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Nondifferentiable second order symmetric duality in multiobjective programming
Applied Mathematics Letters 18 (2005) 721–728 www.elsevier.com/locate/aml
Nondifferentiable second order symmetric duality in multiobjective programming I. Ahmad∗, Z. Husain Department of Mathematics, Aligarh Muslim University, Aligarh-202 002, India Received 21 April 2004; accepted 3 May 2004
Abstract A pair of Mond–Weir type nondifferentiable multiobjective second order symmetric dual programs is formulated and symmetric duality theorems are established under the assumptions of second order F-pseudoconvexity/ F-pseudoconcavity. © 2005 Elsevier Ltd. All rights reserved. Keywords: Symmetric duality; Multiobjective programming; Second order F-pseudoconvexity; Efficient solutions
1. Introduction Symmetric duality in mathematical programming was introduced by Dorn [8], who defined a program and its dual to be symmetric if the dual of the dual is the original problem. Chandra and Husain [5] studied a pair of symmetric dual nondifferentiable programs by assuming convexity/concavity of the scalar function f (x, y). Subsequently, Chandra et al. [3] presented another pair of symmetric dual nondifferentiable programs weakening convexity/concavity assumptions to pseudoconvexity/pseudoconcavity. Weir and Mond [18] discussed symmetric duality in multiobjective programming by using the concept of efficiency. Chandra and Prasad [6] presented a pair of multiobjective programming problems by associating a vector valued infinite game to this pair. Gulati et al. [10] also established duality results for multiobjective symmetric dual problems without non-negativity constraints. ∗ Corresponding author.
E-mail address: [email protected] (I. Ahmad). 0893-9659/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2004.05.010
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I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728
Mangasarian [12] considered a nonlinear program and discussed second order duality under certain inequalities. Mond [14] assumed rather simple inequalities. Bector and Chandra [2] defined the functions satisfying the inequalities in [14] to be bonvex/boncave. Mangasarian [12, p. 609] and Mond [14, p. 93] have indicated possible computational advantages of second order duals over the first order duals. An alternative approach to higher order duality is given in [17]. Mishra [13] and Gulati et al. [9] studied single objective second order symmetric duality for Wolfe and Mond–Weir type models. Kim et al. [11] presented a pair of multiobjective second order symmetric dual problems and established duality results under convexity. Recently, Yang and Hou [19] applied invexity to multiobjective second order symmetric dual problems of [11] omitting non-negativity constraints but with an additional assumption on the invexity. In this paper, we consider a pair of nondifferentiable multiobjective second order symmetric dual programs of Mond–Weir type. We prove weak, strong and converse duality theorems under second order F-pseudoconvexity/F-pseudoconcavity. 2. Notations and preliminaries The following convention for inequalities will be used: If x, u ∈ R n , then x u ⇔ x i u i , i = 1, 2, . . . , n; x ≥ u ⇔ x u and x = u; x > u ⇔ x i > u i , i = 1, 2, . . . , n. If g(x, y) is a real valued twice differentiable function of x and y, where x ∈ R n and y ∈ R m , then x g(x, ¯ y¯ ) and y g(x, ¯ y¯ ) denote the gradient vectors with respect to first and second variable evaluated at (x, ¯ y¯ ) respectively. Also x x g(x, ¯ y¯ ) and yy g(x, ¯ y¯ ) are, respectively, the n ×n and m ×m symmetric Hessian matrices with respect to first and second variable evaluated at (x, ¯ y¯ ). Consider the following multiobjective programming problem: (P) Minimize f (x) subject to x ∈ X, where f : R n → R k and X ⊂ R n . Definition 2.1. A point x¯ ∈ X is said to be weak efficient for (P) if there exists no other x ∈ X with f (x) < f (x). ¯ Definition 2.2. A point x¯ ∈ X is said to be an efficient solution of (P) if there exists no other x ∈ X such that f (x) ≤ f (x¯ ). Definition 2.3. A functional F : X × X × R n −→ R (where X ⊆ R n ) is sublinear in its third component, if for all x, u ∈ X , (i) F (x, u; a1 + a2 ) F (x, u; a1 ) + F (x, u; a2 ) for all a1 , a2 ∈ R n ; and (ii) F (x, u; αa) = α F (x, u; a) for all α ∈ R+ , and for all a ∈ R n . For notational convenience, we write Fx,u (a) = F (x, u; a). Definition 2.4. A real valued twice differentiable function g(., y) : X × Y → R is said to be second order F-pseudoconvex at u ∈ X with respect to p ∈ R n , if there exists a sublinear functional F : X × X × R n → R such that 1 Fx,u (x g(u, y) + x x g(u, y) p) 0 ⇒ g(x, y) g(u, y) − p t x x g(u, y) p. 2
I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728
723
A real valued twice differentiable function g is second order F-pseudoconcave if −g is second order F-pseudoconvex. We shall make use of the following generalized Schwartz inequality: 1
1
x t Ay (x t Ax) 2 (y t Ay) 2
where x, y ∈ R n , and A ∈ R n × R n is a positive semidefinite matrix. Equality holds if for some λ 0, Ax = λAy. 3. Mond–Weir type second order symmetric duality We present the following pair of second order nondifferentiable multiobjective problems with k-objectives and establish weak, strong and converse duality theorems. (MP): Minimize K (x, y, w, p) = (K 1 (x, y, w, p), K 2 (x, y, w, p), . . . , K k (x, y, w, p)) subject to
k
λi [ y f i (x, y) − Ci wi + yy f i (x, y) pi ] 0
(1)
i=1
yt
k
λi [ y f i (x, y) − Ci wi + yy f i (x, y) pi ] 0
(2)
i=1
wit Ci wi 1, i = 1, 2, . . . , k λ>0 x 0.
