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Higher-order symmetric duality for a class of multiobjective fractional programming problems

Journal of Inequalities and Applications20122012:142

https://doi.org/10.1186/1029-242X-2012-142

  • Received: 28 December 2011
  • Accepted: 20 June 2012
  • Published:

Abstract

In this paper, a pair of nondifferentiable multiobjective fractional programming problems is formulated. For a differentiable function, we introduce the definition of higher-order (F, α, ρ, d)-convexity, which extends some kinds of generalized convexity, such as second order F-convexity and higher-order F -convexity. Under the higher-order (F, α, ρ, d)-convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems.

Mathematics Subject Classification (2010) 90C29; 90C30; 90C46.

Keywords

  • Higher-order symmetric duality
  • multiobjective fractional programming
  • higher-order (F, α, ρ, d)-convexity.

Introduction

Symmetric duality in nonlinear programming in which the dual of the dual is the primal was introduced by Dorn [1]. The notion of symmetric duality was developed significantly by Dantzig et al. [2], and the Wolfe dual models presented in [2]. Mond [3] presented a slightly different pair of symmetric dual nonlinear programs and obtained more generalized duality results than that of Dantzig et al. [2]. Mond and Weir [4] then gave another pair of symmetric dual nonlinear programs in which a weaker convexity assumption was imposed on involved functions. Later, Mond and Weir [5], Weir and Mond [6] as well as Gulati et al. [7] generalized single objective symmetric duality to multiobjective case.

Chandra et al. [8] first formulated a pair of symmetric dual fractional programs with certain convexity hypothesis. Pandey [9] introduced second-order η-invex function for multiobjective fractional programming problem and established weak and strong duality theorems. Yang et al. [10] discussed a class of nondifferentiable multiobjective fractional programming problems, and proved duality theorems under the assumptions of invex (pseudoinvex, pseudoincave) functions. Higher-order duality in nonlinear programs have been studied by some researchers. Mangasarian [11] formulated a class of higher-order dual problems for the nonlinear programming problem by introducing twice differentiable functions. Mond and Zhang [12] obtained duality results for various higher-order dual programming problems under higher-order invexity assumptions. Under invexity-type conditions, such as higher-order type I, higher-order pseudo-type I, and higher-order quasi-type I conditions, Mishra and Rueda [13] gave various duality results. Recently, Chen [14] also discussed the duality theorems under higher-order F-convexity (F-pseudo-convexity, F-quasi-convexity) for a pair of multiobjective nondifferentiable program. But, up to now, there is not sufficient literatures dealing with higher-order fractional symmetric duality.

In this paper, we first formulate a pair of nondifferentiable multiobjective fractional pro-gramming problems. For a differentiable function h: R n ×R n → R, we introduce the definition of higher-order (F, α, ρ, d)-convexity, which extends some kinds of generalized convexity, such as second order F-convexity in [15] and higher-order F -convexity in [14]. Under the higher-order (F, α, ρ, d)- convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems.

Preliminaries

Let R n be the n-dimensional Euclidean space and let R + n be its non-negative orthant. The following conventions for vectors in R n will be used:
x < y if and only if y - x int  R n ; x y if and only if y - x R + n \ { 0 } ; x y if and only if y - x R + n ; x y is the negation of x y .

For a real-valued twice differentiable function h(x, y) defined on an open set in R n × R m , denote by x h ( x ¯ , y ¯ ) the gradient vector of h with respect to x at ( x ¯ , y ¯ ) , x x h ( x ¯ , y ¯ ) the hessian matrix with respect to x at ( x ¯ , y ¯ ) . Similarly, y h ( x ¯ , y ¯ ) , x y h ( x ¯ , y ¯ ) and y y h ( x ¯ , y ¯ ) are also defined.

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 } .
A support function, being convex and everywhere finite, has a subdifferential, that is, there exists a z R n such that
s ( y | C ) s ( x | C ) + z T ( y - x ) , x C .
The subdifferential of s(x|C) is given by
s ( x | C ) = { z C : z T x = s ( x | C ) } .
For a convex set D R n , the normal cone to D at a point x D is defined by
N D ( x ) = { y R n : y T ( z - x ) 0 , z D } .

When C is a compact convex set, y N C (x) if and only if s(y|C) = x T y, or equivalently, x s(y|C).

