ON SYMMETRIC DUALITY IN

J. Appl. Math. & Computing Vol. 19(2005), No. 1 - 2, pp. 371 - 384

ON SYMMETRIC DUALITY IN NONDIFFERENTIABLE MATHEMATICAL PROGRAMMING WITH F-CONVEXITY I. AHMAD∗ AND Z. HUSAIN

Abstract. Usual symmetric duality results are proved for Wolfe and Mond -Weir type nondifferentiable nonlinear symmetric dual programs under Fconvexity F-concavity and F-pseudoconvexity F-pseudoconcavity assumptions. These duality results are then used to formulate Wolfe and MondWeir type nondifferentiable minimax mixed integer dual programs and symmetric duality theorems are established. Moreover, nondifferentiable fractional symmetric dual programs are studied by using the above programs. AMS Mathematics Subject Classification : 90C30, 90C11, 90C20, 49N15. Key words and phrases : Symmetric duality, nondifferentiable programming, minimax programming, integer programming, F-convexity.

1. Introduction The study of symmetric duality for nonlinear programs has been of much interest in the recent past. Dantzig et al. [6], Mond [11] and Bazaara and Goode [2] formulated a pair of symmetric dual programs involving a scalar function K(x, y), x ∈ Rn , y ∈ Rm under the condition that K(., y) is convex for each y and K(x, .) is concave for each x. Later, Mond and Weir [13] presented a different pair of symmetric dual nonlinear programs which allows the weakening of the convexity-concavity hypotheses for K(x, y) to pseudoconvexity-pseudoconcavity. Balas [1] generalized the symmetric duality results of Dantzig et al. [6] by constraining some of the primal and dual variables to belong to the arbitrary sets of integers. Kumar et al. [10] extended the integer programming symmetric duality results of Balas [1] assuming pseudoconvexity-pseudoconcavity of the kernel function K(x, y). As a follow up, Gulati et al. [7] presented two distinct pairs of nondifferentiable minimax mixed integer dual programs and established symmetric and self duality theorems. Recently, Husain and Jabeen [8] discussed Received June 2, 2004. Revised February 5, 2005. ∗ Corresponding author. c

2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 371

372

I. Ahmad and Z. Husain

the mixed type duality results for nondifferentiable nonlinear programming problems. In this paper, we study nondifferentiable symmetric duality under F-convexity F-concavity/F-pseudoconvexity F-pseudoconcavity for Wolfe and Mond-Weir type models respectively. These duality results are then used to investigate duality for minimax version of nondifferentiable dual programs wherein some of the primal and dual variables are constrained to belong to some arbitrary sets. Nondifferentiable fractional symmetric duality results have also been discussed. The symmetric duality results presented in this paper subsume most of the already known results in the literature. 2. Notations and preliminaries n Let Rn denotes the n-dimensional Euclidean space and R+ be its nonnegative orthant. For reader’s convenience, we write the following definitions of the generalized F-convexity from Chandra et al. [4]:

Definition 1. A functional F : X × X × Rn −→ R (where X ⊆ Rn ) is sublinear if for all x, x ¯ ∈ X, (A) Fx,¯x (a + b) ≤ Fx,¯x (a) + Fx,¯x (b), for all a, b ∈ Rn , (B) Fx,¯x (αa) = αFx,¯x (a), for all α ∈ R, α ≥ 0, and a ∈ Rn . From(B), it follows that Fx,¯x (0) = 0. Let K(x, y) be real valued twice differentiable function defined on an open set in Rn × Rm . Let 5x K(x, y) and 5y K(x, y) denote the partial derivatives of K with respect to x and y respectively. Also let 52x K(x, y) denotes the Hessian matrix of K. 5xy K(x, y), 5yx K(x, y) and 52y K(x, y) are defined similarly. Definition 2. A function K(., y) is said to be F-convex at x ¯ for fixed y ∈ Y if x, y)) K(x, y) − K(¯ x, y) ≥ Fx,¯x (5x K(¯ for all x ∈ X and for some arbitrary sublinear functional F. Definition 3. A function K(x, .) is said to be F-concave at y¯ for fixed x ∈ X if K(x, y¯) − K(x, y) ≥ Fy,¯y (− 5y K(x, y¯)) for all y ∈ Y and for some arbitrary sublinear functional F. Definition 4. A function K(., y) is said to be F-pseudoconvex at x¯ for fixed y ∈ Y if Fx,¯x (5x K(¯ x, y)) ≥ 0 ⇒ K(x, y) ≥ K(¯ x, y) for all x ∈ X and for some arbitrary sublinear functional F.