(3) (4) (5)
(MD): Maximize G(u, v, z, r ) = (G 1 (u, v, z, r ), G 2 (u, v, z, r ), . . . , G k (u, v, z, r )) subject to
k
λi [x f i (u, v) + Bi z i + x x f i (u, v)ri ] 0
(6)
i=1
ut
k
λi [x f i (u, v) + Bi z i + x x f i (u, v)ri ] 0
(7)
i=1
z it Bi z i 1, i = 1, 2, . . . , k λ>0 v 0.
(8) (9) (10)
where 1
K i (x, y, w, p) = f i (x, y) + (x t Bi x) 2 − y t Ci wi −
1 t p f i (x, y) pi , 2 i yy
1 1 G i (u, v, z, r ) = f i (u, v) − (v t Ci v) 2 + u t Bi z i − rit x x f i (u, v)ri , 2
λi ∈ R, pi ∈ R m , ri ∈ R n , i = 1, 2, . . . , k, and f i , i = 1, 2, . . . , k are thrice differentiable functions from R n × R m to R, Bi and Ci , i = 1, 2, . . . , k are positive semidefinite matrices. Also we take p = ( p1 , p2 , . . . , pk ), r = (r1 , r2 , . . . , rk ), w = (w1 , w2 , . . . , wk ) and z = (z 1 , z 2 , . . . , z k ).
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Remark. If k = 1, then (MP) and (MD) become the nondifferentiable second order symmetric dual programs of Ahmad and Husain [1]. Theorem 3.1 (Weak Duality). Let (x, y, λ, w, p) be feasible for (MP) and (u, v, λ, z, r ) feasible for (MD). Assume that (i) ki=1 λi [ f i (., v) + (.)t Bi z i ] is second order F-pseudoconvex at u, (ii) ki=1 λi [ f i (x, .) − (.)t Ci wi ] is second order F-pseudoconcave at y, (iii) Fx,u (ξ ) + u t ξ 0, for ξ ∈ R n , and (iv) Fv,y (η) + y t η 0, for η ∈ R m . Then K (x, y, w, p) ≤ G(u, v, z, r ). Proof. By taking ξ = ki=1 λi (x f i (u, v) + Bi z i + x x f i (u, v)ri ), we have k Fx,u λi (x f i (u, v) + Bi z i + x x f i (u, v)ri ) i=1
−u
k
t
λi (x f i (u, v) + Bi z i + x x f i (u, v)ri ) 0 (by hypothesis (iii) and (7)),
i=1
which by second order F-pseudoconvexity of ki=1 λi [ fi (., v) + (.)t Bi z i ] at u yields k k 1 λi [ fi (x, v) + x t Bi z i ] λi f i (u, v) + u t Bi z i − rit x x f i (u, v)ri . 2 i=1 i=1
(11)
On taking η = − ki=1 λi ( y f i (x, y) − Ci wi + yy f i (x, y) pi ), we have k Fv,y − λi ( y f i (x, y) − Ci wi + yy f i (x, y) pi ) i=1 k
y
t
λi y f i (x, y) − Ci wi + yy f i (x, y) pi
0 (by hypothesis (iv) and (2)),
i=1
which by second order F-pseudoconcavity of ki=1 λi [ f i (x, .) − (.)t Ci wi ] at y gives k k 1 t t t λi [ fi (x, v) − v Ci wi ] λi f i (x, y) − y Ci wi − pi yy f i (x, y) pi . 2 i=1 i=1 Combining inequalities (11) and (12), we get k k 1 λi [x t Bi z i + v t Ci wi ] λi f i (u, v) + u t Bi z i − rit x x f i (u, v)ri 2 i=1 i=1 1 t t − f i (x, y) + y Ci wi + pi yy f i (x, y) pi . 2
(12)
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Applying the Schwartz inequality, (3) and (8), we obtain k 1 1 t t t 2 λi f i (x, y) + (x Bi x) − y Ci wi − pi yy f i (x, y) pi 2 i=1 k 1 1 t t t λi f i (u, v) − (v Ci v) 2 + u Bi z i − ri x x f i (u, v)ri . 2 i=1 Hence K (x, y, w, p) ≤ G(u, v, z, r ). ¯ w, Theorem 3.2 (Strong Duality). Let f be thrice differentiable on R n × R m and (x, ¯ y¯ , λ, ¯ p), ¯ a weak efficient solution for (MP), and λ = λ¯ fixed in (MD). Assume that (i) yy f i is nonsingular for all i = 1, 2, . . . , k, (ii) the matrix ki=1 λ¯ i ( yy f i p¯ i ) y is positive or negative definite, and
