Minimax mixed integer symmetric duality for multiobjective variational ...
European Journal of Operational Research 177 (2007) 71–82 www.elsevier.com/locate/ejor
Continuous Optimization
Minimax mixed integer symmetric duality for multiobjective variational problems I. Ahmad *, Z. Husain Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India Received 3 June 2004; accepted 22 June 2005 Available online 15 February 2006
Abstract A Mond–Weir type multiobjective variational mixed integer symmetric dual program over arbitrary cones is formulated. Applying the separability and generalized F-convexity on the functions involved, weak, strong and converse duality theorems are established. Self duality theorem is proved. A close relationship between these variational problems and static symmetric dual minimax mixed integer multiobjective programming problems is also presented. 2006 Elsevier B.V. All rights reserved. Keywords: Multiobjective symmetric duality; Variational problem; Mixed integer programming; Efficient solutions; Generalized F-convexity
1. Introduction The concept of symmetric duality was introduced and developed by Dorn [8] and Dantzig et al. [6]. Bazaraa and Goode [2] generalized the results in [6] to arbitrary cones. Mond and Weir [19] presented two pairs of symmetric duals multiobjective programming problems for efficient solutions and established appropriate duality results with the nonnegative orthant as the cone. Nanda and Das [20] presented the symmetric dual fractional programming problem for arbitrary cones assuming the functions to be pseudoinvex. Das and Nanda [7] studied symmetric duality in multiobjective programming with cone constraints. Subsequently, Kim et al. [14] derived symmetric duality results for multiobjective programs under pseudoinvex functions and arbitrary cones.
*
Corresponding author. E-mail address: [email protected] (I. Ahmad).
0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.06.070
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Chandra and Abha [3] and Chandra and Kumar [4] pointed out some logical shortcomings in the formulations of the duals and the proofs of the duality theorems of Das and Nanda [7], Kim et al. [14] and Nanda and Das [20] respectively, and observed that these results are highly restricted as they are not valid even for convex case. Recently, Suneja et al. [21] formulated a pair of multiobjective symmetric dual programs over arbitrary cones involving cone-convex functions. Balas [1] generalized the results of Dantzig et al. [6] by constraining some of the primal and dual variables to belong to arbitrary sets of integers and thus introduced minimax symmetric dual programs. Mishra and Das [17] generalized the results of Balas [1] to arbitrary cones. Later on, Mishra et al. [16] extended these results for convex cone domains involving pseudoconvexity/pseudoconcavity. Kumar et al. [15] formulated a modified pair of minimax symmetric dual programs on the lines of Mond and Weir [19]. Recently, Kim and Song [13] formulated two pairs of multiobjective mixed integer symmetric dual programs for arbitrary cones and established duality results. Mond and Hanson [18] extended symmetric duality to variational problems, giving continuous analogues of the results of [6]. Kim and Lee [12] presented a pair of multiobjective symmetric dual variational problems and discussed duality results for efficient solutions assuming invexity. In [11], Gulati et al. constructed a different pair of multiobjective symmetric dual variational programs in which duality results are obtained for properly efficient solutions under pseudoconvexity/pseudoconcavity assumptions. Motivated from the work of Balas [1], Gulati et al. [10] established symmetric duality results for Wolfe and Mond–Weir type single objective minimax mixed integer symmetric variational programs. In [5], Chen extended Wolfe type minimax mixed integer symmetric variational programs in [10] to multiobjective case over arbitrary cones and proved appropriate duality theorems in order to relate these programs. The purpose of this paper is to study Mond–Weir type minimax mixed integer symmetric dual programs for multiobjective variational programming problems involving arbitrary cones and to establish weak, strong, converse and self duality theorems under F-pseudoconvexity/F-pseudoconcavity assumptions on the functions involved.
2. Notations and preliminaries Let I = [a, b] be a real interval, x : I ! Rn and y : I ! Rm are differentiable functions having derivatives x_ and y_ , respectively. Let U Rn1 and V Rm1 be two arbitrary sets of integers and C 1 Rn2 and C 2 Rm2 be closed convex cones with nonempty interiors. Let f i ðt; x; x_ ; y; y_ Þ ¼ f i ðt; x1 ; x_ 1 ; x2 ; x_ 2 ; y 1 ; y_ 1 ; y 2 ; y_ 2 Þ, i = 1, 2, . . . , k be twice continuously differentiable function with respect to x2 ; x_ 2 ; y 2 ; y_ 2 , where x1 2 U, y1 2 V, x2 2 C1, y2 2 C2. Superscripts denote vector components and subscripts denote partial derivatives. fxi2 ; fx_i2 ; fyi2 and fy_i2 denote gradient vectors of f i with respect to x2 ; x_ 2 ; y 2 and y_ 2 . fxi2 x2 ; fx_i2 x_ 2 ; fyi2 y 2 , and fy_i2 y_ 2 denote the Hessian matrices of f i with respect to x2 ; x_ 2 ; y 2 and y_ 2 . Other Hessian matrices fxi2 y 2 ; fxi2 y_ 2 ; fx_i2 y 2 ; fx_i2 y_ 2 ; fyi2 x2 ; fy_i2 x2 ; fyi2 x_ 2 and fy_i2 x_ 2 are defined similarly. Denote by X the space of twice continuously differentiable functions x : I ! Rn with norm kxk = kxk1 + kDxk1 + kD2xk1, where the differentiation operator D is given by Z t u ¼ Dx () xðtÞ ¼ a þ uðsÞds; a
where a is a given boundary value. Therefore D dtd except at discontinuities. Denote by Y the space of twice continuously differentiable functions y : I ! Rm with the norm as that of the space X.
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
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Consider the following multiobjective variational problem: Z b Z b 1 k ðVPÞ: Minimize / ðt; xðtÞ; x_ ðtÞÞdt; . . . ; / ðt; xðtÞ; x_ ðtÞÞdt ; a
a
subject to xðaÞ ¼ 0; xðbÞ ¼ 0; x_ ðaÞ ¼ 0; x_ ðbÞ ¼ 0; hðt; xðtÞ; x_ ðtÞÞ 5 0; t 2 I; where / : I · X · X ! Rk and h : I · X · X ! Rm are differentiable functions. Let Q denote the set of all feasible solutions of (VP), i.e., Q ¼ fx 2 X jxðaÞ ¼ 0; xðbÞ ¼ 0; x_ ðaÞ ¼ 0; x_ ðbÞ ¼ 0; hðt; xðtÞ; x_ ðtÞÞ 5 0; t 2 Ig. Definition 1 (Geoffrion [9]). A point x0 2 Q is said to be an efficient solution of (VP), if for all x 2 Q, Z b Z b /i ðt; x0 ðtÞ; x_ 0 ðtÞÞdt = /i ðt; xðtÞ; x_ ðtÞÞdt for all i ¼ 1; 2; . . . ; k a a Z b Z b /i ðt; x0 ðtÞ; x_ 0 ðtÞÞdt ¼ /i ðt; xðtÞ; x_ ðtÞÞdt for all i ¼ 1; 2; . . . ; k. ) a
a
Definition 2. A point x0 2 Q is said to be a weak efficient solution of (VP), if for all x 2 Q, Z b Z b i 0 0 / ðt; x ðtÞ; x_ ðtÞÞdt > /i ðt; xðtÞ; x_ ðtÞÞdt for all i ¼ 1; 2; . . . ; k a a Z b Z b i 0 0 / ðt; x ðtÞ; x_ ðtÞÞdt ¼ /i ðt; xðtÞ; x_ ðtÞÞdt for all i ¼ 1; 2; . . . ; k. ) a
a
Definition 3. A functional F : I · X · X · X · X · Rn ! R is sublinear, if for any x; x_ ; u; u_ 2 X ; _ f1 þ f2 Þ 5 F ðt; x; x_ ; u; u; _ f1 Þ þ F ðt; x; x_ ; u; u; _ f2 Þ, for any f1, f2 2 Rn; and (i) F ðt; x; x_ ; u; u; _ bfÞ ¼ bF ðt; x; x_ ; u; u; _ fÞ, for any b 2 R, b = 0 and f 2 Rn. (ii) F ðt; x; x_ ; u; u; _ 0Þ ¼ 0. From (ii), it follows that F ðt; x; x_ ; u; u; Rb Definition 4. The functional a f ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞdt is said to be F-pseudoconvex in x and x_ for fixed y and y_ , if Z b Z b Z b _ fx ðt; u; u; _ y; yÞ _ Df x_ ðt; u; u; _ y; y_ ÞÞdt = 0 ) _ y; y_ Þdt F ðt; x; x_ ; u; u; f ðt; x; x_ ; y; y_ Þdt = f ðt; u; u; a
a
a
for all x, u : I ! Rn and for some arbitrary sublinear functional F. Rb Definition 5. The functional a f ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞdt is said to be F-pseudoconcave in y and y_ for fixed x and x_ , if Z b Z b Z b _ F ðt; v; v_ ; y; y_ ; ffy ðt; x; x_ ; v; v_ Þ Df y_ ðt; x; x_ ; v; v_ ÞgÞdt = 0 ) f ðt; x; x_ ; v; v_ Þdt 5 f ðt; x; x_ ; y; yÞdt a
for all y, v : I ! Rm and for some arbitrary sublinear functional.
