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Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity
European Journal of Operational Research 181 (2007) 76–85 www.elsevier.com/locate/ejor
Discrete Optimization
Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity Xiuhong Chen a
a,b,*
, Jiangyu Yang
a
Department of Computer Science, Nanjing University of Science and Technology, Nanjing 210094, PR China b School of Information Technology, Southern Yangtze University, Wuxi 214122, PR China Received 6 December 2004; accepted 18 April 2006 Available online 26 July 2006
Abstract A pair of symmetric dual multiobjective variational mixed integer programs for the polars of arbitrary cones are formulated, which some primal and dual variables are constrained to belong to the set of integers. Under the separability with respect to integer variables and partial-invexity assumptions on the functions involved, we prove the weak, strong, converse and self-duality theorems to related minimax efficient solution. These results include some of available results. 2006 Elsevier B.V. All rights reserved. Keywords: Multiobjective variational problems; Mixed integer programming; Symmetric duality; Self-duality; Partial-invexity; Closed convex cone
1. Introduction Symmetric duality for nonlinear programming problems has been studied by many researchers in the past. Dantzig et al. [9], Mond [15] and Bazaraa and Goode [2] gave a pair of symmetric dual programs involving a scale function f(x, y), x 2 Rn, y 2 Rm under the condition that f(Æ, y) is convex for each y and f(x, Æ) is concave for each x. Mond and Weir [18] presented a different pair of symmetric dual nonlinear programs for f(x, y) to pseudoconvexity–pseudoconcavity. Balas [1] examined the symmetric duality results of Dantzig et al. [9] when some primal and dual variables are constrained to belong to some arbitrary set, for example, the set of integer. Kumar et al. [11] were on the lines of Mond and Weir [18] rather than the lines of Dantzig et al. [9] as taken by Mishra et al. [12], and presented a pair of symmetric dual minimax integer programming problems, which some primal and dual variables are constrained to belong to the set of integers. Recently, Chandra and Kumar [5] formulated a pair of symmetric dual nonlinear programming for the polars of arbitrary cones. Kim and Song [10] also presented two pairs of nonlinear multiobjective mixed integer programs for the polars of arbitrary cones, and established the weak, strong and converse duality theorems by using the concept of efficiency. *
Corresponding author. Address: School of Information Technology, Southern Yangtze University, Jiangsu 214122, PR China. E-mail addresses: [email protected], [email protected] (X. Chen).
0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.04.045
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
77
Duality for the multiobjective variational problems has also been of great interest in recent years [3,4,14,16]. When the objective and constraint functions involved are invex, we [6] also proved the duality to related properly efficient solution for a class of multiobjective variational problems. Under the partial-invexity assumptions of scalar function /ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ with x(t) 2 Rn and y(t) 2 Rm, we [7] also studied the symmetric duality and self-duality for a class of multiobjective fractional variational problems. In [8], a Wolfe type multiobjective variational mixed integer symmetric dual program over arbitrary cones was discussed. In this paper, we propose to study another pair of multiobjective mixed integer programs for the polars of arbitrary cones which is in the spirit of Mond and Weir. We prove the weak, strong, converse and self-duality theorems to related the concept of efficient solution. These results generalize some results of [4,5,7,10,11,13,19]. The remainder of the paper is organized as follows. In Section 2, we give some notations and lemmas. In Section 3, we formulate a pair of multiobjective variational mixed integer programs for the polars of arbitrary cones, which is in the spirit of Mond and Weir. Under the separability with respect to the integer variables and the partial-invexity assumptions on the functions involved, we prove the weak duality, strong duality and converse duality, where the reason which the strong and converse duality hold is that the objective functions of two problems are the same one. In Section 4, we discuss the self-duality. 2. Notations and lemmas Let I = [a, b] be a real interval, x : I ! Rn and y : I ! Rm. We constrain some of the components of x and y which belong to arbitrary sets of integer. Suppose that the first n1 components of x and the first m1 components of y (0 6 n1 6 n, 0 6 m1 6 m) are constrained to be integers and the following notations are introduced ðx; yÞ ¼ ðx1 ; x2 ; y 1 ; y 2 Þ; x1 2 U ; y 1 2 V ; x2 2 C 1 ; y 2 2 C 2 ; where U and V are two arbitrary sets of integers vectors in Rn1 and Rm1 , respectively. Let C1 and C2 be two closed convex cones in Rn2 and Rm2 with nonempty interiors, respectively. n = n1 + n2, m = m1 + m2. Let fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ (i = 1, 2, . . . , p) be twice continuously differentiable functions at x2, x_ 2 , y2 and y_ 2 , where x1(t) 2 U, y1(t) 2 V, x2(t) 2 C1, y2(t) 2 C2, t 2 I; x2 : I ! S1 and y2 : I ! S2 with derivative x_ 2 and y_ 2 , respectively; S 1 Rn2 and S 2 Rm2 be open and C1 S1 and C2 S2. Denote by fix2 and fi_x2 the first partial derivatives of fi with respect to x2(t) and x_ 2 ðtÞ, respectively; fix2 x2 the Hessian matrix of fi with respect to x2. Similarly, fiy 2 and fi_y 2 denote the first partial derivatives of fi with respect to y2(t) and y_ 2 ðtÞ, respectively; fix2 x_ 2 , fix2 y 2 , fix2 y_ 2 , fi_x2 x2 , fi_x2 x_ 2 , fi_x2 y 2 , fi_x2 y_ 2 , fiy 2 x2 , fiy 2 x_ 2 , fiy 2 y 2 , fiy 2 y_ 2 , fiy_ 2 x2 , fiy_ 2 x_ 2 , fiy_ 2 y 2 , fiy_ 2 y_ 2 denote the other Hessian matrixes of fi with respect to x2(t), x_ 2 ðtÞ, y2(t) and y_ 2 ðtÞ, respectively, for 1, 2, . . . , p. _ For any r-dimensional vector function Qðt; zðtÞ; z_ ðtÞ; wðtÞ; wðtÞÞ, denote the first partial derivative with _ respect to z(t), z_ ðtÞ, w(t) and wðtÞ by Qz, Qz_ , Qw and Qw_ , respectively, that is, 1 1 0 1 0 1 oQ oQ1 oQ1 oQ oQ1 oQ1 n n 1 2 1 2 oz o_z oz oz o_z o_z C C B oQ B oQ B 12 oQ22 oQn2 C B 12 oQ22 oQn2 C B oz B o_z oz C o_z C oz o_z Qz ¼ B Qz_ ¼ B C ; C ; . . C C B .. .. B .. .. A A @ . . .. @ . . .. r r r r r r oQ oQ oQ oQ oQ oQ ozn rn o_zn rn 1 oz2 z1 o_z2 1 0 oz 0o_oQ 1 1 1 oQ1 oQ1 oQ1 oQ1 own oQ ow_ n ow1 ow2 ow_ 1 ow_ 2 B oQ2 oQ2 B oQ2 oQ2 2 C 2 C C C B 1 B 1 oQ oQ B ow B ow_ own C ow_ n C ow2 ow_ 2 Qw ¼ B ; Qw_ ¼ B C C : .. .. C C B .. .. B .. .. A A @ . . . @ . . . r r r r r r oQ oQ oQ oQ oQ oQ own ow_ n ow1 ow2 ow_ 1 ow_ 2 rn rn Let C(I, Rs) denote the space of piecewise smooth function z(t) with norm kzk = kzk1 + kDzk1, where the differentiation operator D is given by Z t u ¼ Dz () zðtÞ ¼ uðaÞ þ uðsÞ ds; a
where u(a) is a given boundary value. Therefore
d dt
¼ D except at discontinuities.