(iii) the set y f 1 − C1 w¯ 1 + yy f 1 p¯ 1 , y f 2 − C2 w¯ 2 + yy f 2 p¯ 2 , . . . , y f k − Ck w¯ k + yy f k p¯ k is linearly independent, ¯ y¯ ), i = 1, 2, . . . , k. Then (x, ¯ y¯ , λ¯ , z¯ , r¯ = 0) is feasible for (MD), and the two objectives where f i = f i (x, have the same values. Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of (MP) and (MD), then (x, ¯ y¯ , λ¯ , z¯ , r¯ = 0) is an efficient solution for (MD). Proof. Since (x, ¯ y¯ , λ¯ , w, ¯ p) ¯ is a weak efficient solution of (MP), by the Fritz–John conditions [7], there exist α ∈ R k , β ∈ R m , γ ∈ R, ν ∈ R k , δ ∈ R k and ξ ∈ R n such that k 1 αi x f i + Bi z¯ i − ( yy f i p¯ i )x p¯ i 2 i=1 +
k
λ¯ i yx f i + ( yy f i p¯ i )x (β − γ y¯ ) − ξ = 0
i=1 k
αi
i=1
+
1 y f i − Ci w¯ i − ( yy f i p¯ i ) y p¯ i 2
k
λ¯ i [ yy f i + ( yy f i p¯ i ) y ](β − γ y¯ ) − γ
i=1
k
λ¯ i [ y f i − Ci w¯ i + yy f i p¯ i ] = 0
(14)
i=1
(β − γ y¯ )t y f i − Ci w¯ i + yy f i p¯ i − δi = 0, i = 1, 2, . . . , k t¯ i = 1, 2, . . . , k αi Ci y¯ + (β − γ y¯ ) λi Ci = 2νi Ci w¯ i , t i = 1, 2, . . . , k (β − γ y¯ )λ¯ i − αi p¯ i yy f i = 0, 1 2
x¯ t Bi z¯ i = (x¯ t Bi x) ¯ , i = 1, 2, . . . , k k βt λ¯ i ( y f i − Ci w¯ i + yy f i p¯ i ) = 0 i=1
(13)
(15) (16) (17) (18) (19)
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I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728 k
λ¯ i ( y f i − Ci w¯ i + yy f i p¯ i ) = 0
(20)
νi (w¯ it Ci w¯ i − 1) = 0, i = 1, 2, . . . , k t¯ δ λ=0 x¯ t ξ = 0 i = 1, 2, . . . , k z¯ it Bi z¯ i 1, (α, β, γ , ν, δ, ξ ) 0 (α, β, γ , ν, δ, ξ ) = 0.
(21) (22) (23) (24) (25) (26)
γ y¯
i=1
Since λ¯ > 0 and δ 0, (22) implies δ = 0. Consequently, (15) yields (β − γ y¯ )t y f i − Ci w¯ i + yy f i p¯ i = 0, i = 1, 2, . . . , k.
(27)
Since yy f i is nonsingular for i = 1, 2, . . . , k, from (17), it follows that (β − γ y¯ )λ¯ i = αi p¯ i ,
i = 1, 2, . . . , k.
(28)
From (14), we get k k λ¯ i yy f i (β − γ y¯ − γ p¯ i ) (αi − γ λ¯ i )( y f i − Ci w¯ i ) + i=1
+
k
( yy f i p¯ i ) y
i=1
i=1
1 ¯ (β − γ y¯ )λi − αi p¯ i = 0. 2
By using (28), it follows that k k 1 ¯ λ¯ i ( yy f i p¯ i ) y (β − γ y¯ ) = 0. (αi − γ λi )( y f i − Ci w¯ i + yy f i p¯ i ) + 2 i=1 i=1
(29)
Premultiplying (29) by (β − γ y¯ )t and using (27), we obtain (β − γ y¯ )t
k
λ¯ i ( yy f i p¯ i ) y (β − γ y¯ ) = 0,
i=1
which by hypothesis (ii) implies β = γ y¯ . Therefore, from (29), we get yields αi = γ λ¯ i ,
k
i=1 (αi
i = 1, 2, . . . , k.
(30) − γ λ¯ i )( y f i − Ci w¯ i + yy f i p¯ i ) = 0, which by hypothesis (iii) (31)
If γ = 0, then αi = 0, i = 1, 2, . . . , k and from (30), β = 0. Also from (13) and (16), we get ξi = 0, νi = 0, i = 1, 2, . . . , k. Thus (α, β, γ , δ, ν, ξ ) = 0, a contradiction to (26). Hence γ > 0. Since λ¯ i > 0, i = 1, 2, . . . , k, (31) implies αi > 0, i = 1, 2, . . . , k. Using (30) in (28), we get αi p¯ i = 0, i = 1, 2, . . . , k, and hence p¯ i = 0, i = 1, 2, . . . , k. Using (30) and p¯ i = 0, i = 1, 2, . . . , k,
I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728
in (13), it follows that
k
i=1 αi [ x
727
f i + Bi z¯ i ] = ξ , which by (31) gives
k
ξ λ¯ i [x f i + Bi z¯ i ] = 0, γ i=1
(32)
and x¯ t
k
x¯ t ξ = 0. λ¯ i [x f i + Bi z¯ i ] = γ i=1
(33)
Also, from (30), we have y¯ =
β 0. γ
(34)
¯ z¯ , r¯ = 0) is feasible for (MD). Now let Hence from (24) and (32)–(34), (x, ¯ y¯ , λ, and from (16) and (30)
2νi αi
Ci y¯ = aCi w¯ i ,
= a. Then a 0 (35)
which is the condition for equality in the Schwartz inequality. Therefore 1
1
y¯ t Ci w¯ i = ( y¯ t Ci y¯ ) 2 (w¯ it Ci w¯ i ) 2 . 1
In case νi > 0, (21) gives w¯ it Ci w¯ i = 1 and so y¯ t Ci w¯ i = ( y¯ t Ci y¯ ) 2 . In case νi = 0, (35) gives Ci y¯ = 0 1
and so y¯ t Ci w¯ i = ( y¯ t Ci y¯ ) 2 = 0. Thus in either case 1
y¯ t Ci w¯ i = ( y¯ t Ci y¯ ) 2 .
