On Duality with Support Functions for a Multiobjective Fractional
Programming Problem
Indira P. Debnath and S. K. Gupta
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247 667, India
Keywords:
Nonlinear Multiobjective Programming, Higher Order (K × Q)- F-type I, Support Functions, Efficient
Solutions.
Abstract:
In this article, a different class of function called (K × Q)-F-type I has been introduced. Further, we have
formulated a problem over cones and appropriate duality results have been established taking the concerned
functions to be (K×Q)- F-type I. The results which we have put forward in the paper generalizes some of the
known results appeared in the literature.
1 INTRODUCTION
The mathematical programming problems involving
ratio of two functions in the objective function is
called fractional programming problems. Many types
of optimization problems involves this fascinating
subject. Fractional programming problem emerges in
several types of optimization problems (as for exam-
ple, portfolio selection, production, information the-
ory and numerous decision making problems in man-
agement science). It has also been extensively used in
business and economics situation. Further, these type
of problems are also useful in engineering and eco-
nomics where it is often used to maximize a fractional
function for measuring the efficiency or productivity
of a system. (Bector and Chandra, 1987), (Bector
et al., 1993), (Schaible, 1995) etc. have given some
applications of fractional programming problems.
Duality theory happens to be the central concept
in optimization. This theory aids us to experience and
develop new algorithms (numerical algorithms) be-
cause it gives us appropriate stopping rules for a pair
of primal-dual problems. (Dorn, 1960) considered
a convex minimization problem of a differentiable
function subject to some linear constraintsand studied
its duality relations. Over the past few years, many re-
searchers generalized these results to the case of non-
differentiable convex problem (Schechter, 1979) and
differentiable convex problem (Hanson, 1981). As-
suming the functions to be invex, (Hanson, 1981) es-
tablished that KKT conditions are sufficient for op-
timality. Two different types of functions (namely
Type I and Type II) were first presented by (Hanson
and Mond, 1987) for a scalar optimization problem.
(Rueda and Hanson, 1988) extended these functions
to pseudo-Type I and quasi-Type I.
The Type I function for a single objective was
extended to a MOPP (multiobjective programming
problem) by (Kaul et al., 1994), where they defined
the Type I, its different generalizations, thus estab-
lished duality relations for the Wolfe and the Mond-
Weir type model. (Kuk and Tanino, 2003) consid-
ered nonsmooth programming problem and derived
the duality results considering the functions to be gen-
eralized Type I. On the other hand, (Suneja et al.,
2008) presented (F, ρ, σ)-type I functions for the case
of higher order. Further, they considered two dual
models (one Mond-Weir and the other Schaible type
both are in higher order case) and obtained their corre-
sponding dual relations (for multiobjective fractional
programs in nondifferentiable case).
Recently, fractional programming duality has be-
come an interesting topic of research. For a con-
vex nondifferentiable fractional problem, (Bector
et al., 1993) established some optimality conditions
(namely Fritz John and KKT necessary and suffi-
cient optimality criteria) and proved some duality re-
sults. Considering a vectorial optimization problem
over cones, (Bhatia, 2012) discussed the sufficient op-
timality conditions and proved some results (duality
theorems) using cone convex and its generalizations
for the case of higher order. (Slimani and Mishra,
2014) introduced a nonlinear multiple objective frac-
tional programming with inequality constraints and
proved duality results for a Mond-Weir type model
using semilocally V-type I-preinvex functions.
Debnath, I. and Gupta, S.
On Duality with Support Functions for a Multiobjective Fractional Programming Problem.
DOI: 10.5220/0005666001150121
In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 115-121
ISBN: 978-989-758-171-7
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
115
(Kim and Lee, 2009) introduced the nondiffer-
entiable multiobjective problem involving cone con-
straints and hence studied their duality relations using
higher order invexity assumptions. On the other hand,
considering the same problem as in (Kim and Lee,
2009), (Ahmad, 2012) formulated a dual program
(unified and higher order) and discussed some re-
sults in duality (under higher order generalized type-
I functions). In recent times, (Debnath et al., 2015)
constructed a pair of higher order Wolfe type multi-
objective nondifferentiable symmetric dual program
over arbitrary cones and studied duality relations un-
der higher-order K-F convexity assumption.
