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Multiobjective symmetric duality involving cones
European Journal of Operational Research 141 (2002) 471–479 www.elsevier.com/locate/dsw
Continuous Optimization
Multiobjective symmetric duality involving cones S.K. Suneja a, Sunila Aggarwal a, Sonia Davar b
b,*
a Department of Mathematics, Miranda House, University of Delhi, Delhi 110007, India Department of Mathematics, St. Stephen’s College, University of Delhi, Delhi 110007, India
Received 12 October 2000; accepted 20 June 2001
Abstract In this paper, a pair of multiobjective symmetric dual programs over arbitrary cones are formulated for cone-convex functions. Weak, strong, converse and self-duality theorems are proved for these programs. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Multiobjective symmetric duality; Cone-convex functions; Weak minimum; Multiobjective optimization
1. Introduction Symmetric duality in nonlinear programming in which the dual of the dual is the primal was first introduced by Dorn [6]. The notion of symmetric duality was developed significantly by Dantzig et al. [4], Chandra and Husain [3] and Mond and Weir [10]. Dantzig et al. [4] formulated a pair of symmetric dual programs and established duality results for convex/concave functions by taking non-negative orthant as the cone. The same result was generalized by Bazaraa and Goode [1] to arbitrary cones. Mond and Weir [10] presented two pairs of symmetric dual multiobjective programming problems for efficient solutions and obtained symmetric duality results concerning pseudoconvex/pseudoconcave functions and Chandra et al. [2] studied symmetric dual fractional programming problems assuming the function involved to be pseudoconvex/pseudoconcave. Nanda [11] studied symmetric dual problems assuming the functions to be invex with non-negative orthant as the cone. Nanda and Das [12] also studied the symmetric dual fractional programming problem for arbitrary cones assuming the functions to be pseudo-invex. Recently, Kim et al. [7] studied a pair of multiobjective symmetric dual programs for pseudo-invex functions and arbitrary cones. Devi [5] formulated a pair of second-order symmetric dual programs and obtained duality results involving g-bonvex (second-order invex) functions. Mishra [8] formulated a pair of multiobjective second-order symmetric dual nonlinear programming problems under second-order pseudoinvexity assumptions on the functions involved over arbitrary cones and established duality results. Mishra
*
Corresponding author. Tel.: +91-11-643-4969. E-mail address: soniadavar@rediffmail.com (S. Davar).
0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 2 5 8 - 2
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[9] also studied second order symmetric duality under second-order F -convexity, F -concavity, F -pseudoconvexity, F -pseudoconcavity for second-order Wolfe and Mond–Weir type models, respectively. In this paper, we formulate a pair of symmetric dual programs over arbitrary cones and establish weak, strong, converse and self-duality theorems by using cone-convexity. In all the above-mentioned papers including the one by Kim et al. [7], the objective function is optimized with respect to the non-negative orthant. However, in this paper, the objective function is optimized with respect to an arbitrary closed convex cone by assuming the function involved to be cone-convex.
2. Notations and preliminaries Definition 1. Let f : X ! Y , where X is a convex subset of a topological vector space and Y is an ordered locally convex topological vector space. Let K be a closed convex cone in Y with non-empty interior. Then f is called K-convex on X if, for any x1 ; x2 2 X and h 2 ð0; 1Þ, hf ðx1 Þ þ ð1 hÞf ðx2 Þ f ðhx1 þ ð1 hÞx2 Þ 2 K: We consider the following vector minimization problem: ðVPÞ K minimize f ðxÞ subject to gðxÞ 2 Q;
x 2 C;
where C Rn , K and Q are closed convex cones with non-empty interiors in Rp and Rm , respectively. f : Rn ! Rp and g : Rn ! Rm . Let X 0 ¼ fx 2 C : gðxÞ 2 Qg. Definition 2. A point x 2 X 0 is called a weak minimum of (VP) if ðfor all x 2 X 0 Þ
f ðxÞ f ðxÞ 62 int K:
Definition 3. The positive polar cone K of K is defined by K ¼ fz 2 Rp : xT z P 0 for all x 2 Kg: We formulate the following multiobjective symmetric dual problems: ðSPÞ
K minimize subject to
f ðx; yÞ ½y T ry ðkT f Þðx; yÞe ðx; yÞ 2 C1 C2 ; ry ðkT f Þðx; yÞ 2 C2 ; k 2 K ;
e 2 int K;
kT e ¼ 1
and ðSDÞ K maximize subject to
f ðu; vÞ ½uT rx ðkT f Þðu; vÞe ðu; vÞ 2 C1 C2 ; rx ðkT f Þðu; vÞ 2 C1 ; k 2 K ;
e 2 int K;
kT e ¼ 1;
where f : Rn Rm ! Rp is a twice differentiable function, C1 and C2 are closed convex cones with nonempty interiors in Rn and Rm , respectively. C1 and C2 are positive polar cones of C1 and C2 , respectively, K is a closed convex cone in Rp such that int K 6¼ /.