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X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
Consider the following multiobjective programming problem ðMPÞ
minimize gðxÞ subject to hðxÞ 6 0;
where g : C ! Rp, h : C ! Rm and C is a closed convex cone with nonempty interior in Rn. Denote P the set of feasible solutions of (MP). Definition 2.1. A point x 2 P is said to be a Pareto minimum solution, that is, an efficient solution, of (MP) if there exists no other x 2 P such that gðxÞ 6 gðxÞ. Definition 2.2. A point ðx; yÞ is an efficient solution of the following minimax multiobjective programming problem max min x
y
F ðx; yÞ
subject to Gðx; yÞ 6 0; where F : C · C ! Rp, G : C · C ! Rm and C is a closed convex cone with nonempty interior in Rn, if y is a Pareto minimum solution of the inner problem ‘‘miny F ðx; yÞ’’ for each x and x is a Pareto maximum solution of the exterior problem ‘‘maxx F ðx; yÞ’’. Definition 2.3. A cone C* is said to be the polar of C, if C ¼ fzjzT x 6 0; for all x 2 Cg: Definition 2.4. A real valued function /ðt; x1 ; x_ 1 ; x2 ; x_ 2 ; . . . ; xp ; x_ p Þ is said to be separable with respect to x1 and x_ 1 , if there exist real functions nðt; x1 ; x_ 1 Þ and fðt; x2 ; x_ 2 ; . . . ; xp ; x_ p Þ such that /ðt; x1 ; x_ 1 ; x2 ; x_ 2 ; . . . ; xp ; x_ p Þ ¼ nðt; x1 ; x_ 1 Þ þ fðt; x2 ; x_ 2 ; . . . ; xp ; x_ p Þ: The following lemma, which is said to be the generalized form of Fritz–John condition for the vector valued functions proposed by Bazaraa and Goode [2], play a main role during the process of proof of the strong duality. Lemma 2.5. Let K be a convex set with nonempty interior in Rn and let C be a closed convex cone in Rn having a nonempty interior. Let F and G be two vector-valued functions defined on K. If z0 is an efficient solution of the following problem: minimize F ðzÞ subject to GðzÞ 2 C; z 2 K; then there exists a nonzero vector (r0, r) such that T r0 F z ðz0 Þ þ rT Gz ðz0 Þ ðz z0 Þ P 0; 8z 2 K; and r0 P 0, r 2 C*, and rTG(z0) = 0. In order to obtain our main results, we introduce the following definition. _ Definition 2.6. Let S Rn be open set. If there exists a vector function g1 ðt; xðtÞ; x_ ðtÞ; uðtÞ; uðtÞÞ 2 S with g1 = 0 at t if x(t) = u(t), such that for the scalar function hðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ the functional Z b H ðx; x_ ; y; y_ Þ ¼ hðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ dt a
satisfies _ y; yÞ _ P H ðx; x_ ; y; y_ Þ H ðu; u;
Z
b
n o _ _ gT1 hx ðt; uðtÞ; uðtÞ; yðtÞ; y_ ðtÞÞ þ ðDg1 ÞT hx_ ðt; uðtÞ; uðtÞ; yðtÞ; y_ ðtÞÞ dt;
a
then H ðx; x_ ; y; y_ Þ is said to be partially invex in x and x_ on I with respect to g1 for fixed y.
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
If H satisfies _ H ðx; x_ ; v; v_ Þ P H ðx; x_ ; y; yÞ
Z
b
n
79
o gT2 hy ðt; xðtÞ; x_ ðtÞ; vðtÞ; v_ ðtÞÞ þ ðDg2 ÞT hy_ ðt; xðtÞ; x_ ðtÞ; vðtÞ; v_ ðtÞÞ dt;
a
_ vðtÞ; v_ ðtÞÞ 2 S with g2 = 0 at t if y(t) = v(t), then H ðx; x_ ; y; y_ Þ is said to be partially invex in where g2 ðt; yðtÞ; yðtÞ; y and y_ on I with respect to g2 for fixed x. If H is partially invex in x and x_ (or in y and y_ ) on I with respect to g1 (or g2) for fixed y (or for fixed x), then H is said to be partially incave in x and x_ (or in y and y_ ) on I with respect to g1 (or g2) for fixed y (or for fixed x). _ Example 2.7. Let I = [0, 1], S = (1, 1) and g1 ðt; xðtÞ; x_ ðtÞ; uðtÞ; uðtÞÞ 2 S is an arbitrary element satisfying 2 _ 2 þ ð_xðtÞ2 1Þ2 yðtÞ2 þ 1, then the funcg1 = 0 if x(t) = u(t) onR I. Define hðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ ¼ ln½xðtÞ yðtÞ 1 _ tional H ðx; x_ ; y; y_ Þ ¼ 0 hðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞ dt P 0. It is obvious that hx ðt; 0; 1; yðtÞ; y_ ðtÞÞ ¼ 0 and _ is partially invex in x = 0 and x_ ¼ 1 on I with respect to g1 for hx_ ðt; 0; 1; yðtÞ; y_ ðtÞÞ ¼ 0. Thus H ðx; x_ ; y; yÞ fixed y. _ is not partially invex in y = 0 and y_ ¼ 1 on I with respect to Similarly, we can prove that H ðx; x_ ; y; yÞ g2 = (y(t) v(t))2 for fixed x. 3. Symmetric duality results In this section, we consider the following symmetric dual minimax multiobjective variational mixed integer programs for the arbitrary cones Z b ðMSPÞ max min f ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt x1
x2 ;y
subject to
a
xðaÞ ¼ 0 ¼ yðaÞ;
xðbÞ ¼ 0 ¼ yðbÞ;
x_ 2 ðaÞ ¼ 0 ¼ y_ 2 ðaÞ; x1 ðtÞ 2 U ;
ð1Þ
x_ 2 ðbÞ ¼ 0 ¼ y_ 2 ðbÞ;
x2 ðtÞ 2 C 1 ;
y 1 ðtÞ 2 V ;
ð2Þ m2
y 2 ðtÞ 2 R ;
8t 2 I;
ð3Þ
f iy 2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ Dfi_y 2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ 2 C 2 ; i ¼ 1; 2; . . . ; p;
8t 2 I;
ð4Þ
T
ðy 2 ðtÞÞ ½fiy 2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ Dfi_y 2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ P 0; i ¼ 1; 2; . . . ; p;
8t 2 I;
ð5Þ
and ðMSDÞ
Z min max v1
u;v2
b
f ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt
a
subject to uðaÞ ¼ 0 ¼ vðaÞ; uðbÞ ¼ 0 ¼ vðbÞ; u_ 2 ðaÞ ¼ 0 ¼ v_ 2 ðaÞ; u_ 2 ðbÞ ¼ 0 ¼ v_ 2 ðbÞ; u1 ðtÞ 2 U ; u2 ðtÞ 2 Rn2 ; v1 ðtÞ 2 V ; v2 ðtÞ 2 C 2 ; 8t 2 I; fix2 ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ þ Dfi_x2 ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ 2 C 1 ; i ¼ 1; 2; . . . ; p;
8t 2 I;
ð6Þ ð7Þ ð8Þ
ð9Þ
T
ðu2 ðtÞÞ ½fix2 ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ þ Dfi_x2 ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ 6 0; i ¼ 1; 2; . . . ; p;
8t 2 I;
ð10Þ
80
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
where the function f ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is a p-dimensional vector-valued function. In (MSP) and (MSD), if fi does not depend explicitly on t for i = 1, 2, . . . , p, then our problems become the pair of problems given by Kim and Song [10]. Furthermore, if p = 1, C 1 ¼ Rnþ2 and C 2 ¼ Rmþ2 , then our problems again become the pair of problems considered by Mishra and Das [13]. Denote by X and Y the set of feasible solutions of (MSP) and (MSD), respectively. From now on, suppose that (1) the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1 or y1, i = 1, 2, . . . , p. Without loss of generality, fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, that is, fi can be expressed as fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ fi1 ðt; x1 ðtÞÞ þ fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ; (2) the sets of feasible solutions X and Y having the properties that if (x(t), y(t)) 2 X and (u(t), v(t)) 2 Y, then g1 ðt; x2 ðtÞ; x_ 2 ðtÞ; u2 ðtÞ; u_ 2 ðtÞÞ 2 C 1 and g2 ðt; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ, v_ 2 ðtÞÞ 2 C 2 . Remark 3.1. Under the above assumptions, (MSP) can be expressed as the following form: Z b max min ½f 1 ðt; x1 ðtÞÞ þ f 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt x2 ;y
x1
a
subject to ð1Þ–ð5Þ; where T
f 1 ðt; x1 ðtÞÞ ¼ ðf11 ðt; x1 ðtÞÞ; f21 ðt; x1 ðtÞÞ; . . . ; fp1 ðt; x1 ðtÞÞÞ ; f 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ ðf12 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ; f22 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ; . . . ; T