Consider the following multiobjective programming problem (P):
Minimize f ( x ) subject to  g ( x ) 0 , x X ,

where f: R n → R m , g: R n → R l and X R n . Denote by S the set of feasible solutions of (P).

Definition 2.1. (a) A feasible solution x0 is said to be an efficient solution of (P) if there is no other x S such that f(x) ≤ f(x0).
  1. (b)
    A feasible solution x0 is said to be a properly efficient solution of (P) if it is an efficient solution of (P), and there exists a real number M > 0 such that for all i {1, ..., m}, x S, and f i (x) < f i (x0),
    f i ( x 0 ) - f i ( x ) M ( f j ( x ) - f j ( x 0 ) )
     

for some j {1, ..., m} such that f j (x) > f j (x0).

Definition 2.2. A functional F: X × X × R n → R (where X R n ) is sublinear in its third component if for all (x, u) X × X,
F ( x , u ; a 1 + a 2 ) F ( x , u ; a 1 ) + F ( x , u ; a 2 ) for all  a 1 , a 2 R n ; F ( x , u ; α a ) = α F ( x , u ; a ) for all α R + and for all  a R n .

For convenience, we write F x, u (a) = F (x, u, a).

We now introduce higher-order (F, α, ρ, d)-convex function. Where, F: X × X × R n R is a sublinear functional, α: X × X → R+ \ {0}, ρ R and d: X × X → R. Let Φ: XR and h: X × R n R be differentiable real valued functions.

Definition 2.3. Φ is said to be higher-order (F, α, ρ, d)-convex at u X with respect to h if, (x, p) X × R n ,
Φ ( x ) - Φ ( u ) F x , u ( α ( x Φ ( u ) + p h ( u , p ) ) ) + h ( u , p ) - p T p h ( u , p ) + ρ d 2 ( x , u ) .
Remark 2.1. (1) When α = 1, and ρ = 0 or d = 0, the higher-order (F, α, ρ, d)-convexity reduces to higher-order F-convexity in [14].
  1. (2)

    When α = 1, ρ = 0 or d = 0, and h ( u , p ) = 1 2 p T x x Φ ( u ) p , the higher-order (F, α, ρ, d)-convexity reduces to second order F-convexity in [15].

     

we now give an example of higher-order (F, α, ρ, d)-convex function with respect to h(u, p), which is not higher-order F -convex and second order F-convex.

Example 2.1. Let X R, X = {x: x 1}, f: X → R, F: X × X × R → R, h: X × R → R and d: X × X → R given as follows
f ( x ) = x + 2 x + 1 , F x , u ( a ) = | a | ( x - u ) 2 , h ( u , p ) = p u + 1 , d ( x , u ) = x - u .
And let u = 1, ρ = -1, α = 3 4 . Then for all (x, p) X × R
f ( x ) - f ( u ) = x 2 - x x + 1 F x , u 3 4 ( x f ( u ) + p h ( u , p ) ) + h ( u , p ) - p T p h ( u , p ) - d 2 ( x , u ) = - 1 4 ( x - 1 ) 2 .
This implies f(x) is a higher-order (F, α, ρ, d)-convex function with respect to h at u. But when we let x = 2, p = 3 and x = 6, p = 3 respectively, we have
f ( 2 ) - f ( 1 ) = 2 3 < F x , u ( x f ( u ) + p h ( u , p ) ) + h ( u , p ) - p T p h ( u , p ) = 3 4 , f ( 6 ) - f ( 1 ) = 30 7 < F x , u ( x f ( u ) + x x f ( u ) ) - 1 2 p T x x f ( u ) p = 66 4 .
Hence, f is neither a higher-order F-convex function nor a second order F-convex function. From now on, suppose that the sublinear functional F satisfies the following condition:
F x , y ( a ) + a T y 0 , a R + n .
(1)

