AJOR  Vol.8 No.4 , July 2018
Duality Relations for a Class of a Multiobjective Fractional Programming Problem Involving Support Functions
ABSTRACT
In this article, for a differentiable function , we introduce the definition of the higher-order -invexity. Three duality models for a multiobjective fractional programming problem involving nondifferentiability in terms of support functions have been formulated and usual duality relations have been established under the higher-order -invex assumptions.

1. Introduction

Consider the following nonlinear programming problem (P) Minimize f ( x ) subject to g ( x ) 0 , where f : R n R and g : R n R are twice differen- tiable functions. The Mangasarian [1] second-order dualof (P) is (DP) Maximize

f ( u ) y T g ( u ) 1 2 p T 2 [ f ( u ) y T g ( u ) ] p

such that [ f ( u ) y T g ( u ) ] + 2 [ f ( u ) y T g ( u ) ] p = 0

By introducing two differentiable functions H : R n × R n R and K : R n × R n R m , Mangasarian [1] formulated the following higher-order dual of (P): (DP)1 Maximize

f ( u ) y T g ( u ) + H ( u , p ) y T K ( u , p )

such that p H ( u , p ) p [ y T K ( u , p ) ] = 0 , y 0, where p H ( u , p ) denotes the n × 1 gradient of H ( u , p ) with respect to p and p ( y T K ( u , p ) ) denotes the n × 1 , gradient of y T K ( u , p ) with respect to p.

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

G ( x ) = ( f 1 ( x ) g 1 ( x ) , f 2 ( x ) g 2 ( x ) , , f k ( x ) g k (x)) )

subject to

x X 0 = { x X R n : h j ( x ) 0, j M } ,

where f = ( f 1 , f 2 , , f k ) : X R k , g = ( g 1 , g 2 , , g k ) : X R k and h = ( h 1 , h 2 , , h m ) : X R m 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

G ( x ) = ( f 1 ( x ) + S ( x | C 1 ) , f 2 ( x ) + S ( x | C 2 ) , , f k ( x ) + S ( x | C k ) )

subject to x X 0 = { x X R n : g j ( x ) + S ( x | E j ) 0 , j = 1 , 2 , , m } , where f i : X R ( i = 1 , 2 , , k ) and g j : X R ( j = 1 , 2 , , m ) are differentiable func- tions. C i and E j are compact convex sets in R n and S ( x | C i ) ( i = 1 , 2 , , k ) and S ( x | E j ) ( j = 1 , 2 , , 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 ) - invex with respect to differentiable function H : R n × R n R . We also construct a nontrivial numerical example which illustrates the existence of such a function. We then formulate three higher-order dual problems corresponding to the multiobjective nondifferentiable fractional programming problem. Further, we establish usual duality relations for these primal-dual pairs under aforesaid assumptions.

2. Preliminaries

Let X R n be an open set and ϕ : X R , H : X × R n R be differentiable functions. α , β : X × X R + \ { 0 } , η : X × X R n , ρ R n and θ : X × X R n .

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 ,

α ( x , u ) [ ϕ ( x ) ϕ ( u ) ] ( > ) η T ( x , u ) ( ϕ ( u ) + p H ( u , p ) ) + β ( x , u ) [ H ( u , p ) p T p H ( u , p ) ] + ρ θ ( x , u ) 2 .

Example 2.1. Let ϕ : R R be such that ϕ ( x ) = x 4 + x 2 + 1 .

Let

η ( x , u ) = 1 2 ( x 2 + u 2 ) , H ( u , p ) = 2 p ( x + 1 ) 2 .

Also, suppose

α ( x , u ) = 1 , β ( x , u ) = 2 , ρ = 1 , θ ( x , u ) = ( x 2 + u 2 ) 1 2 .

Now,

ξ = α ( x , u ) [ ϕ ( x ) ϕ ( u ) ] η T ( x , u ) ( ϕ ( u ) + p H ( u , p ) ) β ( x , u ) [ H ( u , p ) p T p H ( u , p ) ] ρ θ ( x , u ) 2 .

ξ = ( x 4 + x 2 u 4 u 2 ) 1 2 ( x 2 + u 2 ) [ 4 u 3 + 2 u 2 ( u + 1 ) 2 ] ( x 2 + u 2 )

ξ = x 4 + x 2 (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 , ρ ) -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 H ( u , p ) = 1 2 p T 2 ϕ ( u ) p , α ( x , u ) = 0 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 1 , f 2 , , f k ) : X R k is ( V , α , β , ρ , θ ) -invex if for all i = 1 , 2 , , k , f i is ( V , α i , β i , ρ i , θ i ) -invex.

Definition 2.2. [12] . Let C be a compact convex set in R n . The support function of C is defined by

S ( x | C ) = max { x T y : y C } .

3. Problem Formulation

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

F ( x ) = { f 1 ( x ) + S ( x | C 1 ) g 1 ( x ) S ( x | D 1 ) , f 2 ( x ) + S ( x | C 2 ) g 2 ( x ) S ( x | D 2 ) , , f k ( x ) + S ( x | C k ) g k ( x ) S ( x | D k ) }

subject to x X 0 = { x X R n : h j ( x ) + S ( x | E j ) 0 , j = 1 , 2 , , m } ,

where f = ( f 1 , f 2 , , f k ) : X R k , g = ( g 1 , g 2 , , g k ) : X R k and h = ( h 1 , h 2 , , h m ) : X R m are differentiable on X, f i ( . ) + S ( . | C i ) 0 and g i ( . ) S ( . | D i ) > 0 . Let H i : X × R n R be differentiable functions, C i , D i and E j are compact convex sets in R n , for all i = 1 , 2 , , k , j = 1 , 2 , , m .

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

i = 1 , 2 , , k , f i ( x ) + S ( x | C i ) g i ( x ) S ( x | D i ) f i ( x 0 ) + S ( x 0 | C i ) g i ( x 0 ) S ( x 0 | D i )

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

f r ( x ) + S ( x | C r ) g r ( x ) S ( x | D r ) < f r ( x 0 ) + S ( x 0 | C r ) g r ( x 0 ) S ( x 0 | D r ) .