(36)
Hence 1
K i (x, ¯ y¯ , w, ¯ p¯ = 0) = f i (x, ¯ y¯ ) + (x¯ t Bi x) ¯ 2 − y¯ t Ci w¯ i 1
= f i (x, ¯ y¯ ) − ( y¯ t Ci y¯ ) 2 + x¯ t B z¯ i = G i (x, ¯ y¯ , z¯ , r¯ = 0) (using (18) and (36)). ¯ z¯ , r¯ = 0) is an efficient solution for (MD). Now it follows from Theorem 3.1 that (x, ¯ y¯ , λ, A converse duality theorem may be merely stated as its proof would run analogously to that of Theorem 3.2. Theorem 3.3 (Converse Duality). Let f be thrice differentiable on R n × R m and (u, ¯ v, ¯ λ¯ , z¯ , r¯ ), a weak efficient solution for (MD), and λ = λ¯ fixed in (MP). Assume that (i) x x f i is nonsingular for all i = 1, 2, . . . , k, (ii) the matrix ki=1 λ¯ i (x x f i r¯i )x is positive or negative definite, and (iii) the set {x f 1 + B1 z¯ 1 + x x f 1r¯1 , x f 2 + B2 z¯ 2 + x x f 2r¯2 , . . . , x f k + Bk z¯ k + x x f k r¯k } is linearly independent, where f i = f i (u, ¯ v), ¯ i = 1, 2, . . . , k. Then (u, ¯ v, ¯ λ¯ , w, ¯ p¯ = 0) is feasible for (MP), and the two objectives have the same values. Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of (MP) and (MD), then (u, ¯ v, ¯ λ¯ , w, ¯ p¯ = 0) is an efficient solution for (MP).
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I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728
4. Special cases (i) If Bi = Ci = 0, i = 1, 2, . . . , k, then (MP) and (MD) reduce to the second order multiobjective symmetric dual program studied by Suneja et al. [16] with the omission of non-negativity constraints from (MP) and (MD). If in addition p = r = 0, and k = 1, then we get the first order symmetric dual programs of Chandra et al. [4]. (ii) If we set p = r = 0, in (MP) and (MD), then we obtain a pair of first order symmetric dual nondifferentiable multiobjective programs considered by Mond et al. [15].
References [1] I. Ahmad, Z. Husain, Nondifferentiable second order symmetric duality, Asia-Pacific J. Oper. Res. 22 (2005) (in press). [2] C.R. Bector, S. Chandra, Generalized bonvexity and higher order duality for fractional programming, Opsearch 24 (1987) 143–154. [3] S. Chandra, B.D. Craven, B. Mond, Generalized concavity and duality with a square root term, Optimization 16 (1985) 653–662. [4] S. Chandra, A. Goyal, I. Husain, On symmetric duality in mathematical programming with F-convexity, Optimization 43 (1998) 1–18. [5] S. Chandra, I. Husain, Nondifferentiable symmetric dual programs, Bull. Austral. Math. Soc. 24 (1981) 295–307. [6] S. Chandra, D. Prasad, Symmetric duality in multiobjective programming, J. Austral. Math. Soc. (Ser. B) 35 (1993) 198–206. [7] B.D. Craven, Lagrangian conditions and quasiduality, Bull. Austral. Math. Soc. 16 (1977) 325–339. [8] W.S. Dorn, A symmetric dual theorem for quadratic programs, J. Oper. Res. Soc. Japan 2 (1960) 93–97. [9] T.R. Gulati, I. Ahmad, I. Husain, Second order symmetric duality with generalized convexity, Opsearch 38 (2001) 210–222. [10] T.R. Gulati, I. Husain, A. Ahmed, Multiobjective symmetric duality with invexity, Bull. Austral. Math. Soc. 56 (1997) 25–36. [11] D.S. Kim, Y.B. Yun, H. Kuk, Second order symmetric and self duality in multiobjective programming, Appl. Math. Lett. 10 (1997) 17–22. [12] O.L. Mangasarian, Second and higher order duality in nonlinear programming, J. Math. Anal. Appl. 51 (1975) 607–620. [13] S.K. Mishra, Second order symmetric duality in mathematical programming with F-convexity, European J. Oper. Res. 127 (2000) 507–518. [14] B. Mond, Second order duality for nonlinear programs, Opsearch 11 (1974) 90–99. [15] B. Mond, I. Husain, M.V. Durga Prasad, Duality for a class of nondifferentiable multiobjective programming, Util. Math. 39 (1991) 3–19. [16] S.K. Suneja, C.S. Lalitha, S. Khurana, Second order symmetric duality in multiobjective programming, European. J. Oper. Res. 144 (2003) 492–500. [17] P.S. Unger, A.P. Hunter Jr., The dual of the dual as a linear approximation of the primal, Int. J. Syst. Sci. 12 (1974) 1119–1130. [18] T. Weir, B. Mond, Symmetric and self duality in multiobjective programming, Asia-Pacific J. Oper. Res. 5 (1988) 124–133. [19] X.M. Yang, S.H. Hou, Second order symmetric duality in multiobjective programming, Appl. Math. Lett. 14 (2001) 587–592.