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473
rx ðkT f Þðx; yÞ and ry ðkT f Þðx; yÞ are gradients of ðkT f Þðx; yÞ with respect to x and y, respectively. rxx ðkT f Þðx; yÞ and ryy ðkT f Þðx; yÞ are the Hessian matrices of ðkT f Þðx; yÞ with respect to x and y, respectively.
3. The duality results We now establish the symmetric duality results for (SP) and (SD). Theorem 1 (Weak duality). Let ðx; k; yÞ and ðu; k; vÞ be feasible solutions of (SP) and (SD), respectively. Assume that f ð; yÞ is K-convex with respect to x for fixed y and f ðx; Þ is K-convex with respect to y for fixed x. Then ðf ðu; vÞ ½uT rx ðkT f Þðu; vÞeÞ ðf ðx; yÞ ½y T ry ðkT f Þðx; yÞeÞ 62 int K: Proof. Let, if possible, ðf ðu; vÞ ½uT rx ðkT f Þðu; vÞeÞ ðf ðx; yÞ ½y T ry ðkT f Þðx; yÞeÞ 2 int K: Then k 2 K implies kT ððf ðx; yÞ ½y T ry ðkT f Þðx; yÞeÞ ðf ðu; vÞ ½uT rx ðkT f Þðu; vÞeÞÞ < 0; which gives that kT f ðu; vÞ kT f ðx; yÞ kT ð½uT rx ðkT f Þðu; vÞeÞ þ kT ð½y T ry ðkT f Þðx; yÞeÞ > 0 ) kT f ðu; vÞ kT f ðx; yÞ uT rx ðkT f Þðu; vÞ þ y T ry ðkT f Þðx; yÞ > 0
as kT e ¼ 1:
ð1Þ
Since f ð; yÞ is K-convex with respect to x for fixed y ¼ v, it follows that hf ðx; vÞ þ ð1 hÞf ðu; vÞ f ðhx þ ð1 hÞu; vÞ 2 K; This gives that
f ðx; vÞ f ðu; vÞ
f ðhx þ ð1 hÞu; vÞ f ðu; vÞ h
h 2 ð0; 1Þ:
2 K;
which implies that f ðx; vÞ f ðu; vÞ rx f ðu; vÞT ðx uÞ 2 K ) kT f ðx; vÞ kT f ðu; vÞ P kT rx f ðu; vÞT ðx uÞ
as k 2 K :
ð2Þ
Similarly, since f ðx; Þ is K-convex with respect to y for each fixed x, we get T
kT f ðx; vÞ þ kT f ðx; yÞ P kT ry f ðx; yÞ ðv yÞ; which when added to (2) gives T
T
T
T
kT f ðx; yÞ kT f ðu; vÞ P kT rx f ðu; vÞ x kT rx f ðu; vÞ u kT ry f ðx; yÞ v þ kT ry f ðx; yÞ y; which implies that kT f ðx; yÞ kT f ðu; vÞ xT rx ðkT f Þðu; vÞ þ uT rx ðkT f Þðu; vÞ þ vT ry ðkT f Þðx; yÞ y T ry ðkT f Þðx; yÞ P 0: We also have by feasibility of ðx; k; yÞ and ðu; k; vÞ for (SP) and (SD), respectively, vT ry ðkT f Þðx; yÞ 6 0
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and xT rx ðkT f Þðu; vÞ P 0: Thus we have kT f ðx; yÞ kT f ðu; vÞ þ uT rx ðkT f Þðu; vÞ y T ry ðkT f Þðx; yÞ P 0; which contradicts (1).