On symmetric duality in nondifferentiable mathematical programming

373

Definition 5. A function K(x, .) is said to be F-pseudoconcave at y¯ for fixed x ∈ X if Fy,¯y (− 5y K(x, y¯)) ≥ 0 ⇒ K(x, y¯) ≥ K(x, y) for all y ∈ Y and for some arbitrary sublinear functional F. In the sequel, we require the following notion of separability of a vector function (Balas [1] and Chandra et al. [5]) Definition 6. Let s1 , s2 , · · · , sr be elements of an arbitrary vector space. A real valued function g(s1 , s2 , · · · , sr ) will be called additively separable with respect to s1 if there exist real valued functions α(s1 ) (independent of s2 , s3 , · · · , sr ) and β(s2 , s3 , · · · , sr ) (independent of s1 ) such that g(s1 , s2 , · · · , sr ) = α(s1 ) + β(s2 , s3 , · · · , sr ). Similarly the real valued function g(s1 , s2 , · · · , sr ) will be called multiplicatively separable with respect to s1 if there exist real valued functions α(s1 ) (independent of s2 , s3 , · · · , sr ) and β(s2 , s3 , · · · , sr ) (independent of s1 ) such that g(s1 , s2 , · · · , sr ) = α(s1 )β(s2 , s3 , · · · , sr ). Lemma 1. (Generalized Schwartz Inequality) Let B be a positive semidefinite symmetric matrix of order n. Then for all x, z ∈ Rn , 1

1

xt Bz ≤ (xt Bx) 2 (z t Bz) 2 . 3. Wolfe type symmetric duality We consider the following pair of Wolfe type symmetric dual problems and establish weak and strong duality theorems: Primal (WP) 1

Minimize K(x, y) − y t 5y K(x, y) + (xt Bx) 2 subject to 5y K(x, y) − Cw ≤ 0

(1)

t

w Cw ≤ 1

(2)

x ≥ 0.

(3)

Dual (WD) 1

Maximize K(u, v) − ut 5x K(u, v) − (v t Cv) 2 subject to 5x K(u, v) + Bz ≥ 0 z t Bz ≤ 1

(4) (5)

v ≥ 0,

(6)

374


where B and C are positive semidefinite matrices of order n and m. Also w and z are respectively the m and n dimensional vectors. Theorem 1. (Weak Duality) Let K(., y)+(.)t Bz be F-convex in x and K(x, .)− (.)t Cw be F-concave in y and for all feasible (x, y, w, u, v, z) to (WP) and (WD): n (i) Fx,u (ξ) + ut ξ ≥ 0, f or ξ ∈ R+ and t m (ii) Fv,y (η) + y η ≥ 0, f or η ∈ R+ . Then inf (WP) ≥ sup (WD). Proof. By F-convexity of K(., y) + (.)t Bz, we have K(x, v) + xt Bz − K(u, v) − ut Bz ≥ Fx,u (5x K(u, v) + Bz).

(7)

t

The F-concavity of K(x, .) − (.) Cw gives K(x, y) − y t Cw − K(x, v) + v t Cw ≥ Fv,y (− 5y K(x, y) + Cw).

(8)

Adding inequalities (7) and (8), we get K(x, y) − K(u, v) + xt Bz − ut Bz − y t Cw + v t Cw ≥ Fx,u (5x K(u, v) + Bz) + Fv,y (− 5y K(x, y) + Cw).

(9)

On taking ξ = 5x K(u, v) + Bz and η = − 5y K(x, y) + Cw, the assumptions (i) and (ii) respectively reduce to Fx,u (5x K(u, v) + Bz) ≥ −ut (5x K(u, v) + Bz). and Fv,y (− 5y K(x, y) + Cw) ≥ −y t (− 5y K(x, y) + Cw). These inequalities together with (9) yield K(x, y) − K(u, v) + xt Bz − ut Bz − y t Cw + v t Cw ≥ −ut 5x K(u, v) − ut Bz + y t 5y K(x, y) − y t Cw. Or K(x, y) − y t 5y K(x, y) + xt Bz ≥ K(u, v) − ut 5x K(u, v) − v t Cw, which in view of Schwartz inequality, (2) and (5) gives 1