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

Theorem 3.1. For some t, if f t ( . ) + ( . ) T z t and ( g t ( . ) ( . ) T v t ) are higher- order ( V , α t , β t , ρ t , θ t ) -invex at u with respect to H t ( u , p ) for same η ( x , u ) . Then, the fractional function ( f t ( . ) + ( . ) T z t g t ( . ) ( . ) T v t ) is higher-order ( V , α ¯ t , β ¯ t , ρ ¯ t , θ ¯ t ) -invex at u with respect to H ¯ t ( u , p ) , where

α ¯ t ( x , u ) = ( g t ( x ) x T v t g t ( u ) u T v t ) α t ( x , u ) , β ¯ t ( x , u ) = β t ( x , u ) ,

θ ¯ t ( x , u ) = θ t ( x , u ) ( 1 g t ( u ) u T v t + f t ( u ) + u T z t ( g t ( u ) u T v t ) 2 ) 1 2 , ρ ¯ t ( x , u ) = ρ t ( x , u )

and

H ¯ t ( u , p ) = ( 1 g t ( u ) u T v t + f t ( u ) + u T z t ( g t ( u ) u T v t ) 2 ) H t ( u , p ) .

Theorem 3.2. In Theorem 3.1,if either ( g t ( . ) ( . ) T v t ) is strictly higher- order ( V , α t , β t , ρ t , θ t ) -invex at u with respect to H t ( u , p ) and ( f t ( . ) ( . ) T z t ) > 0 or ( f t ( . ) ( . ) T z t ) is strictly higher-order ( V , α t , β t , ρ t , θ t ) - invex at u with respect to H t ( u , p ) , then ( f t ( . ) + ( . ) T z t g t ( . ) ( . ) T z t ) is strictly higher- order ( V , α ¯ t , β ¯ t , ρ ¯ t , θ ¯ t ) -invex at u X with respect to H ¯ t ( u , p ) .

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 λ ¯ i > 0 , μ ¯ j R m , z ¯ i R n , v ¯ i R n and w ¯ j R m , i = 1 , 2 , , k , j = 1 , 2 , , m , such that

i = 1 k λ ¯ i ( f i ( x ¯ ) + x ¯ T z ¯ i g i ( x ¯ ) x ¯ T v ¯ i ) + j = 1 m μ ¯ j ( h j ( x ¯ ) + x ¯ T w ¯ j ) = 0 , (1)

j = 1 m μ ¯ j ( h j ( x ¯ ) + x ¯ T w ¯ j ) = 0, (2)

x ¯ T z ¯ i = S ( x ¯ | C i ) , z ¯ i C i , i = 1 , 2 , , k , (3)

x ¯ T v ¯ i = S ( x ¯ | D i ) , v ¯ i D i , i = 1,2, , k , (4)

x ¯ T w ¯ j = S ( x ¯ | E j ) , w ¯ j E j , j = 1 , 2 , , m , (5)

λ ¯ i > 0 , i = 1 , 2 , , k , μ ¯ j 0 , j = 1 , 2 , , m . (6)

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

i = 1 k λ i ( f i ( u ) + u T z i g i ( u ) u T v i ) + j = 1 m μ j ( h j ( u ) + u T w j ) = 0 , (7)

j = 1 m μ j ( h j ( u ) + u T w j ) = 0, (8)

u T z i = S ( u | C i ) , z i C i , i = 1 , 2 , , k , (9)

u T v i = S ( u | D i ) , v i D i , i = 1 , 2 , , k , (10)

u T w j = S ( u | E j ) , w j E j , j = 1 , 2 , , m , (11)

λ ¯ i > 0 , i = 1 , 2 , , k , μ ¯ j 0 , j = 1 , 2 , , m . (12)

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

1) ( f i ( . ) + ( . ) T z i ) and ( g i ( . ) ( . ) T v i ) be higher-order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) - invex at u with respect to H i ( u , p ) ,

2) ( h j ( . ) + ( . ) T w j ) be higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u with res- pect to G j ( u , p ) ,

3) i = 1 k λ i ρ ¯ i 1 θ ¯ i 1 ( x , u ) 2 + j = 1 m μ j ρ j 2 θ j 2 ( x , u ) 2 0 ,

4) i = 1 k λ i ( p H ¯ i ( u , p ) ) + j = 1 m μ j ( p G j ( u , p ) ) = 0 , i = 1 k λ i ( H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ) 0 and j = 1 m μ j ( G j ( u , p ) p T p G j ( u , p ) ) 0 ,

5) α i 1 ( x , u ) = α j 2 ( x , u ) = β i 1 ( x , u ) = β j 2 ( x , u ) = α ( x , u ) ,

where

α ¯ i ( x , u ) = ( g i ( x ) x T v i g i ( u ) u T v i ) α i ( x , u ) , β ¯ i ( x , u ) = β i ( x , u ) ,

θ ¯ i ( x , u ) = θ i ( x , u ) ( 1 g i ( u ) u T v i + f i ( u ) + u T z i ( g i ( u ) u T v i ) 2 ) 1 2

and ρ ¯ i ( x , u ) = ρ i ( x , u ) .

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

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

f i ( x ) + S ( x | C i ) g i ( x ) S ( x | D i ) f i ( u ) + S ( u | C i ) g i ( u ) S ( u | D i ) , forall i = 1 , 2 , , k

and

f r ( x ) + S ( x | C r ) g r ( x ) S ( x | D r ) < f r ( u ) + S ( u | C r ) g r ( u ) S ( u | D r ) , for some r = 1 , 2 , , k ,

which implies

f i ( x ) + x T z i g i ( x ) x T v i f i ( x ) + S ( x | C i ) g i ( x ) S ( x | D i ) f i ( u ) + S ( u | C i ) g i ( u ) S ( u | D i ) = f i ( u ) + u T z i g i ( u ) u T v i , forall i = 1 , 2 , , k (13)

and

f r ( x ) + x T z r g r ( x ) x T v r f r ( x ) + S ( x | C r ) g r ( x ) S ( x | D r ) < f r ( u ) + S ( u | C r ) g r ( u ) S ( u | D r ) = f r ( u ) + u T z r g r ( u ) u T v r , for some r = 1 , 2 , , k . (14)