The above weak duality theorem is illustrated with the help of the following example: Example 1. Let C1 ¼ Rþ , C2 ¼ Rþ , K ¼ fðx; yÞ : x 6 y 6 x; x P 0g and define a function f : R R ! R2 as f ðx; yÞ ¼ ðf1 ðx; yÞ; f2 ðx; yÞÞ, where f1 ðx; yÞ ¼ x3 y 3 ;
f2 ðx; yÞ ¼ x3 :
Then, clearly f ð; yÞ is K-convex with respect to x for fixed y and f ðx; Þ is K-convex with respect to y for fixed x. Also, problems (SP) and (SD) can be written as ðSPÞ
K minimize subject to
ðx3 y 3 ; x3 Þ þ 3k1 y 3 ðe1 ; e2 Þ x P 0; y P 0; 3k1 y 2 P 0; k ¼ ðk1 ; k2 Þ 2 K ¼ K; k1 e 1 þ k2 e 2 ¼ 1
e ¼ ðe1 ; e2 Þ 2 int K;
and ðSDÞ K maximize subject to
ðu3 v3 ; u3 Þ 3ðk1 þ k2 Þu3 ðe1 ; e2 Þ u P 0; v P 0; 3ðk1 þ k2 Þu2 P 0; k ¼ ðk1 ; k2 Þ 2 K ¼ K; k1 e1 þ k2 e2 ¼ 1:
e ¼ ðe1 ; e2 Þ 2 int K;
Let ðx; k; yÞ and ðu; k; vÞ be feasible solutions to (SP) and (SD), respectively. Now by K-convexity of the function f ð; yÞ with respect to x, we have T
f ðx; vÞ f ðu; vÞ rx f ðu; vÞ ðx uÞ 2 K; which gives that ðx3 u3 3u2 ðx uÞ; x3 u3 3u2 ðx uÞÞ 2 K: By the K-convexity of f ðx; Þ with respect to y, we get ðv3 y 3 3y 2 ðv yÞ; 0Þ 2 K: Hence for k ¼ ðk1 ; k2 Þ 2 K ¼ K, we get ðk1 þ k2 Þðx3 u3 3u2 ðx uÞÞ P 0 and k1 ðv3 y 3 3y 2 ðv yÞÞ P 0;
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475
that is, ðk1 þ k2 Þðx3 þ 2u3 3u2 xÞ P 0 and k1 ðv3 þ 2y 3 3y 2 vÞ P 0; which on adding yield ðk1 þ k2 Þðx3 þ 2u3 Þ þ k1 ðv3 þ 2y 3 Þ P ðk1 þ k2 Þð3u2 xÞ þ k1 3y 2 v P 0; where the last inequality follows on account of feasibility conditions of the problems (SP) and (SD). Thus, we have ðk1 þ k2 Þðx3 u3 Þ þ ðk1 þ k2 Þð3u3 Þ þ k1 ðv3 y 3 Þ þ k1 ð3y 3 Þ P 0:
ð3Þ
We are to show that for any e ¼ ðe1 ; e2 Þ 2 int K with k1 e1 þ k2 e2 ¼ 1, ððu3 v3 ; u3 Þ 3ðk1 þ k2 Þu3 ðe1 ; e2 ÞÞ ððx3 y 3 ; x3 Þ þ 3k1 y 3 ðe1 ; e2 ÞÞ 62 int K: Let ððu3 v3 ; u3 Þ 3ðk1 þ k2 Þu3 ðe1 ; e2 ÞÞ ððx3 y 3 ; x3 Þ þ 3k1 y 3 ðe1 ; e2 ÞÞ 2 int K: Then for k ¼ ðk1 ; k2 Þ, we have k1 ðu3 v3 x3 þ y 3 3ðk1 þ k2 Þu3 e1 3k1 y 3 e1 Þ þ k2 ðu3 x3 3ðk1 þ k2 Þu3 e2 3k1 y 3 e2 Þ > 0; which gives that ðk1 þ k2 Þðu3 x3 Þ þ k1 ðy 3 v3 Þ 3ðk1 þ k2 Þu3 3k1 y 3 > 0 or ðk1 þ k2 Þðx3 u3 Þ þ k1 ðv3 y 3 Þ þ 3ðk1 þ k2 Þu3 þ 3k1 y 3 < 0; which is a contradiction to (3). In order to prove the strong duality theorem, we now obtain necessary optimality conditions for a point to be a weak minimum of (VP). Lemma 1. If x is a weak minimum of (VP), then 9a 2 K and b 2 Q not both zero such that T
T
ða T rf ðx Þ þ b T rgðx Þ Þðx x Þ P 0
8x 2 C
and b T gðx Þ ¼ 0: Proof. We shall show that the system F ðxÞ 2 intðK QÞ has no solution x 2 C; where T
T
F ðxÞ ¼ ðrf ðx Þ ðx x Þ; rgðx Þ ðx x Þ þ gðx ÞÞ: Let, if possible, there be a solution x 2 C. Then rf ðx ÞT ðx x Þ 2 int K
