 Open Access
 Total Downloads : 241
 Authors : G. V. Sarada Devi
 Paper ID : IJERTV2IS3764
 Volume & Issue : Volume 02, Issue 03 (March 2013)
 Published (First Online): 28032013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Second Order (b, F) – Convexity in Multiobjective Fractional Programming
G. V. Sarada Devi
Associate Professor In Mathematics,A.J Kalasala Machilipatnam,Krishna(Dt),A.P
Abstract second order duality theorems for multiobjective fractional programming problems under the assumption of second order (b, F) convexity.
Key Words Multiobjective fractional programming, second order (b,F) convexity.
Introduction:
Many researchers have used proper efficiency and efficiency to establish optimality conditions and duality results for multiobjective programming problems under different assumptions of convexity. A second order dual for a non linear programming problem was introduced by Mangasarian and established duality results for non linear programming problems. MOND introduced the concept of second order convex functions and proved second order duality under the assumptions of second order convexity on the functions involved. MOND and Zhang established various duality results for multiobjective programming problems involving second order vinvex functions. Zhang and MOND
introduced second order Fconvex functions as a generalization of Fconvex functions in 1982 and obtained various second order duality results for multiobective nonlinear programming problems under the assumptions of second order Fconvexity. In 2003 Suneja et al obtained duality results for multiobjective programming under the assumption of bonvexity and related functions. In 2004 Ahmed obtained optimality conditions and mixed duality results for non differentiable programming problems. In 2006 Ahmed and Hussain [OPSEARCH] obtained second order Mond Weir type dual results for multiobjective programming problems under the
assumption of the second order F,, p,d – convexity on the function involved. The study of
the second order duality is significant due to the computational advantage over first order duality as it provides tighten bounds for the value of the objective functions when approximations are used.
But, they not consider the recent developed concept like multiobjective fractional programming under second order (b, F) convexity.
In this paper a new class of functions namely, second order (b, F) convex functions which is as extension of (b, F) convex functions in previous chapter and second order Fconvex functions. Then, we derive sufficient optimality conditions for proper efficiency and obtain second order duality theorems for multiobjective fractional programming problems under the assumption of second order (b, F) convex
Preliminaries:
Let X be an open convex subset of Rn and R+ denote the set of all positive real numbers. Let us
assume that
h : x R,f : x Rk
and g:
Rn Rm where
f f1……………fk
and g=
(g1.gm) h=(p, p..hk) are differentiable functions on X fi and gi : X R
i=1,2,.k and j=1,2,.m
Let F be a function defined by
for all
F: XXRn R
and the functions bo(x, u) and Co (x, u): XX R
and
bi x, u and Cj x, u: XX R
for all i=1,2.k, J = 1,2,..m
Consider the following multiobjective fractional programming problem
(FP) minimize
fi x hi x
, i=1,2,k
Subject to g(x) 0 x x
Where fi : x R i 1, 2,………..k
and g : x Rm
Where g=(g1.gm) and h=(p, p, hk) are differentiable function on X.
Let p x x; gj x 0, j 1, 2….m. That is, p is the set of all feasible solutions for the problems (F.P).
Definition:
A function F: XXRn R is said to be sublinear in its third argument if for each
x, u x
Fx, u :a b Fx, u;aFx, u;b
For all a, bRn and
Fx, u;a Fx, u : aforall 0 in R and a Rn
Definition:
A feasible point x0 is said to be efficient for (FP) if there exits no other feasible point x in
(FP) such that
fi x
f x0
, i=1,2,.K and
i
i
h x h x0
i i
i i
fi x
i i
x
x
0
0
fi for some r 1, 2,…..k
h x
h x0
Definition: A feasible point x0 is said to be a properly efficient solution of (FP). It is efficient and
if there exists a scalar M>0 such that, for each i1, 2,…..k and for all feasible x of (FP)
satisfying
fi x
f x0
, have
i
i
h x h x0
f x0
i
i
i i
f x
f x
f x0
i
i
i
i
i
i
i
i i
i i
h x0 h
x
M i
h
x h
x0
fore some r such that f
r x fr
x0
We need the following theorem for proving sufficient optimality conditions for proper efficiency and duality theorems which can be found in 1999.
Second Order (b, F) Convex Functions Definition:
The function h is said to be second order (bo, F) Convex at ux
with respect to
bo x, u
is for all x X and pRn
b x, u fi x fi u 1 PT2 fi u p F x, u : fi u 2 fi u p
o h x h u 2 h u h u h u
i i i i i
The function h is said to be strictly second order (bo, F)convex at ux with respect bo(x, u) if
for all x X
, x u
and
pRn
b x, u fi x fi u 1 pT2 fi u p F x, u; fi u 2 fi u p
o h x h u 2 h u h u h u
i i i
i i
Necessary Optimality Conditions:
0
0
r
r
Assume that x0 is an efficient solution for (FP) at which a constraint qualification is satisfied for each Fpr x , r 1, 2,…..k
p
p
where F
r
x0 , minimize f
x
subject to
fi x
f x0
i i
i i
for all i r x p
i
i
h x
h x0
then
y0 0 in Rm such that
K
K
i
i
fi
x0
y0g
x0 0
i1
h x0
j j
m
m
j1
y g
y g
j j
j j
0 x0 0, j 1, 2,….m
Sufficient optimality conditions :
Let x0 be feasible for (FP) and there
yj 0
in R,
jIx0
and
p0 Rn such that
k f x0
k f
x0
i1
i
j i
j i
0
0
hi x
JIX0
y0g
x0 2
i1
i
0
0
hi x
j j
j j
jIx0
y0.2g
x0 p0 0
j
j
j j
j j
0 0
0 0
Where I(x0) = j: g x0 0. If f is second order (b0, F) convex at x0 with respect to b0(x, x0) with bo(x, x0)>0 and each g , jIx0 is second order C , F convex at x0
with respect
Cj x, x , then x is a properly efficient solution for FP provided that
j
j
T T
T T
P0 2 x0 p0 0 and P0 2g x0 p0 0 for all
jIx0
Proof: Let x be a feasible solution of (FP) Now, since x0 is feasible for (FP) and
yj 0 and
P0 2g x0 p0 0 , for all
P0 2g x0 p0 0 , for all
T
T
j
j
jIx and by the second order (Cj F) convexity of gi at x0,
for all
jIx0 and since
y0 0, jIx0 and F is sub linear, we have
j
j
F x x0 ; y0g x0 2 y0g x0 p0 0
j j
j j
(9.4)
0
0
0
0
i j j
jIx
jIx
Now, by the sub linearity of F and from (9.3)
We have
k f x0 k
f x0
F x, x0 : i 2
i p0 Fx, x0 : y0g x0
y02g x0 p0 )
i1
h x0
i1
h x0
j j j j
0
i i jIx
Now from (9.4) and (9.5) by the second order (b0, F) convexity since
i
i
h
h
i
i
p0T 2
f x0
p0 0 and b (x, x0) > 0
0
0
x0
We can conclude that
i i
i i
i
i
fi x
f x0
h x
h x0
Thus x0 is an optimal solution (FP). Hence X0 is a properly efficient solution for (FP)
Hence the theorem.
Duality Theorems:
Let j, be a subset of M 1, 2,….m and J2=M/J1 consider the following second order dual for (FP).
FDMaximize fi u y g
u 1 PT2 fi u y g
u p
j j
j j
hi u
u
j j
h
h
2
2
i
i
Subject to
fi u 2 fi u p y g
u 2 y g
up 0
hi u
gi u
j j i i
y g u 1 pT2 y g
u 0
j j 2 i i
yj 0
j=1,2..m
Weak Duality Theorem:
Let x be feasible for (FP) and (u, y, p) be feasible for (FD)
If fi hi