1

K(x, y) − y t 5y K(x, y) + (xt Bx) 2 ≥ K(u, v) − ut 5x K(u, v) − (v t Cv) 2 , and hence inf (WP) ≥ sup (WD). Theorem 2. (Strong Duality) Let K : Rn × Rm → R be twice differentiable. Suppose that the hypotheses of Theorem 1 are satisfied, and (¯ x, y¯, w) ¯ be an optimal solution for (WP) such that 52y K(¯ x, y¯) is nonsingular. Then there exists


375

z¯ such that (¯ x, y¯, z¯) is an optimal solution for (WD) and the two objectives are equal. Proof. The proof of this theorem is same as given by Gulati et al. [7, Theorem 6]. 4. Mond-Weir type symmetric duality In this section, we consider the following pair of Mond-Weir type problems: Primal (MP) 1

Minimize K(x, y) − y t Cw + (xt Bx) 2 subject to 5y K(x, y)−Cw ≤ 0 t

(10)

t

y 5y K(x, y)−y Cw ≥ 0

(11)

wt Cw ≤ 1

(12)

x ≥ 0.

(13)

Dual (MD) 1

Maximize K(u, v) + ut Bz − (v t Cv) 2 subject to 5x K(u, v)+Bz ≥ 0 t

(14)

t

(15)

t

(16)

u 5x K(u, v) + u Bz ≤ 0 z Bz ≤ 1

v ≥ 0. (17) where B and C are positive semidefinite matrices of order n and m. Also w and z are respectively the m and n dimensional vectors. Theorem 3. (Weak Duality) Let K(., y) + (.)t Bz be F-pseudoconvex in x and K(x, .) − (.)t Cw be F-pseudoconcave in y and for all feasible (x, y, w, u, v, z) to (MP) and (MD): n (i) Fx,u (ξ) + ut ξ ≥ 0, f or ξ ∈ R+ and t m (ii) Fv,y (η) + y η ≥ 0, f or η ∈ R+ . Then inf (MP) ≥ sup (MD). Proof. On taking ξ = 5x K(u, v) + Bz, we have Fx,u (5x K(u, v) + Bz) ≥ −ut 5x K(u, v) − ut Bz ≥ 0, by ((i) and (15)) which by F-pseudoconvexity of K(., y) + (.)t Bz gives K(x, v) + xt Bz ≥ K(u, v) + ut Bz.

376


Using Schwartz inequality and (16), we obtain 1

K(x, v) + (xt Bx) 2 ≥ K(u, v) + ut Bz. On taking η = − 5y K(x, y) + Cw, we have

(18)

Fv,y (− 5y K(x, y) + Cw) ≥ y t 5y K(x, y) − y t Cw ≥ 0 by ((ii) and (11)), which by F-pseudoconcavity of K(x, .) − (.)t Cw yields K(x, y) − y t Cw ≥ K(x, v) − v t Cw. This alongwith Schwartz inequality and (12), gives 1

K(x, y) − y t Cw ≥ K(x, v) − (v t Cv) 2 . Combining (18) and (19), we have 1

(19)

1

K(x, y) − y t Cw + (xt Bx) 2 ≥ K(u, v) + ut Bz − (v t Cv) 2 , and hence inf (MP) ≥ sup (MD). Theorem 4. (Strong Duality) Let K : Rn × Rm → R be twice differentiable. Suppose that the hypotheses of Theorem 3 are satisfied, and (¯ x, y¯, w) ¯ is an optimal solution for (MP) such that 52y K(¯ x, y¯) is positive or negative definite, and 5y K(¯ x, y¯) − C w ¯ 6= 0. Then there exists a z¯ such that (¯ x, y¯, z¯) is optimal for (MD) and the two objectives are equal. Proof. The proof of this theorem is same as given by Chandra et al. [3].