a
a
74
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Definition 6. A cone C i is said to be polar of Ci for i = 1, 2, if C i ¼ fwjwT x 5 0; for all x 2 C i g. The following concept of separability is required in the sequel which has been extensively used by Balas [1] and Gulati et al. [10]. Let zr, r = 1, 2, . . . , p, be piecewise smooth functions belonging to arbitrary space Zr, r = 1, 2, . . . , p, equipped with norm kzrk = kzrk1 + kDzrk1. A function Lðt; z1 ðtÞ; z_ 1 ðtÞ; z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞÞ will be called additively separable with respect to ðz1 ðtÞ; z_ 1 ðtÞÞ if there exist functions Mðt; z1 ðtÞ; z_ 1 ðtÞÞ (independent of z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞ) and N ðt; z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞÞ (independent of z1 ðtÞ; z_ 1 ðtÞ) such that Lðt; z1 ðtÞ; z_ 1 ðtÞ; z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞÞ ¼ Mðt; z1 ðtÞ; z_ 1 ðtÞÞ þ N ðt; z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞÞ. The following form of Fritz John necessary conditions proposed by Suneja et al. [21] is required to prove the strong and converse duality theorems. Lemma 1. Let P be a convex set with nonempty interior in Rn and suppose that C is a closed convex cone in Rm having a nonempty interior. Let R and S be two vector valued functions defined on P. If z0 is a weak efficient solution of the following problem: Minimize RðzÞ ¼ ðR1 ðzÞ; R2 ðzÞ; . . . ; Rk ðzÞÞ; subject to SðzÞ 2 C; z 2 P ; then there exists a nonzero vector (r0, r) such that
rT0 Rz ðz0 Þ þ rT S z ðz0 Þ ðz z0 Þ = 0; for each z 2 P
and rTS(z0) = 0, r0 = 0, r 2 C*.
3. Mond–Weir type mixed integer symmetric duality In this section, we constrain some of the primal and dual variables to belong to the arbitrary sets of integers, i.e., U and V. Suppose that the first n1 (0 5 n1 5 n) components of x belong to U and the first m1 (0 5 m1 5 m) components of y belong to V. So we write (x, y) = (x1, x2, y1, y2), where x1 ¼ ðx1 ; x2 ; . . . ; xn1 Þ 2 U , and y 1 ¼ ðy 1 ; y 2 ; . . . ; y m1 Þ 2 V ; x2 and y2 being the remaining components of x and y such that x2 2 C1 and y2 2 C2. Let X1 and Y1 be the arbitrary sets of continuously differentiable functions x1 : I ! Rn1 and 1 y : I ! Rm1 ð0 5 n1 5 n; 0 5 m1 5 mÞ equipped with the norms which are prescribed for the spaces X and Y. Partitioning vector functions x and y as above, we introduce the following multiobjective variational minimax mixed integer symmetric dual program: Primal ðMPÞ
Z Max Min x1
¼
x2 ;y
Z a
b
_ f ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞdt
a
b
_ f 1 ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞdt; ...;
Z
b a
_ f k ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞdt ;
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
subject to xðaÞ ¼ 0 ¼ xðbÞ; x_ ðaÞ ¼ 0 ¼ x_ ðbÞ; x1 ðtÞ 2 U ;
75
yðaÞ ¼ 0 ¼ yðbÞ; y_ ðaÞ ¼ 0 ¼ y_ ðbÞ;
y 1 ðtÞ 2 V ;
x2 ðtÞ 2 C 1 ;
T
t 2 I;
T
½ðk f Þy 2 ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ Dðk f Þy_ 2 ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ 2 C 2 ; y 2 ðtÞT ½ðkT f Þy 2 ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ DðkT f Þy_ 2 ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ = 0; k > 0. Z
Dual ðMDÞ
b
_ f ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt
Min Maxu;v2 v1
¼
Z
a b
_ f 1 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt; . . . ;
a
Z
b
_ f k ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt ;
a
subject to uðaÞ ¼ 0 ¼ uðbÞ;
vðaÞ ¼ 0 ¼ vðbÞ;
_ _ uðaÞ ¼ 0 ¼ uðbÞ;
v_ ðaÞ ¼ 0 ¼ v_ ðbÞ;
u1 ðtÞ 2 U ;
v1 ðtÞ 2 V ;
v2 ðtÞ 2 C 2 ;
t 2 I;
_ _ vðtÞ; v_ ðtÞÞ DðkT f Þx_ 2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ 2 C 1 ; ½ðkT f Þx2 ðt; uðtÞ; uðtÞ; _ _ u2 ðtÞT ½ðkT f Þx2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ DðkT f Þx_ 2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ 5 0; k > 0. We shall denote by G and H, the sets of feasible solutions of the primal and dual multiobjective variational problems (MP) and (MD), respectively. Theorem 1 (Weak duality). Let (x, y, k) 2 G and (u, v, k) 2 H. Assume that (i) (ii) (iii) (iv) (v) (vi)
f i ðt; x; x_ ; y; y_ Þ, i = 1, 2, . . . , k is twice differentiable with respect to ðx2 ; x_ 2 Þ and ðy 2 ; y_ 2 Þ, respectively, f i ðt; x; x_ ; y; y_ Þ, i = 1, 2, . . . , k is additively separable with respect to ðx1 ; x_ 1 Þ or ðy 1 ; y_ 1 Þ, Rb T _ k f ðt; x; x_ ; y; yÞdt is F-pseudoconvex in ðx2 ; x_ 2 Þ for each ðx1 ; x_ 1 ; y; y_ Þ on I, Rab T _ k f ðt; x; x_ ; y; yÞdt is F-pseudoconcave in ðy 2 ; y_ 2 Þ for each ðx; x_ ; y 1 ; y_ 1 Þ on I, a T F ðt; x2 ; x_ 2 ; u2 ; u_ 2 ; n2 Þ þ ðu2 Þ n2 = 0, for all x2 ; x_ 2 ; u2 ; u_ 2 2 C 1 and n2 2 C 1 and T F ðt; v2 ; v_ 2 ; y 2 ; y_ 2 ; g2 Þ þ ðy 2 Þ g2 = 0, for all v2 ; v_ 2 ; y 2 ; y_ 2 2 C 2 and g2 2 C 2 .
Then Z
b
f ðt; x; x_ ; y; y_ Þdt i
a
Z
b
_ v; v_ Þdt. f ðt; u; u;
a
Proof. Let g ¼ Max Min x1
x2 ;y
Z
b a
_ f ðt; x; x_ ; y; yÞdtjðx; y; kÞ 2 G
76
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
and h ¼ Min Max v1
Z
u;v2
b
_ v; v_ Þdtjðu; v; kÞ 2 H . f ðt; u; u; a
_ i = 1, 2, . . . , k is additively separable with respect to ðx1 ; x_ 1 Þ or ðy 1 ; y_ 1 Þ (say with respect Since f i ðt; x; x_ ; y; yÞ, 1 1 to ðx ; x_ Þ), it follows that: _ f i ðt; x; x_ ; y; y_ Þ ¼ f1i ðt; x1 ; x_ 1 Þ þ f2i ðt; x2 ; x_ 2 ; y; yÞ; Therefore g can be written as Z b Z 1 1 f1 ðt; x ; x_ Þdt þ g ¼ Max Min x2 ;y
x1
a
or g ¼ Max Min where Z
b
2
1
f1 ðt; x ; x_ Þdt ¼ a
f2 ðt; x ; x_ ; y; y_ Þdtjðx; y; kÞ 2 G
f1 ðt; x1 ; x_ 1 Þdt þ /ðy 1 Þjx1 2 U ; y 1 2 V ; Z
b
f11 ðt; x1 ; x_ 1 Þdt;
Z
a
and /ðy 1 Þ ¼ Min
2
a
b 1
b a
Z
y1
x1
i ¼ 1; 2; . . . ; k.
Z
x2 ;y 2
b
f12 ðt; x1 ; x_ 1 Þdt; . . . ;
a
b
f2 ðt; x2 ; x_ 2 ; y; y_ Þdt ¼
Z
a
Z
b
f1k ðt; x1 ; x_ 1 Þdt
a
b
_ f21 ðt; x2 ; x_ 2 ; y; yÞdt; ...;
Z
a
b
_ f2k ðt; x2 ; x_ 2 ; y; yÞdt
a
subject to x2 ðaÞ ¼ 0 ¼ x2 ðbÞ; 2
y 2 ðaÞ ¼ 0 ¼ y 2 ðbÞ;
2
2
x_ ðaÞ ¼ 0 ¼ x_ ðbÞ; 1
1
x 2 U;
y 2V;
T
ð1Þ
2
y_ ðaÞ ¼ 0 ¼ y_ ðbÞ;
ð2Þ
2
ð3Þ
x 2 C1;
½ðk f2 Þy 2 ðt; x ; x_ 2 ; y; y_ Þ DðkT f2 Þy_ 2 ðt; x2 ; x_ 2 ; y; y_ Þ 2 C 2 ; 2
2 T
T
2
T
2
2
ð4Þ
2
ðy Þ ½ðk f2 Þy 2 ðt; x ; x_ ; y; y_ Þ Dðk f2 Þy_ 2 ðt; x ; x_ ; y; y_ Þ = 0;
ð5Þ
k > 0.