yjg j
is second order (bo, F) convex at u with respect to b0(x, u) with bo (x,
u)>0 and yjgj is second order (CoF) convex at u with respect to Co(x, u) then
fi x
fi u y g
u 1 PT2 fi u y g u p
h x h u j j 2 h u j j
i i i
Proof:
Now since
i yigi is second order (bo, F) convex at u . We have
f
f
hi
b x, u fi x y g
x fi u y g
u 1 pT2 fi u y g
up
o h
x j j
h u i i
2 h
u j j
i i i
f u
F x, u; i y g
u 2 fi u y g
up
h
u j j
h u j j
i i
Now, since x is feasible and (u, y, p) is feasible for (FD) and since yjgj is second order (Co, F) convex at u with respect to C0(x, u) we have.
j j j j
j j j j
Fx, u :y g
u 2y g
up 0
Now, from and since (u, y, p) is feasible for (FD) and F is sublinear
We have
Fxu : fi u y g
u 2 fi u y g
up 0
h
h
h
h
i
i
u j j
i
u j j
bo x, u 0 and
yjgj x 0
we have
fi x
fi x y g
u 1 PT2 fi u y g u p.
h x h u j j 2 h u j j
i i i
Strong Duality Theorem:
Assume that x0 an efficient solution for (FP) at which a constraint qualification is satisfied for
each Fpr (x0)
x 1, 2,……k. Then, there exists
y0 Rm
such that x0 , y0 , p0 0is feasible
solution for (FP) and the corresponding objective functions values of (FP) and (FD) are equal. If
the conditions of weak duality Theorem holds then (x0, y0, p0) = 0 is properly efficient solution for (FD).
Proof:
By the theorem necessary optimal condition
y0 0 in Rm such that (x0, y0) satisfies (1)
and (2). Therefore (x0, y0, p0 =0) is feasible for FD and the objective value of the problem FP at x0 and the objective value of (FD) at (x0, y0, p0 = 0) are equal.
Suppose that (x0, y0, p0 =0) is not efficient for (FD) then there exists a feasible (u, y, p) for (FD) such that
f x0
f u
1 f
u
i i

y g
pT2 i y g u p
j
j
h x0 h u
i
i
i
j ju
2 hi u
j
f x0
f u
1 f
u
i i y g
j j
j j
u
pT2 i y g u p
h x0 h u
i
i
i
2 hi u
j j
Which contridcts the theorem weak duality theorem. Thus (x0, y0, p0 =0) is an efficient solution for (FD). Suppose that (x0, y0, p0 =0) is not properly efficient for FD. Then for every M>0, there exists a feasible solution (u, y, p) of FD and an index i such that
fi u


yjgj u
f x0
i
i
0
0
M fr x f u y g
M fr x f u y g
r j j
hi u
For all r satisfying
hi x
r
r
f x0
i j
i j
fr u

y g
u 0 whenever
r
r
h x0
hr u
fi u y g
u f
x0 0 . This means hat
hi u
j j i
fi u


y g
u
f x0
can be made arbitary large. fi yg
is second order (b , F)
i i
i i
i
i
h u j j
h x0
h j i o
i
i
convex at u with respect to bo (x, u) with b0(x, u) > 0. Since (u, y, p) is feasible for FD and F is sublinear we can conclude that.
2
2
Fx, u;yjgj u yjgj up 0
Now since x0 is feasible for (FP) and (u, y, p) is feasible for (FD) and by the second order (Co,F) convexity of yj, gj at u we have.
j j j j
j j j j
Fx0 , u :y g
u 2 y g
up 0
Which contradicts Thus (x0, y0, p0=0) is properly efficient solution for FD.
Hence the theorem
Acknowledgements
This author is thankful to the anonymous referee for the useful comments
References

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