Higher-order symmetric duality

In the section, we consider the following multiobjective fractional symmetric dual problems: (MFP) Minimize L(x, y, p) = (L1(x, y, p1), ..., L k (x, y, p k )) T subject to
i = 1 k λ i ( y f i ( x , y ) - z i + p i H i ( x , y , p i ) ) - L i ( x , y , p i ) ( y g i ( x , y ) + r i + p i G i ( x , y , p i ) ) 0 , y T i = 1 k λ i ( y f i ( x , y ) - z i + p i H i ( x , y , p i ) ) - L i ( x , y , p i ) ( y g i ( x , y ) + r i + p i G i ( x , y , p i ) ) 0 , λ > 0 , λ T e = 1 , z i D i , r i F i , i = 1 , k .
(MFD) Maximize M(u, v, q) = (M1(u, v, q1),..., M k (u, v, q k )) T subject to
i = 1 k λ i ( x f i ( u , v ) + w i + q i Φ i ( u , v , q i ) ) - M i ( u , v , q i ) ( x g i ( u , v ) - t i + q i Ψ i ( u , v , q i ) ) 0 , u T i = 1 k λ i ( x f i ( u , v ) + w i + q i Φ i ( u , v , q i ) ) - M i ( u , v , q i ) ( x g i ( u , v ) - t i + q i Ψ i ( u , v , q i ) ) 0 , λ > 0 , λ T e = 1 , w i C i , t i E i , i = 1 , k .
where
L i ( x , y , p i ) = f i ( x , y ) + s ( x | C i ) - y T z i + H i ( x , y , p i ) - p i T p i H i ( x , y , p i ) g i ( x , y ) - s ( x | E i ) + y T r i + G i ( x , y , p i ) - p i T p i G i ( x , y , p i ) , M i ( u , v , q i ) = f i ( u , v ) - s ( v | D i ) + u T w i + Φ i ( u , v , q i ) - q i T q i Φ i ( u , v , q i ) g i ( u , v ) + s ( v | F i ) - u T t i + Ψ i ( u , v , q i ) - q i T q i Ψ i ( u , v , q i ) ,

f i : R n × R m R; g i : R n × R m R; H i , G i : R n × R m R and Φ i , Ψ i : R n × R m × R n R are twice differentiable functions for all i = 1 ..., k. C i , E i are compact convex sets in R n , and D i , F i are compact convex sets in R m , i = 1, ..., k. e = (1, ..., 1) T R k . p i R m , q i R n , i = 1, ..., k, p = (p1, ..., p k ), q = (q1, ..., q k ). It is assumed that in the feasible regions the numerators are nonnegative and denominators are positive.

We let S = (S1, ..., S k ) T , W = (W1, ..., W k ) T R k . Then we can express the programs (MFP) and (MFD) equivalently as:

(MFP) S Minimize S subject to
( f i ( x , y ) + s ( x | C i ) - y T z i + H i ( x , y , p i ) - p i T p i H i ( x , y , p i ) ) - S i ( g i ( x , y ) - s ( x | E i ) + y T r i + G i ( x , y , p i ) - p i T p i G i ( x , y , p i ) ) = 0 , i = 1 , , k ,
(2)
i = 1 k λ i ( y f i ( x , y ) - z i + p i H i ( x , y , p i ) ) - S i ( y g i ( x , y ) + r i + p i G i ( x , y , p i ) ) 0 ,
(3)
y T i = 1 k λ i ( y f i ( x , y ) - z i + p i H i ( x , y , p i ) ) - S i ( y g i ( x , y ) + r i + p i G i ( x , y , p i ) ) 0 , λ > 0 , λ T e = 1 , z i D i , r i F i , i = 1 , k .
(4)
(MFD) W Maximize W subject to
( f i ( u , v ) - s ( v | D i ) + u T w i + Φ i ( u , v , q i ) - q i T q i Φ i ( u , v , q i ) ) - W i ( g i ( u , v ) + s ( v | F i ) - u T t i + Ψ i ( u , v , q i ) - q i T q i Ψ i ( u , v , q i ) ) = 0 , i = 1 , , k ,
(5)
i = 1 k λ i ( x f i ( u , v ) + w i + q i Φ i ( u , v , q i ) ) - W i ( x g i ( u , v ) - t i + q i Ψ i ( u , v , q i ) ) 0 ,
(6)
u T i = 1 k λ i ( x f i ( u , v ) + w i + q i Φ i ( u , v , q i ) ) - W i ( x g i ( u , v ) - t i + q i Ψ i ( u , v , q i ) ) 0 , λ > 0 , λ T e = 1 , w i C i , t i E i , i = 1 , k .
(7)

Now we can prove weak, strong and converse duality theorems for (MFP) S and (MFD) W , but equally apply to (MFP) and (MFD).

