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ISSN Online: 2160-8849 ISSN Print: 2160-8830

DOI: 10.4236/ajor.2018.84017 Jul. 3, 2018 294 American Journal of Operations Research

Duality Relations for a Class of a Multiobjective

Fractional Programming Problem Involving

Support Functions

Vandana

1

, Ramu Dubey

2

, Deepmala

3

, Lakshmi Narayan Mishra

4,5*

, Vishnu Narayan Mishra

6

1Department of Management Studies, Indian Institute of Technology Madras, Chennai, India

2Department of Mathematics, Central University of Haryana, Pali, India

3Mathematics Discipline, PDPM-Indian Institute of Information Technology, Design and Manufacturing,

Jabalpur, India

4Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India

5L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opposite-Industrial Training Institute (I.T.I.), Faizabad, India

6Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, India

Abstract

In this article, for a differentiable function H R R: n× n R, we introduce the definition of the higher-order

(

V, , , ,α β ρ d

)

-invexity. Three duality models

for a multiobjective fractional programming problem involving nondifferen- tiability in terms of support functions have been formulated and usual duality relations have been established under the higher-order

(

V, , , ,α β ρ d

)

-invex

assumptions.

Keywords

Efficient Solution, Support Function, Multiobjective Fractional Programming, Generalized Invexity

1. Introduction

Consider the following nonlinear programming problem (P) Minimize f x

( )

subject to g x

( )

≤0, where f R: nR and g R: n R are twice differen- tiable functions. The Mangasarian [1] second-order dualof (P) is (DP) Maximize

( )

T

( )

1 T 2

( )

T

( )

2

f uy g up ∇ f uy g u p

such that f u

( )

y g uT

( )

+ ∇2f u

( )

y g u pT

( )

=0

   

*Corresponding author. How to cite this paper: Vandana, Dubey,

R., Deepmala, Mishra, L.N. and Mishra, V.N. (2018) Duality Relations for a Class of a Multiobjective Fractional Programming Problem Involving Support Functions. Ame- rican Journal of Operations Research, 8, 294-311.

https://doi.org/10.4236/ajor.2018.84017

Received: April 21, 2018 Accepted: June 30, 2018 Published: July 3, 2018

Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/ajor.2018.84017 295 American Journal of Operations Research

By introducing two differentiable functions H R R: n× n R and

: n n m

K R R× →R , Mangasarian [1] formulated the following higher-order dual

of (P): (DP)1 Maximize

( )

T

( )

(

,

)

T

(

,

)

f uy g u +H u py K u p

such that

(

,

)

T

(

,

)

0,

pH u p py K u p

∇ − ∇ = y≥0,where ∇pH u p

(

,

)

denotes

the n×1 gradient of H u p

(

,

)

with respect to p and

(

T

(

,

)

)

p y K u p

denotes

the n×1, gradient of y K u pT

(

,

)

with respect to p.

Further, Egudo [2] studied the following multiobjective fractional program- ming problem: (MFPP) Minimize

( )

1

( )

( )

2

( )

( )

( )

( )

1 2

, , , k k

f x f x f x

G x

g x g x g x

 

= 

 

subject to

( )

{

}

0 n: 0, ,

j

x X∈ = x X∈ ⊂R h xj M

where f =

(

f f1, , ,2 fk

)

:XR gk, , , ,=

(

g g1 2 gk

)

:XRk

and h=

(

h h1, , ,2 hm

)

:XRm are differentiable on X. Also, he discussed duality results for Mond-Weir and Schaible type dual programs under generalized convexity.

For the nondifferentiable multiobjective programming problem: (MPP) Mini- mize

( )

(

1

( )

(

| 1

) ( )

, 2

(

| 2

)

, , k

( )

(

| k

)

)

G x = f x +S x C f x +S x C f x +S x C

subject to 0

{

n:

( )

(

|

)

0, 1,2, ,

}

,

j j

x X∈ = x X∈ ⊂R g x +S x Ej= m where

(

)

: 1,2, ,

i

f XR i= k and g Xj: →R j 1,2, ,

(

= m

)

are differentiable func-

tions. Ci and Ej are compact convex sets in Rn and

(

| i

)(

1,2, ,

)

S x C i= k and S x E

(

| j

)

1,2, ,

(

j= m

)

denote the support func-

tions of compact convex sets, various researchers have worked. Gulati and Agarwal [3] introduced the higher-order Wolfe-type dual model of (MPP) and proved duality theorems under higher-order

(

F, , ,ρ ρ d

)

-type I-assump-

tions.

In last several years, various optimality and duality results have been obtained for multiobjective fractional programming problems. In Chen [4], multiobjective fractional problem and its duality theorems have been considered under higher- order

(

F, , ,α ρ d

)

-convexity. Later on, Suneja et al.[5] discussed higher-order

Mond-Weir and Schaible type nondifferentiable dual programs and their duality theorems under higher-order

(

F, ,ρ σ

)

-type I-assumptions. Several researchers

have also worked in this directions such as ([6] [7]).

In this paper, we first introduce the definition of higher-order

(

V, , , ,α β ρ d

)

-

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DOI: 10.4236/ajor.2018.84017 296 American Journal of Operations Research

establish usual duality relations for these primal-dual pairs under aforesaid assumptions.