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

i = 1 k λ i ( f i ( x ) + x T z i g i ( x ) x T v i f i ( u ) + u T z i g i ( u ) u T v i ) < 0. (15)

From Theorem 3.1, for each i , 1 i k , ( f i ( . ) + ( . ) T z i g i ( . ) ( . ) T v i )

is higher-order ( V , α ¯ i 1 , β ¯ i 1 , ρ ¯ i 1 , θ ¯ i 1 ) -invex at u X 0 with respect to H ¯ i ( u , p ) , we have

α ¯ i 1 ( x , u ) [ f i ( x ) + x T z i g i ( x ) x T v i f i ( u ) + u T z i g i ( u ) u T v i ]

η T ( x , u ) [ ( f i ( u ) + u T z i g i ( u ) u T v i ) + p H ¯ i ( u , p ) ] + β ¯ i 1 ( x , u ) [ H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ] + ρ ¯ i 1 θ ¯ i 1 ( x , u ) 2 . (16)

where

α ¯ i ( x , u ) = ( g i ( x ) x T v i g i ( u ) u T v i ) α i ( x , u ) , β ¯ i ( x , u ) = β i ( x , u ) ,

θ ¯ i ( x , u ) = θ i ( x , u ) ( 1 g i ( u ) u T v i + f i ( u ) + u T z i ( g i ( u ) u T v i ) 2 ) 1 2 , ρ ¯ i ( x , u ) = ρ i ( x , u )

and H ¯ i ( u , p ) = ( 1 g i ( u ) u T v i + f i ( u ) + u T z i ( g i ( u ) u T v i ) 2 ) H i ( u , p ) .

By hypothesis 2), we get

α j 2 ( x , u ) [ h j ( x ) + x T w j ( h j ( u ) + u T w j ) ] η T ( x , u ) [ ( h j ( u ) + u T w j ) + p G j ( u , p ) ] + β j 2 ( x , u ) [ G j ( u , p ) p T p G j ( u , p ) ] + ρ j 2 θ j 2 ( x , u ) 2 . (17)

Adding the two inequalities after multiplying (16) by λ i and (17) by μ j , we obtain

i = 1 k λ i α ¯ i 1 ( x , u ) [ f i ( x ) + x T z i g i ( x ) x T v i f i ( u ) + u T z i g i ( u ) u T v i ] + j = 1 m μ j α j 2 ( x , u ) [ h j ( x ) + x T w j ( h j ( u ) + u T w j ) ] η T ( x , u ) i = 1 k λ i [ ( f i ( u ) + u T z i g i ( u ) u T v i ) + p H ¯ i ( u , p ) ]

+ η T ( x , u ) j = 1 m μ j [ ( h j ( u ) + u T w j ) + p G j ( u , p ) ] + i = 1 k λ i β ¯ i ( x , u ) [ H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ] + j = 1 m μ j β j 2 ( x , u ) [ G j ( u , p ) p T p G j ( u , p ) ] + i = 1 k λ i ρ ¯ i 1 θ ¯ i 1 ( x , u ) 2 + j = 1 m μ j ρ j 2 θ j 2 ( x , u ) 2 . (18)

Using hypothesis 3)-4), we get

i = 1 k λ i [ f i ( x ) + x T z i g i ( x ) x T v i f i ( u ) + u T z i g i ( u ) u T v i ] + j = 1 m μ j [ h j ( x ) + x T w j ( h j ( u ) + u T w j ) ] η T ( x , u ) i = 1 k λ i ( f i ( u ) + u T z i g i ( u ) u T v i ) + η T ( x , u ) j = 1 m μ j ( h j ( u ) + u T w j ) . (19)

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

i = 1 k λ i [ f i ( x ) + x T z i g i ( x ) x T v i f i ( u ) + u T z i g i ( u ) u T v i ] + j = 1 m μ j [ h j ( x ) + x T w j ] 0. (20)

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

i = 1 k λ i ( f i ( x ) + x T z i g i ( x ) x T v i ) i = 1 k λ i ( f i ( u ) + u T z i g i ( u ) u T v i ) .

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

[ f 1 ( u ) + u T z 1 g 1 ( u ) u T v 1 + j = 1 m μ j ( h j ( u ) + u T w j ) + ( H ¯ 1 ( u , p ) p T p H ¯ 1 ( u , p ) ) + j = 1 m μ j ( G j ( u , p ) p T p G j ( u , p ) ) , , f k ( u ) + u T z k g k ( u ) u T v k + j = 1 m μ j ( h j ( u ) + u T w j ) + ( H ¯ k ( u , p ) p T p H ¯ k ( u , p ) ) + j = 1 m μ j ( G j ( u , p ) p T p G j ( u , p ) ) ]

subject to

i = 1 k λ i ( f i ( u ) + u T z i g i ( u ) u T v i ) + j = 1 m μ j ( h j ( u ) + u T w j ) + i = 1 k λ i p H ¯ i ( u , p ) + j = 1 m μ j p G j ( u , p ) = 0 , (21)

z i C i , v i D i , w j E j , i = 1 , 2 , , k , j = 1 , 2 , , m ,

μ j 0 , λ i > 0 , i = 1 k λ i = 1 , i = 1 , 2 , , k , j = 1 , 2 , , m .

Let Z 0 be feasible solution for (MFD)1.

Theorem 4.1. (Weak duality theorem). Let x X 0 and ( u , z , v , μ , λ , w , p ) Z 0 . Suppose that

1) for any i = 1 , 2 , , k , ( f i ( . ) + ( . ) T z i ) and ( g i ( . ) ( . ) T v i ) are higher- order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) -invex at u with respect to H i ( u , p ) ,

2) for any j = 1 , 2 , , m , ( h j ( . ) + ( . ) T w j ) is higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u with respect to G j ( u , p ) ,

3) i = 1 k λ i ρ ¯ i 1 θ ¯ i 1 ( x , u ) 2 + j = 1 m μ j ρ j 2 θ j 2 ( x , u ) 2 0.

