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Higher-order symmetric duality for a class of multiobjective fractional programming problems
Journal of Inequalities and Applications volume 2012, Article number: 142 (2012)
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 be its non-negative orthant. The following conventions for vectors in Rn will be used:
For a real-valued twice differentiable function h(x, y) defined on an open set in Rn × Rm, denote by the gradient vector of h with respect to x at the hessian matrix with respect to x at . Similarly, are also defined.
Let C be a compact convex set in Rn. The support function of C is defined by
A support function, being convex and everywhere finite, has a subdifferential, that is, there exists a z ∈ Rn such that
The subdifferential of s(x|C) is given by
For a convex set D ⊂ Rn, the normal cone to D at a point x ∈ D is defined by
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):
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).
-
(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),
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,
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 × Rn → R is a sublinear functional, α: X × X → R+ \ {0}, ρ ∈ R and d: X × X → R. Let Φ: X → R and h: X × Rn → 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 × Rn,
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].
-
(2)
When α = 1, ρ = 0 or d = 0, and , 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
And let u = 1, ρ = -1, . Then for all (x, p) ∈ X × R
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
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:
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
(MFD) Maximize M(u, v, q) = (M1(u, v, q1),..., M k (u, v, q k ))T subject to
where
f i : R n × R m → R; g i : Rn × Rm → R; H i , G i : Rn × Rm → 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 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
(MFD) W Maximize W subject to
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 -convex at y with respect to -H i (x, y, p i ), g i (x, .) + (.)T r i be higher-order -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:
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
Using the convexity assumptions of f i (., v) + (.)T w i and -(g i (., v) - (.)T t i ) at u, we have
Since F is a sublinear functional and λ > 0, W ≧ 0, α > 0, from (10) and the above two inequalities, we have
Since vT r i ≦ s(v|F i ), from (5) and (11), we have
On the other hand, from (3), (4) and sublinear functional K satisfies condition (1), we obtain
Using the convexity assumptions of -f i (x, .) + (.)T z i and g i (x, .) + (.)T r i at y, we have
Since K is a sublinear functional, and λ > 0, S ≧ 0, , from (13) and the above two inequalities, it holds
Since xT t i ≦ s(x|E i ), from (2) and (14) we have
Adding the above inequality and (12), we get
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
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 be a properly efficient solution of (MFP) S , and fix in (MFD) W . Suppose that
(a)
(b) For all i ∈ {1, ..., k},
(c) (i) for , i = 1, ..., k and is nonsingular for all i = 1, ..., k,
-
(ii)
is positive definite and for all i = 1, ..., k, or is negative definite and for all i = 1, ..., k.
-
(iii)
is linearly independent.
Then , and there exist and , i = 1, ..., k such that is a feasible solution of (MFD) W . Furthermore, if the hypotheses in Theorem 3.1 are satisfied, then is a properly efficient solution of (MFD) W , and the two objective values are equal.
Proof. Since 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 , i = 1, ..., k such that
Since , and μ ≧ 0, (24) implies μ = 0. Consequently, (18) yields
By assumption (i) and (19), we have
Multiplying (16) by left, from (27) and (28) we have
Since , from (28) and the above equation, we have
Which by assumption (ii), we can obtain
Using (29) in (28), we have , i = 1, ..., k. This implies that when β i ≠ 0, for all i ∈ {1, ..., k}. Hence, by assumption (1), we get
Combining this with (16), (28) and (29), it follows that
which by assumption (iii), it yields
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 , from (30) we get β > 0. Hence , i = 1, ..., k implies , i = 1, ..., k. Using (28), (29) and the fact , i = 1, ..., k in (15), by assumption (a), we get
combining this with (30) and δ > 0, , it holds
which yields
On the other hand, by assumption (a) and (2) we get
Since β > 0, by (20) and (29) we get , i = 1, ..., k. This implies
Assumption (b) implies . By (21), we similarly have , i = 1, ..., k. This implies
Combining (25), (33), (34) and (35), we get
combining this with (31) and (32), by assumption (a), is a feasible solution of (MFD) W .
Under the assumptions of Theorem 3.1, if is not an efficient solution of (MFD) W , then there exists other feasible solution , of (MFD) W such that . Since is a feasible solution of (MFP) S , by Theorem 3.1, we have , hence the contradiction implies is an efficient solution of (MFD) W .
If is not a properly efficient solution of (MFD) W , then there exists other feasible solution of (MFD) W such that for an index i ∈ {1, ..., k} and any real number M > 0, for j satisfying whenever This implies 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 be a properly efficient solution of (MFD) W , and fix in (MFP) S . Suppose that
-
(a)
(b) For all i ∈ {1, ..., k},
(c) (i) , for , i = 1, ..., k, and is nonsingular for all i = 1, ..., k, and
-
(ii)
is positive definite and for all i = 1, ..., k, or is negative definite and for all i = 1, ..., k.
-
(iii)
is linearly independent.
Then , and there exist and , i = 1, ..., k such that is a feasible solution of (MFP) S . Furthermore, if the hypotheses in Theorem 3.1 are satisfied, then is a properly efficient solution of (MFP) S , and the two objective values are equal. □
Remark 3.1.(1) If k = 1, , , , and , then (MFP) S and (MFD) W becomes the problems considered by Hou and Yang [17].
-
(2)
If k = 1, , and , then (MFP) S and (MFD) W becomes the problems considered by Mishra [18].
-
(3)
If , and for all i {1, ..., k}, then (MFP) S and (MFD) W becomes the problems considered by Chen [14].
-
(4)
If , , , 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].
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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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DOI: https://doi.org/10.1186/1029-242X-2012-142