Theorem 3.1 (Weak duality). Let (x, y, S, z1, ..., z k , r1, ..., r k , λ, p) be feasible for (MFD) S and let (u, v, W, w1, ..., w k , t1 ..., t k , λ, q) be feasible for (MFD) W . Let i {1, ..., k}, f i (., v) + (.) T w i be higher-order (F, α, ρ i , d i )-convex at u with respect to Φ i (u, v, q i ), - (g i (., v) - (.) T t i ) be higher-order (F, α, ρ, d i )-convex at u with respect to -Ψ i (u, v, q i ), - (f i (x, .) - (.) T z i ) be higher-order ( K , α ¯ , ρ ¯ i , d ¯ i ) -convex at y with respect to -H i (x, y, p i ), g i (x, .) + (.) T r i be higher-order ( K , α ¯ , ρ ¯ i , d ¯ i ) -convex at y with respect to G i (x, y, p i ), where sublinear functional F: R n × R n × R n → R and K: R m × R m × R m → R satisfy the condition (1). If the following conditions hold:
g i ( x , v ) + v T r i - s ( x | E i ) > 0 , i = 1 , , k ,
(8)
i = 1 k λ i ( ( 1 + W i ) ρ i d i 2 ( x , u ) + ( 1 + S i ) ρ ¯ i d ¯ i 2 ( v , y ) ) 0 .
(9)

Then S W.

Proof. Since (u, v, W, w1, ..., w k , t1 ..., t k , λ, q) is feasible for (MFD) W , from (6), (7) and F satisfies condition (1), it follows that
F x , u i = 1 k λ i [ ( x f i ( u , v ) + w i + q i Φ i ( u , v , q i ) ) - W i ( x g i ( u , v ) - t i + q i Ψ i ( u , v , q i ) ) ] 0 .
(10)
Using the convexity assumptions of f i (., v) + (.) T w i and -(g i (., v) - (.) T t i ) at u, we have
f i ( x , v ) + x T w i - f i ( u , v ) - u T w i F x , u ( α ( x f i ( u , v ) + w i + q i Φ i ( u , v , q i ) ) ) + Φ i ( u , v , q i ) - q i T q i Φ i ( u , v , q i ) + ρ i d i 2 ( x , u ) , - g i ( x , v ) + x T t i + g i ( u , v ) - u T t i F x , u ( α ( - x g i ( u , v ) + t i - q i Ψ i ( u , v , q i ) ) ) - Ψ i ( u , v , q i ) + q i T q i Φ i ( u , v , q i ) + ρ i d i 2 ( x , u ) .
Since F is a sublinear functional and λ > 0, W 0, α > 0, from (10) and the above two inequalities, we have
i = 1 k λ i ( f i ( x , v ) + x T w i - f i ( u , v ) - u T w i - Φ i ( u , v , q i ) + q i T q i Φ i ( u , v , q i ) ) + i = 1 k λ i W i ( g i ( u , v ) + v T r i - u T t i + Ψ i ( u , v , q i ) - q i T q i Ψ i ( u , v , q i ) ) + i = 1 k λ i W i ( x T t i - g i ( x , v ) - v T r i ) i = 1 k λ i ( 1 + W i ) ρ i d i 2 ( x , u ) .
(11)
Since v T r i s(v|F i ), from (5) and (11), we have
i = 1 k λ i [ ( f i ( x , v ) + x T w i - s ( v | D i ) ) + W i ( x T t i - v T r i - g i ( x , v ) ) ] i = 1 k λ i ( 1 + W i ) ρ i d i 2 ( x , u ) .
(12)
On the other hand, from (3), (4) and sublinear functional K satisfies condition (1), we obtain
K v , y - i = 1 k λ i ( y f i ( x , y ) - z i + p i H i ( x , y , p i ) ) - S i ( y g i ( x , y ) + r i + p i G i ( x , y , p i ) ) 0 .
(13)
Using the convexity assumptions of -f i (x, .) + (.) T z i and g i (x, .) + (.) T r i at y, we have
- f i ( x , v ) + v T z i + f i ( x , y ) - y T z i K v , y ( α ¯ ( - y f i ( x , y ) + z i - p i H i ( x , y , p i ) ) ) - H i ( x , y , p i ) + p i T p i H i ( x , y , p i ) + ρ ¯ i d ¯ i 2 ( v , y ) , g i ( x , v ) + v T r i - g i ( x , y ) - y T r i K v , y ( α ¯ ( y g i ( x , y ) + r i + p i G i ( x , y , p i ) ) ) + G i ( x , y , p i ) - p i T p i G i ( x , y , p i ) + ρ ¯ i d ¯ i 2 ( v , y ) .
Since K is a sublinear functional, and λ > 0, S 0, α ¯ > 0 , from (13) and the above two inequalities, it holds
i = 1 k λ i ( - f i ( x , v ) + v T z i + f i ( x , y ) - y T z i + H i ( x , y , p i ) - p i T p i H i ( x , y , p i ) ) + i = 1 k λ i S i ( - g i ( x , y ) + x T t i - y T r i - G i ( x , y , p i ) + p i T p i G i ( x , y , p i ) ) + i = 1 k λ i S i ( g i ( x , v ) + v T r i - x T t i ) i = 1 k λ i ( 1 + S i ) ρ ¯ i d ¯ i 2 ( v , y ) .
(14)
Since x T t i s(x|E i ), from (2) and (14) we have
i = 1 k λ i [ ( - f i ( x , v ) + v T z i - s ( x | C i ) ) + S i ( g i ( x , v ) + v T r i - x T t i ) ] i = 1 k λ i ( 1 + S i ) ρ ¯ i d ¯ i 2 ( v , y ) .
Adding the above inequality and (12), we get
i = 1 k λ i ( v T z i - s ( v | D i ) + x T w i - s ( x | C i ) ) + i = 1 k λ i ( S i - W i ) ( g i ( x , v ) + v T r i - x T t i ) i = 1 k λ i ( ρ i d i 2 ( x , u ) ( 1 + W i ) + ρ ¯ i d ¯ i 2 ( v , y ) ( 1 + S i ) ) .
Since λ i > 0, v T z i - s(v|D i ) + x T w i - s(x|C i ) 0, i = 1, ..., k, by (9) it yields
i = 1 k λ i ( S i - W i ) ( g i ( x , v ) + v T r i - x T t i ) 0 .