2. Preliminaries

Let X Rn be an open set and φ:X R H X R, : × nR be differentiable functions. α β, :X X× →R+\ 0

{ }

, η:X X× →Rn, ρ ∈Rn and

:X X Rn

θ × → .

Definition 2.1.

φ

is said to be (strictly) higher-order

(

V, , , ,α β ρ θ

)

-invex at u with respect to H u p

(

,

)

, if there exist

η α β ρ

, , , and θ such that, for any x X and p R n,

( ) ( ) ( ) ( )

( )

(

( )

(

)

)

( ) (

)

(

)

( )

T

2 T

, , ,

, , , , .

p

p

x u x u x u u H u p

x u H u p p H u p x u

α φ φ η φ

β ρ θ

− > ≥ ∇ + ∇

 

 

 

+ − ∇ +

Example 2.1. Let

φ

:RR be such that φ

( )

x =x4+x2+1.

Let

( )

, 1

(

2 2

)

, ,

(

)

2

(

1

)

2

2

x u x u H u p p x

η = + = − + .

Also, suppose

( )

x u, 1, ,

( )

x u 2, 1, ,

( )

x u

(

x2 u2 2

)

1

α = β = ρ= − θ = + .

Now,

( ) ( ) ( )

( )

(

( )

(

)

)

( ) (

)

(

)

( )

T

2 T

, , ,

, , , , .

p

p

x u x u x u u H u p

x u H u p p H u p x u

ξ α φ φ η φ

β ρ θ

=  − − ∇ + ∇

 

− ∇

(

4 2 4 2

) (

1 2 2

)

4 3 2 2

(

1

)

2

(

2 2

)

2

x x u u x u u u u x u

ξ = + ++ + +

 

4 2

x x

ξ = + (at u=0).

0, x R

≥ ∀ ∈ .

Hence,

φ

is higher-order

(

V, , , ,α β ρ θ

)

-invex at u=0 with respect to

(

,

)

H u p . Remark 2.1.

1) If H u p

(

,

)

=0, then the Definition 2.1 reduces to

(

V

)

-invex function introduced by Kuk et al. [8].

2) If H u p

(

,

)

=0 and

ρ

=0, then the Definition 2.1 becomes that of

V-invexity introduced by Jeyakumar and Mond [9].

3) If

(

,

)

1 T 2

( )

, ,

( )

0

2

H u p = p ∇φ u p α x u = and

ρ

=0, then above definition yields in η-bonvexity given by Pandey [10].

4) If

β

=1, then the Definition 2.1 reduced in

(

V, , ,α ρ θ

)

-invex given by

Gulati and Geeta [11].

A differentiable function f =

(

f f1, , ,2 fk

)

:XRk is

(

V, , , ,α β ρ θ

)

-invex if for all i=1,2, , k, fi is

(

V, , , ,α β ρ θi i i i

)

-invex.

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DOI: 10.4236/ajor.2018.84017 297 American Journal of Operations Research

function of C is defined by

(

|

)

max

{

T :

}

.

S x C = x y y C

3. Problem Formulation

Consider the multiobjective programming problem with support function given as: (MFP) Minimize

( )

1

( )

( )

(

(

1

)

)

2

( )

( )

(

(

2

)

)

( )

( )

(

(

)

)

1 1 2 2

| | , | , ,

| | |

k k

k k

f x S x C

f x S x C f x S x C

F x

g x S x D g x S x D g x S x D

 + + + 

 

= 

 

 

subject to 0

{

n:

( )

(

|

)

0, 1,2, ,

}

,

j j

x X∈ = x X∈ ⊂R h x +S x Ej= m

where f =

(

f f1, , ,2 fk

)

:XRk, g=

(

g g1, , ,2 gk

)

:XRk and

(

1, , ,2 m

)

: m

h= h h h XR are differentiable on X, fi

( )

. +S C

(

.| i

)

≥0 and

( )

.

(

.|

)

0

i i

gS D > . Let : n

i

H X R× →R be differentiable functions, C Di, i and Ej are compact convex sets in Rn, for all i=1,2, , , 1,2, , k j= m.

Definition 3.1. [3]. A point x0X0 is said to be an efficient solution (or Pareto optimal) of (MFP), if there exists no x X 0 such that for every

1,2, ,

i= k,

( )

( ) (

(

)

)

( ) (

( ) (

)

)

0 0

0 0

| |

| |

i i

i i

i i i i

f x S x C

f x S x C

g x S x D g x S x D

+ +

− −

and for some r=1,2, , k,

( )

(

)

( ) (

)

( ) (

( ) (

)

)

0 0

0 0

| |

| |

r r

r r

r r r r

f x S x C

f x S x C

g x S x D g x S x D

+ +

<

− − .

We now state theorems 3.1-3.2, whose proof follows on the lines [13].

Theorem 3.1. For some t, if ft

( ) ( )

. + .Tzt and −

(

gt

( ) ( )

. − .Tvt

)

are higher- order

(

V, , , ,α β ρ θt t t t

)

-invex at u with respect to H u pt

(

,

)

for same η

( )

x u, .