and
ðrgðx ÞT ðx x Þ þ gðx ÞÞ 2 int Q:
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Now, for 0 < a < 1, we have gðx þ aðx x ÞÞ ¼ gðx Þ þ rgðx ÞT aðx x Þ þ oðaÞ T
¼ aðgðx Þ þ rgðx Þ ðx x ÞÞ þ ð1 aÞgðx Þ þ oðaÞ 2 Q; where lim oðaÞ ¼ 0:
a!0þ
Also for 0 < a < 1, f ðx þ aðx x ÞÞ ¼ f ðx Þ þ rf ðx ÞT aðx x Þ þ oðaÞ ) f ðx þ aðx x ÞÞ f ðx Þ T
¼ rf ðx Þ aðx x Þ þ oðaÞ 2 int K; where lim oðaÞ ¼ 0:
a!0þ
This contradicts the fact that f assumes its weak minimum at x . Thus by the Alternative Theorem, 9a 2 K and b 2 Q not both zero such that T
T
a T rf ðx Þ ðx x Þ þ b T ðrgðx Þ ðx x Þ þ gðx ÞÞ P 0
for all x 2 C:
For x ¼ x , the above relation gives b T gðx Þ P 0: Since gðx Þ 2 Q and b 2 Q , we get b T gðx Þ 6 0: Thus b T gðx Þ ¼ 0: Therefore, we obtain T
T
T
ða rf ðx Þ þ b T rgðx Þ Þðx x Þ P 0 and b T gðx Þ ¼ 0:
T
Theorem 2 (Strong duality). Let ðx; k; y Þ be a weak minimum for (SP). Fix k ¼ k in (SD). If ryy ðk f Þðx; yÞ is negative definite and the set fry fi ðx; yÞ : i ¼ 1; 2; . . . ; pg is linearly independent, then ðx; k; yÞ is feasible for (SD), and the objective values of (SP) and (SD) are equal. Furthermore, under the assumptions of Theorem 1, ðx; k; yÞ is a weak maximum of (SD). Proof. Since ðx; k; yÞ is a weak minimum for (SP), from Lemma 1, there exist r0 2 K ; r 2 ðC2 Þ ¼ C2 ; ðr0 ; rÞ 6¼ 0 such that for each ðx; yÞ 2 C1 C2 and k 2 K , T
T
T
½r0T rx f ðx; yÞ þ ðr ðr0T eÞyÞ ryx ðk f Þðx; yÞðx xÞ T
þ ½ðr0 ðr0T eÞkÞT ry f ðx; yÞT þ ðr ðr0T eÞyÞT ryy ðk f Þðx; yÞðy yÞ þ ½ðr ðr0T eÞyÞT ry f ðx; yÞðk kÞ P 0
ð4Þ
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477
and T
rT ry ðk f Þðx; yÞ ¼ 0:
ð5Þ
We claim that r0 6¼ 0. Letting x ¼ x 2 C1 and k ¼ k 2 K in inequality (4), we get for each y 2 C2 , T
T
T
T
½ðr0 ðr0T eÞkÞ ry f ðx; yÞ þ ðr ðr0T eÞyÞ ryy ðk f Þðx; yÞðy yÞ P 0:
ð6Þ
If r0 ¼ 0 2 K , putting y ¼ r þ y 2 C2 , we get T
rT ryy ðk f Þðx; yÞr P 0:
ð7Þ
T
Since the matrix ryy ðk f Þ ðx; yÞ is negative definite, we get r ¼ 0. This is not possible since ðr0 ; rÞ 6¼ 0. Thus r0 6¼ 0. Now substituting x ¼ x and y ¼ y in inequality (4), we get T
½ðr ðr0T eÞyÞ ry f ðx; yÞðk kÞ P 0
for each k 2 K :
ð8Þ
This implies that T
ðr ðr0T eÞyÞ ry f ðx; yÞ ¼ 0; which gives that T
ry f ðx; yÞ ðr ðr0T eÞyÞ ¼ 0:
ð9Þ
Replacing y by r=ðr0T eÞ 2 C2 in inequality (6) and using Eq. (9), we obtain T
T
ðr ðr0T eÞyÞ ryy ðk f Þðx; yÞðr ðr0T eÞyÞ P 0: T
As the matrix ryy ðk f Þðx; yÞ is negative definite, we get r ¼ ðr0T eÞy:
ð10Þ
Using the result in inequality (6), we get T
T
ðr0 ðr0T eÞkÞ ry f ðx; yÞ ¼ 0:
ð11Þ
Now as fry fi ðx; yÞ; i ¼ 1; 2; . . . :; pg is linearly independent, we have r0 ¼ ðr0T eÞk:
ð12Þ