4) α ¯ i 1 ( x , u ) = α j 2 ( x , u ) = β i 1 ( x , u ) = β j 2 ( x , u ) = α ( x , u ) , i = 1 , 2 , , k , j = 1 , 2 , , m ,

where α ¯ t ( x , u ) = ( g t ( x ) x T v t g t ( u ) u T v t ) α t ( x , u ) , β ¯ t ( x , u ) = β t ( x , u ) , θ ¯ t ( x , u ) = θ t ( x , u ) ( 1 g t ( u ) u T v t + f t ( u ) + u T z t ( g t ( u ) u T v t ) 2 ) 1 2 , ρ ¯ t ( x , u ) = ρ t ( x , u ) and H ¯ t ( u , p ) = ( 1 g t ( u ) u T v t + f t ( u ) + u T z t ( g t ( u ) u T v t ) 2 ) H t ( u , p ) .

Then, the following cannot hold

f i ( x ) + S ( x | C i ) g i ( x ) S ( x | D i ) f i ( u ) + u T z i g i ( u ) u T v i + j = 1 m μ j ( h j ( u ) + u T w j ) + ( H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ) + j = 1 m μ j ( G j ( u , p ) p T p G j ( u , p ) ) , forall i = 1 , 2 , , k (22)

and

f r ( x ) + S ( x | C r ) g r ( x ) S ( x | D r ) < f r ( u ) + u T z r g r ( u ) u T v r + j = 1 m μ j ( h j ( u ) + u T w j ) + ( H ¯ r ( u , p ) p T p H ¯ r ( u , p ) ) + j = 1 m μ j ( G j ( u , p ) p T p G j ( u , p ) ) , for some r = 1 , 2 , , k . (23)

Proof. Suppose that (22) and (23) hold, then using λ i > 0 , i = 1 k λ i = 1 , x T z i S ( x | C i ) , x T v i S ( x | D i ) , i = 1 , 2 , , k , we have

i = 1 k λ i ( f i ( x ) + x T z i g i ( x ) x T v i ) < i = 1 k λ i ( f i ( u ) + u T z i g i ( u ) u T v i ) + j = 1 m μ j ( h j ( u ) + u T w j ) + i = 1 k λ i ( H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ) + j = 1 m μ j ( G j ( u , p ) p T p G j ( u , p ) ) . (24)

From hypothesis 1) and Theorem 3.1, for i = 1 , 2 , , k , ( f i ( . ) + ( . ) T z i g i ( . ) ( . ) T v i )

is higher-order ( V , α ¯ i 1 , β ¯ i 1 , ρ ¯ i 1 , θ ¯ i 1 ) -invex at u with respect to H ¯ i ( u , p ) , we get

α ¯ i 1 ( x , u ) [ f i ( x ) + x T z i g i ( x ) x T v i f i ( u ) + u T z i g i ( u ) u T v i ] η T ( x , u ) [ ( f i ( u ) + u T z i g i ( u ) u T v i ) + p H ¯ i ( u , p ) ] + β ¯ i 1 ( x , u ) [ H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ] + ρ ¯ i 1 θ ¯ i 1 ( x , u ) 2 . (25)

For any j = 1 , 2 , , m , ( h j ( . ) + ( . ) T w j ) is higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u with respect to G j ( u , p ) , we have

α j 2 ( x , u ) [ h j ( x ) + x T w j ( h j ( u ) + u T w j ) ] η T ( x , u ) [ ( h j ( u ) + u T w j ) + p G j ( u , p ) ] + β j 2 ( x , u ) [ G j ( u , p ) p T p G j ( u , p ) ] + ρ j 2 θ j 2 ( x , u ) 2 . (26)

Adding the two inequalities after multiplying (25) by λ i and (26) by μ j , we obtain

i = 1 k λ i α ¯ i 1 ( x , u ) [ f i ( x ) + x T z i g i ( x ) x T v i f i ( u ) + u T z i g i ( u ) u T v i ] + j = 1 m μ j α j 2 ( x , u ) [ h j ( x ) + x T w j ( h j ( u ) + u T w j ) ] η T ( x , u ) i = 1 k λ i [ ( f i ( u ) + u T z i g i ( u ) u T v i ) + p H ¯ i ( u , p ) ]

+ η T ( x , u ) j = 1 m μ j [ ( h j ( u ) + u T w j ) + p G j ( u , p ) ] + i = 1 k λ i β ¯ i ( x , u ) [ H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ] + j = 1 m μ j β j 2 ( x , u ) [ G j ( u , p ) p T p G j ( u , p ) ] + i = 1 k λ i ρ ¯ i 1 θ ¯ i 1 ( x , u ) 2 + j = 1 m μ j ρ j 2 θ j 2 ( x , u ) 2 . (27)

Using hypothesis 3) and (21), we get

i = 1 k λ i α ¯ i 1 ( x , u ) [ f i ( x ) + x T z i g i ( x ) x T v i f i ( u ) + u T z i g i ( u ) u T v i ] + j = 1 m μ j α j 2 ( x , u ) [ h j ( x ) + x T w j ( h j ( u ) + u T w j ) ] i = 1 k λ i β ¯ i 1 ( x , u ) [ H ¯ i ( u , p ) + p T p H ¯ i ( u , p ) ] + j = 1 m μ j β j 2 ( x , u ) [ G j ( u , p ) p T p G j ( u , p ) ] . (28)

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

i = 1 k λ i ( f i ( x ) + x T z i g i ( x ) x T v i ) i = 1 k λ i ( f i ( u ) + u T z i g i ( u ) u T v i ) + j = 1 m μ j ( h j ( u ) + u T w j ) + i = 1 k λ i ( H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ) + j = 1 m μ j ( G j ( u , p ) p T p G j ( u , p ) ) .