By assumptions (8), we have g i (x, v)+v T r i -x T t i > 0, i = 1, ..., k. Since λ > 0, it follows that S W. □

Theorem 3.2 (Strong duality). Let ( x ¯ , y ¯ , S ¯ , z ¯ 1 , , z ¯ k , r ¯ 1 , , r ¯ k , λ ¯ , p ¯ ) be a properly efficient solution of (MFP) S , and fix λ = λ ¯ in (MFD) W . Suppose that

(a)

x H i ( x ¯ , y ¯ , 0 ) = x G i ( x ¯ , y ¯ , 0 ) = 0 , q i Φ i ( x ¯ , y ¯ , 0 ) = q i Ψ i ( x ¯ , y ¯ , 0 ) = 0 , H i ( x ¯ , y ¯ , 0 ) = G i ( x ¯ , y ¯ , 0 ) = 0 , Φ i ( x ¯ , y ¯ , 0 ) = Ψ i ( x ¯ , y ¯ , 0 ) = 0 , y H i ( x ¯ , y ¯ , 0 ) = y G i ( x ¯ , y ¯ , 0 ) = 0 , p i H i ( x ¯ , y ¯ , 0 ) = p i G i ( x ¯ , y ¯ , 0 ) = 0 , i = 1 , , k .

(b) For all i {1, ..., k},
f i ( x ¯ , y ¯ ) + s ( x ¯ | C i ) - y ¯ T z ¯ i + H i ( x ¯ , y ¯ , p ¯ i ) - p ¯ i T p i H i ( x ¯ , y ¯ , p ¯ i ) > 0 .
(c) (i) p i p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i p i G i ( x ¯ , y ¯ , p ¯ i ) 0 for p ¯ i = 0 , i = 1, ..., k and p i p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i p i G i ( x ¯ , y ¯ , p ¯ i ) is nonsingular for all i = 1, ..., k,
  1. (ii)

    i = 1 k λ ¯ i ( y y f i ( x ¯ , y ¯ ) - S ¯ i y y g i ( x ¯ , y ¯ ) ) is positive definite and p ¯ i T ( ( y H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i y G i ( x ¯ , y ¯ , p ¯ i ) ) - ( p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) 0 for all i = 1, ..., k, or i = 1 k λ ¯ i ( y y f i ( x ¯ , y ¯ ) - S ¯ i y y g i ( x ¯ , y ¯ ) ) is negative definite and p ¯ i T ( ( y H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i y G i ( x ¯ , y ¯ , p ¯ i ) ) - ( p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) 0 for all i = 1, ..., k.

