Second Order (b, F) – Convexity in Multiobjective Fractional Programming

DOI : 10.17577/IJERTV2IS3764

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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 v-invex functions. Zhang and MOND

introduced second order F-convex functions as a generalization of F-convex functions in 1982 and obtained various second order duality results for multiobective nonlinear programming problems under the assumptions of second order F-convexity. 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 F-convex 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

1. Lai, H.C. and H.O., C.P. (1986), Duality theorem of non-differentiable convex multiobjective programming, Journal of optimization theory and Applications, 50, No.3, 407-420.

2. Mond, B. and Weir, T. (1981), Generalized concavity and duality in generalized concavity in optimization and economics. Academic Press, San Diego, 263-279.