
, then by lemma 2.2.1, it follows thatx
*is not aproperly efficient solution of
(MFP
t*)
.Therefore for every
M
0
, there exists a feasiblex
for(MFP
*)
t and hence
(MFP)
and an indexi
such that( ( )
f x
i
t g x
i* i( ))
( ( )
f x
i *
t g x
i* i( ))
*And
M
x
g
t
x
f
x
g
t
x
f
x
g
t
x
f
x
g
t
x
f
j j j j
j j
i i i i
i
i
))
(
)
(
(
))
(
)
(
(
))
(
)
(
(
))
(
)
(
(
* *
* *
* *
* *
For all
j
satisfying(
f
j(
x
)
t
*jg
j(
x
))
(
f
j(
x
*)
t
*jg
j(
x
*))
That is
t g x
i* i( )
f x
i( )
M f x
(
j( )
t g x
*j j( ))
for all j satisfying *( )
( )
0
j j j
f x
t g x
Thus
t g x
i* i( )
f x
i( )
can be made large and hence for
i*
0
, we get the inequality* *
1
(
( )
( ))
0
k
i i i i i
t g x
f x
(16) Now proceeding on similar lines in Theorem 3, we get a contradiction to (16). Thereforex
*must be properlyThe above result can also be proved under the assumption that for each feasible
u
, , ,
t
,1
( ( )
( ))
k
i i i i i
f x
t g x
is semilocally pseudoconvex and * 1 mj j j
h
is semilocally quasiconvex.5. Conclusion: In this paper the problem under consideration is multiobjective non-linear fractional programming problem involving semilocally convex and related functions. We have discussed the interrelation between the solution sets involving properly efficient solutions of multiobjective fractional programming and corresponding scalar fractional programming problem. Necessary and sufficient optimality conditions are obtained for efficient and properly efficient solutions. Some duality results are established for multiobjective Shaible type dual.
References
[1] Bector, C. R. and Chandra, S.;”Multiobjective fractional programming: A parametric approach”, Research report faculty of
management, University of Monitoba,1987.
[2] Bector, C. R., Chandra, S. and Bector M.K.; “Sufficient optimality conditions for a quasiconvex programming problem”, Journal of optimization theory and applications,Vol.59, 202-221,1988.
[3] Bector, C. R., Chandra, S. and Bector M. K. ; “Generalized fractional programming duality: A parametric approach”, Journal of Journal of optimization theory and applications,Vol.60, 243-260,1989.
[4] Dinkelbech, W.; “On non-linear fractional programming”, Management Science,Vol.13, 492-498, 1967.
[5] Geoffrion, A. M.; “Proper efficiency and the theory of vector maximization”, Journal of Mathematical Analysis and Applications, Vol. 22,618-630,1968.
[6] Jagannathan, R.; “Duality for non-linear fractional programming”, Zeitschrift-
Fur
, Operations Research,Vol.17,1-3,1973.[7] Kaul, R. N. and Kaur, S.; “Generalized convex functions properties, optimality and duality”, Technical Report, SOL, Vol. 84, Standfort University, California, 1984.
[8] Kaul, R. N. and Lyall, V.; “A note on non-linear fractional vector maximization”, Operational Research, Vol. 26,108-121, 1989. [9] Kaul, R. N., Lyall, V. and Kaur, S.; “Semilocally pseudolinearity and efficiency”, European Journal of Operational Research , Vol.
36,402-409,1988.
[10] Mangasarian, O. L.; “Non-linear Programming”, Mc-Graw-Hill, New York,1969.
[11] Mangasarian, O. L.; “Non-linear fractional Programming”, Journal of Operational Research, Society of Japan, Vol. 12,1-10 1969. [12] Suneja, S. K. and Gupta, S.; “Duality in non-linear programming involving semilocally convex and related functions”, Journal of
Optimization, Vol. 28, 17-29, 1993.