5. Minimax mixed integer programming As in Balas [1] and Gulati et al. [7], we constrain some of the components of the vector variables x ∈ Rn and y ∈ Rm to belong to arbitrary sets of integers U ⊂ Rn1 and V ⊂ Rm1 respectively, where 0 ≤ n1 ≤ n and 0 ≤ m1 ≤ m. So we write (x, y) = (x1 , x2 , y 1 , y 2 ), where x1 = (x1 , x2 , · · · , xn1 ) ∈ U and y 1 = (y1 , y2 , · · · , ym1 ) ∈ V, x2 and y 2 being the remaining components of x and y respectively. We consider the following pair of nondifferentiable mixed integer symmetric dual programs introduced by Gulati et al. [7]. Primal (SP) t

t

1

Maxx1 Minx2 ,y,w K(x, y) − (y 2 ) 5y2 K(x, y) + ((x2 ) Bx2 ) 2 subject to 5y2 K(x, y) − Cw ≤ 0

(20)


377

wt Cw ≤ 1

(21)

x1 ∈ U, y 1 ∈ V x2 ≥ 0.

(22)

Dual (SD) 1

Minv1 Maxu,v2 ,z K(u, v) − (u2 )t 5x2 K(u, v) − ((v 2 )t Cv 2 ) 2 subject to 5x2 K(u, v) + Bz ≥ 0

(23)

z t Bz ≤ 1

(24)

u1 ∈ U, v 1 ∈ V v 2 ≥ 0, (25) where B and C are positive semidefinite matrices of order n − n1 and m − m1 respectively. The above pair becomes the pair considered by Kim and Song [9] with the omission of the nonnegative constraints x ≥ 0 and v ≥ 0 from the primal and dual problems respectively. Theorem 5. (Symmetric Duality) Let (¯ x, y¯, w) ¯ be an optimal solution for (SP). Also, let (i) K(x, y) be separable with respect to x1 or y 1 , (ii) K(x, y) be twice differentiable in x2 and y 2 , (iii) K(x, y) + (x2 )t Bz be F-convex in x2 for every (x1 , y, z) and K(x, y) − (y 2 )t Cw be F-concave in y 2 for every (x, y 1 , w), (iv) 52y2 K 2 (¯ x, y¯) be nonsingular, n2 (v) Fx2 ,u2 (ξ 2 ) + (u2 )t ξ 2 ≥ 0, for ξ 2 ∈ R+ and m2 2 2 t 2 2 (vi) Fv2 ,y2 (η ) + (y ) η ≥ 0, for η ∈ R+ , for all (x, y, w, u, v, z) feasible for (SP) and (SD). Then there exists a z¯ such that (¯ x, y¯, z¯) is an optimal solution for (SD) and the two optimal values are equal. Proof. Let h 12 i t t : (x, y, w) ∈ Q q = Maxx1 Minx2 ,y,w K(x, y) − (y 2 ) 5y2 K(x, y) + (x2 ) Bx2 and

h i 1 s = Minv1 Maxu,v2 ,z K(u, v) − (u2 )t 5x2 K(u, v) − ((v 2 )t Cv 2 ) 2 : (u, v, z) ∈ S , where Q and S are feasible regions of primal (SP) and dual (SD) respectively. Since K(x, y) is additively separable with respect to x1 or y 1 (say with respect to x1 ), it follows that K(x, y) = K 1 (x1 ) + K 2 (x2 , y).

(26)

378


Therefore 5y2 K(x, y) = 5y2 K 2 (x2 , y) and q can be written as h q = Maxx1 Minx2 ,y,w K 1 (x1 ) + K 2 (x2 , y) − (y 2 )t 5y2 K 2 (x2 , y) 1

+((x2 )t Bx2 ) 2 : 5y2 K 2 (x2 , y) − Cw ≤ 0, wt Cw ≤ 1, x2 ≥ 0, x1 ∈ U, i y1 ∈ V h = Maxx1 Miny1 Minx2 ,y2 ,w K 1 (x1 ) + K 2 (x2 , y) − (y 2 )t 5y2 K 2 (x2 , y) 1

+((x2 )t Bx2 ) 2 : 5y2 K 2 (x2 , y) − Cw ≤ 0, wt Cw ≤ 1, x2 ≥ 0, x1 ∈ U, i y1 ∈ V . Or

h i q = Maxx1 Miny1 K 1 (x1 ) + φ(y 1 ) : x1 ∈ U, y 1 ∈ V ,

where h 1 φ(y 1 ) = Minx2 ,y2 ,w K 2 (x2 , y) − (y 2 )t 5y2 K 2 (x2 , y) + ((x2 )t Bx2 ) 2 : i 5y2 K 2 (x2 , y) − Cw ≤ 0, wt Cw ≤ 1, x2 ≥ 0 . Similarly

h i s = Minv1 Maxu1 K 1 (u1 ) + ψ(v 1 ) : u1 ∈ U, v 1 ∈ V ,

where h 1 ψ(v 1 ) = Maxu2 ,v2 ,z K 2 (u2 , v) − (u2 )t 5x2 K 2 (u2 , v) − ((v 2 )t Cv 2 ) 2 : i 5x2 K 2 (u2 , v) + Bz ≥ 0, z t Bz ≤ 1, v 2 ≥ 0 .