Nondifferentiable second order symmetric duality in multiobjective programming I. Ahmad∗, Z. Husain Department of Mathematics, Aligarh Muslim University, Aligarh-202 002, India Received 21 April 2004; accepted 3 May 2004
Abstract A pair of Mond–Weir type nondifferentiable multiobjective second order symmetric dual programs is formulated and symmetric duality theorems are established under the assumptions of second order F-pseudoconvexity/ F-pseudoconcavity. © 2005 Elsevier Ltd. All rights reserved. Keywords: Symmetric duality; Multiobjective programming; Second order F-pseudoconvexity; Efficient solutions
1. Introduction Symmetric duality in mathematical programming was introduced by Dorn [8], who defined a program and its dual to be symmetric if the dual of the dual is the original problem. Chandra and Husain [5] studied a pair of symmetric dual nondifferentiable programs by assuming convexity/concavity of the scalar function f (x, y). Subsequently, Chandra et al. [3] presented another pair of symmetric dual nondifferentiable programs weakening convexity/concavity assumptions to pseudoconvexity/pseudoconcavity. Weir and Mond [18] discussed symmetric duality in multiobjective programming by using the concept of efficiency. Chandra and Prasad [6] presented a pair of multiobjective programming problems by associating a vector valued infinite game to this pair. Gulati et al. [10] also established duality results for multiobjective symmetric dual problems without non-negativity constraints. ∗ Corresponding author.
E-mail address: [email protected] (I. Ahmad). 0893-9659/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2004.05.010
722
I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728
Mangasarian [12] considered a nonlinear program and discussed second order duality under certain inequalities. Mond [14] assumed rather simple inequalities. Bector and Chandra [2] defined the functions satisfying the inequalities in [14] to be bonvex/boncave. Mangasarian [12, p. 609] and Mond [14, p. 93] have indicated possible computational advantages of second order duals over the first order duals. An alternative approach to higher order duality is given in [17]. Mishra [13] and Gulati et al. [9] studied single objective second order symmetric duality for Wolfe and Mond–Weir type models. Kim et al. [11] presented a pair of multiobjective second order symmetric dual problems and established duality results under convexity. Recently, Yang and Hou [19] applied invexity to multiobjective second order symmetric dual problems of [11] omitting non-negativity constraints but with an additional assumption on the invexity. In this paper, we consider a pair of nondifferentiable multiobjective second order symmetric dual programs of Mond–Weir type. We prove weak, strong and converse duality theorems under second order F-pseudoconvexity/F-pseudoconcavity. 2. Notations and preliminaries The following convention for inequalities will be used: If x, u ∈ R n , then x u ⇔ x i u i , i = 1, 2, . . . , n; x ≥ u ⇔ x u and x = u; x > u ⇔ x i > u i , i = 1, 2, . . . , n. If g(x, y) is a real valued twice differentiable function of x and y, where x ∈ R n and y ∈ R m , then x g(x, ¯ y¯ ) and y g(x, ¯ y¯ ) denote the gradient vectors with respect to first and second variable evaluated at (x, ¯ y¯ ) respectively. Also x x g(x, ¯ y¯ ) and yy g(x, ¯ y¯ ) are, respectively, the n ×n and m ×m symmetric Hessian matrices with respect to first and second variable evaluated at (x, ¯ y¯ ). Consider the following multiobjective programming problem: (P) Minimize f (x) subject to x ∈ X, where f : R n → R k and X ⊂ R n . Definition 2.1. A point x¯ ∈ X is said to be weak efficient for (P) if there exists no other x ∈ X with f (x) < f (x). ¯ Definition 2.2. A point x¯ ∈ X is said to be an efficient solution of (P) if there exists no other x ∈ X such that f (x) ≤ f (x¯ ). Definition 2.3. A functional F : X × X × R n −→ R (where X ⊆ R n ) is sublinear in its third component, if for all x, u ∈ X , (i) F (x, u; a1 + a2 ) F (x, u; a1 ) + F (x, u; a2 ) for all a1 , a2 ∈ R n ; and (ii) F (x, u; αa) = α F (x, u; a) for all α ∈ R+ , and for all a ∈ R n . For notational convenience, we write Fx,u (a) = F (x, u; a). Definition 2.4. A real valued twice differentiable function g(., y) : X × Y → R is said to be second order F-pseudoconvex at u ∈ X with respect to p ∈ R n , if there exists a sublinear functional F : X × X × R n → R such that 1 Fx,u (x g(u, y) + x x g(u, y) p) 0 ⇒ g(x, y) g(u, y) − p t x x g(u, y) p. 2
I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728
723
A real valued twice differentiable function g is second order F-pseudoconcave if −g is second order F-pseudoconvex. We shall make use of the following generalized Schwartz inequality: 1
1
x t Ay (x t Ax) 2 (y t Ay) 2
where x, y ∈ R n , and A ∈ R n × R n is a positive semidefinite matrix. Equality holds if for some λ 0, Ax = λAy. 3. Mond–Weir type second order symmetric duality We present the following pair of second order nondifferentiable multiobjective problems with k-objectives and establish weak, strong and converse duality theorems. (MP): Minimize K (x, y, w, p) = (K 1 (x, y, w, p), K 2 (x, y, w, p), . . . , K k (x, y, w, p)) subject to
k
λi [ y f i (x, y) − Ci wi + yy f i (x, y) pi ] 0
(1)
i=1
yt
k
λi [ y f i (x, y) − Ci wi + yy f i (x, y) pi ] 0
(2)
i=1
wit Ci wi 1, i = 1, 2, . . . , k λ>0 x 0.