The paper is arranged as we describe. In section
2, a class of (K × Q)-F-type I function has been put
forward and further some definitions and terminolo-
gies have been given. In the section 3, we have for-
mulated a dual model (Mond-Weir model) for a frac-
tional problem over arbitrary cones (for nondifferen-
tiable multiobjective case) and proved duality rela-
tions considering the concerned functions as higher
order (K × Q)-F-type I. Section 3 contains the con-
clusion.
2 SOME BASIC DEFINITIONS
Consider the following MOPP (multiobjective pro-
gramming problem):
(VP) K-minimize φ(x)
subject to ψ(x) Q,
x X R
n
,
where X R
n
be open and φ : X R
k
, ψ : X R
m
defines vector valued differentiable functions,
K R
k
and Q R
m
denotes the closed con-
vex pointed cones having non-null interiors. Let
X
0
= {x X : ψ(x) Q} denotes the feasible set.
Definition 2.1 (Agarwal et al., 2010). A point
¯x X
0
is a weak efficient solution of (VP) if there
exists no x X
0
such that
φ( ¯x) φ(x) intK.
Definition 2.2 (Agarwal et al., 2010). A point ¯x X
0
is an efficient solution of (VP) if there exists no x X
0
such that
φ( ¯x) φ(x) K\{0}.
Definition 2.3 (Gupta et al., 2012). The positive dual
cone K
of K is defined by
K
= {y : x
T
y 0, for all x K}.
Definition 2.4. For all (x, u) X × X, a functional
H : X ×X ×R
n
R is called sublinear in respect with
the third component, if
(i) H(x, u;b
1
+ b
2
) H(x, u;b
1
) + H(x, u;b
2
) for all
b
1
, b
2
R
n
,
(ii) H(x, u;βb) = βH(x, u;b), for all β R
+
and for
all b R
n
.
Clearly, H(x, u;0) = 0.
Definition 2.5. Let F : X × X × R
n
R be called a
functional which is sublinear with respect to the third
variable. Also, let H : X × R
n
R
k
, G : X × R
n
R
m
be differentiable functions. Then the function (φ, ψ)
will be called higher-order (K × Q)-F type I at u R
n
in respect with the functions H and G, if for each
x X
0
, p
i
, q
j
R
n
, (i = 1, 2, ..., k, j = 1, 2, ..., m), we
have
φ
1
(x) φ
1
(u) F(x, u;
x
φ
1
(u) +
p
1
H
1
(u, p
1
))
H
1
(u, p
1
) + p
T
1
[
p
1
H
1
(u, p
1
)], ..., φ
k
(x) φ
k
(u)
F(x, u;
x
φ
k
(u) +
p
k
H
k
(u, p
k
)) H
k
(u, p
k
) +
p
T
k
[
p
k
H
k
(u, p
k
)]
K.
and
ψ
1
(u) F(x, u;
x
ψ
1
(u) +
q
1
G
1
(u, q
1
))
G
1
(u, q
1
) + q
T
1
q
1
G
1
(u, q
1
), ..., ψ
m
(u)
F(x, u;
x
ψ
m
(u) +
q
m
G
m
(u, q
m
)) G
m
(u, q
m
) +
q
T
m
q
m
G
m
(u, q
m
)
Q.
Definition 2.6 (Gupta et al., 2012). Let ϕ be a
convex set in R
n
which is also compact. The support
function of ϕ is given as
τ(x|ϕ) = max{x
T
y : y ϕ}.
The subdifferentiable of τ(x|ϕ) is defined by
∂τ(x|ϕ) = {z ϕ : z
T
x = τ(x|ϕ)}.
We now present the following problem (KP)
(multiobjective fractional programming problem)
over arbitrary cones containing support functions.