     
  2. (iii)

    { y f i ( x ¯ , y ¯ ) - z ¯ i + p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i ( y g i ( x ¯ , y ¯ ) + r ¯ i + p i G i ( x ¯ , y ¯ , p ¯ i ) ) : i = 1 , , k } is linearly independent.

     

Then p ¯ = 0 , and there exist w ¯ i C i and t ¯ i E i , i = 1, ..., k such that ( x ¯ , y ¯ , S ¯ , w ¯ 1 , . . . , w ¯ k , t ¯ 1 , , t ¯ k , λ ¯ , q ¯ = 0 ) is a feasible solution of (MFD) W . Furthermore, if the hypotheses in Theorem 3.1 are satisfied, then ( x ¯ , y ¯ , S ¯ , w ¯ 1 , . . . , w ¯ k , t ¯ 1 , , t ¯ k , λ ¯ , q ¯ = 0 ) is a properly efficient solution of (MFD) W , and the two objective values are equal.

Proof. Since ( x ¯ , y ¯ , S ¯ , z ¯ 1 , , z ¯ k , r ¯ 1 , , r ¯ k , λ ¯ , p ¯ ) is a properly efficient solution of (MFP) S , by the Fritz John type necessary optimality conditions [16], there exist α R k , β R k , γ R m , δ R, μ R k and w ¯ i R n , t ¯ i R n , i = 1, ..., k such that
i = 1 k β i ( ( x f i ( x ¯ , y ¯ ) + w ¯ i + x H i ( x ¯ , y ¯ , p ¯ i ) ) - S ¯ i ( x g i ( x ¯ , y ¯ ) - t ¯ i + x G i ( x ¯ , y ¯ , p ¯ i ) ) ) + ( γ - δ y ¯ ) T i = 1 k λ ¯ i ( y x f i ( x ¯ , y ¯ ) - S ¯ i y x g i ( x ¯ , y ¯ ) ) + i = 1 k ( p i x H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i x G i ( x ¯ , y ¯ , p ¯ i ) ) T ( ( γ - δ y ¯ ) λ ¯ i - β i p ¯ i ) = 0 ,
(15)
i = 1 k ( β i - δ λ ¯ i ) ( y f i ( x ¯ , y ¯ ) - z i + p i H i ( x ¯ , y ¯ , p ¯ i ) ) - S ¯ i ( y g i ( x ¯ , y ¯ ) + r ¯ i + p i G i ( x ¯ , y ¯ , p ¯ i ) ) + i = 1 k β i ( ( y H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i y G i ( x ¯ , y ¯ , p ¯ i ) ) - ( p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) + i = 1 k λ ¯ i ( ( y y f i ( x ¯ , y ¯ ) - S ¯ i y y g i ( x ¯ , y ¯ ) ) T ( γ - δ y ¯ ) ) + i = 1 k ( p i y H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i y G i ( x ¯ , y ¯ , p ¯ i ) ) T ( - β i p ¯ i + ( γ - δ y ¯ ) λ ¯ i ) = 0 ,
(16)
α i - β i ( g i ( x ¯ , y ¯ ) - s ( x ¯ | E i ) + y ¯ T r ¯ i + G i ( x ¯ , y ¯ , p ¯ i ) - p ¯ i T p i G i ( x ¯ , y ¯ , p ¯ i ) ) - ( γ - δ y ¯ ) T ( λ ¯ i ( y g i ( x ¯ , y ¯ ) + r ¯ i + p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) = 0 , i = 1 , , k ,
(17)
( γ - δ y ¯ ) T ( y f i ( x ¯ , y ¯ ) - z ¯ i + p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i ( y g i ( x ¯ , y ¯ ) + r ¯ i + p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) - μ i = 0 , i = 1 , , k ,
(18)
( λ ¯ i ( γ - δ y ¯ ) - β i p ¯ i ) T ( p i p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i p i G i ( x ¯ , y ¯ , p ¯ i ) ) = 0 , i = 1 , , k ,
(19)
β i y ¯ + ( γ - δ y ¯ ) λ ¯ i N D i ( z ¯ i ) , i = 1 , , k ,
(20)
β i S ¯ i y ¯ + λ ¯ i S ¯ i ( γ - δ y ¯ ) N F i ( r ¯ i ) , i = 1 , , k ,
(21)
γ T i = 1 k λ ¯ i ( y f i ( x ¯ , y ¯ ) - z ¯ i + p i H i ( x ¯ , y ¯ , p ¯ i ) ) - S ¯ i ( y g i ( x ¯ , y ¯ ) + r ¯ i + p i G i ( x ¯ , y ¯ , p ¯ i ) ) = 0 ,
(22)
δ y ¯ T i = 1 k λ ¯ i ( y f i ( x ¯ , y ¯ ) - z ¯ i + p i H i ( x ¯ , y ¯ , p ¯ i ) ) - S ¯ i ( y g i ( x ¯ , y ¯ ) + r ¯ i + p i G i ( x ¯ , y ¯ , p ¯ i ) ) = 0 ,
(23)
μ T λ ¯ = 0 ,
(24)
w ¯ i C i , t ¯ i E i , x ¯ T t ¯ i = s ( x ¯ | E i ) , x ¯ T w ¯ i = s ( x ¯ | C i ) , i = 1 , , k ,
(25)