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 i = 1 , 2 , , k , j = 1 , 2 , , m ,

H ¯ i ( u ¯ , 0 ) = 0 , G j ( u ¯ , 0 ) = 0 , p H ¯ i ( u ¯ , 0 ) = 0 , p G j ( u ¯ , 0 ) = 0 , (29)

then there exist λ ¯ R k , μ ¯ R m , z ¯ i R n , v ¯ i R n and w ¯ j R n , i = 1 , 2 , , k , j = 1 , 2 , , 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 λ ¯ R k , μ ¯ R m , z ¯ i R n , v ¯ i R n and w ¯ j R n , i = 1 , 2 , , k , j = 1 , 2 , , m , such that

i = 1 k λ ¯ i ( f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ) + j = 1 m μ ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) = 0 , (30)

j = 1 m μ ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) = 0 , (31)

u ¯ T z ¯ i = S ( u ¯ | C i ) , u ¯ T v ¯ i = S ( u ¯ | D i ) , u ¯ T w ¯ j = S ( u ¯ | E j ) , (32)

z ¯ i C i , v ¯ i D i , w ¯ j E j , (33)

λ ¯ i > 0 , i = 1 k λ ¯ i = 1 , μ ¯ j 0 , i = 1 , 2 , , k , j = 1 , 2 , , m . (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

f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i + j = 1 m μ ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) f i ( u ) + u T z i g i ( u ) u T v i + j = 1 m μ j ( h j ( u ) + u T w j ) , forall i = 1 , 2 , , k

and

f r ( u ¯ ) + u ¯ T z ¯ r g r ( u ¯ ) u ¯ T v ¯ r + j = 1 m μ ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) < f r ( u ) + u T z r g r ( u ) u T v r + j = 1 m μ j ( h j ( u ) + u T w j ) , for some r = 1 , 2 , , k .

By equation (31), we obtain

f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i f i ( u ) + u T z i g i ( u ) u T v i + j = 1 m μ j ( h j ( u ) + u T w j ) , forall i = 1 , 2 , , k

and

f r ( u ¯ ) + u ¯ T z ¯ r g r ( u ¯ ) u ¯ T v ¯ r < f r ( u ) + u T z r g r ( u ) u T v r + j = 1 m μ j ( h j ( u ) + u T w j ) , for some r = 1 , 2 , , k .

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 ¯ ) Z 0 . Let

1) i = 1 k λ ¯ i ( f i ( x ¯ ) + x ¯ T z ¯ i g i ( x ¯ ) x ¯ T v ¯ i ) i = 1 k λ ¯ i ( f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ) + j = 1 m μ ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) + i = 1 k λ ¯ i ( H ¯ i ( u ¯ , p ¯ ) p ¯ T p H ¯ i ( u ¯ , p ¯ ) ) + j = 1 m μ ¯ j ( G j ( u ¯ , p ¯ ) p ¯ T p G j ( u ¯ , p ¯ ) ) ,

2) for any i = 1 , 2 , , k , ( f i ( . ) + ( . ) T z ¯ i ) be strictly higher-order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) -invex at u ¯ with respect to H i ( u ¯ , p ¯ ) and ( g i ( . ) + ( . ) T v ¯ i ) be higher-order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) -invex at u ¯ with respect to H i ( u ¯ , p ¯ ) ,

3) for any j = 1 , 2 , , m , ( h j ( . ) + ( . ) T w j ) be higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u ¯ with respect to G j ( u ¯ , p ¯ ) ,

4) i = 1 k λ ¯ i ρ ¯ i 1 θ ¯ i 1 ( x ¯ , u ¯ ) 2 + j = 1 m μ ¯ j ρ j 2 θ j 2 ( x ¯ , u ¯ ) 2 0.

5) α ¯ i 1 ( x ¯ , u ¯ ) = α j 2 ( x ¯ , u ¯ ) = β i 1 ( x ¯ , u ¯ ) = β j 2 ( x ¯ , u ¯ ) = α ( x ¯ , u ¯ ) , i = 1 , 2 , , k , j = 1 , 2 , , m .

Then, x ¯ = u ¯ .

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

α ¯ i 1 ( x ¯ , u ¯ ) [ f i ( x ¯ ) + x ¯ T z ¯ i g i ( x ¯ ) x ¯ T v ¯ i f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ] > η T ( x ¯ , u ¯ ) [ ( f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ) + p H ¯ i ( u ¯ , p ¯ ) ] + β ¯ i 1 ( x ¯ , u ¯ ) [ H ¯ i ( u ¯ , p ¯ ) p ¯ T p H ¯ i ( u ¯ , p ¯ ) ] + ρ ¯ i 1 θ ¯ i 1 ( x ¯ , u ¯ ) 2 . (35)

For any j = 1 , 2 , , m , ( h j ( . ) + ( . ) T w j ) is higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u with respect to G j ( u ¯ , p ¯ ) , we have

α j 2 ( x ¯ , u ¯ ) [ h j ( x ¯ ) + x ¯ T w ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) ] η T ( x ¯ , u ¯ ) [ ( h j ( u ¯ ) + u ¯ T w ¯ j ) + p G j ( u ¯ , p ¯ ) ] + β j 2 ( x ¯ , u ¯ ) [ G j ( u ¯ , p ¯ ) p ¯ T p G j ( u ¯ , p ¯ ) ] + ρ j 2 θ j 2 ( x ¯ , u ¯ ) 2 . (36)

Adding the two inequalities after multiplying (35) by λ ¯ i and (36) by μ ¯ j , we obtain

i = 1 k λ ¯ i α ¯ i 1 ( x ¯ , u ¯ ) [ f i ( x ¯ ) + x ¯ T z ¯ i g i ( x ¯ ) x ¯ T v ¯ i f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ] + j = 1 m μ ¯ j α j 2 ( x ¯ , u ¯ ) [ h j ( x ¯ ) + x ¯ T w ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) ] > η T ( x ¯ , u ¯ ) i = 1 k λ ¯ i [ ( f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ) p H i ( u ¯ , p ¯ ) ]