fp2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞÞ : Let ðMPÞ /ðy 1 Þ ¼ min
Z
x2 ;y 2
b
f 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞdt
a
subject to ð1Þ and ð2Þ; x2 ðtÞ 2 C 1 ; y 2 ðtÞ 2 Rm2 ; and; fiy22 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ Df 2iy_ 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ
ð11Þ 2
C 2 ;
i ¼ 1; 2; . . . ; p; 8t 2 I; ðy 2 ðtÞÞ
T
ð12Þ
½fiy22 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ Df 2i_y 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ
i ¼ 1; 2; . . . ; p; 8t 2 I:
P 0; ð13Þ
So, (MSP) may become Z b max min f1 ðt; x1 ðtÞÞ dt þ /1 ðy 1 Þ : x1 2 U ; y 1 2 V : y1
x1
a
Similarly, (MSD) can also be written as Z b min max f 1 ðt; u1 ðtÞÞ dt þ w1 ðv1 Þ : u1 2 U ; v1 2 V ; v1
u1
a
where ðMDÞ
wðv1 Þ ¼ max u2 ;v2
subject to
Z
b
f 2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt
a
ð6Þ and ð7Þ; u2 ðtÞ 2 Rn2 ; v2 ðtÞ 2 C 2 ;
and;
ð14Þ
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
81
fix22 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ þ Df 2iu_ 2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ 2 C 1 ; i ¼ 1; 2; . . . ; p; ðu2 ðtÞÞ
T
8t 2 I;
ð15Þ
½fix22 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ
P 0;
þ
Df 2i_x2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð16Þ
Thus, under the above separable assumption of f with respect to x1 or y1, we only discuss the symmetric duality between (MP) and (MD). Theorem 3.2 (Weak duality). Let (x(t), y(t)) and (u(t), v(t)) be feasible solutions of (MSP) and (MSD), respectively. R b 2 Assume that the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, and f ðt; ; ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt is partial-invex in x2 and a i R b x_ 2 for each y1(t) and y2(t) on I with respect to the same g1 ¼ g1 ðt; x2 ðtÞ; x_ 2 ðtÞ; u2 ðtÞ; u_ 2 ðtÞÞ 2 C 1 , i = 1, 2, . . . , p; a fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; ; ; Þ dt is partial-incave in y1, y2 and y_ 2 for each x2(t) on I with respect to g2 ¼ g2 ðt; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ 2 C 2 . Then the following inequalities cannot simultaneously hold: (I) for all i 2 {1, 2, . . . , p} Z b Z fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt 6 a
b
fi ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt; a
(II) for at least one j 2 {1, 2, . . . , p} Z b Z fj ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ < a
b
fj ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt: a
Proof. Under the separable assumption, from the above remark, for each feasible solution (x(t), y(t)) of (MSP) and each feasible solution (u(t), v(t)) of (MSD), it is sufficient to prove that the following inequalities cannot simultaneously hold: (I 0 ) for all i 2 {1, 2, . . . , p} Z b Z 2 fi ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt 6 a
b
fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt; a
(II 0 ) for at least one j 2 {1, 2, . . . , p} Z b Z 2 fj ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt < a
b
fj2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt:
a
Rb 2 y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt with respect to g1 2 C1, for each k 2 K ¼ From the partial-invexity of a fi ðt; ; ; P k ¼ ðk1 ; k2 ; . . . ; kp Þ : ki P 0; i ¼ 1; 2; . . . ; p; pi¼1 ki ¼ 1 , we have Z b Z b p p X X 2 ki fi ðt; x2 ðtÞ; x_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt ki fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt a
i¼1
¼
p X
Z
a
b
ki a
i¼1
¼
½fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt
a
p X
Z
a
b
ki
i¼1
P
i¼1
Z
b
gT1
h i T gT1 fix22 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ þ ðDg1 Þ fi_2x2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt
" p X i¼1
ki fix22 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ
D
p X i¼1
# ki fi_2x2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ
dt
82
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
(by integrating by parts, and when t = a and t = b, x2(t) = u2(t), so it follows that g1 = 0) ðby g1 2 C 1 ; ð15Þ and k 2 KÞ
P 0:
Similarly, from the incavity of p X
Z
Rb a
ð17Þ
f2i ðt; x2 ðtÞ; x_ 2 ðtÞ; ; ; Þ dt with respect to g2 2 C2, we get
b
fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt 6
ki
p X
a
i¼1
Z
b
fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt:
ki
ð18Þ
a
i¼1
From (17) and (18), we have p X i¼1
Z
b
fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt
ki
P
a
The proof is complete.
p X
Z
b
fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt:
ki a
i¼1
h
Remark 3.3. Because the integer variables x1 are separated from the objective functions, the proof of Theorem 3.2 only consider the weak duality between (MP) and (MD). The same reason is suitable to Theorems 3.5 and 3.7. _ Example 3.4. Let I = [0, 1], C1 = C2 = (1, 1), g1 ¼ g1 ðt; xðtÞ; x_ ðtÞ; uðtÞ; uðtÞÞ 2 C 1 and g2 ¼ g2 ðt; yðtÞ; y_ ðtÞ; vðtÞ; v_ ðtÞÞ 2 C 2 are arbitrary elements satisfying g1 = 0 if x(t) = u(t) and g2 = 0 if y(t) = v(t) on I, respectively. Define f1 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ x1 ðtÞ þ y 1 ðtÞ þ ln½x2 ðtÞ2 þ ð_x2 ðtÞ2 1Þ2 þ 1 ln½y 2 ðtÞ2 þ ð_y 2 ðtÞ2 1Þ2 þ 1; f2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ x1 ðtÞ þ y 1 ðtÞ þ
x2 ðtÞ
2
1 þ x2 ðtÞ
2
2
þ ð_x2 ðtÞ 1Þ 2
y 2 ðtÞ
2 2
2
1 þ y 2 ðtÞ
2
ð_y 2 ðtÞ 1Þ ;
Rb then the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, and a fi2 ðt; ; ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt is partial-invex in x2 = 0 and x_ 2 ¼ 1 for each y1(t) and y2(t) on I with respect to the same Rb g1; a fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; ; ; Þ dt is partial-incave in y2 = 0 and y_ 2 ¼ 1 for each x2(t) and y1(t) on I with respect to g2, i = 1, 2. We can verify that ðx1 ðtÞ 2 U ; x2 ðtÞ ¼ 0; x_ 2 ðtÞ ¼ 1; y 1 ðtÞ 2 V ; y 2 ðtÞ ¼ 0; y_ 2 ðtÞ ¼ 1Þ is a feasible solution of (MP) and (MD), respectively. And the objective values of (MSP) and (MSD) are equal. Thus the conclusion in Theorem 3.2 holds. Theorem 3.5 (Strong duality). Let ðxðtÞ; yðtÞÞ be an efficient solution of (MSP). Assume that the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, i = 1, 2, . . . , p, and t 2 I. If the following conditions hold: (I) fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is twice differentiable at x2, x_ 2 , y2 and y_ 2 , i = 1, 2, . . . , p; (II) for all i 2 {1, 2, . . . , p} and ri ðtÞ 2 Rm2 n o ðri ðtÞÞT ðfiy22 y 2 Df 2iy_ 2 y 2 Þ þ D½ðri ðtÞÞT ðfiy22 y_ 2 Df 2iy_ 2 y_ 2 Þ ri ðtÞ P 0 ) ri ðtÞ ¼ 0; t 2 I; p
(III) the set ffiy22 Df 2iy_ 2 gi¼1 is linearly independent; then ðxðtÞ; yðtÞÞ is a feasible solution of (MSD) and the objective values of (MSP) and (MSD) are equal. Furthermore, if the partial-invexity/partial-incavity conditions in Theorem 3.2 are also satisfied, then ðxðtÞ; yðtÞÞ is an efficient solution of (MSD).