Then, the fractional function

( ) ( )

( ) ( )

T

T

. .

. .

t t

t t

f z

g v

+

 

  is higher-order

(

V, , , ,t t t t

)

α β ρ θ

-invex at u with respect to H u pt

(

,

)

, where

( )

, t

( )

( )

TT t

( )

,

t t

t t

g x x v

x u x u

g u u v

α =  − α

  , βt

( )

x u, =βt

( )

x u, ,

( )

( )

( )

(

( )

( )

)

1 2 T

T T 2

1

, , t t

t t

t t t t

f u u z

x u x u

g u u v g u u v

θ =θ  + + 

 − 

 

, ρt

( )

x u, =ρt

( )

x u,

and

(

)

( )

( )

( )

(

)

(

)

T

T T 2

1

, t t ,

t t

t t t t

f u u z

H u p H u p

g u u v g u u v

+

 

= +

 − 

 

.

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DOI: 10.4236/ajor.2018.84017 298 American Journal of Operations Research

order

(

V, , , ,α β ρ θt t t t

)

-invex at u with respect to H u pt

(

,

)

and

( ) ( )

(

T

)

. . 0

t t

fz > or

(

ft

( ) ( )

. − .Tzt

)

is strictly higher-order

(

V, , , ,α β ρ θt t t t

)

- invex at u with respect to H u pt

(

,

)

, then

( ) ( )

( ) ( )

T

T

. .

. .

t t

t t

f z

g z

+

 

  is strictly higher-

order

(

V, , , ,α β ρ θt t t t

)

-invex at u X∈ with respect to H u pt

(

,

)

.

Theorem 3.3 (Necessary Condition) [14]. Assume that x is an efficient solution of (MFP) and the Slater’s constraint qualification is satisfied on X. Then

there exist 0, , , m n n

i j R z R v Ri i

λ > µ ∈ ∈ ∈

and m, 1,2, , , 1,2, ,

j

wR i= k j= m, such that

( )

( )

(

( )

)

T

T T

1 1 0,

k m

i i

i j j j

i i i j

f x x z

h x x w

g x x v

λ µ

= =

 + 

∇+ ∇ + =

 

(1)

( )

(

T

)

=1 0,

m

j j j

j h x x w

µ + =

(2)

(

)

T | , , 1,2, , ,

i i i i

x z S x C z C i= ∈ = k (3)

(

)

T | , , 1,2, , ,

i i i i

x v S x D v D i= ∈ = k (4)

(

)

T | , , 1,2, , ,

j j j j

x w =S x E wE j= m (5)

0, 1,2, , , 0, 1,2, , .

i i k j j m

λ > = µ ≥ = (6)

Theorem 3.4. (Sufficient Condition). Let u be a feasible solution of (MFP). Then, there exist λ >i 0, 1,2, ,i= k and µj≥0, 1,2, ,j= m, such that

( )

( )

(

( )

)

T

T T

1 1 0,

k m

i i

i j j j

i i i j

f u u z

h u u w

g u u v

λ µ

= =

 + 

∇+ ∇ + =

 

(7)

( )

(

T

)

=1 0,

m

j j j

j h u u w

µ + =

(8)

(

)

T | , , 1,2, , ,

i i i i

u z S u C z C i= ∈ = k (9)

(

)

T | , , 1,2, , ,

i i i i

u v S u D v D i= ∈ = k (10)

(

)

T | , , 1,2, , ,

j j j j

u w =S u E wE j= m (11)

0, 1,2, , , 0, 1,2, , .

i i k j j m

λ > = µ ≥ = (12)

Let, for i=1,2, , , 1,2, , k j= m, 1)

(

fi

( ) ( )

. + .Tzi

)

and

(

( ) ( )

)

T

. .

i i

g v

− − be higher-order

(

, , , ,1 1 1 1

)

i i i i

V α β ρ θ -

invex at u with respect to H u pi

(

,

)

,

2)

(

hj

( ) ( )

. + .Twj

)

be higher-order

(

V, , , ,α β ρ θ2j 2j 2j 2j

)

-invex at u with res- pect to G u pj

(

,

)

,

3) 1 1

( )

2 2 2

( )

2

1 , 1 , 0,

k m

i i i j j j

i x u j x u

λ ρ θ µ ρ θ

= =

+ ≥

4)

(

(

)

)

(

(

)

)

1 , 1 , 0

k m

i p i j p j

i H u p j G u p

λ µ

= =

∇ + ∇ =

(6)

DOI: 10.4236/ajor.2018.84017 299 American Journal of Operations Research

(

)

(

)

(

T

)

1 , , 0

k

i i p i

i H u p p H u p

λ

=

− ∇ ≥

and

(

(

)

T

(

)

)

1 , , 0

m

j j p j

j G u p p G u p

µ

=

− ∇ ≥

,

5) 1

( )

, 2

( )

, 1

( )

, 2

( )

,

( )

, ,

i x u j x u i x u j x u x u

α =α =β =β =α

where

( )

, i

( )

( )

TT i

( )

,

i i

i i

g x x v

x u x u

g u u v

α =  − α

  , βi

( )

x u, =βi

( )

x u, ,

( )

( )

( )

( )

( )

(

)

1 2 T

T T 2

1

, , i i

i i

i i i i

f u u z

x u x u

g u u v g u u v

θ =θ  + + 

 − 

 

and ρi

( )

x u, =ρi

( )

x u, .