(27)

(28)

(29)

(30)

For any given y 1 , programs, (28) and (30) are a pair of symmetric dual nondifferentiable programs considered in section 3, and hence in view of various hypotheses made here, Theorem 1 and 2 of Section 3 become applicable. Therefore for y 1 = y¯1 , φ(¯ y 1 ) = ψ(¯ y 1 ). It remains to show that (¯ x, y¯, z¯) is optimal for (SD). The proof of this part follows similar lines of the proof of Theorem 1 of Gulati et al. [7]. We now consider the following pair of Mond-Weir type nondifferentiable mixed integer symmetric dual programs: Primal (SP1) t

t

1

Maxx1 Minx2 ,y,w K(x, y) − (y 2 ) Cw + ((x2 ) Bx2 ) 2 subject to

5y2 K(x, y) − Cw ≤ 0 2 t

(31)

2 t

(y ) 5y2 K(x, y) − (y ) Cw ≥ 0

(32)

wt Cw ≤ 1

(33)


379

x1 ∈ U, y 1 ∈ V x2 ≥ 0.

(34)

Dual (SD1) t

t

1

Minv1 Maxu,v2 ,z K(u, v) + (u2 ) Bz − ((v 2 ) Cv 2 ) 2 subject to 5x2 K(u, v) + Bz ≥ 0 (u2 )t 5x2 K(u, v) + (u2 )t Bz ≤ 0 z t Bz ≤ 1 1 u ∈ U, v 1 ∈ V v 2 ≥ 0.

(35) (36) (37) (38)

Theorem 6. (Symmetric Duality) Let (¯ x, y¯, w) ¯ be an optimal solution for (SP1). Also, let (i) K(x, y) be separable with respect to x1 or y 1 , (ii) K(x, y) be twice differentiable in x2 and y 2 , (iii) K(x, y) + (x2 )t Bz be F-pseudoconvex in x2 for every (x1 , y, z) and K(x, y) −(y 2 )t Cw be F-pseudoconcave in y 2 for every (x, y 1 , w), (iv) 52y2 K 2 (¯ x, y¯) be positive or negative definite, (v) 5y2 K 2 (¯ x, y¯) − C w ¯ 6= 0, n2 (vi) Fx2 ,u2 (ξ 2 ) + (u2 )t ξ 2 ≥ 0, for ξ 2 ∈ R+ , and m2 2 2 t 2 2 (vii) Fv2 ,y2 (η ) + (y ) η ≥ 0, for η ∈ R+ , for all (x, y, w, u, v, z) feasible for (SP1) and (SD1). Then there exists a z¯ such that (¯ x, y¯, z¯) is an optimal solution for (SD1) and the two objectives are equal. Proof. Let h i 1 t t q = Maxx1 Minx2 ,y,w K(x, y) − (y 2 ) Cw + ((x2 ) Bx2 ) 2 : (x, y, w) ∈ Q and h i 1 t t s = Minv1 Maxu,v2 ,z K(u, v) + (u2 ) Bz − ((v 2 ) Cv 2 ) 2 : (u, v, z) ∈ S , where Q and S are feasible regions of primal (SP1) and dual (SD1) respectively. Since K(x, y) is additively separable with respect to x1 or y 1 (say with respect to x1 ), it follows that K(x, y) = K 1 (x1 ) + K 2 (x2 , y). Therefore 5y2 K(x, y) = 5y2 K 2 (x2 , y) and q can be written as h 1 q = Maxx1 Minx2 ,y,w K 1 (x1 ) + K 2 (x2 , y) − (y 2 )t Cw + ((x2 )t Bx2 ) 2 : 5y2 K 2 (x2 , y) − Cw ≤ 0, (y 2 )t 5y2 K 2 (x2 , y) − (y 2 )t Cw ≥ 0, i wt Cw ≤ 1, x2 ≥ 0, x1 ∈ U, y 1 ∈ V .