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83
Proof. For given y1, (MP) and (MD) become a pair of symmetric dual problems. Since ðxðtÞ; yðtÞÞ is an efficient solution of (MSP), then ðx2 ðtÞ; y 2 ðtÞÞ is also an efficient solution of (MP) for y 1 ðtÞ. By Lemma 2.5 and Theorem 1 in [3], there exists a vector a 2 Rp; bi(t) 2 C2, ci(t) 2 R, i = 1, 2, . . . , p, t 2 I, such that for each x2(t) 2 C1 and y 2 ðtÞ 2 Rm2 n T aT ðfx22 Df 2x_ 2 Þ þ ðbi ðtÞ ci ðtÞy 2 ðtÞÞ ðfiy22 x2 Df 2iy_ 2 x2 Þ o D½ðbi ðtÞ ci ðtÞy 2 ðtÞÞT ðfiy22 x_ 2 Df 2iy_ 2 x_ 2 Þ ðx2 ðtÞ x2 ðtÞÞ P 0; i ¼ 1; 2; . . . ; p; 8t 2 I; n T T aT ðfy22 Df 2y_ 2 Þ ci ðtÞðfiy22 Df 2iy_ 2 Þ þ ðbi ðtÞ ci ðtÞy 2 ðtÞÞ ðfiy22 y 2 Df 2iy_ 2 y 2 Þ o T D½ðbi ðtÞ ci ðtÞy 2 ðtÞÞ ðfiy22 y_ 2 Df 2iy_ 2 y_ 2 Þ ðy 2 ðtÞ y 2 ðtÞÞ P 0; i ¼ 1; 2; . . . ; p; T
ðbi ðtÞÞ
ðfiy22
8t 2 I;
Df 2i_y 2 Þ
¼ 0;
ci ðtÞðy 2 ðtÞÞ ðfiy22 Df 2i_y 2 Þ ¼ 0; bi ðtÞ 2 C 2 ;
ða; bi ðtÞ; ci ðtÞÞ 6¼ 0;
ð20Þ
i ¼ 1; 2; . . . ; p; 8t 2 I;
T
a P 0;
ð19Þ
ð21Þ
i ¼ 1; 2; . . . ; p; 8t 2 I;
ci ðtÞ P 0;
ð22Þ
i ¼ 1; 2; . . . ; p; 8t 2 I;
ð23Þ
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð24Þ
From (21) and (22), we obtain ðbi ðtÞ ci ðtÞy 2 ðtÞÞT ðfiy22 Df 2iy_ 2 Þ ¼ 0;
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð25Þ
Let y 2 ðtÞ ¼ bi ðtÞ ðci ðtÞ 1Þy 2 ðtÞ 2 Rm2 in (20) and using (25), we get fðbi ðtÞ ci ðtÞy 2 ðtÞÞT ðfiy22 y 2 Df 2i_y 2 y 2 Þ D½ðbi ðtÞ ci ðtÞy 2 ðtÞÞT ðfiy22 y_ 2 Df 2iy_ 2 y_ 2 Þgðbi ðtÞ ci ðtÞy 2 ðtÞÞ P 0; i ¼ 1; 2; . . . ; p; 8t 2 I: From the condition (II), we have bi ðtÞ ¼ ci ðtÞy 2 ðtÞ;
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð26Þ
If ci(t) = 0, i = 1, 2, . . . , p, then by (26), bi(t) = 0, i = 1, 2, . . . , p, and so (20) becomes aT ðfy22 Df 2y_ 2 Þðy 2 ðtÞ y 2 ðyÞÞ P 0;
8t 2 I:
ð27Þ jth
z}|{ T Let y 2 ðtÞ ¼ y 2 ðtÞ þ ej 2 Rm2 and y 2 ðtÞ ¼ y 2 ðtÞ ej 2 Rm2 in (27), where ej ¼ ð0; . . . ; 0; 1 ; 0; . . . ; 0Þ 2 Rm2 , j = 1, 2, . . . , p, we get
aT fy22 Df 2y_ 2 ¼ 0: p
Since ff2iy 2 Df2iy_ 2 gi¼1 is linearly independent, a = 0, which contradicts (24). Thus ci(t) > 0, i = 1, 2, . . . , p, t 2 I, and so y 2 ðtÞ 2 C 2 . Furthermore, we obtain from (20) and condition (III) that a ¼ P ðtÞ; 8t 2 I; ð28Þ ith z}|{ where P ðtÞ ¼ ð0; . . . ; 0; ci ðtÞ; 0; . . . ; 0ÞT 2 Rp , i = 1, 2, . . . , p. Using (26), (28) and ci(t) > 0, t 2 I, (19) becomes T
ðfix22 Df 2i_x2 Þ ðx2 ðtÞ x2 ðtÞÞ P 0;
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð29Þ
Since x2(t) 2 C1 is arbitrary and C1 is a closed convex cone, x2 ðtÞ þ x2 ðtÞ 2 C 1 and 2xðtÞ 2 C 1 , and so the above inequality yields fix22 þ Df 2ix2 2 C 1 ; T
ðx2 ðtÞÞ ½fix22 þ
Df 2ix2
i ¼ 1; 2; . . . ; p; t 2 I; 6 0;
i ¼ 1; 2; . . . ; p; t 2 I:
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Thus, ðxðtÞ; yðtÞÞ is a feasible solution for (MSD). It is obvious that the values of the objective function of (MSP) and (MSD) are equal at ðxðtÞ; yðtÞÞ. So, ðxðtÞ; yðtÞÞ is an efficient solution of (MSD) from Theorem 3.2. We can also give an example to show that Theorem 3.5 holds. Remark 3.6. Generally, the strong duality does not hold for the integer linear programming. However, it holds between (MSP) and (MSD) because their objective functions are the same one. It is similar to the lines of the proof in Theorem 3.5 that we can prove the following converse duality theorem. Theorem 3.7 (Converse duality). Let ðuðtÞ; vðtÞÞ be an efficient solution of (MSD). Assume that the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, i = 1, 2, . . . , p, and t 2 I. If the following conditions hold: (I) fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ is twice differentiable at u2, u_ 2 , v2 and v_ 2 ; (II) for i 2 {1, 2, . . . , p} and ri ðtÞ 2 Rn2 n o T T ðri ðtÞ ðfix22 x2 þ Df 2i_x2 x2 Þ þ D½ðri ðtÞÞ ðfix22 x_ 2 þ Df 2i_x2 x_ 2 Þ ri ðtÞ 6 0 ) ri ðtÞ ¼ 0;
t 2 I;
p
(III) the set ffix22 þ Df 2i_x2 gi¼1 is linearly independent; then ðuðtÞ; vðtÞÞ is a feasible solution of (MSP) and the objective values of (MSD) and (MSP) are equal. Furthermore, if the partial-invexity/partial-incavity conditions in Theorem 3.2 are also satisfied, then ðuðtÞ; vðtÞÞ is an efficient solution of (MSP). 4. Self-duality In this section, under the so-called skew symmetric assumption on the functions involved, we discuss a class of self-dual problems and give the self-duality. First, we introduce the following definition. Definition 4.1. The function h : I Rn1 Rn2 Rn2 Rn1 Rn2 Rn2 7! R is said to be skew symmetric if for all x and y in the domain of h hðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ hðt; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞ; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞÞ;
t 2 I;
where x1(t) 2 U, y1(t) 2 U, x2(t) 2 C and y2(t) 2 C for all t 2 I, and U is an arbitrary sets of integers in Rn1 , C is a closed convex cone in Rn2 with non-empty interior, n1 + n2 = n. It is easy to obtain the following self-dual theorem. Theorem 4.2 (Self-duality). If fi is skew symmetric function in (MSP), i = 1, 2, . . . , p, then (MSP) is self-dual, that is, the dual problem of (MSP) is itself. Furthermore, the feasibility of (x(t), y(t)) for (MSP) implies the feasibility of (y(t), x(t)) for (MSD), and the converse. Theorem 4.3. Under the conditions of Theorems 3.5, 3.7 and 4.2, if ðxðtÞ; yðtÞÞ is an efficient solution for (MSP), then ðyðtÞ; xðtÞÞ is an efficient solution for (MSD), and the common optimal value is 0, and the converse. In the above, we have discussed the symmetric duality between (MSP) and (MSD). Here, our conclusions include the some results of [2,5,9–13,15,17,19]. Furthermore, we can introduce the partial quasi-invexity and pseudo-invexity, and establish the symmetric duality. Here, we omit the discussion in these cases. Acknowledgements This research is supported in part by the National Natural Science Foundation of China (60472060) and the Natural Science Foundation of Jiangsu High School (03KJB110012, 00KJD110001). Authors also thank anonymous referees for their helpful comments and suggestions.