ð6Þ
Similarly h can be written as Z b 1 1 1 1 1 f1 ðt; u ; u_ Þdt þ wðv Þju 2 U ; v 2 V ; h ¼ Min Max v1
u1
where wðv1 Þ ¼ Max u2 ;v2
a
Z
b
f2 ðt; u2 ; u_ 2 ; v; v_ Þdt ¼
Z
a
b
f21 ðt; u2 ; u_ 2 ; v; v_ Þdt; . . . ;
Z
a
b
f2k ðt; u2 ; u_ 2 ; v; v_ Þdt ;
a
subject to u2 ðaÞ ¼ 0 ¼ u2 ðbÞ; 2
v2 ðaÞ ¼ 0 ¼ v2 ðbÞ;
2
2
u_ ðaÞ ¼ 0 ¼ u_ ðbÞ; 1
1
u 2 U;
2
v_ ðaÞ ¼ 0 ¼ v_ ðbÞ;
ð8Þ
2
v 2V;
T
ð7Þ
2
v 2 C2;
ð9Þ T
2
2
2
½ðk f2 Þx2 ðt; u ; u_ ; v; v_ Þ Dðk f2 Þx_ 2 ðt; u ; u_ ; v; v_ Þ 2 2 T
T
2
2
T
2
2
C 1 ;
ðu Þ ½ðk f2 Þx2 ðt; u ; u_ ; v; v_ Þ Dðk f2 Þx_ 2 ðt; u ; u_ ; v; v_ Þ 5 0; k > 0.
ð10Þ ð11Þ ð12Þ
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
77
Let (x, y, k) 2 G and (u, v, k) 2 H. In order to prove the theorem, it is sufficient to prove that / (y1) i w(v1), for a given y1. On taking n2 ¼ ðkT f2 Þx2 DðkT f2 Þx_ 2 ; we have F ½t; x2 ; x_ 2 ; u2 ; u_ 2 ; ðkT f2 Þx2 DðkT f2 Þx_ 2 = ðu2 ÞT ½ðkT f2 Þx2 DðkT f2 Þx_ 2 = 0 ðby hypothesis (v) and (11)Þ; which implies Z b F ½t; x2 ; x_ 2 ; u2 ; u_ 2 ; ðkT f2 Þx2 DðkT f2 Þx_ 2 dt = 0. a
Rb This in view of F-pseudoconvexity of a kT f2 dt in ðx2 ; x_ 2 Þ yields Z b Z b T 2 2 k f2 ðt; x ; x_ ; v; v_ Þdt = kT f2 ðt; u2 ; u_ 2 ; v; v_ Þdt. a
ð13Þ
a
By taking g2 ¼ fðkT f2 Þy 2 DðkT f2 Þy_ 2 g, we get T
F ½t; v2 ; v_ 2 ; y 2 ; y_ 2 ; fðkT f2 Þy 2 DðkT f2 Þy_ 2 g = ðy 2 Þ ½ðkT f2 Þy 2 DðkT f2 Þy_ 2 = 0 ðby hypothesis (vi) and (5)Þ; Z b F ½t; v2 ; v_ 2 ; y 2 ; y_ 2 ; fðkT f2 Þy 2 DðkT f2 Þy_ 2 gdt = 0; ) a
Rb which by F-pseudoconcavity of a kT f2 dt in ðy 2 ; y_ 2 Þ gives Z b Z b kT f2 ðt; x2 ; x_ 2 ; v; v_ Þdt 5 kT f2 ðt; x2 ; x_ 2 ; y; y_ Þdt. a
ð14Þ
a
Adding the inequalities (13) and (14), we obtain Z b Z b T 2 2 _ k f2 ðt; x ; x_ ; y; yÞdt = kT f2 ðt; u2 ; u_ 2 ; v; v_ Þdt. a
Hence Z a
a
b 2
2
_ f2 ðt; x ; x_ ; y; yÞdt i
Z
b
f2 ðt; u2 ; u_ 2 ; v; v_ Þdt.
a
Theorem 2 (Strong duality). Let ðxðtÞ; y ðtÞ; kÞ be an efficient solution of (MP), and fixed k ¼ k in (MD). Assume that _ (i) f i ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞ, i = 1, 2, . . . , k is twice differentiable at x2 ðtÞ; x_ 2 ðtÞ; y 2 ðtÞ and y_ 2 ðtÞ, i _ (ii) f ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞ, i = 1, 2, . . . , k is additively separable with respect to ðx1 ðtÞ; x_ 1 ðtÞÞ or 1 1 ðy ðtÞ; y_ ðtÞÞ, T T (iii) the system ½ðr1 ðtÞ y 2 r2 Þ fð kT f2 Þy 2 y 2 DðkT f2 Þy_ 2 y 2 g Dfðr1 ðtÞ y 2 r2 Þ fðkT f2 Þy 2 y_ 2 DðkT f2 Þy_ 2 y_ 2 gg 2 2 ðr1 ðtÞ y r2 Þ ¼ 0 ) ðr1 ðtÞ y r2 Þ ¼ 0, for every r1(t) 2 C2, t 2 I, and (iv) the set fðf2y1 2 Df 12y_ 2 Þ; . . . ; ðf2yk 2 Df k2_y 2 Þg is linearly independent. Then ðxðtÞ; y ðtÞ; kÞ is a feasible solution of (MD) and the objective values of (MP) and (MD) are equal. Furthermore, if the assumptions of Theorem 1 are satisfied, then ðxðtÞ; y ðtÞ; kÞ is an efficient solution of (MD).
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Proof. Since ðxðtÞ; y ðtÞ; kÞ is an efficient solution of (MP), hence it is weak efficient, then by Lemma 1, there exist r0 2 Rk, r1(t) 2 C2, r2 2 R and d 2 Rk such that rT0 ½f2x2 Df 2_x2 ðx2 ðtÞ x2 ðtÞÞ þ ½ðr1 ðtÞ y 2 r2 ÞT ððkT f2 Þy 2 x2 T f2 Þ 2 2 Þ Dfðr1 ðtÞ y 2 r2 ÞT ðð Dðk kT f2 Þ 2 2 DðkT f2 Þ y_ x
y x_
2 y_ 2 x_ 2 Þgðx ðtÞ
x2 ðtÞÞ=0;
for all x2 ðtÞ 2 C 1 and t 2 I; T T ðr0 kr2 Þ ½f2y 2 Df 2_y 2 þ ½ðr1 ðtÞ y 2 r2 Þ ððkT f2 Þy 2 y 2 DðkT f2 Þy_ 2 y 2 Þ T Dfðr1 ðtÞ y 2 r2 Þ ðð kT f2 Þ 2 2 Dð kT f2 Þ 2 2 Þg ¼ 0; 8t 2 I; y y_
2
y_ y_
T
ð15Þ ð16Þ
ðr1 ðtÞ y r2 Þ ðf2y 2 Df 2y_ 2 Þ d ¼ 0; 8t 2 I; kT f2 Þ 2 Dð kT f2 Þ 2 Þ ¼ 0; 8t 2 I; r1 ðtÞT ðð
ð17Þ
r2 y ðð kT f2 Þy 2 Dð kT f2 Þy_ 2 Þ ¼ 0; k ¼ 0; dT
ð19Þ
ð18Þ
y_
y
2
r0 = 0; r2 = 0; d = 0; ðr0 ; r1 ðtÞ; r2 ; dÞ 6¼ 0.
8t 2 I
ð20Þ ð21Þ ð22Þ
t 2 I;
As k > 0, it follows from (20) that d = 0. Therefore (17) reduces to T ðr1 ðtÞ y 2 r2 Þ ðf2y 2 Df 2y_ 2 Þ ¼ 0
8t 2 I.
ð23Þ
2
Multiplying (16) by ðr1 ðtÞ y r2 Þ and using (23), we get T T ½ðr1 ðtÞ y 2 r2 Þ ðð kT f2 Þy 2 y 2 Dð kT f2 Þy_ 2 y 2 Þ Dfðr1 ðtÞ y 2 r2 Þ ððkT f2 Þy 2 y_ 2 DðkT f2 Þy_ 2 y_ 2 Þgðr1 ðtÞ y 2 r2 Þ
¼ 0; which by hypothesis (iii) gives r1 ðtÞ ¼ y 2 r2 .
ð24Þ
From (16) and (24) T ðr0 kr2 Þ ðf2y 2 Df 2y_ 2 Þ ¼ 0.