Then, u is an efficient solution of (MFP).

Proof. Suppose u is not an efficient solution of (MFP). Then there exists 0

x X such that

( )

(

)

( )

(

||

)

( )

( )

(

(

||

)

)

, for all 1,2, ,

i i i i

i i i i

f x S x C f u S u C

i k

g x S x D g u S u D

+ +

≤ =

− −

and

( )

(

)

( )

(

||

)

( )

( )

(

(

||

)

)

, for some 1,2, , ,

r r r r

r r r r

f x S x C f u S u C

r k

g x S x D g u S u D

+ +

< =

− −

which implies

( )

( )

( )

( ) (

(

)

)

( )

( ) (

(

)

)

( )

( )

T

T

T

T

| |

| |

, for all 1,2, ,

i i i i

i i

i i i i

i i

i i

i i

f x S x C f u S u C

f x x z

g x S x D g u S u D

g x x v

f u u z

i k

g u u v

+ +

+

≤ ≤

− −

+

= =

(13)

and

( )

( )

( )

( )

(

(

)

)

( )

( )

(

(

)

)

( )

( )

T

T

T

T

| |

| |

, for some 1,2, , .

r r r r r r

r r r r

r r

r r

r r

f x x z f x S x C f u S u C

g x S x D g u S u D

g x x v

f u u z

r k

g u u v

+ + +

≤ <

− −

+

= =

(14)

Since λ >i 0, 1,2, ,i= k, inequalities (13) and (14) gives

( )

( )

( )

( )

T T

T T

1 0.

k

i i i i

i

i i i i i

f x x z f u u z

g x x v g u u v

λ

=

 + + 

− <

 

 

(15)

From Theorem 3.1, for each i, 1≤ ≤i k,

( ) ( )

( ) ( )

T

T

. .

. .

i i

i i

f z

g v

+

 

 

is higher-order

(

, , , ,1 1 1 1

)

i i i i

V α β ρ θ -invex at u X 0 with respect to

(

,

)

i

H u p , we have

( ) ( )

( )

T

( )

( )

T

1

T T

, i i i i i

i i i i

f x x z f u u z

x u

g x x v g u u v

α  + − + 

− −

 

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DOI: 10.4236/ajor.2018.84017 300 American Journal of Operations Research

( )

( )

( )

(

)

( )

(

)

(

)

( )

T T T 2

1 T 1 1

, ,

, , , , .

i i

p i

i i

i i p i i i

f u u z

x u H u p

g u u v

x u H u p p H u p x u

η

β ρ θ

  +   ≥ ∇+ ∇  −         + − ∇ + (16) where

( )

, i

( )

( )

TT i

( )

,

i i

i i

g x x v

x u x u

g u u v

α =  − α

  , βi

( )

x u, =βi

( )

x u, ,

( )

( )

( )

( )

( )

(

)

1 2 T

T T 2

1

, , i i

i i

i i i i

f u u z

x u x u

g u u v g u u v

θ =θ  + + 

 − 

 

, ρi

( )

x u, =ρi

( )

x u,

and

(

)

( )

( )

( )

(

)

(

)

T

T T 2

1

, i i ,

i i

i i i i

f u u z

H u p H u p

g u u v g u u v

+    = + −  −    .

By hypothesis 2), we get

( ) ( )

(

( )

)

( )

(

( )

)

(

)

( )

(

)

(

)

( )

2 T T

T T

2

2 T 2 2

,

, ,

, , , , .

j j j j j

j j p j

j j p j j j

x u h x x w h u u w

x u h u u w G u p

x u G u p p G u p x u

α

η

β ρ θ

 + − +      ≥ ∇ + + ∇   + − ∇ + (17)

Adding the two inequalities after multiplying (16) by λi and (17) by µj, we obtain

( ) ( )

( )

( )

( )

( ) ( )

(

( )

)

( )

( )

( )

(

)

T T 1 T T 1

2 T T

1 T T T 1 , , , , k

i i i i

i i

i i i i i

m

j j j j j j

j

k

i i

i p i

i i i

f x x z f u u z

x u

g x x v g u u v

x u h x x w h u u w

f u u z

x u H u p

g u u v

λα µ α η λ = = =  + +  −         + + − +   +   ≥ ∇+ ∇  −      

( )

(

( )

)

(

)

( )

(

)

(

)

( )

(

)

(

)

( )

( )

T T 1 T 1 2 T 1 2 2

1 1 2 2

1 1 , , , , , , , , , , . m

j j j p j

j k

i i i p i

i m

j j j p j

j

k m

i i i j j j

i j

x u h u u w G u p

x u H u p p H u p

x u G u p p G u p

x u x u

η µ

λ β

µ β

λ ρ θ µ ρ θ

= = = = =   + ∇ + + ∇   + − ∇   + − ∇ + +

(18)