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Discrete Optimization
Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity Xiuhong Chen a
a,b,*
, Jiangyu Yang
a
Department of Computer Science, Nanjing University of Science and Technology, Nanjing 210094, PR China b School of Information Technology, Southern Yangtze University, Wuxi 214122, PR China Received 6 December 2004; accepted 18 April 2006 Available online 26 July 2006
Abstract A pair of symmetric dual multiobjective variational mixed integer programs for the polars of arbitrary cones are formulated, which some primal and dual variables are constrained to belong to the set of integers. Under the separability with respect to integer variables and partial-invexity assumptions on the functions involved, we prove the weak, strong, converse and self-duality theorems to related minimax efficient solution. These results include some of available results. 2006 Elsevier B.V. All rights reserved. Keywords: Multiobjective variational problems; Mixed integer programming; Symmetric duality; Self-duality; Partial-invexity; Closed convex cone
1. Introduction Symmetric duality for nonlinear programming problems has been studied by many researchers in the past. Dantzig et al. [9], Mond [15] and Bazaraa and Goode [2] gave a pair of symmetric dual programs involving a scale function f(x, y), x 2 Rn, y 2 Rm under the condition that f(Æ, y) is convex for each y and f(x, Æ) is concave for each x. Mond and Weir [18] presented a different pair of symmetric dual nonlinear programs for f(x, y) to pseudoconvexity–pseudoconcavity. Balas [1] examined the symmetric duality results of Dantzig et al. [9] when some primal and dual variables are constrained to belong to some arbitrary set, for example, the set of integer. Kumar et al. [11] were on the lines of Mond and Weir [18] rather than the lines of Dantzig et al. [9] as taken by Mishra et al. [12], and presented a pair of symmetric dual minimax integer programming problems, which some primal and dual variables are constrained to belong to the set of integers. Recently, Chandra and Kumar [5] formulated a pair of symmetric dual nonlinear programming for the polars of arbitrary cones. Kim and Song [10] also presented two pairs of nonlinear multiobjective mixed integer programs for the polars of arbitrary cones, and established the weak, strong and converse duality theorems by using the concept of efficiency. *
Corresponding author. Address: School of Information Technology, Southern Yangtze University, Jiangsu 214122, PR China. E-mail addresses: [email protected], [email protected] (X. Chen).
0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.04.045
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77
Duality for the multiobjective variational problems has also been of great interest in recent years [3,4,14,16]. When the objective and constraint functions involved are invex, we [6] also proved the duality to related properly efficient solution for a class of multiobjective variational problems. Under the partial-invexity assumptions of scalar function /ðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ with x(t) 2 Rn and y(t) 2 Rm, we [7] also studied the symmetric duality and self-duality for a class of multiobjective fractional variational problems. In [8], a Wolfe type multiobjective variational mixed integer symmetric dual program over arbitrary cones was discussed. In this paper, we propose to study another pair of multiobjective mixed integer programs for the polars of arbitrary cones which is in the spirit of Mond and Weir. We prove the weak, strong, converse and self-duality theorems to related the concept of efficient solution. These results generalize some results of [4,5,7,10,11,13,19]. The remainder of the paper is organized as follows. In Section 2, we give some notations and lemmas. In Section 3, we formulate a pair of multiobjective variational mixed integer programs for the polars of arbitrary cones, which is in the spirit of Mond and Weir. Under the separability with respect to the integer variables and the partial-invexity assumptions on the functions involved, we prove the weak duality, strong duality and converse duality, where the reason which the strong and converse duality hold is that the objective functions of two problems are the same one. In Section 4, we discuss the self-duality. 2. Notations and lemmas Let I = [a, b] be a real interval, x : I ! Rn and y : I ! Rm. We constrain some of the components of x and y which belong to arbitrary sets of integer. Suppose that the first n1 components of x and the first m1 components of y (0 6 n1 6 n, 0 6 m1 6 m) are constrained to be integers and the following notations are introduced ðx; yÞ ¼ ðx1 ; x2 ; y 1 ; y 2 Þ; x1 2 U ; y 1 2 V ; x2 2 C 1 ; y 2 2 C 2 ; where U and V are two arbitrary sets of integers vectors in Rn1 and Rm1 , respectively. Let C1 and C2 be two closed convex cones in Rn2 and Rm2 with nonempty interiors, respectively. n = n1 + n2, m = m1 + m2. Let fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ (i = 1, 2, . . . , p) be twice continuously differentiable functions at x2, x_ 2 , y2 and y_ 2 , where x1(t) 2 U, y1(t) 2 V, x2(t) 2 C1, y2(t) 2 C2, t 2 I; x2 : I ! S1 and y2 : I ! S2 with derivative x_ 2 and y_ 2 , respectively; S 1 Rn2 and S 2 Rm2 be open and C1 S1 and C2 S2. Denote by fix2 and fi_x2 the first partial derivatives of fi with respect to x2(t) and x_ 2 ðtÞ, respectively; fix2 x2 the Hessian matrix of fi with respect to x2. Similarly, fiy 2 and fi_y 2 denote the first partial derivatives of fi with respect to y2(t) and y_ 2 ðtÞ, respectively; fix2 x_ 2 , fix2 y 2 , fix2 y_ 2 , fi_x2 x2 , fi_x2 x_ 2 , fi_x2 y 2 , fi_x2 y_ 2 , fiy 2 x2 , fiy 2 x_ 2 , fiy 2 y 2 , fiy 2 y_ 2 , fiy_ 2 x2 , fiy_ 2 x_ 2 , fiy_ 2 y 2 , fiy_ 2 y_ 2 denote the other Hessian matrixes of fi with respect to x2(t), x_ 2 ðtÞ, y2(t) and y_ 2 ðtÞ, respectively, for 1, 2, . . . , p. _ For any r-dimensional vector function Qðt; zðtÞ; z_ ðtÞ; wðtÞ; wðtÞÞ, denote the first partial derivative with _ respect to z(t), z_ ðtÞ, w(t) and wðtÞ by Qz, Qz_ , Qw and Qw_ , respectively, that is, 1 1 0 1 0 1 oQ oQ1 oQ1 oQ oQ1 oQ1 n n 1 2 1 2 oz o_z oz oz o_z o_z C C B oQ B oQ B 12 oQ22 oQn2 C B 12 oQ22 oQn2 C B oz B o_z oz C o_z C oz o_z Qz ¼ B Qz_ ¼ B C ; C ; . . C C B .. .. B .. .. A A @ . . .. @ . . .. r r r r r r oQ oQ oQ oQ oQ oQ ozn rn o_zn rn 1 oz2 z1 o_z2 1 0 oz 0o_oQ 1 1 1 oQ1 oQ1 oQ1 oQ1 own oQ ow_ n ow1 ow2 ow_ 1 ow_ 2 B oQ2 oQ2 B oQ2 oQ2 2 C 2 C C C B 1 B 1 oQ oQ B ow B ow_ own C ow_ n C ow2 ow_ 2 Qw ¼ B ; Qw_ ¼ B C C : .. .. C C B .. .. B .. .. A A @ . . . @ . . . r r r r r r oQ oQ oQ oQ oQ oQ own ow_ n ow1 ow2 ow_ 1 ow_ 2 rn rn Let C(I, Rs) denote the space of piecewise smooth function z(t) with norm kzk = kzk1 + kDzk1, where the differentiation operator D is given by Z t u ¼ Dz () zðtÞ ¼ uðaÞ þ uðsÞ ds; a
where u(a) is a given boundary value. Therefore
d dt
¼ D except at discontinuities.