( α , β , γ , δ , μ ) 0 , ( α , γ , δ , μ ) 0 .
(26)
Since λ ¯ > 0 , and μ 0, (24) implies μ = 0. Consequently, (18) yields
( γ - δ y ¯ ) T y f i ( x ¯ , y ¯ ) - z ¯ i + p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i ( y g i ( x ¯ , y ¯ ) + r ¯ i + p i G i ( x ¯ , y ¯ , p ¯ i ) ) = 0 , i = 1 , , k .
(27)
By assumption (i) and (19), we have
λ ¯ i ( γ - δ y ¯ ) = β i p ¯ i , i = 1 , . . . , k .
(28)
Multiplying (16) ( γ - δ y ¯ ) by left, from (27) and (28) we have
( γ - δ y ¯ ) T i = 1 k β i ( ( y H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i y G i ( x ¯ , y ¯ , p ¯ i ) ) - ( p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) + ( γ - δ y ¯ ) T i = 1 k λ ¯ i ( y y f i ( x ¯ , y ¯ ) - S ¯ i y y g i ( x ¯ , y ¯ ) ) ( γ - δ y ¯ ) = 0 .
Since λ ¯ > 0 , from (28) and the above equation, we have
i = 1 k β i 2 λ ¯ i p ¯ i T ( ( y H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i y G i ( x ¯ , y ¯ , p ¯ i ) ) - ( p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) + ( γ - δ y ¯ ) T i = 1 k λ ¯ i ( y y f i ( x ¯ , y ¯ ) - S ¯ i y y g i ( x ¯ , y ¯ ) ) ( γ - δ y ¯ ) = 0 .
Which by assumption (ii), we can obtain
γ - δ y ¯ = 0 .
(29)
Using (29) in (28), we have β i p ¯ i = 0 , i = 1, ..., k. This implies that p ¯ i = 0 when β i ≠ 0, for all i {1, ..., k}. Hence, by assumption (1), we get
i = 1 k β i ( ( y H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i y G i ( x ¯ , y ¯ , p ¯ i ) ) - ( p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) = 0 .
Combining this with (16), (28) and (29), it follows that
i = 1 k ( β i - δ λ ¯ i ) ( y f i ( x ¯ , y ¯ ) - z ¯ i + p i H i ( x ¯ , y ¯ , p ¯ i ) - S ¯ i ( y g i ( x ¯ , y ¯ ) + r ¯ i + p i G i ( x ¯ , y ¯ , p ¯ i ) ) ) = 0 ,
which by assumption (iii), it yields
β i - δ λ ¯ i = 0 , i = 1 , , k .
(30)
We claim that δ ≠ 0, otherwise, from (29) and (30) we get β = 0, γ = 0. Using (29) in (17), we get α = 0. This contradicts with (26). Hence δ = 0. Since λ ¯ > 0 , from (30) we get β > 0. Hence β i p ¯ i = 0 , i = 1, ..., k implies p ¯ i = 0 , i = 1, ..., k. Using (28), (29) and the fact p ¯ i = 0 , i = 1, ..., k in (15), by assumption (a), we get
i = 1 k β i ( ( x f i ( x ¯ , y ¯ ) + w ¯ i ) - S ¯ i ( x g i ( x ¯ , y ¯ ) - t ¯ i ) ) = 0 ,
combining this with (30) and δ > 0, λ ¯ > 0 , it holds
i = 1 k λ ¯ i ( ( x f i ( x ¯ , y ¯ ) + w ¯ i ) - S ¯ i ( x g i ( x ¯ , y ¯ ) - t ¯ i ) ) = 0 ,
(31)
which yields
x ¯ T i = 1 k λ ¯ i ( ( x f i ( x ¯ , y ¯ ) + w ¯ i ) - S ¯ i ( x g i ( x ¯ , y ¯ ) - t ¯ i ) ) = 0 .
(32)
On the other hand, by assumption (a) and (2) we get
( f i ( x ¯ , y ¯ ) + s ( x ¯ | C i ) - y ¯ T z ¯ i ) - S ¯ i ( g i ( x ¯ , y ¯ ) - s ( x ¯ | E i ) + y ¯ T r ¯ i ) = 0 , i = 1 , , k .
(33)
Since β > 0, by (20) and (29) we get y ¯ N D i ( z ¯ i ) , i = 1, ..., k. This implies
y ¯ T z ¯ i = s ( y ¯ | D i ) , i = 1 , , k .
(34)
Assumption (b) implies S ¯ > 0 . By (21), we similarly have y ¯ N F i ( r ¯ i ) , i = 1, ..., k. This implies
y ¯ T r ¯ i = s ( y ¯ | F i ) , i = 1 , , k .
(35)
Combining (25), (33), (34) and (35), we get
( f i ( x ¯ , y ¯ ) + x ¯ T w ¯ i - s ( y ¯ | D i ) ) - S ¯ i g i ( x ¯ , y ¯ ) - x ¯ T t ¯ i + s ( y ¯ | F i ) = 0 , i = 1 , , k ,