(KP) K-minimize
h
φ
1
(x) + τ(x|C
1
)
ψ
1
(x) τ(x|D
1
)
, ...,
φ
k
(x) + τ(x|C
k
)
ψ
k
(x) τ(x|D
k
)
i
subject to
h
ϕ
j
(x) + τ(x|M
j
)
i
Q, j = 1, 2, ..., m.
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
116
where
φ : R
n
R
k
, ψ : R
n
R
k
and ϕ : R
n
R
m
are
continuously differentiable functions. Assume that
φ
i
(.) + τ(.|C
i
) 0 and ψ
i
(.) τ(.|D
i
) > 0. C
i
, D
i
and
M
j
denotes the compact convex sets in R
n
and their
respective support functions are denoted by τ(x|C
i
),
τ(x|D
i
) and τ(x|M
j
).
Throughout the paper, the following notations
have been used:
P(.) + (.)
T
z
Q(.) (.)
T
v
=
n
φ
1
(.) + (.)
T
z
1
ψ
1
(.) (.)
T
v
1
,
φ
2
(.) + (.)
T
z
2
ψ
2
(.) (.)
T
v
2
, ...,
φ
k
(.) + (.)
T
z
k
ψ
k
(.) (.)
T
v
k
o
R(.) + (.)
T
w =
n
ϕ
1
(.) + (.)
T
w
1
, ϕ
2
(.) + (.)
T
w
2
, ...,
ϕ
m
(.) + (.)
T
w
m
o
.
H =
H
1
(u, p
1
), H
2
(u, p
2
), ..., H
k
(u, p
k
)
.
G =
G
1
(u, q
1
), G
2
(u, q
2
), ..., G
k
(u, q
m
)
.
In the following section, we associate dual model
for the primal problem (KP) and establish duality
relations between them.
3 MOND-WEIR TYPE DUAL
MODEL
Consider the following Mond-Weir type higher order
dual program of the problem (KP):
(MPD) K-maximize
h
φ
1
(u) + u
T
z
1
ψ
1
(u) u
T
v
1
, ...,
φ
k
(u) + u
T
z
k
ψ
k
(u) u
T
v
k
i
subject to
k
i=1
λ
i
h
φ
i
(u) + u
T
z
i
ψ
i
(u) u
T
v
i
i
+
m
j=1
µ
j
h
(ϕ
j
(u)+u
T
w
j
)
i
+
k
i=1
λ
i
p
i
H
i
(u, p
i
) +
m
j=1
µ
j
q
j
G
j
(u, q
j
) = 0. (1)
m
j=1
µ
j
h
(ϕ
j
(u)+u
T
w
j
)+G
j
(u, q
j
)q
T
j
q
j
G
j
(u, q
j
)
i
0.
(2)
k
i=1
λ
i
h
H
i
(u, p
i
) p
T
i
p
i
H
i
(u, p
i
)
i
0. (3)
z
i
C
i
, v
i
D
i
, w
j
M
j
, (i = 1, 2, ..., k,
j = 1, 2, ..., m),
λ intK
, µ intQ
, (λ, µ) 6= (0, 0).
Remark 3.1. If we consider K = R
k
+
and Q = R
m
+
,
the above discussed Mond-Weir model reduces to the
model studied in (Suneja et al., 2008).
Next, we will prove duality results between (KP) and
(MPD).
Theorem 3.1 (Weak Duality Theorem). Con-
sider x be a member of the feasible set for (KP) and
(u, v, w, λ, µ, z, p, q) belongs to the feasible set for
(MPD). Suppose that
(i)
h
P(.) + (.)
T
z
Q(.) (.)
T
v
, R(.) + (.)
T
w
i
is higher order (K ×
Q)-F-type I at u in respect with functions H and
G, where H : X × R
n
R
k
and G : X × R
n
R
m
are differentiable functions,
(ii) R
k
+
K and R
m
+
Q.
Then
φ
1
(u) + u
T
z
1
ψ
1
(u) u
T
v
1
, ...,
φ
k
(u) + u
T
z
k
ψ
k
(u) u
T
v
k
φ
1
(x) + τ(x|C
1
)
ψ
1
(x) τ(x|D
1
)
, ...,
φ
k
(x) + τ(x|C
k
)
ψ
k
(x) τ(x|D
k
)
6∈ K\{0}.