Using Eqs. (10) and (12), we obtain from inequality (4) T
rx ðk f Þðx; yÞT ðx xÞ P 0
for each x 2 C1 :
ð13Þ
Since for each x 2 C1 , x 2 C1 , x þ x 2 C1 as C1 is a closed convex cone, inequality (13) gives T
T
xT rx ðk f Þðx; yÞ P 0 ) rx ðk f Þðx; yÞ 2 C1 : Thus ðx; k; yÞ is feasible for (SD). Putting x ¼ 0 and x ¼ x in inequalities (13) and (14) respectively, we get T
xT rx ðk f Þðx; yÞ ¼ 0 and substituting Eq. (10) in Eq. (5), we get T
y T ry ðk f Þðx; yÞ ¼ 0:
ð14Þ
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Thus (SP) and (SD) have equal objective values at ðx; k; yÞ. We shall now show that ðx; k; yÞ is a weak maximum for (SD), otherwise there would exist a feasible solution ðu; k; vÞ such that T
T
ðf ðu; vÞ ½uT rx ðk f Þðu; vÞeÞ ðf ðx; yÞ ½xT rx ðk f Þðx; yÞeÞ 2 int K: As T
T
xT rx ðk f Þðx; yÞ ¼ y T ry ðk f Þðx; yÞ; we get T
T
ðf ðu; vÞ ½uT rx ðk f Þðu; vÞeÞ ðf ðx; yÞ ½y T ry ðk f Þðx; yÞeÞ 2 int K; which contradicts the weak duality theorem. T
Theorem 3 (Converse duality). Let ðu; k; vÞ be a weak maximum of (SD). Fix k ¼ k in (SP). If rxx ðk f Þðu; vÞ is positive definite and the set frx fi ðu; vÞ; i ¼ 1; . . . ; pg is linearly independent, then ðu; k; vÞ is feasible for (SP), and the objective values of (SP) and (SD) are equal. Also under the assumptions of Theorem 1, ðu; k; vÞ is a weak minimum for (SP). Proof. It follows on the lines of Theorem 2.
4. Self-duality An optimization problem is said to be self-dual if, when the dual is written in the form of the primal, the new problem so obtained is the same as the primal. We now assume that m ¼ n, f ðx; yÞ ¼ f ðy; xÞ, that is, f is skew-symmetric and C1 ¼ C2 . Rewriting the dual problem (SD) as a minimization problem: ðSD0 Þ
K minimize subject to
f ðu; vÞ þ ½uT rx ðkT f Þðu; vÞe ðu; vÞ 2 C1 C2 ; rx ðkT f Þðu; vÞ 2 C1 ; k 2 K ;
e 2 int K;
kT e ¼ 1:
Since rx f ðu; vÞ ¼ ry f ðv; uÞ, the problem ðSD0 Þ reduces to K minimize
f ðv; uÞ ½uT ry ðkT f Þðv; uÞe
subject to
ry ðkT f Þðv; uÞ 2 C1 ; k 2 K ;
e 2 int K;
kT e ¼ 1:
This shows that ðSD0 Þ is formally identical to (SP), that is, the objective and constraint functions are identical. Therefore, this problem is self-dual. Consequently, the feasibility of ðx; k; yÞ for (SP) implies the feasibility of ðy; k; xÞ for (SD) and conversely. Theorem 4 (Self-duality). Assume that (SP) is self-dual and that the conditions of Theorem 1 are satisfied. If T ðx; k; yÞ is a weak minimum for (SP), and ryy ðk f Þðx; yÞ is negative definite and the set fry fi ðx; yÞ : i ¼ 1; 2; . . . ; pg is linearly independent, then ðy; k; xÞ is a weak minimum and weak maximum, respectively, for both (SP) and (SD), and the common optimal value is 0.
