Duality Relations for a Class of a Multiobjective Fractional Programming Problem Involving Support Functions

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.

• 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 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.

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