Since fðf2y1 2 Df 12_y 2 Þ; . . . ; ðf2yk 2 Df k2_y 2 Þg is linearly independent, then the above equation implies 2. r0 ¼ kr
ð25Þ
If r2 = 0, then from (24) and (25), we obtain r1(t) = r0 = 0. Hence (r0, r1(t), r2, d) = 0, contradicting (22). Thus r2 > 0, Eq. (24) yields y 2 ¼
r1 ðtÞ 2 C2. r2
Now, (15) along with (24) and (25) gives kr2 ½f2x2 Df 2 ðx2 ðtÞ x2 ðtÞÞ = 0. 2_x
Since r2 > 0, the above inequality implies T f2 Þ 2 Dð kT f2 Þ 2 ðx2 ðtÞ x2 ðtÞÞ = 0. ½ðk x
ð26Þ
x_
Let x2(t) 2 C1. Then x2 ðtÞ þ x2 ðtÞ 2 C 1 and so (26) shows that for every x2(t) 2 C1 T f2 Þ 2 Dð ½ðk kT f2 Þ 2 x2 ðtÞ = 0 i.e., ½ð kT f2 Þ 2 DðkT f2 Þ 2 2 C . x
x_
x
x_
1
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79
Also, by letting x2(t) = 0 and x2 ðtÞ ¼ 2x2 ðtÞ simultaneously in the inequality (26), we get x2 ðtÞT ½ð kT f2 Þx2 Dð kT f2 Þx_ 2 ¼ 0. Thus ðxðtÞ; y ðtÞ; kÞ is a feasible solution of (MD) and the objective functional values are equal. By Theorem 1, ðxðtÞ; y ðtÞ; kÞ is an efficient solution of (MD). h A converse duality theorem may be stated as its proof would run analogously to that of Theorem 2. Theorem 3 (Converse duality). Let ð uðtÞ; vðtÞ; kÞ be an efficient solution of (MD), and fixed k ¼ k in (MP). Assume that _ (i) f i ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ, i = 1, 2, . . . , k is twice differentiable at u2 ðtÞ; u_ 2 ðtÞ; v2 ðtÞ and v_ 2 ðtÞ, i _ (ii) f ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ, i = 1, 2, . . . , k is additively separable with respect to ðu1 ðtÞ; u_ 1 ðtÞÞ or 1 1 ðv ðtÞ; v_ ðtÞÞ, T T (iii) the system ½ðr1 ðtÞ u2 r2 Þ fð kT f2 Þx2 x2 DðkT f2 Þx_ 2 x2 g Dfðr1 ðtÞ u2 r2 Þ fðkT f2 Þx2 x_ 2 DðkT f2 Þx_ 2 x_ 2 gg 2 2 ðr1 ðtÞ u r2 Þ ¼ 0 ) ðr1 ðtÞ u r2 Þ ¼ 0, for every r1(t) 2 C1, t 2 I, and (iv) the set fðf2x1 2 Df 12_x2 Þ; . . . ; ðf2xk 2 Df k2_x2 Þg is linearly independent. Then ð uðtÞ; vðtÞ; kÞ is a feasible solution of (MP) and the objective values of (MP) and (MD) are equal. Furthermore, if the assumptions of Theorem 1 are satisfied, then ðuðtÞ; vðtÞ; kÞ is an efficient solution of (MP).
4. Self duality A mathematical programming problem is said to be self dual if it is formally identical with its dual, that is, the dual can be recast in the form of the primal. If we assume (i) C2 = C1, and _ (ii) f i ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ : I Rn Rn Rn Rn ! R, i = 1, 2, . . . , k to be skew symmetric, that is, _ _ f i ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ ¼ f i ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ; i ¼ 1; 2; . . . ; k then we shall show that the programs (MP) and (MD) are self duals. By recasting the dual problem (MD) as maxmin problem, we have Z b _ f ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt Dual ðMDÞ Max Min v1
u;v2
Z ¼
a b 1
_ f ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt; . . . ;
Z
a
b
_ _ f ðt; uðtÞ; uðtÞ; vðtÞ; vðtÞÞdt ; k
a
subject to uðaÞ ¼ 0 ¼ uðbÞ;
vðaÞ ¼ 0 ¼ vðbÞ;
_ _ uðaÞ ¼ 0 ¼ uðbÞ;
v_ ðaÞ ¼ 0 ¼ v_ ðbÞ;
u1 ðtÞ 2 U ; T
v1 ðtÞ 2 V ;
v2 ðtÞ 2 C 2 ;
t 2 I; T
_ _ ½ðk f Þx2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ Dðk f Þx_ 2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ 2 C 1 ; _ _ vðtÞ; v_ ðtÞÞ DðkT f Þx_ 2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ = 0; u2 ðtÞT ½ðkT f Þx2 ðt; uðtÞ; uðtÞ; k > 0.
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_ Since f i ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ, i = 1, 2, . . . , k is skew symmetric, we have _ _ vðtÞ; v_ ðtÞÞ ¼ fyi2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ; fxi2 ðt; uðtÞ; uðtÞ;
i ¼ 1; 2; . . . ; k
_ _ fx_i2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ ¼ fy_i2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ;
i ¼ 1; 2; . . . ; k.
and
The above problem becomes Dual ðMDÞ
Z
b
_ Maxv1 Minu;v2 f ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞdt a Z b Z 1 _ f ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞdt; ...; ¼ a
_ f ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞdt ; k
a
subject to uðaÞ ¼ 0 ¼ uðbÞ; _ _ uðaÞ ¼ 0 ¼ uðbÞ; u1 ðtÞ 2 U ;
b
vðaÞ ¼ 0 ¼ vðbÞ; v_ ðaÞ ¼ 0 ¼ v_ ðbÞ;
v1 ðtÞ 2 V ;
T
v2 ðtÞ 2 C 1 ;
t 2 I;
T
_ _ Dðk f Þy_ 2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ 2 C 1 ; ½ðk f Þy 2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ _ _ DðkT f Þy_ 2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ = 0; u2 ðtÞT ½ðkT f Þy 2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ k > 0. is formally identical to (MP); that is, the objective and the constraint functions of We observe that ðMDÞ (MP) and ðMDÞ are identical. Therefore (MP) is a self dual. It can easily seen that the feasibility of (x(t), y(t), k) for (MP) implies the feasibility of (y(t), x(t), k) for (MD), and conversely. _ Theorem 4 (Self duality). Assume that f i ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞ, i = 1, 2, . . . , k is skew symmetric. Then (MP) is a self dual. Also, if (MP) and (MD) are dual variational problems and ðxðtÞ; y ðtÞ; kÞ is a joint efficient solution, then so is ðy ðtÞ; xðtÞ; kÞ and the common objective functional value is 0. Proof. Since ðxðtÞ; y ðtÞ; kÞ is a joint efficient solution of (MP) and (MD), the objective functional values are equal to Z b f ðt; xðtÞ; x_ ðtÞ; y ðtÞ; y_ ðtÞÞdt. a
is feasible for (MP) if and only if ðy ðtÞ; xðtÞ; kÞ is feasible for (MD). ThereFrom self duality, ðxðtÞ; y ðtÞ; kÞ fore efficiency of ðxðtÞ; y ðtÞ; kÞ for (MP) implies efficiency of ðy ðtÞ; xðtÞ; kÞ for (MD) and vice versa. Hence the objective functional values are equal to Z b f ðt; y ðtÞ; y_ ðtÞ; xðtÞ; x_ ðtÞÞdt. a
Therefore Z b Z _ _ f ðt; xðtÞ; xðtÞ; y ðtÞ; y ðtÞÞdt ¼ a
f ðt; y ðtÞ; y_ ðtÞ; xðtÞ; x_ ðtÞÞdt ¼
a
Thus we have Z b f ðt; xðtÞ; x_ ðtÞ; y ðtÞ; y_ ðtÞÞdt ¼ 0. a
b
Z a
b
f ðt; xðtÞ; x_ ðtÞ; y ðtÞ; y_ ðtÞÞdt.
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5. Static symmetric dual multiobjective programs If the time dependency in (MP) and (MD) is relaxed, then they reduce to the following static minimax mixed integer symmetric dual problems: ðMP1Þ:
Max Min f ðx; yÞ; x1
x2 ;y
subject to x1 2 U ;
y1 2 V ;
x2 2 C 1 ;
ðkT f Þy 2 ðx; yÞ 2 C 2 ; T
ðy 2 Þ ðkT f Þy 2 ðx; yÞ = 0; k > 0. ðMD1Þ:
Min Max f ðu; vÞ v1
u;v2
subject to u1 2 U ;
v1 2 V ;
v2 2 C 2 ;
ðkT f Þx2 ðu; vÞ 2 C 1 ; ðu2 ÞT ðkT f Þx2 ðu; vÞ 5 0; k > 0. The above problems (MP1) and (MD1) are the Mond–Weir type mixed integer symmetric dual programs considered in [13], with the omission of kte = 1, as this is not required for the duality theorems to hold.
Acknowledgements The authors wish to thank the referee for several valuable suggestions which have considerably improved the presentation of the paper.
References [1] E. Balas, Minimax and duality for linear and nonlinear mixed integer programming, in: J. Abadie (Ed.), Integer and Nonlinear Programming, North-Holland, Amsterdam, 1970, pp. 385–417. [2] M.S. Bazaraa, J.J. Goode, On symmetric duality in nonlinear programming, Operations Research 21 (1973) 1–9. [3] S. Chandra, Abha, Technical note on symmetric duality in multiobjective programming: Some remarks on recent results, European Journal of Operational Research 124 (2000) 651–654. [4] S. Chandra, V. Kumar, A note on pseudoinvexity and symmetric duality, European Journal of Operational Research 105 (1998) 626–629. [5] X. Chen, Minimax and symmetric duality for a class of multiobjective variational mixed integer programming problems, European Journal of Operational Research 154 (2004) 71–83. [6] G.B. Dantzig, E. Eisenberg, R.W. Cottle, Symmetric dual nonlinear programs, Pacific Journal of Mathematics 15 (1965) 809–812. [7] L.N. Das, S. Nanda, Symmetric dual multiobjective programming, European Journal of Operational Research 97 (1997) 167–171. [8] W.S. Dorn, A symmetric dual theorem for quadratic programs, Journal of Operations Research Society of Japan 2 (1960) 93–97. [9] A.M. Geoffrion, Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications 22 (1968) 613–630. [10] T.R. Gulati, I. Ahmad, I. Husain, Symmetric duality for minimax variational problems, Mathematical Methods of Operations Research 48 (1998) 81–95.