Proof. We will prove the result by contradic-
tion. Suppose,
φ
1
(u) + u
T
z
1
ψ
1
(u) u
T
v
1
, ...,
φ
k
(u) + u
T
z
k
ψ
k
(u) u
T
v
k
φ
1
(x) + τ(x|C
1
)
ψ
1
(x) τ(x|D
1
)
, ...,
φ
k
(x) + τ(x|C
k
)
ψ
k
(x) τ(x|D
k
)
K\{0}.
Since λ intK
, we have
k
i=1
λ
i
h
φ
i
(u) + u
T
z
i
ψ
i
(v) u
T
v
i
φ
i
(x) + τ(x|C
i
)
ψ
i
(x) τ(x|D
i
)
i
> 0.
(4)
Now, since x
T
z
i
τ(x|C
i
), x
T
v
i
τ(x|D
i
), we obtain
h
φ
i
(x) + τ(x|C
i
)
ψ
i
(x) τ(x|D
i
)
φ
i
(x) + x
T
z
i
ψ
i
(x) x
T
v
i
i
0.
Using hypothesis (ii), we have K
R
k
+
λ
intK
intR
k
+
which yields λ > 0. Therefore, the
above inequality implies,
k
i=1
λ
i
φ
i
(x) + τ(x|C
i
)
ψ
i
(x) τ(x|D
i
)
φ
i
(x) + x
T
z
i
ψ
i
(x) x
T
v
i
0. (5)
Further, on adding (4) and (5), we have
k
i=1
λ
i
h
φ
i
(x) + x
T
z
i
ψ
i
(x) x
T
v
i
φ
i
(u) + u
T
z
i
ψ
i
(v) u
T
v
i
i
< 0. (6)
On Duality with Support Functions for a Multiobjective Fractional Programming Problem
117
By assumption (i) (since
h
P(.) + (.)
T
z
Q(.) (.)
T
v
, R(.)+(.)
T
w
i
is higher order (K × Q)-F-type I at u in respect with
functions H and G), we thus have
φ
1
(x) + x
T
z
1
ψ
1
(x) x
T
v
1
φ
1
(u) + u
T
z
1
ψ
1
(u) u
T
v
1
F
x, u;
φ
1
(u) + u
T
z
1
ψ
1
(u) u
T
v
1
+ H
1
(u, p
1
)
H
1
(u, p
1
) +
p
T
1
p
1
H
1
(u, p
1
), ...,
φ
k
(x) + x
T
z
k
ψ
k
(x) x
T
v
k
φ
k
(u) + u
T
z
k
ψ
k
(u) u
T
v
k
F
x, u;
φ
k
(u) + u
T
z
k
ψ
k
(u) u
T
v
k
+ H
k
(u, p
k
)
H
k
(u, p
k
)
+p
T
k
p
k
H
k
(u, p
k
)
K, (7)
and
ϕ
1
(u) u
T
w
1
F
x, u;(ϕ
1
(u) + u
T
w
1
)
+
q
1
G
1
(u, q
1
)
G
1
(u, q
1
) + q
T
1
q
1
G
1
(u, q
1
),
..., ϕ
m
(u) u
T
w
m
F
x, u;(ϕ
m
(u) + u
T
w
m
) +
q
m
G
m
(u, q
m
)
G
m
(u, q
m
)
+q
T
m
q
m
G
m
(u, q
m
)
Q. (8)
From (7) and λ intK
, we obtain
k
i=1
λ
i
h
φ
i
(x) + x
T
z
i
ψ
i
(x) x
T
v
i
φ
i
(u) + u
T
z
i
ψ
i
(u) u
T
v
i
F
x, u;
φ
i
(u) + u
T
z
i
ψ
i
(u) u
T
v
i
+
p
i
H
i
(u, p
i
)
H
i
(u, p
i
) + p
T
i
p
i
H
i
(u, p
i
)
i
0.
