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

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.

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.

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 × R^m, 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) = x^T y, or equivalently, x ∈ ∂s(y|C).

Consider the following multiobjective programming problem (P):

where f: Rⁿ → R^m, g: Rⁿ → R^l 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:

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 × R^m → R; H_i, G_i: Rⁿ × 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 Rⁿ, 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 ∈ 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 ∈ R^k. 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: R^m × R^m × R^m → 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 v^T 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 x^T t_i ≦ s(x|E_i), from (2) and (14) we have

Adding the above inequality and (12), we get

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

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 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 ∈ R^k, β ∈ R^k, γ ∈ R^m, δ ∈ R, μ ∈ R^k 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 λ^T e = 1 in (MFP)_S and (MFD)_W, then the two problems reduce to the problems considered by Yang et al. [19].

The authors declare that they have no competing interests.

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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