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479
Proof. By Theorem 2, ðx; k; yÞ is a weak maximum for (SD), and the optimal values of (SP) and (SD) are equal to f ðx; yÞ. Using self-duality ðy; k; xÞ is feasible for both (SP) and (SD), and using Theorems 1 and 2, we get that it is optimal for both the problems. Now as f is skew-symmetric, we have f ðx; yÞ ¼ f ðy; xÞ: Hence f ðx; yÞ ¼ f ðy; xÞ ¼ f ðx; yÞ; and so f ðx; yÞ ¼ f ðy; xÞ ¼ 0:
5. Conclusion A pair of symmetric dual programs have been formulated by considering the optimization with respect to an arbitrary cone under the assumption of cone-convexity. The results may be further generalized by relaxing the condition of cone-convexity to cone-pseudoconvexity.
References [1] M.S. Bazaraa, J.J. Goode, On symmetric duality in nonlinear programming, Operations Research 21 (1973) 1–9. [2] S. Chandra, B.D. Craven, B. Mond, Symmetric dual fractional programming, Zeitschrift fur Operations Research 24 (1984) 59– 64. [3] S. Chandra, I. Husain, Symmetric dual non-differentiable programs, Bulletin of the Australian Mathematical Society 24 (1981) 295–307. [4] G.B. Dantzig, E. Eisenberg, R.W. Cottle, Symmetric dual nonlinear programming, Pacific Journal of Mathematics 15 (1965) 809– 812. [5] G. Devi, Symmetric duality for nonlinear programming problem involving g-bonvex functions, European Journal of Operational Research 104 (1998) 615–621. [6] W.S. Dorn, A symmetric dual theorem for quadratic programs, Journal of the Operations Research Society of Japan 2 (1960) 93– 97. [7] D.S. Kim, Y.B. Yun, W.J. Lee, Multiobjective symmetric duality with cone constraints, European Journal of Operational Research 107 (1998) 686–691. [8] S.K. Mishra, Multiobjective second order symmetric duality with cone constraints, European Journal of Operational Research 126 (2000) 675–682. [9] S.K. Mishra, Second order symmetric duality in mathematical programming with F -convexity, European Journal of Operational Research 127 (2000) 507–518. [10] B. Mond, T. Weir, Symmetric duality for nonlinear multiobjective programming, in: S. Kumar (Ed.), Recent Development in Mathematical Programming, Gordan and Breach, London, 1991, pp. 137–153. [11] S. Nanda, Invex generalizations of some duality results, Opsearch 25 (2) (1998) 105–111. [12] S. Nanda, L.N. Das, Pseudo-invexity and symmetric duality in nonlinear fractional programming, European Journal of Operational Research 73 (1994) 577–582.
Continuous Optimization
Multiobjective symmetric duality involving cones S.K. Suneja a, Sunila Aggarwal a, Sonia Davar b
b,*
a Department of Mathematics, Miranda House, University of Delhi, Delhi 110007, India Department of Mathematics, St. Stephen’s College, University of Delhi, Delhi 110007, India
Received 12 October 2000; accepted 20 June 2001
Abstract In this paper, a pair of multiobjective symmetric dual programs over arbitrary cones are formulated for cone-convex functions. Weak, strong, converse and self-duality theorems are proved for these programs. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Multiobjective symmetric duality; Cone-convex functions; Weak minimum; Multiobjective optimization
1. Introduction Symmetric duality in nonlinear programming in which the dual of the dual is the primal was first introduced by Dorn [6]. The notion of symmetric duality was developed significantly by Dantzig et al. [4], Chandra and Husain [3] and Mond and Weir [10]. Dantzig et al. [4] formulated a pair of symmetric dual programs and established duality results for convex/concave functions by taking non-negative orthant as the cone. The same result was generalized by Bazaraa and Goode [1] to arbitrary cones. Mond and Weir [10] presented two pairs of symmetric dual multiobjective programming problems for efficient solutions and obtained symmetric duality results concerning pseudoconvex/pseudoconcave functions and Chandra et al. [2] studied symmetric dual fractional programming problems assuming the function involved to be pseudoconvex/pseudoconcave. Nanda [11] studied symmetric dual problems assuming the functions to be invex with non-negative orthant as the cone. Nanda and Das [12] also studied the symmetric dual fractional programming problem for arbitrary cones assuming the functions to be pseudo-invex. Recently, Kim et al. [7] studied a pair of multiobjective symmetric dual programs for pseudo-invex functions and arbitrary cones. Devi [5] formulated a pair of second-order symmetric dual programs and obtained duality results involving g-bonvex (second-order invex) functions. Mishra [8] formulated a pair of multiobjective second-order symmetric dual nonlinear programming problems under second-order pseudoinvexity assumptions on the functions involved over arbitrary cones and established duality results. Mishra
*
Corresponding author. Tel.: +91-11-643-4969. E-mail address: soniadavar@rediffmail.com (S. Davar).