Using hypothesis 3)-4), we get

( )

( )

( )

( )

( )

(

( )

)

( )

( )

( )

( )

(

( )

)

T T T T T T 1 1 T

T T T

T

1 1

, , .

k m

i i i i

i j j j j j

i i i i i j

k m

i i

i j j j

i i i j

f x x z f u u z

h x x w h u u w

g x x v g u u v

f u u z

x u x u h u u w

g u u v

λ µ

η λ η µ

(8)

DOI: 10.4236/ajor.2018.84017 301 American Journal of Operations Research

Further, using (7)-(8), therefore

( )

( )

( )

( )

( )

T T

T

T T

1 1 0.

k m

i i i i

i j j j

i i i i i j

f x x z f u u z

h x x w

g x x v g u u v

λ µ

= =

 + + 

 

− + + ≥

 

− −

 

 

(20)

Since x is feasible solution for (MFP), it follows that

( )

( )

( )

( )

T T

T T

1 1 .

k k

i i i i

i i

i i i i i i

f x x z f u u z

g x x v g u u v

λ λ

= =

 +   + 

   

 

   

This contradicts (15). Therefore, u is an efficient solution of (MFP).

4. Duality Model-I

Consider the following dual (MFD)1 of (MFP): (MFD)1 Maximize

( )

( )

(

( )

)

(

(

)

(

)

)

(

)

(

)

(

)

( )

( )

(

( )

)

(

(

)

(

)

)

(

)

(

)

(

)

T

1 1 T T

1 1

T 1

1 1

T

1 T

T T

T 1

T

1

, ,

, , , ,

, ,

, ,

m

j j j p

j m

j j p j

j

m

k k

j j j k p k

j

k k

m

j j p j

j

f u u z h u u w H u p p H u p

g u u v

G u p p G u p

f u u z

h u u w H u p p H u p

g u u v

G u p p G u p

µ

µ

µ

µ

=

=

=

=

 +

+ + + − ∇

 − 

+ − ∇

+

+ + + − ∇

+ − ∇

subject to

( )

( )

(

( )

)

(

)

(

)

T

T T

1 1

1 , 1 , 0,

k m

i i

i j j j

i i i j

k m

i p i j p j

i j

f u u z

h u u w g u u v

H u p G u p

λ µ

λ µ

= =

= =

 + 

∇+ ∇ +

 

+ ∇ + ∇ =

(21)

, , , 1,2, , , 1,2, , ,

i i i i j j

z C v D w∈ ∈ ∈E i= k j= m

1

0, 0, k 1, 1,2, , , 1,2, , .

j i i

i i k j m

µ λ λ

=

≥ >

= = =

Let Z0 be feasible solution for (MFD)

1.

Theorem 4.1. (Weak duality theorem). Let x X 0 and

(

u z v, , , , , ,µ λ w p

)

Z0. Suppose that

1) for any i=1,2, , k,

(

fi

( ) ( )

. + .Tzi

)

and

(

( ) ( )

)

T

. .

i i

g v

− − are higher-

order

(

, , , ,1 1 1 1

)

i i i i

V α β ρ θ -invex at u with respect to H u pi

(

,

)

,

2) for any j=1,2, , m,

(

hj

( ) ( )

. + .Twj

)

is higher-order

(

V, , , ,α β ρ θ2j 2j 2j 2j

)

-invex at u with respect to G u pj

(

,

)

,

3) 1 1

( )

2 2 2

( )

2

1 , 1 , 0.

k m

i i i j j j

i x u j x u

λ ρ θ µ ρ θ

= =

+ ≥

4) 1

( )

, 2

( )

, 1

( )

, 2

( )

,

( )

, , 1,2, , ,

i x u j x u i x u j x u x u i k

α =α =β =β =α ∀ = 

1,2, , ,j=  m

where t

( )

, t

( )

( )

TT t t

( )

,

t t

g x x v

x u x u

g u u v

α =  − α

(9)

DOI: 10.4236/ajor.2018.84017 302 American Journal of Operations Research

( )

( )

( )

( )

( )

(

)

1 2 T

T T 2

1

, , t t

t t

t t t t

f u u z

x u x u

g u u v g u u v

θ =θ  + + 

 − 

 

, ρt

( )

x u, =ρt

( )

x u, and

(

)

( )

( )

( )

(

)

(

)

T

T T 2

1

, t t ,

t t

t t t t

f u u z

H u p H u p

g u u v g u u v

+    = + −  −    .