+ η T ( x ¯ , u ¯ ) j = 1 m μ ¯ j [ ( h j ( u ¯ ) + u ¯ T w ¯ j ) + p G j ( u ¯ , p ¯ ) ] + i = 1 k λ ¯ i β ¯ i 1 ( x ¯ , u ¯ ) [ H ¯ i ( u ¯ , p ¯ ) p ¯ T p H ¯ i ( u ¯ , p ¯ ) ] + j = 1 m μ ¯ j β j 2 ( x ¯ , u ¯ ) [ G j ( u ¯ , p ¯ ) p ¯ T p G j ( u ¯ , p ¯ ) ] + i = 1 k λ ¯ i ρ ¯ i 1 θ ¯ i 1 ( x ¯ , u ¯ ) 2 + j = 1 m μ ¯ j ρ j 2 θ j 2 ( x ¯ , u ¯ ) 2 . (37)

Using hypothesis 3) and (21), we get

i = 1 k λ ¯ i α ¯ i 1 ( x ¯ , u ¯ ) [ f i ( x ¯ ) + x ¯ T z ¯ i g i ( x ¯ ) x ¯ T v ¯ i f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ] + j = 1 m μ ¯ j α j 2 ( x ¯ , u ¯ ) [ h j ( x ¯ ) + x ¯ T w ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) ] > i = 1 k λ ¯ i β ¯ i 1 ( x ¯ , u ¯ ) [ H ¯ i ( u ¯ , p ¯ ) p ¯ T p H ¯ i ( u ¯ , p ¯ ) ] + j = 1 m μ ¯ j β j 2 ( x ¯ , u ¯ ) [ G j ( u ¯ , p ¯ ) p ¯ T p G j ( u ¯ , p ¯ ) ] . (38)

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

i = 1 k λ ¯ i ( f i ( x ¯ ) + x ¯ T z ¯ i g i ( x ¯ ) x ¯ T v ¯ i ) > i = 1 k λ ¯ i ( f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ) + j = 1 m μ ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) + i = 1 k λ ¯ i ( H ¯ i ( u ¯ , p ¯ ) p ¯ T p H ¯ i ( u ¯ , p ¯ ) ) + j = 1 m μ ¯ j ( G j ( u ¯ , p ¯ ) p ¯ T p G j ( u ¯ , p ¯ ) ) .

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

5. Duality Model-II

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

[ f 1 ( u ) + u T z 1 g 1 ( u ) u T v 1 + j = 1 m μ j ( h j ( u ) + u T w j ) , , f k ( u ) + u T z k g k ( u ) u T v k + j = 1 m μ j ( h j ( u ) + u T w j ) ]

subject to

i = 1 k λ i ( f i ( u ) + u T z i g i ( u ) u T v i ) + j = 1 m μ j ( h j ( u ) + u T w j ) + i = 1 k λ i p H i ( u , p ) + j = 1 m μ j p G j ( u , p ) = 0 , (39)

i = 1 k λ i ( H i ( u , p ) p T p H i ( u , p ) ) + j = 1 m μ j ( G j ( u , p ) p T p G j ( u , p ) ) 0 , (40)

z i C i , v i D i , w j E j , i = 1 , 2 , , k , j = 1 , 2 , , m , (41)

μ j 0 , λ i > 0 , i = 1 k λ i = 1 , i = 1 , 2 , , k , j = 1 , 2 , , m . (42)

Let P 0 be the feasible solution for (MFD)2.

Theorem 5.1. (Weak duality theorem). Let x X 0 and ( u , z , v , y , λ , w , p ) P 0 . Let for i = 1 , 2 , , k , j = 1 , 2 , , m ,

1) ( f i ( . ) + ( . ) T z i g i ( . ) ( . ) T v i ) be higher-order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) -invex at u with res- pect to H i ( u , p ) ,

2) ( h j ( . ) + ( . ) T w j ) be higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u with res- pect to G j ( u , p ) ,

3) i = 1 k λ i ρ i 1 θ i 1 ( x , u ) 2 + j = 1 m μ j ρ j 2 θ j 2 ( x , u ) 2 0.

4) α i 1 ( x , u ) = α j 2 ( x , u ) = β ( x , u ) = β j 2 ( x , u ) = α ( x , u ) .

Then the following cannot hold

f i ( x ) + S ( x | C i ) g i ( x ) S ( x | D i ) f i ( u ) + u T z i g i ( u ) u T v i + j = 1 m μ j ( h j ( u ) + u T w j ) , i = 1 , 2 , , k (43)

and

f r ( x ) + S ( x | C r ) g r ( x ) S ( x | D r ) < f r ( u ) + u T z r g r ( u ) u T v r + j = 1 m μ j ( h j ( u ) + u T w j ) , for some r = 1 , 2 , , k . (44)

Proof. The proof follows on the lines of Theorem 4.1.

Theorem 5.2 (Strong duality theorem). If u ¯ X 0 is an efficient solution of (MFP) and the Slater’s constraint qualification hold. Also, if for any i = 1 , 2 , , k , j = 1 , 2 , , m ,

H i ( u ¯ , 0 ) = 0 , G j ( u ¯ , 0 ) = 0 , p H i ( u ¯ , 0 ) = 0 , p G j ( u ¯ , 0 ) = 0 , (45)

then there exist λ ¯ R k , μ ¯ R m , z ¯ i R n , v ¯ i R n and w ¯ j R n , i = 1 , 2 , , k , j = 1 , 2 , , m , such that ( u , z ¯ , v ¯ , μ ¯ , λ ¯ , w ¯ , p ¯ = 0 ) is a feasible solution of (MFD)2 and the objective function values of (MFP) and (MFD)2 are equal. Furthermore, if the conditions of Theorem 5.1 hold for all feasible solu- tions of (MFP) and (MFD)2 then, ( u , z ¯ , v ¯ , μ ¯ , λ ¯ , w ¯ , p ¯ = 0 ) is an efficient solution of (MFD)2.

Proof. The proof follows on the lines of Theorem 4.2.