380


Or

h i q = Maxx1 Miny1 K 1 (x1 ) + φ(y 1 ) : x1 ∈ U, y 1 ∈ V ,

where h 1 φ(y 1 ) = Minx2 ,y2 ,w K 2 (x2 , y) − (y 2 )t Cw + ((x2 )t Bx2 ) 2 : 5y2 K 2 (x2 , y) − Cw ≤ 0, (y 2 )t 5y2 K 2 (x2 , y) − (y 2 )t Cw ≥ 0, i wt Cw ≤ 1, x2 ≥ 0 . Similarly

(39)

i h s = Minv1 Maxu1 K 1 (u1 ) + ψ(v 1 ) : u1 ∈ U, v 1 ∈ V ,

where h 1 ψ(v 1 ) = Maxu2 ,v2 ,z K 2 (u2 , v) + (u2 )t Bz − ((v 2 )t Cv 2 ) 2 : 5x2 K 2 (u2 , v) + Bz ≥ 0, (u2 )t 5x2 K 2 (u2 , v) + (u2 )t Bz ≤ 0, i z t Bz ≤ 1, v 2 ≥ 0 .

(40)

For any given y 1 , programs (39) and (40) are a pair of symmetric dual nondifferentiable programs considered in section 4, and hence in view of the assumptions made here Theorem 3 and 4 become applicable. Therefore for y 1 = y¯1 , we have φ(¯ y 1 ) = ψ(¯ y 1 ). The remainder of the proof of this theorem is the same as that of Theorem 5. 6. Nondifferentiable symmetric fractional programs We consider the following pair of fractional programs formulated by Mond et al. [12] and show that the duality results of section 4 can be applied to study the symmetric duality for the programs (FP) and (FD): 1

(FP) Minimize

P (x, y, z) K(x, y) + (xt Bx) 2 − y t Cz = 1 Q(x, y, r) G(x, y) − (xt Dx) 2 + y t Er

subject to h i h i Q(x, y, r) 5y K(x, y) − Cz − P (x, y, z) 5y G(x, y) + Er ≤ 0 n h i h io y t Q(x, y, r) 5y K(x, y) − Cz − P (x, y, z) 5y G(x, y) + Er ≥ 0 z t Cz ≤ 1 t

r Er ≤ 1 x ≥ 0.

(41) (42) (43) (44) (45)


381

1

(FD) Maximize

K(u, v) − (v t Bv) 2 + ut Bw L(u, v, w) = 1 H(u, v, s) G(u, v) + (v t Ev) 2 − ut Ds

subject to h i h i H(u, v, s) 5x K(u, v) + Bw − L(u, v, w) 5x G(u, v) − Ds ≥ 0 n h i h io ut H(u, v, s) 5x K(u, v) + Bw − L(u, v, w) 5x G(u, v) − Ds ≤ 0

(46) (47)

wt Bw ≤ 1

(48)

st Ds ≤ 1

(49)

v ≥ 0.

(50)

Here it is assumed that in the region defined from (41) to (45) and (46) to (50) P ≥ 0, Q > 0, L ≥ 0 and H > 0. Lemma 2. (i) If K(., y) + (.)t Bw is F-convex and G(., y) − (.)t Ds is F-concave, then P (., y, z) is F-pseudoconvex. Q(., y, r) (ii) If K(x, .) + (.)t Bw is F-concave and G(x, .) − (.)t Ds is F-convex, then P (x, ., z) is F-pseudoconcave. Q(x, ., r) Proof. The F-convexity of K(., y) + (.)t Bw and F-concavity of G(., y) − (.)t Ds give the following inequalities

Now

K(x, y) + xt Bw − K(u, y) − ut Bw ≥ Fx,u (5x K(u, y) + Bw)

(51)

G(u, y) − ut Ds − G(x, y) + xt Ds ≥ Fx,u (− 5x G(u, y) + Ds).