combining this with (31) and (32), by assumption (a), ( x ¯ , y ¯ , S ¯ , w ¯ 1 , . . . , w ¯ k , t ¯ 1 , , t ¯ k , λ ¯ , q ¯ = 0 ) is a feasible solution of (MFD) W .

Under the assumptions of Theorem 3.1, if ( x ¯ , y ¯ , S ¯ , w ¯ 1 , . . . , w ¯ k , t ¯ 1 , , t ¯ k , λ ¯ , q ¯ = 0 ) is not an efficient solution of (MFD) W , then there exists other feasible solution ( u , v , W , w 1 , , w k , t 1 , , t k , λ ¯ , q ) , of (MFD) W such that S ¯ W . Since ( x ¯ , y ¯ , S ¯ , z ¯ 1 , , z ¯ k , r ¯ 1 , , r ¯ k , λ ¯ , p ¯ ) is a feasible solution of (MFP) S , by Theorem 3.1, we have S ¯ W , hence the contradiction implies ( x ¯ , y ¯ , S ¯ , w ¯ 1 , . . . , w ¯ k , t ¯ 1 , , t ¯ k , λ ¯ , q ¯ = 0 ) is an efficient solution of (MFD) W .

If ( x ¯ , y ¯ , S ¯ , w ¯ 1 , . . . , w ¯ k , t ¯ 1 , , t ¯ k , λ ¯ , q ¯ = 0 ) is not a properly efficient solution of (MFD) W , then there exists other feasible solution ( u , v , W , w 1 , , w k , t 1 , , t k , λ ¯ , q ) of (MFD) W such that for an index i {1, ..., k} and any real number M > 0, W i - S ¯ i > M ( S ¯ j - W j ) for j satisfying S ¯ j > W j whenever W i > S ¯ i This implies W i > S ¯ i can be made arbitrarily large and this contradicts with Theorem 3.1. And it is easy to find that the two objective values are equal. □

Theorem 3.3 (Strict converse duality). Let ( u ¯ , v ¯ , W ¯ , w ¯ 1 , , w ¯ k , t ¯ 1 , , t ¯ k , λ ¯ , q ¯ ) be a properly efficient solution of (MFD) W , and fix λ