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

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.

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: Rn ×Rn → 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 Rn be the n-dimensional Euclidean space and let R + n be its non-negative orthant. The following conventions for vectors in Rn 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 Rn × Rm, 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 Rn. 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 Rn 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 Rn, 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) = xT 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: Rn → Rm, g: Rn → Rl and X Rn. 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 × Rn → R (where X Rn) 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 × RnR is a sublinear functional, α: X × X → R+ \ {0}, ρ R and d: X × X → R. Let Φ: XR and h: X × RnR 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 × Rn,

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 : Rn × RmR; H i , G i : Rn × RmR 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 Rn, and D i , F i are compact convex sets in Rm, i = 1, ..., k. e = (1, ..., 1)T Rk. p i Rm, q i Rn, 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 Rk. 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, .) - (.)Tz 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: Rn × Rn × Rn → R and K: Rm × Rm × Rm → 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 vT 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 xT 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, vT z i - s(v|D i ) + xT 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)+vT r i -xT 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 Rk, β Rk, γ Rm, δ R, μ Rk 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= λ ¯ in (MFP) S . Suppose that

  1. (a)

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

(b) For all i {1, ..., k},

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 ) > 0 .

(c) (i) q i q i Φ i ( u ¯ , v ¯ , q ¯ i ) - W ¯ i q i q i Ψ i ( u ¯ , v ¯ , q ¯ i ) 0 , for q ¯ i = 0 , i = 1, ..., k, and q i q i Φ i ( u ¯ , v ¯ , q ¯ i ) - W ¯ i q i q i Ψ i ( u ¯ , v ¯ , q ¯ i ) is nonsingular for all i = 1, ..., k, and

  1. (ii)

    i = 1 k λ ¯ i ( x x f i ( u ¯ , v ¯ ) - W ¯ i x x g i ( u ¯ , v ¯ ) ) is positive definite and q ¯ i T ( ( x Φ i ( u ¯ , v ¯ , q ¯ i ) - W ¯ i x Ψ i ( u ¯ , v ¯ , q ¯ i ) ) - ( q i Φ i ( u ¯ , v ¯ , q ¯ i ) - W ¯ i q i Ψ i ( u ¯ , v ¯ , q ¯ i ) ) ) 0 for all i = 1, ..., k, or i = 1 k λ ¯ i ( x x f i ( u ¯ , v ¯ ) - W ¯ i x x g i ( u ¯ , v ¯ ) ) is negative definite and q ¯ i T ( ( x Φ i ( u ¯ , v ¯ , q ¯ i ) - W ¯ i x Ψ i ( u ¯ , v ¯ , q ¯ i ) ) - ( q i Φ i ( u ¯ , v ¯ , q ¯ i ) - W ¯ i q i Ψ i ( u ¯ , v ¯ , q ¯ i ) ) ) 0 for all i = 1, ..., k.

  2. (iii)

    { 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 ) ) : i = 1 , , k } is linearly independent.

Then q ¯ = 0 , and there exist z ¯ i D i and r ¯ i F i , i = 1, ..., k such that ( u ¯ , v ¯ , W ¯ , z ¯ 1 , . . . , z ¯ k , r ¯ 1 , , r ¯ k , λ ¯ , p ¯ = 0 ) is a feasible solution of (MFP) S . Furthermore, if the hypotheses in Theorem 3.1 are satisfied, then ( u ¯ , v ¯ , W ¯ , z ¯ 1 , . . . , z ¯ k , r ¯ 1 , , r ¯ k , λ ¯ , p ¯ = 0 ) is a properly efficient solution of (MFP) S , and the two objective values are equal. □

Remark 3.1.(1) If k = 1, H 1 ( x , y , p 1 ) = 1 2 p 1 T y y f 1 ( x , y ) p 1 , g 1 ( x , y ) -s ( x | E 1 ) + y T r 1 + G 1 ( x , y , p 1 ) - p 1 T p 1 G 1 ( x , y , p 1 ) =1, Φ 1 ( u , v , q 1 ) = 1 2 q 1 T x x f 1 ( u , v ) q 1 , and g 1 ( u , v ) +s ( v | F 1 ) - u T t 1 + Ψ 1 ( u , v , q 1 ) - q 1 T q 1 Ψ 1 ( u , v , q 1 ) =1, then (MFP) S and (MFD) W becomes the problems considered by Hou and Yang [17].

  1. (2)

    If k = 1, g 1 ( x , y ) -s ( x | E 1 ) + y T r 1 + G 1 ( x , y , p 1 ) - p 1 T p 1 G 1 ( x , y , p 1 ) =1, and g 1 ( u , v ) +s ( v | F 1 ) - u T t 1 + Ψ 1 ( u , v , q 1 ) - q 1 T q 1 Ψ 1 ( u , v , q 1 ) =1, then (MFP) S and (MFD) W becomes the problems considered by Mishra [18].

  2. (3)

    If 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 ) =1, and 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 ) = 1 for all i {1, ..., k}, then (MFP) S and (MFD) W becomes the problems considered by Chen [14].

