Research

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

Gao Ying

Author Affiliations

Department of Mathematics, Chongqing Normal University, Chongqing 400047, China

Journal of Inequalities and Applications 2012, 2012:142 doi:10.1186/1029-242X-2012-142

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

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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: 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 NC(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 fi(x) < fi(x0),

for some j ∈ {1, ..., m} such that fj(x) > fj (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 Fx, 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:

(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), ..., Lk(x, y, pk))T subject to

(MFD) Maximize M(u, v, q) = (M1(u, v, q1),..., Mk(u, v, qk))T subject to

where

fi: Rn × Rm R; gi: Rn × Rm R; Hi, Gi: Rn × Rm R and Φi, Ψi: Rn × Rm × Rn R are twice differentiable functions for all i = 1 ..., k. Ci, Ei are compact convex sets in Rn, and Di, Fi are compact convex sets in Rm, i = 1, ..., k. e = (1, ..., 1)T Rk. pi Rm, qi Rn, i = 1, ..., k, p = (p1, ..., pk), q = (q1, ..., qk). It is assumed that in the feasible regions the numerators are nonnegative and denominators are positive.

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

(MFP)S Minimize S subject to

(2)

(3)

(4)

(MFD)W Maximize W subject to

(5)

(6)

(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, ..., zk, r1, ..., rk, λ, p) be feasible for (MFD)S and let (u, v, W, w1, ..., wk, t1 ..., tk, λ, q) be feasible for (MFD)W . Let ∀i ∈ {1, ..., k}, fi(., v) + (.)T wi be higher-order (F, α, ρi, di)-convex at u with respect to Φi(u, v, qi), - (gi(., v) - (.)T ti) be higher-order (F, α, ρ, di)-convex at u with respect to -Ψi(u, v, qi), - (fi(x, .) - (.)Tzi) be higher-order -convex at y with respect to -Hi(x, y, pi), gi(x, .) + (.)T ri be higher-order -convex at y with respect to Gi(x, y, pi), where sublinear functional F: Rn × Rn × Rn → R and K: Rm × Rm × Rm → R satisfy the condition (1). If the following conditions hold:

(8)

(9)

Then S W.

Proof. Since (u, v, W, w1, ..., wk, t1 ..., tk, λ, q) is feasible for (MFD)W, from (6), (7) and F satisfies condition (1), it follows that

(10)

Using the convexity assumptions of fi(., v) + (.)T wi and -(gi(., v) - (.)T ti) 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

(11)

Since vT ri s(v|Fi), from (5) and (11), we have

(12)

On the other hand, from (3), (4) and sublinear functional K satisfies condition (1), we obtain

(13)

Using the convexity assumptions of -fi(x, .) + (.)T zi and gi(x, .) + (.)T ri at y, we have

Since K is a sublinear functional, and λ > 0, S ≧ 0, , from (13) and the above two inequalities, it holds

(14)

Since xT ti s(x|Ei), from (2) and (14) we have

Adding the above inequality and (12), we get

Since λi > 0, vT zi - s(v|Di) + xT wi - s(x|Ci) ≦ 0, i = 1, ..., k, by (9) it yields

By assumptions (8), we have gi(x, v)+vT ri -xT ti > 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

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

Since , and μ ≧ 0, (24) implies μ = 0. Consequently, (18) yields

(27)

By assumption (i) and (19), we have

(28)

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

(29)

Using (29) in (28), we have , i = 1, ..., k. This implies that when βi ≠ 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

(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 , 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

(31)

which yields

(32)

On the other hand, by assumption (a) and (2) we get

(33)

Since β > 0, by (20) and (29) we get , i = 1, ..., k. This implies

(34)

Assumption (b) implies . By (21), we similarly have , i = 1, ..., k. This implies

(35)

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].

### Competing interests

The authors declare that they have no competing interests.

### 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).

### References

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

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

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

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

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

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

7. Gulati, TR, Husain, I, Ahmed, A: Multiobjective symmetric duality with invexity. Bull Aust Math Soc. 56, 25–36 (1997). Publisher Full Text

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

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

10. Yang, XM, Wang, SY, Dneg, XT: Symmetric duality for a class of multiobjective fractional programming problems. J Math Anal Appl. 274, 279–295 (2002). Publisher Full Text

11. Mangasarian, OL: Second-and higher-order duality in nonlinear programming. J Math Anal Appl. 51, 607–620 (1975). Publisher Full Text

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

13. Mishra, SK, Rurda, NG: Higher-order generalized invexity and duality in mathematical programming. J Math Anal Appl. 247, 173–182 (2000). Publisher Full Text

14. Chen, XH: Higher-order symmetric duality in nondifferentiable multiobjective programming problems. J Math Anal Appl. 290, 423–435 (2004). Publisher Full Text

15. Mishra, SK: Second order symmetric duality in mathematical programming with F--convexity. Eur J Oper Res. 127, 507–518 (2000). Publisher Full Text

16. Craven, BD: Lagrange conditions and quasiduality. Bull Austral Math Soc. 16, 325–339 (1977). Publisher Full Text

17. Hou, SH, Yang, XM: On second-order symmetric duality in nondifferentiable programming. J Math Anal Appl. 255, 491–498 (2001). Publisher Full Text

18. Mishra, SK: Nondifferentiable higher-order symmetric duality in mathematical programming with generalized invexity. Eur J Oper Res. 167, 28–34 (2005). Publisher Full Text

19. Yang, XM, Yang, XQ, Teo, KL, Hou, SH: Second order symmetric duality in nondifferentiable multiobjective programming with F-convexity. Eur J Oper Res. 164, 406–416 (2005). Publisher Full Text