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[11] T.R. Gulati, I. Husain, A. Ahmed, Symmetric duality for multiobjective variational problems, Journal of Mathematical Analysis and Applications 210 (1997) 22–38. [12] D.S. Kim, W.J. Lee, Symmetric duality for multiobjective variational problems with invexity, Journal of Mathematical Analysis and Applications 218 (1998) 34–48. [13] D.S. Kim, Y.R. Song, Minimax and symmetric duality for nonlinear multiobjective mixed integer programming, European Journal of Operational Research 128 (2001) 435–446. [14] D.S. Kim, Y.B. Yun, W.J. Lee, Multiobjective symmetric duality with cone constraints, European Journal of Operational Research 107 (1998) 686–691. [15] V. Kumar, I. Husain, S. Chandra, Symmetric duality for minimax nonlinear mixed integer programming, European Journal of Operational Research 80 (1995) 425–430. [16] M.S. Mishra, D. Acharya, S. Nanda, On a pair of nonlinear mixed integer programming problem, European Journal of Operational Research 19 (1985) 98–103. [17] B.K. Mishra, C. Das, On minimax and symmetric duality for a nonlinear mixed integer programming problem, Opsearch 17 (1980) 1–11. [18] B. Mond, M.A. Hanson, Symmetric duality for variational problems, Journal of Mathematical Analysis and Applications 23 (1968) 161–172. [19] B. Mond, T. Weir, Symmetric duality for nonlinear multiobjective programming, in: S. Kumar (Ed.), Recent Developments in Mathematical Programming, Gordon and Breach, London, 1991, pp. 137–153. [20] S. Nanda, L.N. Das, Pseudoinvexity and symmetric duality in nonlinear fractional programming, European Journal of Operational Research 73 (1994) 577–582. [21] S.K. Suneja, S. Aggarwal, S. Davar, Multiobjective symmetric duality involving cones, European Journal of Operational Research 141 (2002) 471–479.
Continuous Optimization
Minimax mixed integer symmetric duality for multiobjective variational problems I. Ahmad *, Z. Husain Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India Received 3 June 2004; accepted 22 June 2005 Available online 15 February 2006
Abstract A Mond–Weir type multiobjective variational mixed integer symmetric dual program over arbitrary cones is formulated. Applying the separability and generalized F-convexity on the functions involved, weak, strong and converse duality theorems are established. Self duality theorem is proved. A close relationship between these variational problems and static symmetric dual minimax mixed integer multiobjective programming problems is also presented. 2006 Elsevier B.V. All rights reserved. Keywords: Multiobjective symmetric duality; Variational problem; Mixed integer programming; Efficient solutions; Generalized F-convexity
1. Introduction The concept of symmetric duality was introduced and developed by Dorn [8] and Dantzig et al. [6]. Bazaraa and Goode [2] generalized the results in [6] to arbitrary cones. Mond and Weir [19] presented two pairs of symmetric duals multiobjective programming problems for efficient solutions and established appropriate duality results with the nonnegative orthant as the cone. Nanda and Das [20] presented the symmetric dual fractional programming problem for arbitrary cones assuming the functions to be pseudoinvex. Das and Nanda [7] studied symmetric duality in multiobjective programming with cone constraints. Subsequently, Kim et al. [14] derived symmetric duality results for multiobjective programs under pseudoinvex functions and arbitrary cones.
*
Corresponding author. E-mail address: [email protected] (I. Ahmad).
0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.06.070
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Chandra and Abha [3] and Chandra and Kumar [4] pointed out some logical shortcomings in the formulations of the duals and the proofs of the duality theorems of Das and Nanda [7], Kim et al. [14] and Nanda and Das [20] respectively, and observed that these results are highly restricted as they are not valid even for convex case. Recently, Suneja et al. [21] formulated a pair of multiobjective symmetric dual programs over arbitrary cones involving cone-convex functions. Balas [1] generalized the results of Dantzig et al. [6] by constraining some of the primal and dual variables to belong to arbitrary sets of integers and thus introduced minimax symmetric dual programs. Mishra and Das [17] generalized the results of Balas [1] to arbitrary cones. Later on, Mishra et al. [16] extended these results for convex cone domains involving pseudoconvexity/pseudoconcavity. Kumar et al. [15] formulated a modified pair of minimax symmetric dual programs on the lines of Mond and Weir [19]. Recently, Kim and Song [13] formulated two pairs of multiobjective mixed integer symmetric dual programs for arbitrary cones and established duality results. Mond and Hanson [18] extended symmetric duality to variational problems, giving continuous analogues of the results of [6]. Kim and Lee [12] presented a pair of multiobjective symmetric dual variational problems and discussed duality results for efficient solutions assuming invexity. In [11], Gulati et al. constructed a different pair of multiobjective symmetric dual variational programs in which duality results are obtained for properly efficient solutions under pseudoconvexity/pseudoconcavity assumptions. Motivated from the work of Balas [1], Gulati et al. [10] established symmetric duality results for Wolfe and Mond–Weir type single objective minimax mixed integer symmetric variational programs. In [5], Chen extended Wolfe type minimax mixed integer symmetric variational programs in [10] to multiobjective case over arbitrary cones and proved appropriate duality theorems in order to relate these programs. The purpose of this paper is to study Mond–Weir type minimax mixed integer symmetric dual programs for multiobjective variational programming problems involving arbitrary cones and to establish weak, strong, converse and self duality theorems under F-pseudoconvexity/F-pseudoconcavity assumptions on the functions involved.
2. Notations and preliminaries Let I = [a, b] be a real interval, x : I ! Rn and y : I ! Rm are differentiable functions having derivatives x_ and y_ , respectively. Let U Rn1 and V Rm1 be two arbitrary sets of integers and C 1 Rn2 and C 2 Rm2 be closed convex cones with nonempty interiors. Let f i ðt; x; x_ ; y; y_ Þ ¼ f i ðt; x1 ; x_ 1 ; x2 ; x_ 2 ; y 1 ; y_ 1 ; y 2 ; y_ 2 Þ, i = 1, 2, . . . , k be twice continuously differentiable function with respect to x2 ; x_ 2 ; y 2 ; y_ 2 , where x1 2 U, y1 2 V, x2 2 C1, y2 2 C2. Superscripts denote vector components and subscripts denote partial derivatives. fxi2 ; fx_i2 ; fyi2 and fy_i2 denote gradient vectors of f i with respect to x2 ; x_ 2 ; y 2 and y_ 2 . fxi2 x2 ; fx_i2 x_ 2 ; fyi2 y 2 , and fy_i2 y_ 2 denote the Hessian matrices of f i with respect to x2 ; x_ 2 ; y 2 and y_ 2 . Other Hessian matrices fxi2 y 2 ; fxi2 y_ 2 ; fx_i2 y 2 ; fx_i2 y_ 2 ; fyi2 x2 ; fy_i2 x2 ; fyi2 x_ 2 and fy_i2 x_ 2 are defined similarly. Denote by X the space of twice continuously differentiable functions x : I ! Rn with norm kxk = kxk1 + kDxk1 + kD2xk1, where the differentiation operator D is given by Z t u ¼ Dx () xðtÞ ¼ a þ uðsÞds; a
where a is a given boundary value. Therefore D dtd except at discontinuities. Denote by Y the space of twice continuously differentiable functions y : I ! Rm with the norm as that of the space X.
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Consider the following multiobjective variational problem: Z b Z b 1 k ðVPÞ: Minimize / ðt; xðtÞ; x_ ðtÞÞdt; . . . ; / ðt; xðtÞ; x_ ðtÞÞdt ; a
a
subject to xðaÞ ¼ 0; xðbÞ ¼ 0; x_ ðaÞ ¼ 0; x_ ðbÞ ¼ 0; hðt; xðtÞ; x_ ðtÞÞ 5 0; t 2 I; where / : I · X · X ! Rk and h : I · X · X ! Rm are differentiable functions. Let Q denote the set of all feasible solutions of (VP), i.e., Q ¼ fx 2 X jxðaÞ ¼ 0; xðbÞ ¼ 0; x_ ðaÞ ¼ 0; x_ ðbÞ ¼ 0; hðt; xðtÞ; x_ ðtÞÞ 5 0; t 2 Ig. Definition 1 (Geoffrion [9]). A point x0 2 Q is said to be an efficient solution of (VP), if for all x 2 Q, Z b Z b /i ðt; x0 ðtÞ; x_ 0 ðtÞÞdt = /i ðt; xðtÞ; x_ ðtÞÞdt for all i ¼ 1; 2; . . . ; k a a Z b Z b /i ðt; x0 ðtÞ; x_ 0 ðtÞÞdt ¼ /i ðt; xðtÞ; x_ ðtÞÞdt for all i ¼ 1; 2; . . . ; k. ) a
a
Definition 2. A point x0 2 Q is said to be a weak efficient solution of (VP), if for all x 2 Q, Z b Z b i 0 0 / ðt; x ðtÞ; x_ ðtÞÞdt > /i ðt; xðtÞ; x_ ðtÞÞdt for all i ¼ 1; 2; . . . ; k a a Z b Z b i 0 0 / ðt; x ðtÞ; x_ ðtÞÞdt ¼ /i ðt; xðtÞ; x_ ðtÞÞdt for all i ¼ 1; 2; . . . ; k. ) a
a
Definition 3. A functional F : I · X · X · X · X · Rn ! R is sublinear, if for any x; x_ ; u; u_ 2 X ; _ f1 þ f2 Þ 5 F ðt; x; x_ ; u; u; _ f1 Þ þ F ðt; x; x_ ; u; u; _ f2 Þ, for any f1, f2 2 Rn; and (i) F ðt; x; x_ ; u; u; _ bfÞ ¼ bF ðt; x; x_ ; u; u; _ fÞ, for any b 2 R, b = 0 and f 2 Rn. (ii) F ðt; x; x_ ; u; u; _ 0Þ ¼ 0. From (ii), it follows that F ðt; x; x_ ; u; u; Rb Definition 4. The functional a f ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞdt is said to be F-pseudoconvex in x and x_ for fixed y and y_ , if Z b Z b Z b _ fx ðt; u; u; _ y; yÞ _ Df x_ ðt; u; u; _ y; y_ ÞÞdt = 0 ) _ y; y_ Þdt F ðt; x; x_ ; u; u; f ðt; x; x_ ; y; y_ Þdt = f ðt; u; u; a
a
a
for all x, u : I ! Rn and for some arbitrary sublinear functional F. Rb Definition 5. The functional a f ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞdt is said to be F-pseudoconcave in y and y_ for fixed x and x_ , if Z b Z b Z b _ F ðt; v; v_ ; y; y_ ; ffy ðt; x; x_ ; v; v_ Þ Df y_ ðt; x; x_ ; v; v_ ÞgÞdt = 0 ) f ðt; x; x_ ; v; v_ Þdt 5 f ðt; x; x_ ; y; yÞdt a
for all y, v : I ! Rm and for some arbitrary sublinear functional.