(3) (4) (5)
(MD): Maximize G(u, v, z, r ) = (G 1 (u, v, z, r ), G 2 (u, v, z, r ), . . . , G k (u, v, z, r )) subject to
k
λi [x f i (u, v) + Bi z i + x x f i (u, v)ri ] 0
(6)
i=1
ut
k
λi [x f i (u, v) + Bi z i + x x f i (u, v)ri ] 0
(7)
i=1
z it Bi z i 1, i = 1, 2, . . . , k λ>0 v 0.
(8) (9) (10)
where 1
K i (x, y, w, p) = f i (x, y) + (x t Bi x) 2 − y t Ci wi −
1 t p f i (x, y) pi , 2 i yy
1 1 G i (u, v, z, r ) = f i (u, v) − (v t Ci v) 2 + u t Bi z i − rit x x f i (u, v)ri , 2
λi ∈ R, pi ∈ R m , ri ∈ R n , i = 1, 2, . . . , k, and f i , i = 1, 2, . . . , k are thrice differentiable functions from R n × R m to R, Bi and Ci , i = 1, 2, . . . , k are positive semidefinite matrices. Also we take p = ( p1 , p2 , . . . , pk ), r = (r1 , r2 , . . . , rk ), w = (w1 , w2 , . . . , wk ) and z = (z 1 , z 2 , . . . , z k ).
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Remark. If k = 1, then (MP) and (MD) become the nondifferentiable second order symmetric dual programs of Ahmad and Husain [1]. Theorem 3.1 (Weak Duality). Let (x, y, λ, w, p) be feasible for (MP) and (u, v, λ, z, r ) feasible for (MD). Assume that (i) ki=1 λi [ f i (., v) + (.)t Bi z i ] is second order F-pseudoconvex at u, (ii) ki=1 λi [ f i (x, .) − (.)t Ci wi ] is second order F-pseudoconcave at y, (iii) Fx,u (ξ ) + u t ξ 0, for ξ ∈ R n , and (iv) Fv,y (η) + y t η 0, for η ∈ R m . Then K (x, y, w, p) ≤ G(u, v, z, r ). Proof. By taking ξ = ki=1 λi (x f i (u, v) + Bi z i + x x f i (u, v)ri ), we have k Fx,u λi (x f i (u, v) + Bi z i + x x f i (u, v)ri ) i=1
−u
k
t
λi (x f i (u, v) + Bi z i + x x f i (u, v)ri ) 0 (by hypothesis (iii) and (7)),
i=1
which by second order F-pseudoconvexity of ki=1 λi [ fi (., v) + (.)t Bi z i ] at u yields k k 1 λi [ fi (x, v) + x t Bi z i ] λi f i (u, v) + u t Bi z i − rit x x f i (u, v)ri . 2 i=1 i=1
(11)
On taking η = − ki=1 λi ( y f i (x, y) − Ci wi + yy f i (x, y) pi ), we have k Fv,y − λi ( y f i (x, y) − Ci wi + yy f i (x, y) pi ) i=1 k
y
t
λi y f i (x, y) − Ci wi + yy f i (x, y) pi
0 (by hypothesis (iv) and (2)),
i=1
which by second order F-pseudoconcavity of ki=1 λi [ f i (x, .) − (.)t Ci wi ] at y gives k k 1 t t t λi [ fi (x, v) − v Ci wi ] λi f i (x, y) − y Ci wi − pi yy f i (x, y) pi . 2 i=1 i=1 Combining inequalities (11) and (12), we get k k 1 λi [x t Bi z i + v t Ci wi ] λi f i (u, v) + u t Bi z i − rit x x f i (u, v)ri 2 i=1 i=1 1 t t − f i (x, y) + y Ci wi + pi yy f i (x, y) pi . 2
(12)
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725
Applying the Schwartz inequality, (3) and (8), we obtain k 1 1 t t t 2 λi f i (x, y) + (x Bi x) − y Ci wi − pi yy f i (x, y) pi 2 i=1 k 1 1 t t t λi f i (u, v) − (v Ci v) 2 + u Bi z i − ri x x f i (u, v)ri . 2 i=1 Hence K (x, y, w, p) ≤ G(u, v, z, r ). ¯ w, Theorem 3.2 (Strong Duality). Let f be thrice differentiable on R n × R m and (x, ¯ y¯ , λ, ¯ p), ¯ a weak efficient solution for (MP), and λ = λ¯ fixed in (MD). Assume that (i) yy f i is nonsingular for all i = 1, 2, . . . , k, (ii) the matrix ki=1 λ¯ i ( yy f i p¯ i ) y is positive or negative definite, and
(iii) the set y f 1 − C1 w¯ 1 + yy f 1 p¯ 1 , y f 2 − C2 w¯ 2 + yy f 2 p¯ 2 , . . . , y f k − Ck w¯ k + yy f k p¯ k is linearly independent, ¯ y¯ ), i = 1, 2, . . . , k. Then (x, ¯ y¯ , λ¯ , z¯ , r¯ = 0) is feasible for (MD), and the two objectives where f i = f i (x, have the same values. Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of (MP) and (MD), then (x, ¯ y¯ , λ¯ , z¯ , r¯ = 0) is an efficient solution for (MD). Proof. Since (x, ¯ y¯ , λ¯ , w, ¯ p) ¯ is a weak efficient solution of (MP), by the Fritz–John conditions [7], there exist α ∈ R k , β ∈ R m , γ ∈ R, ν ∈ R k , δ ∈ R k and ξ ∈ R n such that k 1 αi x f i + Bi z¯ i − ( yy f i p¯ i )x p¯ i 2 i=1 +