0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 2 5 8 - 2
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[9] also studied second order symmetric duality under second-order F -convexity, F -concavity, F -pseudoconvexity, F -pseudoconcavity for second-order Wolfe and Mond–Weir type models, respectively. In this paper, we formulate a pair of symmetric dual programs over arbitrary cones and establish weak, strong, converse and self-duality theorems by using cone-convexity. In all the above-mentioned papers including the one by Kim et al. [7], the objective function is optimized with respect to the non-negative orthant. However, in this paper, the objective function is optimized with respect to an arbitrary closed convex cone by assuming the function involved to be cone-convex.
2. Notations and preliminaries Definition 1. Let f : X ! Y , where X is a convex subset of a topological vector space and Y is an ordered locally convex topological vector space. Let K be a closed convex cone in Y with non-empty interior. Then f is called K-convex on X if, for any x1 ; x2 2 X and h 2 ð0; 1Þ, hf ðx1 Þ þ ð1 hÞf ðx2 Þ f ðhx1 þ ð1 hÞx2 Þ 2 K: We consider the following vector minimization problem: ðVPÞ K minimize f ðxÞ subject to gðxÞ 2 Q;
x 2 C;
where C Rn , K and Q are closed convex cones with non-empty interiors in Rp and Rm , respectively. f : Rn ! Rp and g : Rn ! Rm . Let X 0 ¼ fx 2 C : gðxÞ 2 Qg. Definition 2. A point x 2 X 0 is called a weak minimum of (VP) if ðfor all x 2 X 0 Þ
f ðxÞ f ðxÞ 62 int K:
Definition 3. The positive polar cone K of K is defined by K ¼ fz 2 Rp : xT z P 0 for all x 2 Kg: We formulate the following multiobjective symmetric dual problems: ðSPÞ
K minimize subject to
f ðx; yÞ ½y T ry ðkT f Þðx; yÞe ðx; yÞ 2 C1 C2 ; ry ðkT f Þðx; yÞ 2 C2 ; k 2 K ;
e 2 int K;
kT e ¼ 1
and ðSDÞ K maximize subject to
f ðu; vÞ ½uT rx ðkT f Þðu; vÞe ðu; vÞ 2 C1 C2 ; rx ðkT f Þðu; vÞ 2 C1 ; k 2 K ;
e 2 int K;
kT e ¼ 1;
where f : Rn Rm ! Rp is a twice differentiable function, C1 and C2 are closed convex cones with nonempty interiors in Rn and Rm , respectively. C1 and C2 are positive polar cones of C1 and C2 , respectively, K is a closed convex cone in Rp such that int K 6¼ /.
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rx ðkT f Þðx; yÞ and ry ðkT f Þðx; yÞ are gradients of ðkT f Þðx; yÞ with respect to x and y, respectively. rxx ðkT f Þðx; yÞ and ryy ðkT f Þðx; yÞ are the Hessian matrices of ðkT f Þðx; yÞ with respect to x and y, respectively.