Then, the following cannot hold

( )

(

)

( )

(

)

( )

( )

(

( )

)

(

(

)

(

)

)

(

)

(

)

(

)

T T T T 1 T 1 | | , ,

, , , for all 1,2, ,

i i

i i

m

i i

j j j i p i

j

i i

m

j j p j

j

f x S x C

g x S x D

f u u z

h u u w H u p p H u p

g u u v

G u p p G u p i k

µ µ = = + − + ≤ + + + − ∇ − + − ∇ =

(22) and

( )

(

)

( )

(

)

( )

( )

(

( )

)

(

(

)

(

)

)

(

)

(

)

(

)

T T T T 1 T 1 | | , ,

, , , for some 1,2, , .

r r

r r

m

r r

j j j r p r

j

r r

m

j j p j

j

f x S x C

g x S x D

f u u z

h u u w H u p p H u p

g u u v

G u p p G u p r k

µ µ = = + − +

< + + + − ∇

+ − ∇ =

(23)

Proof. Suppose that (22) and (23) hold, then using λ >i 0,

1 1 k i i λ = =

,

(

)

T | i i

x z S x C, T

(

|

)

i i

x v S x D, i=1,2, , k, we have

( )

( )

( )

( )

(

( )

)

(

)

(

)

(

)

(

)

(

)

(

)

T T T T T

1 1 1

T 1 T 1 , , , , .

k k m

i i i i

i i j j j

i i i i i i j

k

i i p i

i m

j j p j

j

f x x z f u u z h u u w

g x x v g u u v

H u p p H u p

G u p p G u p

λ λ µ

λ µ = = = = =  +   + 

< + +

            + − ∇ + − ∇

(24)

From hypothesis 1) and Theorem 3.1, for i=1,2, , k,

( ) ( )

( ) ( )

T T . . . . i i i i f z g v+       

is higher-order

(

, , , ,1 1 1 1

)

i i i i

V α β ρ θ -invex at u with respect to H u pi

(

,

)

, we get

( ) ( )

( )

( )

( )

( )

( )

( )

(

)

( )

(

)

(

)

( )

T T 1 T T T T T 2

1 T 1 1

,

, ,

, , , , .

i i i i

i

i i i i

i i

p i

i i

i i p i i i

f x x z f u u z

x u

g x x v g u u v

f u u z

x u H u p

g u u v

x u H u p p H u p x u

α

η

β ρ θ

 + +  −         +   ≥ ∇+ ∇  −         + − ∇ + (25)

(10)

DOI: 10.4236/ajor.2018.84017 303 American Journal of Operations Research

( ) ( )

(

( )

)

( )

(

( )

)

(

)

( )

(

)

(

)

( )

2 T T

T T

2

2 T 2 2

,

, ,

, , , , .

j j j j j

j j p j

j j p j j j

x u h x x w h u u w

x u h u u w G u p

x u G u p p G u p x u

α

η

β ρ θ

 + − +      ≥ ∇ + + ∇   + − ∇ + (26)

Adding the two inequalities after multiplying (25) by λi and (26) by µj, we obtain

( ) ( )

( )

( )

( )

( ) ( )

(

( )

)

( )

( )

( )

(

)

T T 1 T T 1

2 T T

1 T T T 1 , , , , k

i i i i

i i

i i i i i

m

j j j j j j

j

k

i i

i p i

i i i

f x x z f u u z

x u

g x x v g u u v

x u h x x w h u u w

f u u z

x u H u p

g u u v

λα µ α η λ = = =  + +  −   − −       + + − +   +   ≥ ∇+ ∇  −      

( )

(

( )

)

(

)

( )

(

)

(

)

( )

(

)

(

)

( )

( )

T T 1 T 1 2 T 1 2 2

1 1 2 2

1 1 , , , , , , , , , , . m

j j j p j

j k

i i i p i

i m

j j j p j

j

k m

i i i j j j

i j

x u h u u w G u p

x u H u p p H u p

x u G u p p G u p

x u x u

η µ

λ β

µ β

λ ρ θ µ ρ θ

= = = = =   + ∇ + + ∇   + − ∇   + − ∇ + +

(27)

Using hypothesis 3) and (21), we get

( ) ( )

( )

( )

( )

( ) ( )

(

( )

)

( )

(

)

(

)

( )

(

)

(

)

T T 1 T T 1

2 T T

1 1 T 1 2 T 1 , , , , , , , , . k

i i i i

i i

i i i i i

m

j j j j j j

j k

i i i p i

i m

j j j p j

j

f x x z f u u z

x u

g x x v g u u v

x u h x x w h u u w

x u H u p p H u p

x u G u p p G u p

λα µ α λ β µ β = = = =  + +  −   − −       + + − +   ≥ + ∇   + − ∇

(28)

Finally, using hypothesis 4) and x is feasible solution for (MFP), it follows that

( )

( )

( )

( )

(

( )

)

(

)

(

)

(

)

(

)

(

)

(

)

T T T T T

1 1 1

T 1 T 1 , , , , .

k k m

i i i i

i i j j j

i i i i i i j

k

i i p i

i m

j j p j

j

f x x z f u u z

h u u w

g x x v g u u v

H u p p H u p

G u p p G u p

λ λ µ

λ µ = = = = =  +   +  ≥ + +             + − ∇ + − ∇

This contradicts Equation (24). Hence, the result.