k
λ¯ i yx f i + ( yy f i p¯ i )x (β − γ y¯ ) − ξ = 0
i=1 k
αi
i=1
+
1 y f i − Ci w¯ i − ( yy f i p¯ i ) y p¯ i 2
k
λ¯ i [ yy f i + ( yy f i p¯ i ) y ](β − γ y¯ ) − γ
i=1
k
λ¯ i [ y f i − Ci w¯ i + yy f i p¯ i ] = 0
(14)
i=1
(β − γ y¯ )t y f i − Ci w¯ i + yy f i p¯ i − δi = 0, i = 1, 2, . . . , k t¯ i = 1, 2, . . . , k αi Ci y¯ + (β − γ y¯ ) λi Ci = 2νi Ci w¯ i , t i = 1, 2, . . . , k (β − γ y¯ )λ¯ i − αi p¯ i yy f i = 0, 1 2
x¯ t Bi z¯ i = (x¯ t Bi x) ¯ , i = 1, 2, . . . , k k βt λ¯ i ( y f i − Ci w¯ i + yy f i p¯ i ) = 0 i=1
(13)
(15) (16) (17) (18) (19)
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I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728 k
λ¯ i ( y f i − Ci w¯ i + yy f i p¯ i ) = 0
(20)
νi (w¯ it Ci w¯ i − 1) = 0, i = 1, 2, . . . , k t¯ δ λ=0 x¯ t ξ = 0 i = 1, 2, . . . , k z¯ it Bi z¯ i 1, (α, β, γ , ν, δ, ξ ) 0 (α, β, γ , ν, δ, ξ ) = 0.
(21) (22) (23) (24) (25) (26)
γ y¯
i=1
Since λ¯ > 0 and δ 0, (22) implies δ = 0. Consequently, (15) yields (β − γ y¯ )t y f i − Ci w¯ i + yy f i p¯ i = 0, i = 1, 2, . . . , k.
(27)
Since yy f i is nonsingular for i = 1, 2, . . . , k, from (17), it follows that (β − γ y¯ )λ¯ i = αi p¯ i ,
i = 1, 2, . . . , k.
(28)
From (14), we get k k λ¯ i yy f i (β − γ y¯ − γ p¯ i ) (αi − γ λ¯ i )( y f i − Ci w¯ i ) + i=1
+
k
( yy f i p¯ i ) y
i=1
i=1
1 ¯ (β − γ y¯ )λi − αi p¯ i = 0. 2
By using (28), it follows that k k 1 ¯ λ¯ i ( yy f i p¯ i ) y (β − γ y¯ ) = 0. (αi − γ λi )( y f i − Ci w¯ i + yy f i p¯ i ) + 2 i=1 i=1
(29)
Premultiplying (29) by (β − γ y¯ )t and using (27), we obtain (β − γ y¯ )t
k
λ¯ i ( yy f i p¯ i ) y (β − γ y¯ ) = 0,
i=1
which by hypothesis (ii) implies β = γ y¯ . Therefore, from (29), we get yields αi = γ λ¯ i ,
k
i=1 (αi
i = 1, 2, . . . , k.
(30) − γ λ¯ i )( y f i − Ci w¯ i + yy f i p¯ i ) = 0, which by hypothesis (iii) (31)
If γ = 0, then αi = 0, i = 1, 2, . . . , k and from (30), β = 0. Also from (13) and (16), we get ξi = 0, νi = 0, i = 1, 2, . . . , k. Thus (α, β, γ , δ, ν, ξ ) = 0, a contradiction to (26). Hence γ > 0. Since λ¯ i > 0, i = 1, 2, . . . , k, (31) implies αi > 0, i = 1, 2, . . . , k. Using (30) in (28), we get αi p¯ i = 0, i = 1, 2, . . . , k, and hence p¯ i = 0, i = 1, 2, . . . , k. Using (30) and p¯ i = 0, i = 1, 2, . . . , k,
I. Ahmad, Z. Husain / Applied Mathematics Letters 18 (2005) 721–728
in (13), it follows that
k
i=1 αi [ x
727
f i + Bi z¯ i ] = ξ , which by (31) gives
k
ξ λ¯ i [x f i + Bi z¯ i ] = 0, γ i=1
(32)
and x¯ t
k
x¯ t ξ = 0. λ¯ i [x f i + Bi z¯ i ] = γ i=1
(33)
Also, from (30), we have y¯ =
β 0. γ
(34)
¯ z¯ , r¯ = 0) is feasible for (MD). Now let Hence from (24) and (32)–(34), (x, ¯ y¯ , λ, and from (16) and (30)
2νi αi
Ci y¯ = aCi w¯ i ,
= a. Then a 0 (35)
which is the condition for equality in the Schwartz inequality. Therefore 1
1
y¯ t Ci w¯ i = ( y¯ t Ci y¯ ) 2 (w¯ it Ci w¯ i ) 2 . 1
In case νi > 0, (21) gives w¯ it Ci w¯ i = 1 and so y¯ t Ci w¯ i = ( y¯ t Ci y¯ ) 2 . In case νi = 0, (35) gives Ci y¯ = 0 1
and so y¯ t Ci w¯ i = ( y¯ t Ci y¯ ) 2 = 0. Thus in either case 1
y¯ t Ci w¯ i = ( y¯ t Ci y¯ ) 2 .