  3. (4)

    If 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 ) =1, 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 ) = 1 , H i ( x , y , p i ) = 1 2 p i T y y f i ( x , y ) p i , Φ i ( u , v , q i ) = 1 2 q i T x x f i ( u , v ) q i , for all i {1, ..., k}, and there is not the condition λT e = 1 in (MFP) S and (MFD) W , then the two problems reduce to the problems considered by Yang et al. [19].

References

  1. Dorn WS: A symmetric dual theorem for quadratic programs. J Oper Res Soc Jpn 1960, 2: 93–97.

    Google Scholar 

  2. Dantzig GB, Eisenberg E, Cottle RW: Symmetric dual nonlinear programs. Pacific J Math 1965, 15: 809–812.

    Article  MathSciNet  MATH  Google Scholar 

  3. Mond B: A symmetric dual theorem for nonlinear programs. Q J Appl Math 1965, 23: 265–269.

    MathSciNet  MATH  Google Scholar 

  4. Mond B, Weir T: Generalized concavity and duality. In Generalized Concavity in Optimization and Economics. Edited by: Schaible S, Ziemba WT. Academic Press, New York; 1981.

    Google Scholar 

  5. Mond B, Weir T: Symmetric duality for nonlinear multiobjective programming. In Recent Developments in Mathematical Programming. Edited by: Kumar S. Gordon and Breach Science, London; 1991:137–153.

    Google Scholar 

  6. Weir T, Mond B: Symmetric and self duality in multiple objective programming. Asia Pacific J Oper Res 1988, 4: 124–133.

    MathSciNet  MATH  Google Scholar 

  7. Gulati TR, Husain I, Ahmed A: Multiobjective symmetric duality with invexity. Bull Aust Math Soc 1997, 56: 25–36. 10.1017/S0004972700030707

    Article  MathSciNet  MATH  Google Scholar 

  8. Chandra S, Craven BD, Mond B: Symmetric dual fractional programming. Z Oper Res 1985, 29: 59–64.

    MathSciNet  MATH  Google Scholar 

  9. Pandey S: Duality for multiobjective fractional programming involving generalized η- bonvex functions. Opsearch 1991, 28: 36–43.

    MATH  Google Scholar 

  10. Yang XM, Wang SY, Dneg XT: Symmetric duality for a class of multiobjective fractional programming problems. J Math Anal Appl 2002, 274: 279–295. 10.1016/S0022-247X(02)00299-8

    Article  MathSciNet  Google Scholar 

  11. Mangasarian OL: Second-and higher-order duality in nonlinear programming. J Math Anal Appl 1975, 51: 607–620. 10.1016/0022-247X(75)90111-0

    Article  MathSciNet  Google Scholar 

  12. Mond B, Zhang J: Higher-order invexity and duality in mathematical programming. In Generalized Convexity, Generalized Monotonicity: Recent Results. Edited by: Crouzeix JP, et al. Kluwer, Dordrecht; 1998:357–372.

    Chapter  Google Scholar 

  13. Mishra SK, Rurda NG: Higher-order generalized invexity and duality in mathematical programming. J Math Anal Appl 2000, 247: 173–182. 10.1006/jmaa.2000.6842

    Article  MathSciNet  Google Scholar 

  14. Chen XH: Higher-order symmetric duality in nondifferentiable multiobjective programming problems. J Math Anal Appl 2004, 290: 423–435. 10.1016/j.jmaa.2003.10.004

    Article  MathSciNet  MATH  Google Scholar 

  15. Mishra SK: Second order symmetric duality in mathematical programming with F --convexity. Eur J Oper Res 2000, 127: 507–518. 10.1016/S0377-2217(99)00334-3

    Article  MathSciNet  MATH  Google Scholar 

  16. Craven BD: Lagrange conditions and quasiduality. Bull Austral Math Soc 1977, 16: 325–339. 10.1017/S0004972700023431

    Article  MathSciNet  MATH  Google Scholar 

  17. Hou SH, Yang XM: On second-order symmetric duality in nondifferentiable programming. J Math Anal Appl 2001, 255: 491–498. 10.1006/jmaa.2000.7242

    Article  MathSciNet  MATH  Google Scholar 

  18. Mishra SK: Nondifferentiable higher-order symmetric duality in mathematical programming with generalized invexity. Eur J Oper Res 2005, 167: 28–34. 10.1016/j.ejor.2004.02.024

    Article  MATH  Google Scholar 

  19. Yang XM, Yang XQ, Teo KL, Hou SH: Second order symmetric duality in nondifferentiable multiobjective programming with F -convexity. Eur J Oper Res 2005, 164: 406–416. 10.1016/j.ejor.2003.04.007

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was partially supported by the National Science Foundation of China (11126348), the Education Committee Project Research Foundation of Chongqing (KJ110624, KJ120628), the special fund of Chongqing key laboratory(CSTC,2011KLORSE03) and the Doctoral Foundation of Chongqing Normal University(No.10XLB015).

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Ying, G. Higher-order symmetric duality for a class of multiobjective fractional programming problems. J Inequal Appl 2012, 142 (2012). https://doi.org/10.1186/1029-242X-2012-142

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