Theorem 4.2. (Strong duality theorem). If u X 0 is an efficient solution of (MFP) and the Slater’s constraint qualification holds. Also, if for any

1,2, , , 1,2, ,

(11)

DOI: 10.4236/ajor.2018.84017 304 American Journal of Operations Research

( )

,0 0, ,0

( )

0, ,0

( )

0, ,0

( )

0,

i j p i p j

H u = G u = ∇ H u = ∇ G u = (29)

then there exist k, , , m n n

i i

R R z R v R

λ∈ µ∈ ∈ ∈ and

, 1,2, , , 1,2, ,

n j

wR i= k j= m , such that

(

u z v, , , , , ,µ λ w p=0

)

is a

feasible solution of (MFD)1 and the objective function values of (MFP) and

(MFD)1 are equal. Furthermore, if the hypotheses of Theorem 4.1 hold for all

feasible solutions of (MFP) and (MFD)1 then,

(

u z v, , , , , ,µ λ w p=0

)

is an

efficient solution of (MFD)1.

Proof. Since u is an efficient solution of (MFP) and the Slater’s constraint qualification holds, then by Theorem 3.3, there exist

, , ,

k m n n

i i

R R z R v R

λ∈ µ∈ ∈ ∈ and n, 1,2, , , 1,2, ,

j

wR i= k j= m , such

that

( )

( )

(

( )

)

T

T T

1 1 0,

k m

i i

i j j j

i i i j

f u u z h u u w

g u u v

λ µ

= =

 + 

∇+ ∇ + =

 

(30)

( )

(

T

)

1 0,

m

j j j

j h u u w

µ

=

+ =

(31)

(

)

(

)

(

)

T | , |T , |T ,

i i i i j j

u z S u C u v S u D u w= = =S u E (32)

, , ,

i i i i j j

z C v D w∈ ∈ ∈E (33)

1

0, k 1, 0, 1,2, , , 1,2, , .

i i j

i

i k j m

λ λ µ

=

>

= ≥ = = (34)

Thus,

(

u z v, , , , , ,µ λ w p=0

)

is feasible for (MFD)1 and the objective func-

tion values of (MFP) and (MFD)1 are equal.

We now show that

(

u z v, , , , , ,µ λ w p=0

)

is an efficient solution of (MFD)1.

If not, then there exists

(

u z v′ ′ ′ ′ ′ ′ ′ =, , , , , ,µ λ w p 0

)

of (MFD)1 such that

( )

( )

(

( )

)

( )

( )

(

( )

)

T

T T

1 T

T T

1 , for all 1,2, ,

m

i i

j j j

j

i i

m

i i

j j j

j

i i

f u u z

h u u w

g u u v

f u u z

h u u w i k

g u u v

µ

µ

=

= +

+ +

− ′ + ′ ′

′ ′ ′ ′

≤ + + =

′ − ′ ′

and

( )

( )

(

( )

)

( )

( )

(

( )

)

T

T T

1 T

T T

1 , for some 1,2, , .

m

r r

j j j

j

r r

m

r r

j j j

j

r r

f u u z

h u u w

g u u v

f u u z

h u u w r k

g u u v

µ

µ

=

= +

+ +

− ′ + ′ ′

′ ′ ′ ′

< + + =

′ − ′ ′

By equation (31), we obtain

( )

( )

( )

( )

(

( )

)

T T

T

T T

1 , for all 1,2, ,

m

i i i i

j j j

j

i i i i

f u u z f u u z

h u u w i k

g u u v g u u v

′ ′ ′

+ +

′ ′ ′ ′

≤ + + =

′ ′ ′

− −

and

( )

( )

( )

( )

(

( )

)

T T

T

T T

1 , for some 1,2, , .

m

r r r r

j j j

j

r r r r

f u u z f u u z

h u u w r k

g u u v g u u v

′ ′ ′

+ +

′ ′ ′ ′

< + + =

′ ′ ′

(12)

DOI: 10.4236/ajor.2018.84017 305 American Journal of Operations Research

This contradicts the Theorem 4.1. This complete the result. Theorem 4.3. (Strict converse duality theorem). Let x X 0 and

(

u z v, , , , , ,µ λ w p

)

Z0. Let

1)

( )

( )

(

( )

)

(

)

(

)

(

)

(

)

(

)

(

)

T

T T

1 1 1

T 1

T 1

( ) ( )

, ,

, , ,

T

k k m

i i

i i

i T i j j j

i i i i i i j

k

i i p i

i m

j j p j

j

f u u z

f x x z h u u w

g x x v g u u v

H u p p H u p

G u p p G u p

λ λ µ

λ

µ

= = =

=

=

 + 

 + 

≤  + +

   

+ − ∇

+ − ∇

2) for any i=1,2, , k,

(

( ) ( )

. .T

)

i i

f + z be strictly higher-order

(

, , , ,1 1 1 1

)

i i i i

V α β ρ θ -invex at u with respect to H u pi

(

,

)

and −

(

gi

( ) ( )

. + .Tvi

)

be higher-order

(

, , , ,1 1 1 1

)

i i i i

V α β ρ θ -invex at u with respect to H u pi

(

,

)

,

3) for any j=1,2, , m,

(

hj

( ) ( )

. + .Twj

)

be higher-order

(

, , , ,2 2 2 2

)

j j j j

V α β ρ θ -invex at u with respect to G u pj

(

,

)

,

4) 1 1

(

)

2 2 2

(

)

2

1 , 1 , 0.

k m

i i i j j j

i x u j x u

λ ρ θ µ ρ θ

= =

+ ≥

5) 1

(

,

)

2

(

,

)

1

(

,

)

2

(

,

)

(

, , 1,2, , ,

)

i x u j x u i x u j x u x u i k

α =α =β =β =α ∀ = 

1,2, , .

j=  m

Then, x u= .