78
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
Consider the following multiobjective programming problem ðMPÞ
minimize gðxÞ subject to hðxÞ 6 0;
where g : C ! Rp, h : C ! Rm and C is a closed convex cone with nonempty interior in Rn. Denote P the set of feasible solutions of (MP). Definition 2.1. A point x 2 P is said to be a Pareto minimum solution, that is, an efficient solution, of (MP) if there exists no other x 2 P such that gðxÞ 6 gðxÞ. Definition 2.2. A point ðx; yÞ is an efficient solution of the following minimax multiobjective programming problem max min x
y
F ðx; yÞ
subject to Gðx; yÞ 6 0; where F : C · C ! Rp, G : C · C ! Rm and C is a closed convex cone with nonempty interior in Rn, if y is a Pareto minimum solution of the inner problem ‘‘miny F ðx; yÞ’’ for each x and x is a Pareto maximum solution of the exterior problem ‘‘maxx F ðx; yÞ’’. Definition 2.3. A cone C* is said to be the polar of C, if C ¼ fzjzT x 6 0; for all x 2 Cg: Definition 2.4. A real valued function /ðt; x1 ; x_ 1 ; x2 ; x_ 2 ; . . . ; xp ; x_ p Þ is said to be separable with respect to x1 and x_ 1 , if there exist real functions nðt; x1 ; x_ 1 Þ and fðt; x2 ; x_ 2 ; . . . ; xp ; x_ p Þ such that /ðt; x1 ; x_ 1 ; x2 ; x_ 2 ; . . . ; xp ; x_ p Þ ¼ nðt; x1 ; x_ 1 Þ þ fðt; x2 ; x_ 2 ; . . . ; xp ; x_ p Þ: The following lemma, which is said to be the generalized form of Fritz–John condition for the vector valued functions proposed by Bazaraa and Goode [2], play a main role during the process of proof of the strong duality. Lemma 2.5. Let K be a convex set with nonempty interior in Rn and let C be a closed convex cone in Rn having a nonempty interior. Let F and G be two vector-valued functions defined on K. If z0 is an efficient solution of the following problem: minimize F ðzÞ subject to GðzÞ 2 C; z 2 K; then there exists a nonzero vector (r0, r) such that T r0 F z ðz0 Þ þ rT Gz ðz0 Þ ðz z0 Þ P 0; 8z 2 K; and r0 P 0, r 2 C*, and rTG(z0) = 0. In order to obtain our main results, we introduce the following definition. _ Definition 2.6. Let S Rn be open set. If there exists a vector function g1 ðt; xðtÞ; x_ ðtÞ; uðtÞ; uðtÞÞ 2 S with g1 = 0 at t if x(t) = u(t), such that for the scalar function hðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ the functional Z b H ðx; x_ ; y; y_ Þ ¼ hðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ dt a
satisfies _ y; yÞ _ P H ðx; x_ ; y; y_ Þ H ðu; u;
Z
b
n o _ _ gT1 hx ðt; uðtÞ; uðtÞ; yðtÞ; y_ ðtÞÞ þ ðDg1 ÞT hx_ ðt; uðtÞ; uðtÞ; yðtÞ; y_ ðtÞÞ dt;
a
then H ðx; x_ ; y; y_ Þ is said to be partially invex in x and x_ on I with respect to g1 for fixed y.
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
If H satisfies _ H ðx; x_ ; v; v_ Þ P H ðx; x_ ; y; yÞ
Z
b
n
79
o gT2 hy ðt; xðtÞ; x_ ðtÞ; vðtÞ; v_ ðtÞÞ þ ðDg2 ÞT hy_ ðt; xðtÞ; x_ ðtÞ; vðtÞ; v_ ðtÞÞ dt;
a
_ vðtÞ; v_ ðtÞÞ 2 S with g2 = 0 at t if y(t) = v(t), then H ðx; x_ ; y; y_ Þ is said to be partially invex in where g2 ðt; yðtÞ; yðtÞ; y and y_ on I with respect to g2 for fixed x. If H is partially invex in x and x_ (or in y and y_ ) on I with respect to g1 (or g2) for fixed y (or for fixed x), then H is said to be partially incave in x and x_ (or in y and y_ ) on I with respect to g1 (or g2) for fixed y (or for fixed x). _ Example 2.7. Let I = [0, 1], S = (1, 1) and g1 ðt; xðtÞ; x_ ðtÞ; uðtÞ; uðtÞÞ 2 S is an arbitrary element satisfying 2 _ 2 þ ð_xðtÞ2 1Þ2 yðtÞ2 þ 1, then the funcg1 = 0 if x(t) = u(t) onR I. Define hðt; xðtÞ; x_ ðtÞ; yðtÞ; y_ ðtÞÞ ¼ ln½xðtÞ yðtÞ 1 _ tional H ðx; x_ ; y; y_ Þ ¼ 0 hðt; xðtÞ; x_ ðtÞ; yðtÞ; yðtÞÞ dt P 0. It is obvious that hx ðt; 0; 1; yðtÞ; y_ ðtÞÞ ¼ 0 and _ is partially invex in x = 0 and x_ ¼ 1 on I with respect to g1 for hx_ ðt; 0; 1; yðtÞ; y_ ðtÞÞ ¼ 0. Thus H ðx; x_ ; y; yÞ fixed y. _ is not partially invex in y = 0 and y_ ¼ 1 on I with respect to Similarly, we can prove that H ðx; x_ ; y; yÞ g2 = (y(t) v(t))2 for fixed x. 3. Symmetric duality results In this section, we consider the following symmetric dual minimax multiobjective variational mixed integer programs for the arbitrary cones Z b ðMSPÞ max min f ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt x1
x2 ;y
subject to
a
xðaÞ ¼ 0 ¼ yðaÞ;
xðbÞ ¼ 0 ¼ yðbÞ;
x_ 2 ðaÞ ¼ 0 ¼ y_ 2 ðaÞ; x1 ðtÞ 2 U ;
ð1Þ
x_ 2 ðbÞ ¼ 0 ¼ y_ 2 ðbÞ;
x2 ðtÞ 2 C 1 ;
y 1 ðtÞ 2 V ;
ð2Þ m2
y 2 ðtÞ 2 R ;
8t 2 I;
ð3Þ
f iy 2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ Dfi_y 2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ 2 C 2 ; i ¼ 1; 2; . . . ; p;
8t 2 I;
ð4Þ
T
ðy 2 ðtÞÞ ½fiy 2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ Dfi_y 2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ P 0; i ¼ 1; 2; . . . ; p;
8t 2 I;
ð5Þ
and ðMSDÞ
Z min max v1
u;v2
b
f ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt
a
subject to uðaÞ ¼ 0 ¼ vðaÞ; uðbÞ ¼ 0 ¼ vðbÞ; u_ 2 ðaÞ ¼ 0 ¼ v_ 2 ðaÞ; u_ 2 ðbÞ ¼ 0 ¼ v_ 2 ðbÞ; u1 ðtÞ 2 U ; u2 ðtÞ 2 Rn2 ; v1 ðtÞ 2 V ; v2 ðtÞ 2 C 2 ; 8t 2 I; fix2 ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ þ Dfi_x2 ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ 2 C 1 ; i ¼ 1; 2; . . . ; p;
8t 2 I;
ð6Þ ð7Þ ð8Þ
ð9Þ
T
ðu2 ðtÞÞ ½fix2 ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ þ Dfi_x2 ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ 6 0; i ¼ 1; 2; . . . ; p;
8t 2 I;
ð10Þ
80
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
where the function f ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is a p-dimensional vector-valued function. In (MSP) and (MSD), if fi does not depend explicitly on t for i = 1, 2, . . . , p, then our problems become the pair of problems given by Kim and Song [10]. Furthermore, if p = 1, C 1 ¼ Rnþ2 and C 2 ¼ Rmþ2 , then our problems again become the pair of problems considered by Mishra and Das [13]. Denote by X and Y the set of feasible solutions of (MSP) and (MSD), respectively. From now on, suppose that (1) the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1 or y1, i = 1, 2, . . . , p. Without loss of generality, fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, that is, fi can be expressed as fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ fi1 ðt; x1 ðtÞÞ þ fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ; (2) the sets of feasible solutions X and Y having the properties that if (x(t), y(t)) 2 X and (u(t), v(t)) 2 Y, then g1 ðt; x2 ðtÞ; x_ 2 ðtÞ; u2 ðtÞ; u_ 2 ðtÞÞ 2 C 1 and g2 ðt; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ, v_ 2 ðtÞÞ 2 C 2 . Remark 3.1. Under the above assumptions, (MSP) can be expressed as the following form: Z b max min ½f 1 ðt; x1 ðtÞÞ þ f 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt x2 ;y
x1
a
subject to ð1Þ–ð5Þ; where T
f 1 ðt; x1 ðtÞÞ ¼ ðf11 ðt; x1 ðtÞÞ; f21 ðt; x1 ðtÞÞ; . . . ; fp1 ðt; x1 ðtÞÞÞ ; f 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ ðf12 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ; f22 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ; . . . ; T