Theorem 5.3. (Strict converse duality theorem). Let x ¯ X 0 and ( u ¯ , z ¯ , v ¯ , μ ¯ , λ ¯ , w ¯ , p ¯ ) P 0 . Let i = 1 , 2 , , k , j = 1 , 2 , , m ,

1) i = 1 k λ ¯ i ( f i ( x ¯ ) + x ¯ T z ¯ i g i ( x ¯ ) x ¯ T v ¯ i ) i = 1 k λ ¯ i ( f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ) + j = 1 m μ ¯ j ( h j ( u ¯ ) + u ¯ T w ¯ j ) ,

2) ( f i ( . ) + ( . ) T z ¯ i g i ( . ) ( . ) T v ¯ i ) be strictly higher-order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) -invex at u ¯ with respect to H i ( u ¯ , p ¯ ) ,

3) ( h j ( . ) + ( . ) T w j ) be higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u ¯ with respect to G j ( u ¯ , p ¯ ) ,

4) i = 1 k λ ¯ i ρ i 1 θ i 1 ( x ¯ , u ¯ ) 2 + j = 1 m μ ¯ j ρ j 2 θ j 2 ( x ¯ , u ¯ ) 2 0.

5) α i 1 ( x ¯ , u ¯ ) = α j 2 ( x ¯ , u ¯ ) = β i 1 ( x ¯ , u ¯ ) = β j 2 ( x ¯ , u ¯ ) = α ( x ¯ , u ¯ ) .

Then, x ¯ = u ¯ .

Proof. The proof follows on the lines of Theorem 4.3.

6. Duality Model-III

Consider the following dual (MFD)3 of (MFP): (MFD)3 Maximize

[ f 1 ( u ) + u T z 1 g 1 ( u ) u T v 1 + ( H ¯ 1 ( u , p ) p T p H ¯ 1 ( u , p ) ) , , f k ( u ) + u T z k g k ( u ) u T v k + ( H ¯ k ( u , p ) p T p H ¯ k ( u , p ) ) ]

subject to

i = 1 k λ i ( f i ( u ) + u T z i g i ( u ) u T v i ) + j = 1 m μ j ( h j ( u ) + u T w j ) + i = 1 k λ i p H ¯ i ( u , p ) + j = 1 m μ j p G j ( u , p ) = 0 , (46)

j = 1 m μ j [ h j ( u ) + u T w j + G j ( u , p ) p T p G j ( u , p ) ] 0 , (47)

z i C i , v i D i , w j E j , i = 1 , 2 , , k , j = 1 , 2 , , m , (48)

μ j 0 , λ i > 0 , i = 1 k λ i = 1 , i = 1 , 2 , , k , j = 1 , 2 , , m . (49)

Let S 0 be feasible solution of (MFD)3.

Theorem 6.1. (Weak duality theorem). Let x X 0 and ( u , z , v , μ , λ , w , p ) S 0 . Let i = 1 , 2 , , k , j = 1 , 2 , , m ,

1) ( f i ( . ) + ( . ) T z i ) and ( g i ( . ) ( . ) T v i ) be higher-order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) -invex at u with respect to H i ( u , p ) ,

2) ( h j ( . ) + ( . ) T w j ) be higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u with res- pect to G j ( u , p ) ,

3) i = 1 k λ i ρ ¯ i 1 θ ¯ i 1 ( x , u ) 2 + j = 1 m μ j ρ j 2 θ j 2 ( x , u ) 2 0.

4) α ¯ i 1 ( x , u ) = α j 2 ( x , u ) = β i 1 ( x , u ) = β j 2 ( x , u ) = α ( x , u ) ,

where

α ¯ t ( x , u ) = ( g t ( x ) x T v t g t ( u ) u T v t ) α t ( x , u ) , β ¯ t ( x , u ) = β t ( x , u ) ,

θ ¯ t ( x , u ) = θ t ( x , u ) ( 1 g t ( u ) u T v t + f t ( u ) + u T z t ( g t ( u ) u T v t ) 2 ) 1 2 , ρ ¯ t ( x , u ) = ρ t ( x , u )

and

H ¯ t ( u , p ) = ( 1 g t ( u ) u T v t + f t ( u ) + u T z t ( g t ( u ) u T v t ) 2 ) H t ( u , p ) .

Then, the following cannot hold

f i ( x ) + S ( x | C i ) g i ( x ) S ( x | D i ) f i ( u ) + u T z i g i ( u ) u T v i + ( H ¯ i ( u , p ) p T p H ¯ i ( u , p ) ) , forall i = 1 , 2 , , k (50)

and

f r ( x ) + S ( x | C r ) g r ( x ) S ( x | D r ) < f r ( u ) + u T z r g r ( u ) u T v r + ( H ¯ r ( u , p ) p T p H ¯ r ( u , p ) ) , for some r = 1 , 2 , , k . (51)

Proof. The proof follows on the lines of Theorem 4.1.

Theorem 6.2. (Strong duality theorem). If u ¯ X 0 is an efficient solution of (MFP) and let the Slater’s constraint qualification be satisfied. Also, if for any i = 1 , 2 , , k , j = 1 , 2 , , m ,

H ¯ i ( u ¯ , 0 ) = 0 , G j ( u ¯ , 0 ) = 0 , p H ¯ i ( u ¯ , 0 ) = 0 , p G j ( u ¯ , 0 ) = 0 , (52)

then there exist λ ¯ R k , μ ¯ R m , z ¯ i R n , v ¯ i R n and w ¯ j R n , i = 1 , 2 , , k , j = 1 , 2 , , m , such that ( u , z ¯ , v ¯ , μ ¯ , λ ¯ , w ¯ , p ¯ = 0 ) is a feasible solution of (MFD)3 and the objective function values of (MFP) and (MFD)3 are equal. Furthermore, if the conditions of Theorem 6.1 hold for all feasible solutions of (MFP) and (MFD)3 then, ( u , z ¯ , v ¯ , μ ¯ , λ ¯ , w ¯ , p ¯ = 0 ) is an efficient solution of (MFD)3.

Proof. The proof follows on the lines of Theorem 4.2.