a
a
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I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
Definition 6. A cone C i is said to be polar of Ci for i = 1, 2, if C i ¼ fwjwT x 5 0; for all x 2 C i g. The following concept of separability is required in the sequel which has been extensively used by Balas [1] and Gulati et al. [10]. Let zr, r = 1, 2, . . . , p, be piecewise smooth functions belonging to arbitrary space Zr, r = 1, 2, . . . , p, equipped with norm kzrk = kzrk1 + kDzrk1. A function Lðt; z1 ðtÞ; z_ 1 ðtÞ; z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞÞ will be called additively separable with respect to ðz1 ðtÞ; z_ 1 ðtÞÞ if there exist functions Mðt; z1 ðtÞ; z_ 1 ðtÞÞ (independent of z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞ) and N ðt; z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞÞ (independent of z1 ðtÞ; z_ 1 ðtÞ) such that Lðt; z1 ðtÞ; z_ 1 ðtÞ; z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞÞ ¼ Mðt; z1 ðtÞ; z_ 1 ðtÞÞ þ N ðt; z2 ðtÞ; z_ 2 ðtÞ; . . . ; zp ðtÞ; z_ p ðtÞÞ. The following form of Fritz John necessary conditions proposed by Suneja et al. [21] is required to prove the strong and converse duality theorems. Lemma 1. Let P be a convex set with nonempty interior in Rn and suppose that C is a closed convex cone in Rm having a nonempty interior. Let R and S be two vector valued functions defined on P. If z0 is a weak efficient solution of the following problem: Minimize RðzÞ ¼ ðR1 ðzÞ; R2 ðzÞ; . . . ; Rk ðzÞÞ; subject to SðzÞ 2 C; z 2 P ; then there exists a nonzero vector (r0, r) such that
rT0 Rz ðz0 Þ þ rT S z ðz0 Þ ðz z0 Þ = 0; for each z 2 P
and rTS(z0) = 0, r0 = 0, r 2 C*.
3. Mond–Weir type mixed integer symmetric duality In this section, we constrain some of the primal and dual variables to belong to the arbitrary sets of integers, i.e., U and V. Suppose that the first n1 (0 5 n1 5 n) components of x belong to U and the first m1 (0 5 m1 5 m) components of y belong to V. So we write (x, y) = (x1, x2, y1, y2), where x1 ¼ ðx1 ; x2 ; . . . ; xn1 Þ 2 U , and y 1 ¼ ðy 1 ; y 2 ; . . . ; y m1 Þ 2 V ; x2 and y2 being the remaining components of x and y such that x2 2 C1 and y2 2 C2. Let X1 and Y1 be the arbitrary sets of continuously differentiable functions x1 : I ! Rn1 and 1 y : I ! Rm1 ð0 5 n1 5 n; 0 5 m1 5 mÞ equipped with the norms which are prescribed for the spaces X and Y. Partitioning vector functions x and y as above, we introduce the following multiobjective variational minimax mixed integer symmetric dual program: Primal ðMPÞ
Z Max Min x1
¼
x2 ;y
Z a
b
_ f ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞdt
a
b
_ f 1 ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞdt; ...;
Z
b a
_ f k ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞdt ;
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
subject to xðaÞ ¼ 0 ¼ xðbÞ; x_ ðaÞ ¼ 0 ¼ x_ ðbÞ; x1 ðtÞ 2 U ;
75
yðaÞ ¼ 0 ¼ yðbÞ; y_ ðaÞ ¼ 0 ¼ y_ ðbÞ;
y 1 ðtÞ 2 V ;
x2 ðtÞ 2 C 1 ;
T
t 2 I;
T
½ðk f Þy 2 ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ Dðk f Þy_ 2 ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ 2 C 2 ; y 2 ðtÞT ½ðkT f Þy 2 ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ DðkT f Þy_ 2 ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ = 0; k > 0. Z
Dual ðMDÞ
b
_ f ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt
Min Maxu;v2 v1
¼
Z
a b
_ f 1 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt; . . . ;
a
Z
b
_ f k ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt ;
a
subject to uðaÞ ¼ 0 ¼ uðbÞ;
vðaÞ ¼ 0 ¼ vðbÞ;
_ _ uðaÞ ¼ 0 ¼ uðbÞ;
v_ ðaÞ ¼ 0 ¼ v_ ðbÞ;
u1 ðtÞ 2 U ;
v1 ðtÞ 2 V ;
v2 ðtÞ 2 C 2 ;
t 2 I;
_ _ vðtÞ; v_ ðtÞÞ DðkT f Þx_ 2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ 2 C 1 ; ½ðkT f Þx2 ðt; uðtÞ; uðtÞ; _ _ u2 ðtÞT ½ðkT f Þx2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ DðkT f Þx_ 2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ 5 0; k > 0. We shall denote by G and H, the sets of feasible solutions of the primal and dual multiobjective variational problems (MP) and (MD), respectively. Theorem 1 (Weak duality). Let (x, y, k) 2 G and (u, v, k) 2 H. Assume that (i) (ii) (iii) (iv) (v) (vi)
f i ðt; x; x_ ; y; y_ Þ, i = 1, 2, . . . , k is twice differentiable with respect to ðx2 ; x_ 2 Þ and ðy 2 ; y_ 2 Þ, respectively, f i ðt; x; x_ ; y; y_ Þ, i = 1, 2, . . . , k is additively separable with respect to ðx1 ; x_ 1 Þ or ðy 1 ; y_ 1 Þ, Rb T _ k f ðt; x; x_ ; y; yÞdt is F-pseudoconvex in ðx2 ; x_ 2 Þ for each ðx1 ; x_ 1 ; y; y_ Þ on I, Rab T _ k f ðt; x; x_ ; y; yÞdt is F-pseudoconcave in ðy 2 ; y_ 2 Þ for each ðx; x_ ; y 1 ; y_ 1 Þ on I, a T F ðt; x2 ; x_ 2 ; u2 ; u_ 2 ; n2 Þ þ ðu2 Þ n2 = 0, for all x2 ; x_ 2 ; u2 ; u_ 2 2 C 1 and n2 2 C 1 and T F ðt; v2 ; v_ 2 ; y 2 ; y_ 2 ; g2 Þ þ ðy 2 Þ g2 = 0, for all v2 ; v_ 2 ; y 2 ; y_ 2 2 C 2 and g2 2 C 2 .
Then Z
b
f ðt; x; x_ ; y; y_ Þdt i
a
Z
b
_ v; v_ Þdt. f ðt; u; u;
a
Proof. Let g ¼ Max Min x1
x2 ;y
Z
b a
_ f ðt; x; x_ ; y; yÞdtjðx; y; kÞ 2 G
76
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
and h ¼ Min Max v1
Z
u;v2
b
_ v; v_ Þdtjðu; v; kÞ 2 H . f ðt; u; u; a
_ i = 1, 2, . . . , k is additively separable with respect to ðx1 ; x_ 1 Þ or ðy 1 ; y_ 1 Þ (say with respect Since f i ðt; x; x_ ; y; yÞ, 1 1 to ðx ; x_ Þ), it follows that: _ f i ðt; x; x_ ; y; y_ Þ ¼ f1i ðt; x1 ; x_ 1 Þ þ f2i ðt; x2 ; x_ 2 ; y; yÞ; Therefore g can be written as Z b Z 1 1 f1 ðt; x ; x_ Þdt þ g ¼ Max Min x2 ;y
x1
a
or g ¼ Max Min where Z
b
2
1
f1 ðt; x ; x_ Þdt ¼ a
f2 ðt; x ; x_ ; y; y_ Þdtjðx; y; kÞ 2 G
f1 ðt; x1 ; x_ 1 Þdt þ /ðy 1 Þjx1 2 U ; y 1 2 V ; Z
b
f11 ðt; x1 ; x_ 1 Þdt;
Z
a
and /ðy 1 Þ ¼ Min
2
a
b 1
b a
Z
y1
x1
i ¼ 1; 2; . . . ; k.