Using λ > 0 and the fact that F is sublinear, we
get
k
i=1
λ
i
φ
i
(x) + x
T
z
i
ψ
i
(x) x
T
v
i
φ
i
(u) + u
T
z
i
ψ
i
(u) u
T
v
i
k
i=1
λ
i
h
H
i
(u, p
i
) p
T
i
p
i
H
i
(u, p
i
)
i
+F
x, u;
k
i=1
λ
i
φ
i
(u) + u
T
z
i
ψ
i
(u) u
T
v
i
+
p
i
H
i
(u, p
i
)

.
(9)
Again from (8) and µ intQ
, we have
m
j=1
µ
j
(ϕ
j
(u)+u
T
w
j
)F(x, u;(ϕ
j
(u)+u
T
w
j
)+
q
j
G
j
(u, q
j
)) G
j
(u, q
j
) + q
T
j
q
j
G
j
(u, q
j
)
0.
Now, by assumption (ii), Q
R
m
+
µ intQ
intR
m
+
which yields µ > 0. Therefore, the above
inequality and the sublinearity of F together imply,
m
j=1
µ
j
h
(ϕ
j
(u) + u
T
w
j
) G
j
(u, q
j
)
+ q
T
j
q
j
G
j
(u, q
j
)
i
F
x, u;
m
j=1
µ
j
(ϕ
j
(u)+ u
T
w
j
)+
q
j
G
j
(u, q
j
)

.
(10)
It follows from the sublinearity of F, (9) and (10) that
k
i=1
λ
i
h
φ
i
(x) + x
T
z
i
ψ
i
(x) x
T
v
i
φ
i
(u) + u
T
z
i
ψ
i
(u) u
T
v
i
i
k
i=1
h
H
i
(u, p
i
) p
T
i
p
i
H
i
(u, p
i
)
i
+
F
h
x, u;
k
i=1
λ
i
φ
i
(u) + u
T
z
i
ψ
i
(u) u
T
v
i
+
p
i
H
i
(u, p
i
)
+
m
j=1
µ
j
(ϕ
j
(u) + u
T
w
j
) +
q
j
G
j
(u, q
j
)
i
m
j=1
µ
j
h
(ϕ
j
(u) + u
T
w
j
) G
j
(u, q
j
) +
q
T
j
q
j
G
j
(u, q
j
)
i
.
Finally, using dual constraint (1)-(3), we get
k
i=1
λ
i
h
φ
i
(x) + x
T
z
i
ψ
i
(x) x
T
v
i
φ
i
(u) + u
T
z
i
ψ
i
(u) u
T
v
i
i
0,
which contradicts (6). Hence the result.
Definition 3.1 (Clarke, 1983). The function
g : R
n
R will be called a locally Lipschitz at
x
0
R
n
if k 0 and a neighbourhood δ(x
0
) of x
0
s.
t.
||g(x) g(y)|| k||x y||, x, y δ(x
0
).
Definition 3.2 (Husain and Zabeen, 2005). A locally
Lipschitz function g : R
n
× R
n
R at x
0
R
n
in the
direction t R
n
is said to have generalized directional
derivative if
g
(x
0
;t) = lim
yx
0
sup
p0
+
g(y+ pt) g(y)
p
,
where y R
n
and p > 0.
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
118
Definition 3.3 (Wang, 2005). A function g which is
also locally Lipschitz at x
0
R
n
is said to have gener-
alized gradient or the subdifferential if
g(x
0
) = {ζ R
n
: g
(x
0
;t) hζ,ti, t R
n
}.
Remark 3.2.
(i) For a convex function φ, it will be called locally
Lipschitz at x
0
R
n
, if
∂φ(x
0
) = {ζ R
n
: φ(x) φ(x
0
) (x x
0
)
T
ζ,
x R
n
}.
(ii) If φ at x
0
is continuously differentiable, then φ
at x
0
will be locally Lipschitz, and therefore,
∂φ(x
0
) = {∇φ(x
0
)}.