Theorem 6.3. (Strict converse duality theorem). Let x ¯ X 0 and ( u ¯ , z ¯ , v ¯ , μ ¯ , λ ¯ , w ¯ , p ¯ ) be feasible for (MFD)3. Suppose that:

1)

i = 1 k λ ¯ i ( f i ( x ¯ ) + x ¯ T z ¯ i g i ( x ¯ ) x ¯ T v ¯ i ) i = 1 k λ ¯ i ( f i ( u ¯ ) + u ¯ T z ¯ i g i ( u ¯ ) u ¯ T v ¯ i ) + i = 1 k λ ¯ i ( H ¯ i ( x ¯ , u ¯ ) p ¯ T p H ¯ ( x ¯ , u ¯ ) ) ,

2) for any i = 1 , 2 , , k , ( f i ( . ) + ( . ) T z ¯ i ) be strictly higher-order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) -invex at u ¯ with respect to H i ( u ¯ , p ¯ ) and ( g i ( . ) + ( . ) T v ¯ i ) be higher-order ( V , α i 1 , β i 1 , ρ i 1 , θ i 1 ) -invex at u ¯ with respect to H i ( u ¯ , p ¯ ) ,

3) for any j = 1 , 2 , , m , ( h j ( . ) + ( . ) T w j ) is higher-order ( V , α j 2 , β j 2 , ρ j 2 , θ j 2 ) -invex at u ¯ with respect to G j ( u ¯ , p ¯ ) ,

4) i = 1 k λ ¯ i ρ ¯ i 1 θ ¯ i 1 ( x ¯ , u ¯ ) 2 + j = 1 m μ ¯ j ρ j 2 θ j 2 ( x ¯ , u ¯ ) 2 0.

5) α ¯ i 1 ( x ¯ , u ¯ ) = α j 2 ( x ¯ , u ¯ ) = β i 1 ( x ¯ , u ¯ ) = β j 2 ( x ¯ , u ¯ ) = α ( x ¯ , u ¯ ) , i = 1 , 2 , , k , j = 1 , 2 , , m .

Then, x ¯ = u ¯ .

Proof. The proof follows on the lines of Theorem 4.3.

7. Conclusion

In this paper, we consider a class of non differentiable multiobjective fractional programming (MFP) with higher-order terms in which each numerator and denominator of the objective function contains the support function of a compact convex set. Furthermore, various duality models for higher-order have been formulated for (MFP) and appropriate duality relations have been obtained under higher-order ( V , α , β , ρ , d ) -invexity assumptions.

Acknowledgements

The second author is grateful to the Ministry of Human Resource and Development, India for financial support, to carry this work.

NOTES

*Corresponding author.

Cite this paper
Vandana, &. , Dubey, R. , Deepmala, &. , Mishra, L. , Mishra, V. (2018) Duality Relations for a Class of a Multiobjective Fractional Programming Problem Involving Support Functions. American Journal of Operations Research, 8, 294-311. doi: 10.4236/ajor.2018.84017.
References
[1]   Mangasarian, O.L. (1975) Second and Higher-Order Duality in Nonlinear Programming. Journal of Mathematical Analysis and Applications, 51, 607-620.
https://doi.org/10.1016/0022-247X(75)90111-0

[2]   Egudo, R.R. (1988) Multiobjective Fractional Duality. Bulletin of the Australian Mathematical Society, 37, 367-378.
https://doi.org/10.1017/S0004972700026988

[3]   Gulati, T.R. and Agarwal, D. (2008) Optimality and Duality in Nondifferentiable Multiobjective Mathematical Programming Involving Higher Order (F,α,ρ,d) - Type I Functions. Journal of Applied Mathematics and Computing, 27, 345-364.
https://doi.org/10.1007/s12190-008-0069-9

[4]   Chen, X. (2004) Higher-Order Symmetric Duality in Nondifferentiable Multiobjective Programming Problems. Journal of Mathematical Analysis and Applications, 290, 423-435.
https://doi.org/10.1016/j.jmaa.2003.10.004

[5]   Suneja, S.K., Srivastava, M.K. and Bhatia, M. (2008) Higher Order Duality in Multiobjective Fractional Programming with Support Functions. Journal of Mathematical Analysis and Applications, 347, 8-17.
https://doi.org/10.1016/j.jmaa.2008.05.056

[6]   Dubey, R. and Gupta, S.K. (2016) Duality for a Nondifferentiable Multiobjective Higher-Order Symmetric Fractional Programming Problems with Cone Constraints. Journal of Nonlinear Analysis and Optimization: Theory and Applications, 7, 1-15.

[7]   Gupta, S.K., Dubey, R. and Debnath, I.P. (2017) Second-Order Multiobjective Programming Problems and Symmetric Duality Relations with Gƒ-Bonvexity. OPSEARCH, 54, 365-387.
https://doi.org/10.1007/s12597-016-0280-7

[8]   Kuk, H., Lee, G.M. and Kim, D.S. (1998) Nonsmooth Multiobjective Programs with V-ρ-Invexity. Indian Journal of Pure and Applied Mathematics, 29, 405-412.

[9]   Jeyakumar, V. and Mond, B. (1992) On Generalised Convex Mathematical Programming. Journal of the Australian Mathematical Society Series B, 34, 43-53.
https://doi.org/10.1017/S0334270000007372

[10]   Pandey, S. (1991) Duality for Multiobjective Fractional Programming Involving Generalized η-Bonvex Functions. OPSEARCH, 28, 31-43.

[11]   Gulati, T.R. and Geeta (2011) Duality in Nondifferentiable Multiobjective Fractional Programming Problem with Generalized Invexity. Journal of Applied Mathematics and Computing, 35, 103-118.
https://doi.org/10.1007/s12190-009-0345-3

[12]   Gupta, S.K., Kailey, N. and Sharma, M.K. (2010) Multiobjective Second-Order Nondifferentiable Symmetric Duality Involving (F,α,ρ,d) -Convex Function. Journal of Applied Mathematics and Informatics, 28, 1395-1408.

[13]   Dubey, R., Gupta, S.K. and Khan, M.A. (2015) Optimality and Duality Results for a Nondifferentiable Multiobjective Fractional Programming Problem. Journal of Inequalities and Applications, 354.
https://doi.org/10.1186/s13660-015-0876-0

[14]   Reddy, L.V. and Mukherjee, R.N. (1999) Some Results on Mathematical Programming with Generalized Ratio Invexity. Journal of Mathematical Analysis and Applications, 240, 299-310.
https://doi.org/10.1006/jmaa.1999.6334

 
 
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