Proof. Using hypothesis 2) and Theorem 3.2, we have

(

) ( )

( )

( )

( )

(

)

( )

( )

(

)

(

)

(

)

(

)

(

)

T T

1

T T

T T

T

2

1 T 1 1

,

, ,

, , , , .

i i i i

i

i i i i

i i

p i

i i

i i p i i i

f x x z f u u z

x u

g x x v g u u v

f u u z

x u H u p

g u u v

x u H u p p H u p x u

α

η

β ρ θ

 + + 

 

− −

 

 

  +  

> ∇+ ∇ 

   

 

 

+ − ∇ +

(35)

For any j=1,2, , m,

(

hj

( ) ( )

. + .Twj

)

is higher-order

(

V, , , ,α β ρ θ2j 2j 2j 2j

)

-invex at u with respect to G u pj

(

,

)

, we have

(

) ( )

(

( )

)

(

)

(

( )

)

(

)

(

)

(

)

(

)

(

)

2 T T

T T

2

2 T 2 2

,

, ,

, , , , .

j j j j j

j j p j

j j p j j j

x u h x x w h u u w

x u h u u w G u p

x u G u p p G u p x u

α

η

β ρ θ

 + − + 

 

 

∇ + + ∇

 

+ − ∇ +

(36)

Adding the two inequalities after multiplying (35) by λi and (36) by µj, we obtain

(

) ( )

( )

( )

( )

(

) ( )

(

( )

)

(

)

( )

( )

(

)

T T

1

T T

1

2 T T

1

T T

T 1

,

,

, ,

k

i i i i

i i

i i i i i

m

j j j j j j

j

k

i i

i p i

i i i

f x x z f u u z

x u

g x x v g u u v

x u h x x w h u u w

f u u z

x u H u p

g u u v

λα

µ α

η λ

=

=

=

 + + 

 

 

 

+ + − +

  +  

> ∇− ∇ 

   

 

(13)

DOI: 10.4236/ajor.2018.84017 306 American Journal of Operations Research

(

)

(

( )

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

T T

1

1 T

1

2 T

1

2 2

1 1 2 2

1 1

, ,

, , ,

, , ,

, , .

m

j j j p j

j k

i i i p i

i m

j j j p j

j

k m

i i i j j j

i j

x u h u u w G u p

x u H u p p H u p

x u G u p p G u p

x u x u

η µ

λ β

µ β

λ ρ θ µ ρ θ

=

=

=

= =

 

+ ∇ + + ∇

 

+ − ∇

 

+ − ∇

+ +

(37)

Using hypothesis 3) and (21), we get

(

) ( )

( )

( )

( )

(

) ( )

(

( )

)

(

)

(

)

(

)

(

)

(

)

(

)

T T

1

T T

1

2 T T

1

1 T

1

2 T

1

,

,

, , ,

, , , .

k

i i i i

i i

i i i i i

m

j j j j j j

j k

i i i p i

i m

j j j p j

j

f x x z f u u z

x u

g x x v g u u v

x u h x x w h u u w

x u H u p p H u p

x u G u p p G u p

λα

µ α

λ β

µ β

=

=

=

=

 + + 

 

− −

 

 

 

+ + − +

 

> − ∇

 

+ − ∇

(38)

Finally, using hypothesis 4) and x is feasible solution for (MFP), it follows that

( )

( )

( )

( )

(

( )

)

(

)

(

)

(

)

(

)

(

)

(

)

T T

T

T T

1 1 1

T 1

T 1

, ,

, , .

k k m

i i i i

i i j j j

i i i i i i j

k

i i p i

i m

j j p j

j

f x x z f u u z

h u u w

g x x v g u u v

H u p p H u p

G u p p G u p

λ λ µ

λ

µ

= = =

=

=

 +   + 

> + +

   

 

   

+ − ∇

+ − ∇

This contradicts the hypothesis 1). Hence, the result.

5. Duality Model-II

Consider the following dual (MFD)2 of (MFP): (MFD)2 Maximize

( )

( )

(

( )

)

( )

( )

(

( )

)

T T

1 1 T T

T T

1 1

1 1

, ,

m m

k k

j j j j j j

j k k j

f u u z f u u z

h u u w h u u w

g u u v = µ g u u v

 + + 

+ + + +

 

− −

 

subject to

( )

( )

(

( )

)

(

)

(

)

T

T T

1 1

1 , 1 , 0,

k m

i i

i j j j

i i i j

k m

i p i j p j

i j

f u u z

h u u w g u u v

H u p G u p

λ µ

λ µ

= =

= =

 + 

∇+ ∇ +

 

+ ∇ + ∇ =

(39)

(

)

(

)

(

T

)

(

(

)

T

(

)

)

1 , , 1 , , 0,

k m

i i p i j j p j

i H u p p H u p j G u p p G u p

λ µ

= =

− ∇ + − ∇ ≥

(40)

, , , 1,2, , , 1,2, , ,

i i i i j j

References

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