(52)

P (u, y, z) Fx,u 5x Q(u, y, r) Q(u, y, r)(5x K(u, y) + Bw) − P (u, y, z)(5x G(u, y) − Ds) = Fx,u (Q(u, y, r))2 Fx,u (Q(u, y, r)(5x K(u, y) + Bw) − P (u, y, z)(5xG(u, y) − Ds)) (Q(u, y, r))2 Q(u, y, r)Fx,u (5x K(u, y) + Bw) + P (u, y, z)Fx,u (− 5x G(u, y) + Ds) ≤ (Q(u, y, r))2 (by using (A)), =

382


which alongwith (51) and (52), yields Q(u, y, r) (K(x, y) + xt Bw − K(u, y) − ut Bw) P (u, y, z) Fx,u 5x ≤ Q(u, y, r) (Q(u, y, r))2 P (u, y, z) (G(u, y) − ut Ds − G(x, y) + xt Ds) + , (Q(u, y, r))2 which in view of Schwartz inequality, (48) and (49) gives P (u, y, z) Fx,u 5x Q(u, y, r) 1 1 t 2 Q(u, y, r) K(x, y) + (x Bx) ≤ (Q(u, y, r))2 1 1 t t 2 2 − K(u, y) + (u Bu) + P (u, y, z) − G(x, y) − (x Dx) 1 + G(u, y) − (ut Du) 2 1 1 t t 2 = Q(u, y, r) K(x, y) + (x Bx) − y Cz (Q(u, y, r))2 1 t t 2 · K(u, y) + (u Bu) − y Cz 1 +P (u, y, z) − G(x, y) − (xt Dx) 2 + y t Er 1 . + G(u, y) − (ut Du) 2 + y t Er Or

Fx,u 5x

P (u, y, z) Q(u, y, r)

Q(u, y, r) P (x, y, z) − P (u, y, z) ≤

(Q(u, y, r))2 P (u, y, z) − Q(x, y, r) + Q(u, y, r) +

(Q(u, y, r))2

.

Hence (u,y,z) Fx,u 5x P Q(u,y,r) ≥ 0 ⇒ Q(u, y, r)(P (x, y, z) − P (u, y, z)) + P (u, y, z)(−Q(x, y, r) + Q(u, y, r)) ≥0 (Q(u, y, r))2 i.e.,

P (u, y, z) Fx,u 5x ≥ 0 ⇒ Q(u, y, r)P (x, y, z) − P (u, y, z)Q(x, y, r) ≥ 0, Q(u, y, r)


383

i.e.

P (x, y, z) P (u, y, z) P (u, y, z) Fx,u 5x ≥0⇒ ≥ . Q(u, y, r) Q(x, y, r) Q(u, y, r) P (., y, z) is F-pseudoconvex. Hence Q(., y, r) (ii) similarly the second part of the Lemma 2, can be proved.

In view of Lemma 2, Theorems 3 and 4 can be extended for the problems (FP) and (FD). Symmetric dual nondifferentiable fractional mixed integer programming problems can be formulated by choosing the function to be multiplicatively separable [5] and Theorem 7 of Chandra et al. [4] can be extended to nondifferentiable case. The symmetric duality results presented in this paper can be further generalized to multiobjective case along the lines of Nahak [14]. Acknowledgement The authors wish to thank the referee for his valuable suggestions which have improved the presentation of the paper.

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12. B. Mond, S. Chandra and M. V. Durga Prasad, Symmetric dual nondifferentiable fractional programming, Indian J. Manag. Syst. 3 (1987), 1-10. 13. B. Mond and T. Weir, Generalized concavity and duality, in: Generalized Concavity and Duality in Optimization and Economics, Academic Press, New York (1981), 263-279. 14. C. Nahak, On multiobjective generalized symmetric dual programs with ρ − (η, θ)-invexity, J. Appl. Math. & Computing 5 (1998), 707-714. I. Ahmad is a Lecturer in the Department of Mathematics, Aligarh Muslim University, Aligarh, India. He obtained his M. Sc. in Applied Mathematics, M. Phil and Ph. D in Mathematics from University of Roorkee, Roorkee, India (presently I.I.T. Roorkee). He has published many papers in the area of mathematical programming. Department of Mathematics, Aligarh Muslim University, Aligarh -202 002, India e-mail: [email protected] Z. Husain is a Research Scholar in the Department of Mathematics, Aligarh Muslim University, Aligarh, India. He obtained his M. Sc. from M.J.P. Rohilkhand University, Bareilly, India and M. Phil. from Aligarh Muslim University, Aligarh, India Department of Mathematics, Aligarh Muslim University, Aligarh -202 002, India e-mail: [email protected]