Lemma 3.1. Let us deal with the problem:
(P) K minimize θ(x)
subject to ω(x) Q,
where θ : R
n
R
k
and ω : R
n
R
m
are locally Lips-
chitz functions. For this problem, suppose ¯x be weak
efficient solution, then (0, 0) 6= (λ, µ) intK
×
intQ
in such a way that
0
c
(λ
T
θ+ µ
T
ω)( ¯x), (µ
T
ω)( ¯x) = 0.
Proof. It follows on the lines of (Craven, 1989) and
(Wang et al., 2008).
Proposition 3.1 (Husain and Zabeen, 2005).
Let us suppose that φ
1
: R
n
R and φ
2
: R
n
R at
x
0
R
n
be locally Lipschitz with φ
2
(x) 6= 0. Then
φ
1
φ
2
will be locally Lipschitz at x
0
R
n
.
With the help of the above lemma and the proposi-
tion, we will now prove the following result, between
(KP) and (MPD).
Theorem 3.2 (Strong Duality). If ¯x is a weak effi-
cient of (KP), then (0, 0) 6= (
¯
λ, ¯µ) intK
× intQ
,
¯z
i
C
i
, v
i
D
i
, w
j
M
j
; i belongs to the index set
{1, 2, ..., k}, and j belongs to {1, 2, ..., m} in such
a way that, ( ¯x, ¯v, ¯w,
¯
λ, ¯µ, ¯z, ¯p = 0, ¯q = 0) belongs
to the feasible set for (MPD) and their respective
objective functions possesses same values pro-
vided H( ¯x, 0) = 0, G( ¯x, 0) = 0,
p
i
H
i
( ¯x, 0) = 0,
q
j
G
j
( ¯x, 0) = 0, i = 1, 2, ..., k, j = 1, 2, ..., m.
Moreover, if the supposition of Theorem
3.1 are fulfilled for every feasible solution x
of (KP) and (u, v, w, λ, µ, z, p, q) of (MPD), then
( ¯x, ¯v, ¯w,
¯
λ, ¯µ, ¯z, ¯p = 0, ¯q = 0) is an efficient for (MPD).
Proof. Let ¯x X
0
be a weak efficient of (KP)
and suppose that θ : R
n
R
k
, ϕ : R
n
R
m
be taken
as
θ(x) =
φ
1
(x) + τ(x|C
1
)
ψ
1
(x) τ(x|D
1
)
, ...,
φ
k
(x) + τ(x|C
k
)
ψ
k
(x) τ(x|D
k
)
and ω(x) =
ϕ
1
(x)+ τ(x|M
1
), ..., ϕ
m
(x)+ τ(x|M
m
)
.
The functions τ(x|C
i
), τ(x|D
i
) and τ(x|M
j
),
(i = 1, 2, ..., k, j = 1, 2, ..., m) are locally Lips-
chitz, since each of them are convex. Also, φ , ψ and
ϕ are functions which are continuously differentiable,
hence the above functions are also locally Lipschitz
and as a result, φ
i
(x) + τ(x|C
i
), ψ
i
(x) τ(x|D
i
), i
takes values from 1, 2, ..., k and ϕ
j
(x) + τ(x|M
j
), j
takes values from 1, 2, ..., m are locally Lipschitz.
Using the Proposition 3.1, we thus conclude
that θ(x) and ω(x) are also locally Lipschitz.
Following the Lemma 3.1,
¯
λ intK
and ¯µ intQ
,
(
¯
λ, ¯µ) not equal to (0, 0) in such a way that
0
c
h
k
i=1
¯
λ
i
φ
i
( ¯x) + τ( ¯x|C
i
)
ψ
i
( ¯x) τ( ¯x|D
i
)
+
m
j=1
¯µ
j
ϕ
j
( ¯x)+
τ(¯x|M
j
)
i
and
m
j=1
¯µ
j
ϕ
j
( ¯x) + τ( ¯x|M
j
)
= 0,
which implies
0
k
i=1
¯
λ
i
c
φ
i
( ¯x) + τ( ¯x|C
i
)
ψ
i
( ¯x) τ( ¯x|D
i
)

+
m
j=1
¯µ
j
(∇ϕ
j
( ¯x))
+
m
j=1
¯µ
j
c
S( ¯x|M
j
).