Z
x2 ;y 2
b
f12 ðt; x1 ; x_ 1 Þdt; . . . ;
a
b
f2 ðt; x2 ; x_ 2 ; y; y_ Þdt ¼
Z
a
Z
b
f1k ðt; x1 ; x_ 1 Þdt
a
b
_ f21 ðt; x2 ; x_ 2 ; y; yÞdt; ...;
Z
a
b
_ f2k ðt; x2 ; x_ 2 ; y; yÞdt
a
subject to x2 ðaÞ ¼ 0 ¼ x2 ðbÞ; 2
y 2 ðaÞ ¼ 0 ¼ y 2 ðbÞ;
2
2
x_ ðaÞ ¼ 0 ¼ x_ ðbÞ; 1
1
x 2 U;
y 2V;
T
ð1Þ
2
y_ ðaÞ ¼ 0 ¼ y_ ðbÞ;
ð2Þ
2
ð3Þ
x 2 C1;
½ðk f2 Þy 2 ðt; x ; x_ 2 ; y; y_ Þ DðkT f2 Þy_ 2 ðt; x2 ; x_ 2 ; y; y_ Þ 2 C 2 ; 2
2 T
T
2
T
2
2
ð4Þ
2
ðy Þ ½ðk f2 Þy 2 ðt; x ; x_ ; y; y_ Þ Dðk f2 Þy_ 2 ðt; x ; x_ ; y; y_ Þ = 0;
ð5Þ
k > 0.
ð6Þ
Similarly h can be written as Z b 1 1 1 1 1 f1 ðt; u ; u_ Þdt þ wðv Þju 2 U ; v 2 V ; h ¼ Min Max v1
u1
where wðv1 Þ ¼ Max u2 ;v2
a
Z
b
f2 ðt; u2 ; u_ 2 ; v; v_ Þdt ¼
Z
a
b
f21 ðt; u2 ; u_ 2 ; v; v_ Þdt; . . . ;
Z
a
b
f2k ðt; u2 ; u_ 2 ; v; v_ Þdt ;
a
subject to u2 ðaÞ ¼ 0 ¼ u2 ðbÞ; 2
v2 ðaÞ ¼ 0 ¼ v2 ðbÞ;
2
2
u_ ðaÞ ¼ 0 ¼ u_ ðbÞ; 1
1
u 2 U;
2
v_ ðaÞ ¼ 0 ¼ v_ ðbÞ;
ð8Þ
2
v 2V;
T
ð7Þ
2
v 2 C2;
ð9Þ T
2
2
2
½ðk f2 Þx2 ðt; u ; u_ ; v; v_ Þ Dðk f2 Þx_ 2 ðt; u ; u_ ; v; v_ Þ 2 2 T
T
2
2
T
2
2
C 1 ;
ðu Þ ½ðk f2 Þx2 ðt; u ; u_ ; v; v_ Þ Dðk f2 Þx_ 2 ðt; u ; u_ ; v; v_ Þ 5 0; k > 0.
ð10Þ ð11Þ ð12Þ
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
77
Let (x, y, k) 2 G and (u, v, k) 2 H. In order to prove the theorem, it is sufficient to prove that / (y1) i w(v1), for a given y1. On taking n2 ¼ ðkT f2 Þx2 DðkT f2 Þx_ 2 ; we have F ½t; x2 ; x_ 2 ; u2 ; u_ 2 ; ðkT f2 Þx2 DðkT f2 Þx_ 2 = ðu2 ÞT ½ðkT f2 Þx2 DðkT f2 Þx_ 2 = 0 ðby hypothesis (v) and (11)Þ; which implies Z b F ½t; x2 ; x_ 2 ; u2 ; u_ 2 ; ðkT f2 Þx2 DðkT f2 Þx_ 2 dt = 0. a
Rb This in view of F-pseudoconvexity of a kT f2 dt in ðx2 ; x_ 2 Þ yields Z b Z b T 2 2 k f2 ðt; x ; x_ ; v; v_ Þdt = kT f2 ðt; u2 ; u_ 2 ; v; v_ Þdt. a
ð13Þ
a
By taking g2 ¼ fðkT f2 Þy 2 DðkT f2 Þy_ 2 g, we get T
F ½t; v2 ; v_ 2 ; y 2 ; y_ 2 ; fðkT f2 Þy 2 DðkT f2 Þy_ 2 g = ðy 2 Þ ½ðkT f2 Þy 2 DðkT f2 Þy_ 2 = 0 ðby hypothesis (vi) and (5)Þ; Z b F ½t; v2 ; v_ 2 ; y 2 ; y_ 2 ; fðkT f2 Þy 2 DðkT f2 Þy_ 2 gdt = 0; ) a
Rb which by F-pseudoconcavity of a kT f2 dt in ðy 2 ; y_ 2 Þ gives Z b Z b kT f2 ðt; x2 ; x_ 2 ; v; v_ Þdt 5 kT f2 ðt; x2 ; x_ 2 ; y; y_ Þdt. a
ð14Þ
a
Adding the inequalities (13) and (14), we obtain Z b Z b T 2 2 _ k f2 ðt; x ; x_ ; y; yÞdt = kT f2 ðt; u2 ; u_ 2 ; v; v_ Þdt. a
Hence Z a
a
b 2
2
_ f2 ðt; x ; x_ ; y; yÞdt i
Z
b
f2 ðt; u2 ; u_ 2 ; v; v_ Þdt.
a
Theorem 2 (Strong duality). Let ðxðtÞ; y ðtÞ; kÞ be an efficient solution of (MP), and fixed k ¼ k in (MD). Assume that _ (i) f i ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞ, i = 1, 2, . . . , k is twice differentiable at x2 ðtÞ; x_ 2 ðtÞ; y 2 ðtÞ and y_ 2 ðtÞ, i _ (ii) f ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞ, i = 1, 2, . . . , k is additively separable with respect to ðx1 ðtÞ; x_ 1 ðtÞÞ or 1 1 ðy ðtÞ; y_ ðtÞÞ, T T (iii) the system ½ðr1 ðtÞ y 2 r2 Þ fð kT f2 Þy 2 y 2 DðkT f2 Þy_ 2 y 2 g Dfðr1 ðtÞ y 2 r2 Þ fðkT f2 Þy 2 y_ 2 DðkT f2 Þy_ 2 y_ 2 gg 2 2 ðr1 ðtÞ y r2 Þ ¼ 0 ) ðr1 ðtÞ y r2 Þ ¼ 0, for every r1(t) 2 C2, t 2 I, and (iv) the set fðf2y1 2 Df 12y_ 2 Þ; . . . ; ðf2yk 2 Df k2_y 2 Þg is linearly independent. Then ðxðtÞ; y ðtÞ; kÞ is a feasible solution of (MD) and the objective values of (MP) and (MD) are equal. Furthermore, if the assumptions of Theorem 1 are satisfied, then ðxðtÞ; y ðtÞ; kÞ is an efficient solution of (MD).
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I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
Proof. Since ðxðtÞ; y ðtÞ; kÞ is an efficient solution of (MP), hence it is weak efficient, then by Lemma 1, there exist r0 2 Rk, r1(t) 2 C2, r2 2 R and d 2 Rk such that rT0 ½f2x2 Df 2_x2 ðx2 ðtÞ x2 ðtÞÞ þ ½ðr1 ðtÞ y 2 r2 ÞT ððkT f2 Þy 2 x2 T f2 Þ 2 2 Þ Dfðr1 ðtÞ y 2 r2 ÞT ðð Dðk kT f2 Þ 2 2 DðkT f2 Þ y_ x
y x_
2 y_ 2 x_ 2 Þgðx ðtÞ
x2 ðtÞÞ=0;
for all x2 ðtÞ 2 C 1 and t 2 I; T T ðr0 kr2 Þ ½f2y 2 Df 2_y 2 þ ½ðr1 ðtÞ y 2 r2 Þ ððkT f2 Þy 2 y 2 DðkT f2 Þy_ 2 y 2 Þ T Dfðr1 ðtÞ y 2 r2 Þ ðð kT f2 Þ 2 2 Dð kT f2 Þ 2 2 Þg ¼ 0; 8t 2 I; y y_
2
y_ y_
T
ð15Þ ð16Þ
ðr1 ðtÞ y r2 Þ ðf2y 2 Df 2y_ 2 Þ d ¼ 0; 8t 2 I; kT f2 Þ 2 Dð kT f2 Þ 2 Þ ¼ 0; 8t 2 I; r1 ðtÞT ðð
ð17Þ
r2 y ðð kT f2 Þy 2 Dð kT f2 Þy_ 2 Þ ¼ 0; k ¼ 0; dT
ð19Þ
ð18Þ
y_
y
2
r0 = 0; r2 = 0; d = 0; ðr0 ; r1 ðtÞ; r2 ; dÞ 6¼ 0.
8t 2 I
ð20Þ ð21Þ ð22Þ
t 2 I;
As k > 0, it follows from (20) that d = 0. Therefore (17) reduces to T ðr1 ðtÞ y 2 r2 Þ ðf2y 2 Df 2y_ 2 Þ ¼ 0
8t 2 I.
ð23Þ
2
Multiplying (16) by ðr1 ðtÞ y r2 Þ and using (23), we get T T ½ðr1 ðtÞ y 2 r2 Þ ðð kT f2 Þy 2 y 2 Dð kT f2 Þy_ 2 y 2 Þ Dfðr1 ðtÞ y 2 r2 Þ ððkT f2 Þy 2 y_ 2 DðkT f2 Þy_ 2 y_ 2 Þgðr1 ðtÞ y 2 r2 Þ
¼ 0; which by hypothesis (iii) gives r1 ðtÞ ¼ y 2 r2 .