fp2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞÞ : Let ðMPÞ /ðy 1 Þ ¼ min
Z
x2 ;y 2
b
f 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞdt
a
subject to ð1Þ and ð2Þ; x2 ðtÞ 2 C 1 ; y 2 ðtÞ 2 Rm2 ; and; fiy22 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ Df 2iy_ 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ
ð11Þ 2
C 2 ;
i ¼ 1; 2; . . . ; p; 8t 2 I; ðy 2 ðtÞÞ
T
ð12Þ
½fiy22 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ Df 2i_y 2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ
i ¼ 1; 2; . . . ; p; 8t 2 I:
P 0; ð13Þ
So, (MSP) may become Z b max min f1 ðt; x1 ðtÞÞ dt þ /1 ðy 1 Þ : x1 2 U ; y 1 2 V : y1
x1
a
Similarly, (MSD) can also be written as Z b min max f 1 ðt; u1 ðtÞÞ dt þ w1 ðv1 Þ : u1 2 U ; v1 2 V ; v1
u1
a
where ðMDÞ
wðv1 Þ ¼ max u2 ;v2
subject to
Z
b
f 2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt
a
ð6Þ and ð7Þ; u2 ðtÞ 2 Rn2 ; v2 ðtÞ 2 C 2 ;
and;
ð14Þ
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
81
fix22 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ þ Df 2iu_ 2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ 2 C 1 ; i ¼ 1; 2; . . . ; p; ðu2 ðtÞÞ
T
8t 2 I;
ð15Þ
½fix22 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ
P 0;
þ
Df 2i_x2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð16Þ
Thus, under the above separable assumption of f with respect to x1 or y1, we only discuss the symmetric duality between (MP) and (MD). Theorem 3.2 (Weak duality). Let (x(t), y(t)) and (u(t), v(t)) be feasible solutions of (MSP) and (MSD), respectively. R b 2 Assume that the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, and f ðt; ; ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt is partial-invex in x2 and a i R b x_ 2 for each y1(t) and y2(t) on I with respect to the same g1 ¼ g1 ðt; x2 ðtÞ; x_ 2 ðtÞ; u2 ðtÞ; u_ 2 ðtÞÞ 2 C 1 , i = 1, 2, . . . , p; a fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; ; ; Þ dt is partial-incave in y1, y2 and y_ 2 for each x2(t) on I with respect to g2 ¼ g2 ðt; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ 2 C 2 . Then the following inequalities cannot simultaneously hold: (I) for all i 2 {1, 2, . . . , p} Z b Z fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt 6 a
b
fi ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt; a
(II) for at least one j 2 {1, 2, . . . , p} Z b Z fj ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ < a
b
fj ðt; u1 ðtÞ; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt: a
Proof. Under the separable assumption, from the above remark, for each feasible solution (x(t), y(t)) of (MSP) and each feasible solution (u(t), v(t)) of (MSD), it is sufficient to prove that the following inequalities cannot simultaneously hold: (I 0 ) for all i 2 {1, 2, . . . , p} Z b Z 2 fi ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt 6 a
b
fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt; a
(II 0 ) for at least one j 2 {1, 2, . . . , p} Z b Z 2 fj ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt < a
b
fj2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt:
a
Rb 2 y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt with respect to g1 2 C1, for each k 2 K ¼ From the partial-invexity of a fi ðt; ; ; P k ¼ ðk1 ; k2 ; . . . ; kp Þ : ki P 0; i ¼ 1; 2; . . . ; p; pi¼1 ki ¼ 1 , we have Z b Z b p p X X 2 ki fi ðt; x2 ðtÞ; x_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt ki fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt a
i¼1
¼
p X
Z
a
b
ki a
i¼1
¼
½fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt
a
p X
Z
a
b
ki
i¼1
P
i¼1
Z
b
gT1
h i T gT1 fix22 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ þ ðDg1 Þ fi_2x2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt
" p X i¼1
ki fix22 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ
D
p X i¼1
# ki fi_2x2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ
dt
82
X. Chen, J. Yang / European Journal of Operational Research 181 (2007) 76–85
(by integrating by parts, and when t = a and t = b, x2(t) = u2(t), so it follows that g1 = 0) ðby g1 2 C 1 ; ð15Þ and k 2 KÞ
P 0:
Similarly, from the incavity of p X
Z
Rb a
ð17Þ
f2i ðt; x2 ðtÞ; x_ 2 ðtÞ; ; ; Þ dt with respect to g2 2 C2, we get
b
fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt 6
ki
p X
a
i¼1
Z
b
fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt:
ki
ð18Þ
a
i¼1
From (17) and (18), we have p X i¼1
Z
b
fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt
ki
P
a
The proof is complete.
p X
Z
b
fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ dt:
ki a
i¼1
h
Remark 3.3. Because the integer variables x1 are separated from the objective functions, the proof of Theorem 3.2 only consider the weak duality between (MP) and (MD). The same reason is suitable to Theorems 3.5 and 3.7. _ Example 3.4. Let I = [0, 1], C1 = C2 = (1, 1), g1 ¼ g1 ðt; xðtÞ; x_ ðtÞ; uðtÞ; uðtÞÞ 2 C 1 and g2 ¼ g2 ðt; yðtÞ; y_ ðtÞ; vðtÞ; v_ ðtÞÞ 2 C 2 are arbitrary elements satisfying g1 = 0 if x(t) = u(t) and g2 = 0 if y(t) = v(t) on I, respectively. Define f1 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ x1 ðtÞ þ y 1 ðtÞ þ ln½x2 ðtÞ2 þ ð_x2 ðtÞ2 1Þ2 þ 1 ln½y 2 ðtÞ2 þ ð_y 2 ðtÞ2 1Þ2 þ 1; f2 ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ x1 ðtÞ þ y 1 ðtÞ þ
x2 ðtÞ
2
1 þ x2 ðtÞ
2
2
þ ð_x2 ðtÞ 1Þ 2
y 2 ðtÞ
2 2
2
1 þ y 2 ðtÞ
2
ð_y 2 ðtÞ 1Þ ;
Rb then the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, and a fi2 ðt; ; ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ dt is partial-invex in x2 = 0 and x_ 2 ¼ 1 for each y1(t) and y2(t) on I with respect to the same Rb g1; a fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; ; ; Þ dt is partial-incave in y2 = 0 and y_ 2 ¼ 1 for each x2(t) and y1(t) on I with respect to g2, i = 1, 2. We can verify that ðx1 ðtÞ 2 U ; x2 ðtÞ ¼ 0; x_ 2 ðtÞ ¼ 1; y 1 ðtÞ 2 V ; y 2 ðtÞ ¼ 0; y_ 2 ðtÞ ¼ 1Þ is a feasible solution of (MP) and (MD), respectively. And the objective values of (MSP) and (MSD) are equal. Thus the conclusion in Theorem 3.2 holds. Theorem 3.5 (Strong duality). Let ðxðtÞ; yðtÞÞ be an efficient solution of (MSP). Assume that the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, i = 1, 2, . . . , p, and t 2 I. If the following conditions hold: (I) fi2 ðt; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is twice differentiable at x2, x_ 2 , y2 and y_ 2 , i = 1, 2, . . . , p; (II) for all i 2 {1, 2, . . . , p} and ri ðtÞ 2 Rm2 n o ðri ðtÞÞT ðfiy22 y 2 Df 2iy_ 2 y 2 Þ þ D½ðri ðtÞÞT ðfiy22 y_ 2 Df 2iy_ 2 y_ 2 Þ ri ðtÞ P 0 ) ri ðtÞ ¼ 0; t 2 I; p
(III) the set ffiy22 Df 2iy_ 2 gi¼1 is linearly independent; then ðxðtÞ; yðtÞÞ is a feasible solution of (MSD) and the objective values of (MSP) and (MSD) are equal. Furthermore, if the partial-invexity/partial-incavity conditions in Theorem 3.2 are also satisfied, then ðxðtÞ; yðtÞÞ is an efficient solution of (MSD).