As we know that the support functions are convex, we
have
c
τ(¯x|C
i
) = ∂τ( ¯x|C
i
),
c
τ(¯x|D
i
) = ∂τ( ¯x|D
i
) and
c
τ(¯x|M
j
) = ∂τ( ¯x|M
j
)
Therefore, there exist ¯z
i
∂τ( ¯x|C
i
), ¯v
i
∂τ( ¯x|D
i
) and
¯w
j
∂τ( ¯x|M
j
) just as if
¯x
T
¯z
i
= τ( ¯x|C
i
), ¯x
T
¯v
i
= τ( ¯x|D
i
) and ¯x
T
¯w
j
= τ( ¯x|M
j
),
(11)
Hence,
k
i=1
¯
λ
i
φ
i
( ¯x) + ¯x
T
¯z
i
ψ
i
( ¯x) ¯x
T
¯v
i

+
m
j=1
¯µ
j
(∇ϕ
j
( ¯x)+ ¯x
T
¯w
j
) = 0,
and
m
j=1
¯µ
j
(∇ϕ
j
( ¯x) + ¯x
T
¯w
j
) = 0.
On Duality with Support Functions for a Multiobjective Fractional Programming Problem
119
Using H(¯x, 0) = G( ¯x, 0) = 0,
p
i
H
i
( ¯x, 0) = 0 and
q
j
G
j
( ¯x, 0) = 0, (i = 1, 2, ..., k, j = 1, 2, ..., m), we
obtain (¯x, ¯v, ¯w,
¯
λ, ¯µ, ¯z, ¯p = 0, ¯q = 0) belongs to the
domain feasible for (MPD) and also the respective
values of the objectives are equivalent.
We now claim that for (MPD) ( ¯x, ¯v, ¯w,
¯
λ, ¯µ, ¯z, ¯p =
0, ¯q = 0) is efficient.
On the contrary, let us assume that
( ¯x, ¯v, ¯w,
¯
λ, ¯µ, ¯z, ¯p = 0, ¯q = 0) be efficient for (MPD),
therefore (u, v, w, λ, µ, z, p, q), which is in the feasible
domain for (MPD) such that
h
φ
1
(u)+ u
T
z
1
ψ
1
(u) u
T
v
1
, ...,
φ
k
(u)+ u
T
z
k
ψ
k
(u) u
T
v
k
i
h
φ
1
( ¯x) + ¯x
T
¯z
1
ψ
1
( ¯x) ¯x
T
¯v
1
, ...,
φ
k
( ¯x) + ¯x
T
¯z
k
ψ
k
( ¯x) ¯x
T
¯v
k
i
K\{0}
which using (11) imply
h
φ
1
(u)+ u
T
z
1
ψ
1
(u) u
T
v
1
, ...,
φ
k
(u)+ u
T
z
k
ψ
k
(u) u
T
v
k
i
h
φ
1
( ¯x) + τ( ¯x|C
1
)
ψ
1
( ¯x) τ( ¯x|D
1
)
, ...,
φ
k
( ¯x) + τ( ¯x|C
k
)
ψ
k
( ¯x) τ( ¯x|D
k
)
i
K\{0},
a contradiction to the Theorem 3.1. Therefore,
the required result.
4 CONCLUSIONS
In this paper, we have presented a current class of
higher order (K × Q)- F-type I function. A Mond-
Weir type higher order multiobjective fractional prob-
lem (which is also nondiifferentiable) over cone has
been constructed. Considering this dual program, we
have established the corresponding duality relation un-
der higher order (K × Q)- F-type I function. The re-
sults which we have put forward in this paper are ex-
tension of some previously studied results appearing in
the literature. It is to be noted that, researchers can
further extend our work for different types of duality
problems for fractional problems, such as, mixed type
duality etc.
ACKNOWLEDGEMENTS
The first author is grateful to the Ministry of Human
Resource and Development, India for financial support,
to carry out this work.
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