(36)
Hence 1
K i (x, ¯ y¯ , w, ¯ p¯ = 0) = f i (x, ¯ y¯ ) + (x¯ t Bi x) ¯ 2 − y¯ t Ci w¯ i 1
= f i (x, ¯ y¯ ) − ( y¯ t Ci y¯ ) 2 + x¯ t B z¯ i = G i (x, ¯ y¯ , z¯ , r¯ = 0) (using (18) and (36)). ¯ z¯ , r¯ = 0) is an efficient solution for (MD). Now it follows from Theorem 3.1 that (x, ¯ y¯ , λ, A converse duality theorem may be merely stated as its proof would run analogously to that of Theorem 3.2. Theorem 3.3 (Converse Duality). Let f be thrice differentiable on R n × R m and (u, ¯ v, ¯ λ¯ , z¯ , r¯ ), a weak efficient solution for (MD), and λ = λ¯ fixed in (MP). Assume that (i) x x f i is nonsingular for all i = 1, 2, . . . , k, (ii) the matrix ki=1 λ¯ i (x x f i r¯i )x is positive or negative definite, and (iii) the set {x f 1 + B1 z¯ 1 + x x f 1r¯1 , x f 2 + B2 z¯ 2 + x x f 2r¯2 , . . . , x f k + Bk z¯ k + x x f k r¯k } is linearly independent, where f i = f i (u, ¯ v), ¯ i = 1, 2, . . . , k. Then (u, ¯ v, ¯ λ¯ , w, ¯ p¯ = 0) is feasible for (MP), and the two objectives have the same values. Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of (MP) and (MD), then (u, ¯ v, ¯ λ¯ , w, ¯ p¯ = 0) is an efficient solution for (MP).
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4. Special cases (i) If Bi = Ci = 0, i = 1, 2, . . . , k, then (MP) and (MD) reduce to the second order multiobjective symmetric dual program studied by Suneja et al. [16] with the omission of non-negativity constraints from (MP) and (MD). If in addition p = r = 0, and k = 1, then we get the first order symmetric dual programs of Chandra et al. [4]. (ii) If we set p = r = 0, in (MP) and (MD), then we obtain a pair of first order symmetric dual nondifferentiable multiobjective programs considered by Mond et al. [15].
References [1] I. Ahmad, Z. Husain, Nondifferentiable second order symmetric duality, Asia-Pacific J. Oper. Res. 22 (2005) (in press). [2] C.R. Bector, S. Chandra, Generalized bonvexity and higher order duality for fractional programming, Opsearch 24 (1987) 143–154. [3] S. Chandra, B.D. Craven, B. Mond, Generalized concavity and duality with a square root term, Optimization 16 (1985) 653–662. [4] S. Chandra, A. Goyal, I. Husain, On symmetric duality in mathematical programming with F-convexity, Optimization 43 (1998) 1–18. [5] S. Chandra, I. Husain, Nondifferentiable symmetric dual programs, Bull. Austral. Math. Soc. 24 (1981) 295–307. [6] S. Chandra, D. Prasad, Symmetric duality in multiobjective programming, J. Austral. Math. Soc. (Ser. B) 35 (1993) 198–206. [7] B.D. Craven, Lagrangian conditions and quasiduality, Bull. Austral. Math. Soc. 16 (1977) 325–339. [8] W.S. Dorn, A symmetric dual theorem for quadratic programs, J. Oper. Res. Soc. Japan 2 (1960) 93–97. [9] T.R. Gulati, I. Ahmad, I. Husain, Second order symmetric duality with generalized convexity, Opsearch 38 (2001) 210–222. [10] T.R. Gulati, I. Husain, A. Ahmed, Multiobjective symmetric duality with invexity, Bull. Austral. Math. Soc. 56 (1997) 25–36. [11] D.S. Kim, Y.B. Yun, H. Kuk, Second order symmetric and self duality in multiobjective programming, Appl. Math. Lett. 10 (1997) 17–22. [12] O.L. Mangasarian, Second and higher order duality in nonlinear programming, J. Math. Anal. Appl. 51 (1975) 607–620. [13] S.K. Mishra, Second order symmetric duality in mathematical programming with F-convexity, European J. Oper. Res. 127 (2000) 507–518. [14] B. Mond, Second order duality for nonlinear programs, Opsearch 11 (1974) 90–99. [15] B. Mond, I. Husain, M.V. Durga Prasad, Duality for a class of nondifferentiable multiobjective programming, Util. Math. 39 (1991) 3–19. [16] S.K. Suneja, C.S. Lalitha, S. Khurana, Second order symmetric duality in multiobjective programming, European. J. Oper. Res. 144 (2003) 492–500. [17] P.S. Unger, A.P. Hunter Jr., The dual of the dual as a linear approximation of the primal, Int. J. Syst. Sci. 12 (1974) 1119–1130. [18] T. Weir, B. Mond, Symmetric and self duality in multiobjective programming, Asia-Pacific J. Oper. Res. 5 (1988) 124–133. [19] X.M. Yang, S.H. Hou, Second order symmetric duality in multiobjective programming, Appl. Math. Lett. 14 (2001) 587–592.