ð24Þ
From (16) and (24) T ðr0 kr2 Þ ðf2y 2 Df 2y_ 2 Þ ¼ 0.
Since fðf2y1 2 Df 12_y 2 Þ; . . . ; ðf2yk 2 Df k2_y 2 Þg is linearly independent, then the above equation implies 2. r0 ¼ kr
ð25Þ
If r2 = 0, then from (24) and (25), we obtain r1(t) = r0 = 0. Hence (r0, r1(t), r2, d) = 0, contradicting (22). Thus r2 > 0, Eq. (24) yields y 2 ¼
r1 ðtÞ 2 C2. r2
Now, (15) along with (24) and (25) gives kr2 ½f2x2 Df 2 ðx2 ðtÞ x2 ðtÞÞ = 0. 2_x
Since r2 > 0, the above inequality implies T f2 Þ 2 Dð kT f2 Þ 2 ðx2 ðtÞ x2 ðtÞÞ = 0. ½ðk x
ð26Þ
x_
Let x2(t) 2 C1. Then x2 ðtÞ þ x2 ðtÞ 2 C 1 and so (26) shows that for every x2(t) 2 C1 T f2 Þ 2 Dð ½ðk kT f2 Þ 2 x2 ðtÞ = 0 i.e., ½ð kT f2 Þ 2 DðkT f2 Þ 2 2 C . x
x_
x
x_
1
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
79
Also, by letting x2(t) = 0 and x2 ðtÞ ¼ 2x2 ðtÞ simultaneously in the inequality (26), we get x2 ðtÞT ½ð kT f2 Þx2 Dð kT f2 Þx_ 2 ¼ 0. Thus ðxðtÞ; y ðtÞ; kÞ is a feasible solution of (MD) and the objective functional values are equal. By Theorem 1, ðxðtÞ; y ðtÞ; kÞ is an efficient solution of (MD). h A converse duality theorem may be stated as its proof would run analogously to that of Theorem 2. Theorem 3 (Converse duality). Let ð uðtÞ; vðtÞ; kÞ be an efficient solution of (MD), and fixed k ¼ k in (MP). Assume that _ (i) f i ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ, i = 1, 2, . . . , k is twice differentiable at u2 ðtÞ; u_ 2 ðtÞ; v2 ðtÞ and v_ 2 ðtÞ, i _ (ii) f ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ, i = 1, 2, . . . , k is additively separable with respect to ðu1 ðtÞ; u_ 1 ðtÞÞ or 1 1 ðv ðtÞ; v_ ðtÞÞ, T T (iii) the system ½ðr1 ðtÞ u2 r2 Þ fð kT f2 Þx2 x2 DðkT f2 Þx_ 2 x2 g Dfðr1 ðtÞ u2 r2 Þ fðkT f2 Þx2 x_ 2 DðkT f2 Þx_ 2 x_ 2 gg 2 2 ðr1 ðtÞ u r2 Þ ¼ 0 ) ðr1 ðtÞ u r2 Þ ¼ 0, for every r1(t) 2 C1, t 2 I, and (iv) the set fðf2x1 2 Df 12_x2 Þ; . . . ; ðf2xk 2 Df k2_x2 Þg is linearly independent. Then ð uðtÞ; vðtÞ; kÞ is a feasible solution of (MP) and the objective values of (MP) and (MD) are equal. Furthermore, if the assumptions of Theorem 1 are satisfied, then ðuðtÞ; vðtÞ; kÞ is an efficient solution of (MP).
4. Self duality A mathematical programming problem is said to be self dual if it is formally identical with its dual, that is, the dual can be recast in the form of the primal. If we assume (i) C2 = C1, and _ (ii) f i ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ : I Rn Rn Rn Rn ! R, i = 1, 2, . . . , k to be skew symmetric, that is, _ _ f i ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ ¼ f i ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ; i ¼ 1; 2; . . . ; k then we shall show that the programs (MP) and (MD) are self duals. By recasting the dual problem (MD) as maxmin problem, we have Z b _ f ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt Dual ðMDÞ Max Min v1
u;v2
Z ¼
a b 1
_ f ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞdt; . . . ;
Z
a
b
_ _ f ðt; uðtÞ; uðtÞ; vðtÞ; vðtÞÞdt ; k
a
subject to uðaÞ ¼ 0 ¼ uðbÞ;
vðaÞ ¼ 0 ¼ vðbÞ;
_ _ uðaÞ ¼ 0 ¼ uðbÞ;
v_ ðaÞ ¼ 0 ¼ v_ ðbÞ;
u1 ðtÞ 2 U ; T
v1 ðtÞ 2 V ;
v2 ðtÞ 2 C 2 ;
t 2 I; T
_ _ ½ðk f Þx2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ Dðk f Þx_ 2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ 2 C 1 ; _ _ vðtÞ; v_ ðtÞÞ DðkT f Þx_ 2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ = 0; u2 ðtÞT ½ðkT f Þx2 ðt; uðtÞ; uðtÞ; k > 0.
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_ Since f i ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ, i = 1, 2, . . . , k is skew symmetric, we have _ _ vðtÞ; v_ ðtÞÞ ¼ fyi2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ; fxi2 ðt; uðtÞ; uðtÞ;
i ¼ 1; 2; . . . ; k
_ _ fx_i2 ðt; uðtÞ; uðtÞ; vðtÞ; v_ ðtÞÞ ¼ fy_i2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ;
i ¼ 1; 2; . . . ; k.
and
The above problem becomes Dual ðMDÞ
Z
b
_ Maxv1 Minu;v2 f ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞdt a Z b Z 1 _ f ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞdt; ...; ¼ a
_ f ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞdt ; k
a
subject to uðaÞ ¼ 0 ¼ uðbÞ; _ _ uðaÞ ¼ 0 ¼ uðbÞ; u1 ðtÞ 2 U ;
b
vðaÞ ¼ 0 ¼ vðbÞ; v_ ðaÞ ¼ 0 ¼ v_ ðbÞ;
v1 ðtÞ 2 V ;
T
v2 ðtÞ 2 C 1 ;
t 2 I;
T
_ _ Dðk f Þy_ 2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ 2 C 1 ; ½ðk f Þy 2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ _ _ DðkT f Þy_ 2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ = 0; u2 ðtÞT ½ðkT f Þy 2 ðt; vðtÞ; v_ ðtÞ; uðtÞ; uðtÞÞ k > 0. is formally identical to (MP); that is, the objective and the constraint functions of We observe that ðMDÞ (MP) and ðMDÞ are identical. Therefore (MP) is a self dual. It can easily seen that the feasibility of (x(t), y(t), k) for (MP) implies the feasibility of (y(t), x(t), k) for (MD), and conversely. _ Theorem 4 (Self duality). Assume that f i ðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞ, i = 1, 2, . . . , k is skew symmetric. Then (MP) is a self dual. Also, if (MP) and (MD) are dual variational problems and ðxðtÞ; y ðtÞ; kÞ is a joint efficient solution, then so is ðy ðtÞ; xðtÞ; kÞ and the common objective functional value is 0. Proof. Since ðxðtÞ; y ðtÞ; kÞ is a joint efficient solution of (MP) and (MD), the objective functional values are equal to Z b f ðt; xðtÞ; x_ ðtÞ; y ðtÞ; y_ ðtÞÞdt. a
is feasible for (MP) if and only if ðy ðtÞ; xðtÞ; kÞ is feasible for (MD). ThereFrom self duality, ðxðtÞ; y ðtÞ; kÞ fore efficiency of ðxðtÞ; y ðtÞ; kÞ for (MP) implies efficiency of ðy ðtÞ; xðtÞ; kÞ for (MD) and vice versa. Hence the objective functional values are equal to Z b f ðt; y ðtÞ; y_ ðtÞ; xðtÞ; x_ ðtÞÞdt. a
Therefore Z b Z _ _ f ðt; xðtÞ; xðtÞ; y ðtÞ; y ðtÞÞdt ¼ a
f ðt; y ðtÞ; y_ ðtÞ; xðtÞ; x_ ðtÞÞdt ¼
a
Thus we have Z b f ðt; xðtÞ; x_ ðtÞ; y ðtÞ; y_ ðtÞÞdt ¼ 0. a
b
Z a
b
f ðt; xðtÞ; x_ ðtÞ; y ðtÞ; y_ ðtÞÞdt.
I. Ahmad, Z. Husain / European Journal of Operational Research 177 (2007) 71–82
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5. Static symmetric dual multiobjective programs If the time dependency in (MP) and (MD) is relaxed, then they reduce to the following static minimax mixed integer symmetric dual problems: ðMP1Þ:
Max Min f ðx; yÞ; x1
x2 ;y
subject to x1 2 U ;
y1 2 V ;
x2 2 C 1 ;
ðkT f Þy 2 ðx; yÞ 2 C 2 ; T
ðy 2 Þ ðkT f Þy 2 ðx; yÞ = 0; k > 0. ðMD1Þ:
Min Max f ðu; vÞ v1
u;v2
subject to u1 2 U ;
v1 2 V ;
v2 2 C 2 ;
ðkT f Þx2 ðu; vÞ 2 C 1 ; ðu2 ÞT ðkT f Þx2 ðu; vÞ 5 0; k > 0. The above problems (MP1) and (MD1) are the Mond–Weir type mixed integer symmetric dual programs considered in [13], with the omission of kte = 1, as this is not required for the duality theorems to hold.
Acknowledgements The authors wish to thank the referee for several valuable suggestions which have considerably improved the presentation of the paper.
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