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83
Proof. For given y1, (MP) and (MD) become a pair of symmetric dual problems. Since ðxðtÞ; yðtÞÞ is an efficient solution of (MSP), then ðx2 ðtÞ; y 2 ðtÞÞ is also an efficient solution of (MP) for y 1 ðtÞ. By Lemma 2.5 and Theorem 1 in [3], there exists a vector a 2 Rp; bi(t) 2 C2, ci(t) 2 R, i = 1, 2, . . . , p, t 2 I, such that for each x2(t) 2 C1 and y 2 ðtÞ 2 Rm2 n T aT ðfx22 Df 2x_ 2 Þ þ ðbi ðtÞ ci ðtÞy 2 ðtÞÞ ðfiy22 x2 Df 2iy_ 2 x2 Þ o D½ðbi ðtÞ ci ðtÞy 2 ðtÞÞT ðfiy22 x_ 2 Df 2iy_ 2 x_ 2 Þ ðx2 ðtÞ x2 ðtÞÞ P 0; i ¼ 1; 2; . . . ; p; 8t 2 I; n T T aT ðfy22 Df 2y_ 2 Þ ci ðtÞðfiy22 Df 2iy_ 2 Þ þ ðbi ðtÞ ci ðtÞy 2 ðtÞÞ ðfiy22 y 2 Df 2iy_ 2 y 2 Þ o T D½ðbi ðtÞ ci ðtÞy 2 ðtÞÞ ðfiy22 y_ 2 Df 2iy_ 2 y_ 2 Þ ðy 2 ðtÞ y 2 ðtÞÞ P 0; i ¼ 1; 2; . . . ; p; T
ðbi ðtÞÞ
ðfiy22
8t 2 I;
Df 2i_y 2 Þ
¼ 0;
ci ðtÞðy 2 ðtÞÞ ðfiy22 Df 2i_y 2 Þ ¼ 0; bi ðtÞ 2 C 2 ;
ða; bi ðtÞ; ci ðtÞÞ 6¼ 0;
ð20Þ
i ¼ 1; 2; . . . ; p; 8t 2 I;
T
a P 0;
ð19Þ
ð21Þ
i ¼ 1; 2; . . . ; p; 8t 2 I;
ci ðtÞ P 0;
ð22Þ
i ¼ 1; 2; . . . ; p; 8t 2 I;
ð23Þ
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð24Þ
From (21) and (22), we obtain ðbi ðtÞ ci ðtÞy 2 ðtÞÞT ðfiy22 Df 2iy_ 2 Þ ¼ 0;
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð25Þ
Let y 2 ðtÞ ¼ bi ðtÞ ðci ðtÞ 1Þy 2 ðtÞ 2 Rm2 in (20) and using (25), we get fðbi ðtÞ ci ðtÞy 2 ðtÞÞT ðfiy22 y 2 Df 2i_y 2 y 2 Þ D½ðbi ðtÞ ci ðtÞy 2 ðtÞÞT ðfiy22 y_ 2 Df 2iy_ 2 y_ 2 Þgðbi ðtÞ ci ðtÞy 2 ðtÞÞ P 0; i ¼ 1; 2; . . . ; p; 8t 2 I: From the condition (II), we have bi ðtÞ ¼ ci ðtÞy 2 ðtÞ;
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð26Þ
If ci(t) = 0, i = 1, 2, . . . , p, then by (26), bi(t) = 0, i = 1, 2, . . . , p, and so (20) becomes aT ðfy22 Df 2y_ 2 Þðy 2 ðtÞ y 2 ðyÞÞ P 0;
8t 2 I:
ð27Þ jth
z}|{ T Let y 2 ðtÞ ¼ y 2 ðtÞ þ ej 2 Rm2 and y 2 ðtÞ ¼ y 2 ðtÞ ej 2 Rm2 in (27), where ej ¼ ð0; . . . ; 0; 1 ; 0; . . . ; 0Þ 2 Rm2 , j = 1, 2, . . . , p, we get
aT fy22 Df 2y_ 2 ¼ 0: p
Since ff2iy 2 Df2iy_ 2 gi¼1 is linearly independent, a = 0, which contradicts (24). Thus ci(t) > 0, i = 1, 2, . . . , p, t 2 I, and so y 2 ðtÞ 2 C 2 . Furthermore, we obtain from (20) and condition (III) that a ¼ P ðtÞ; 8t 2 I; ð28Þ ith z}|{ where P ðtÞ ¼ ð0; . . . ; 0; ci ðtÞ; 0; . . . ; 0ÞT 2 Rp , i = 1, 2, . . . , p. Using (26), (28) and ci(t) > 0, t 2 I, (19) becomes T
ðfix22 Df 2i_x2 Þ ðx2 ðtÞ x2 ðtÞÞ P 0;
i ¼ 1; 2; . . . ; p; 8t 2 I:
ð29Þ
Since x2(t) 2 C1 is arbitrary and C1 is a closed convex cone, x2 ðtÞ þ x2 ðtÞ 2 C 1 and 2xðtÞ 2 C 1 , and so the above inequality yields fix22 þ Df 2ix2 2 C 1 ; T
ðx2 ðtÞÞ ½fix22 þ
Df 2ix2
i ¼ 1; 2; . . . ; p; t 2 I; 6 0;
i ¼ 1; 2; . . . ; p; t 2 I:
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Thus, ðxðtÞ; yðtÞÞ is a feasible solution for (MSD). It is obvious that the values of the objective function of (MSP) and (MSD) are equal at ðxðtÞ; yðtÞÞ. So, ðxðtÞ; yðtÞÞ is an efficient solution of (MSD) from Theorem 3.2. We can also give an example to show that Theorem 3.5 holds. Remark 3.6. Generally, the strong duality does not hold for the integer linear programming. However, it holds between (MSP) and (MSD) because their objective functions are the same one. It is similar to the lines of the proof in Theorem 3.5 that we can prove the following converse duality theorem. Theorem 3.7 (Converse duality). Let ðuðtÞ; vðtÞÞ be an efficient solution of (MSD). Assume that the function fi ðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ is separable with respect to x1, i = 1, 2, . . . , p, and t 2 I. If the following conditions hold: (I) fi2 ðt; u2 ðtÞ; u_ 2 ðtÞ; v1 ðtÞ; v2 ðtÞ; v_ 2 ðtÞÞ is twice differentiable at u2, u_ 2 , v2 and v_ 2 ; (II) for i 2 {1, 2, . . . , p} and ri ðtÞ 2 Rn2 n o T T ðri ðtÞ ðfix22 x2 þ Df 2i_x2 x2 Þ þ D½ðri ðtÞÞ ðfix22 x_ 2 þ Df 2i_x2 x_ 2 Þ ri ðtÞ 6 0 ) ri ðtÞ ¼ 0;
t 2 I;
p
(III) the set ffix22 þ Df 2i_x2 gi¼1 is linearly independent; then ðuðtÞ; vðtÞÞ is a feasible solution of (MSP) and the objective values of (MSD) and (MSP) are equal. Furthermore, if the partial-invexity/partial-incavity conditions in Theorem 3.2 are also satisfied, then ðuðtÞ; vðtÞÞ is an efficient solution of (MSP). 4. Self-duality In this section, under the so-called skew symmetric assumption on the functions involved, we discuss a class of self-dual problems and give the self-duality. First, we introduce the following definition. Definition 4.1. The function h : I Rn1 Rn2 Rn2 Rn1 Rn2 Rn2 7! R is said to be skew symmetric if for all x and y in the domain of h hðt; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞ; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞÞ ¼ hðt; y 1 ðtÞ; y 2 ðtÞ; y_ 2 ðtÞ; x1 ðtÞ; x2 ðtÞ; x_ 2 ðtÞÞ;
t 2 I;
where x1(t) 2 U, y1(t) 2 U, x2(t) 2 C and y2(t) 2 C for all t 2 I, and U is an arbitrary sets of integers in Rn1 , C is a closed convex cone in Rn2 with non-empty interior, n1 + n2 = n. It is easy to obtain the following self-dual theorem. Theorem 4.2 (Self-duality). If fi is skew symmetric function in (MSP), i = 1, 2, . . . , p, then (MSP) is self-dual, that is, the dual problem of (MSP) is itself. Furthermore, the feasibility of (x(t), y(t)) for (MSP) implies the feasibility of (y(t), x(t)) for (MSD), and the converse. Theorem 4.3. Under the conditions of Theorems 3.5, 3.7 and 4.2, if ðxðtÞ; yðtÞÞ is an efficient solution for (MSP), then ðyðtÞ; xðtÞÞ is an efficient solution for (MSD), and the common optimal value is 0, and the converse. In the above, we have discussed the symmetric duality between (MSP) and (MSD). Here, our conclusions include the some results of [2,5,9–13,15,17,19]. Furthermore, we can introduce the partial quasi-invexity and pseudo-invexity, and establish the symmetric duality. Here, we omit the discussion in these cases. Acknowledgements This research is supported in part by the National Natural Science Foundation of China (60472060) and the Natural Science Foundation of Jiangsu High School (03KJB110012, 00KJD110001). Authors also thank anonymous referees for their helpful comments and suggestions.
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