3. The duality results We now establish the symmetric duality results for (SP) and (SD). Theorem 1 (Weak duality). Let ðx; k; yÞ and ðu; k; vÞ be feasible solutions of (SP) and (SD), respectively. Assume that f ð; yÞ is K-convex with respect to x for fixed y and f ðx; Þ is K-convex with respect to y for fixed x. Then ðf ðu; vÞ ½uT rx ðkT f Þðu; vÞeÞ ðf ðx; yÞ ½y T ry ðkT f Þðx; yÞeÞ 62 int K: Proof. Let, if possible, ðf ðu; vÞ ½uT rx ðkT f Þðu; vÞeÞ ðf ðx; yÞ ½y T ry ðkT f Þðx; yÞeÞ 2 int K: Then k 2 K implies kT ððf ðx; yÞ ½y T ry ðkT f Þðx; yÞeÞ ðf ðu; vÞ ½uT rx ðkT f Þðu; vÞeÞÞ < 0; which gives that kT f ðu; vÞ kT f ðx; yÞ kT ð½uT rx ðkT f Þðu; vÞeÞ þ kT ð½y T ry ðkT f Þðx; yÞeÞ > 0 ) kT f ðu; vÞ kT f ðx; yÞ uT rx ðkT f Þðu; vÞ þ y T ry ðkT f Þðx; yÞ > 0
as kT e ¼ 1:
ð1Þ
Since f ð; yÞ is K-convex with respect to x for fixed y ¼ v, it follows that hf ðx; vÞ þ ð1 hÞf ðu; vÞ f ðhx þ ð1 hÞu; vÞ 2 K; This gives that
f ðx; vÞ f ðu; vÞ
f ðhx þ ð1 hÞu; vÞ f ðu; vÞ h
h 2 ð0; 1Þ:
2 K;
which implies that f ðx; vÞ f ðu; vÞ rx f ðu; vÞT ðx uÞ 2 K ) kT f ðx; vÞ kT f ðu; vÞ P kT rx f ðu; vÞT ðx uÞ
as k 2 K :
ð2Þ
Similarly, since f ðx; Þ is K-convex with respect to y for each fixed x, we get T
kT f ðx; vÞ þ kT f ðx; yÞ P kT ry f ðx; yÞ ðv yÞ; which when added to (2) gives T
T
T
T
kT f ðx; yÞ kT f ðu; vÞ P kT rx f ðu; vÞ x kT rx f ðu; vÞ u kT ry f ðx; yÞ v þ kT ry f ðx; yÞ y; which implies that kT f ðx; yÞ kT f ðu; vÞ xT rx ðkT f Þðu; vÞ þ uT rx ðkT f Þðu; vÞ þ vT ry ðkT f Þðx; yÞ y T ry ðkT f Þðx; yÞ P 0: We also have by feasibility of ðx; k; yÞ and ðu; k; vÞ for (SP) and (SD), respectively, vT ry ðkT f Þðx; yÞ 6 0
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and xT rx ðkT f Þðu; vÞ P 0: Thus we have kT f ðx; yÞ kT f ðu; vÞ þ uT rx ðkT f Þðu; vÞ y T ry ðkT f Þðx; yÞ P 0; which contradicts (1).
The above weak duality theorem is illustrated with the help of the following example: Example 1. Let C1 ¼ Rþ , C2 ¼ Rþ , K ¼ fðx; yÞ : x 6 y 6 x; x P 0g and define a function f : R R ! R2 as f ðx; yÞ ¼ ðf1 ðx; yÞ; f2 ðx; yÞÞ, where f1 ðx; yÞ ¼ x3 y 3 ;
f2 ðx; yÞ ¼ x3 :
Then, clearly f ð; yÞ is K-convex with respect to x for fixed y and f ðx; Þ is K-convex with respect to y for fixed x. Also, problems (SP) and (SD) can be written as ðSPÞ
K minimize subject to
ðx3 y 3 ; x3 Þ þ 3k1 y 3 ðe1 ; e2 Þ x P 0; y P 0; 3k1 y 2 P 0; k ¼ ðk1 ; k2 Þ 2 K ¼ K; k1 e 1 þ k2 e 2 ¼ 1
e ¼ ðe1 ; e2 Þ 2 int K;
and ðSDÞ K maximize subject to
ðu3 v3 ; u3 Þ 3ðk1 þ k2 Þu3 ðe1 ; e2 Þ u P 0; v P 0; 3ðk1 þ k2 Þu2 P 0; k ¼ ðk1 ; k2 Þ 2 K ¼ K; k1 e1 þ k2 e2 ¼ 1:
e ¼ ðe1 ; e2 Þ 2 int K;
Let ðx; k; yÞ and ðu; k; vÞ be feasible solutions to (SP) and (SD), respectively. Now by K-convexity of the function f ð; yÞ with respect to x, we have T
f ðx; vÞ f ðu; vÞ rx f ðu; vÞ ðx uÞ 2 K; which gives that ðx3 u3 3u2 ðx uÞ; x3 u3 3u2 ðx uÞÞ 2 K: By the K-convexity of f ðx; Þ with respect to y, we get ðv3 y 3 3y 2 ðv yÞ; 0Þ 2 K: Hence for k ¼ ðk1 ; k2 Þ 2 K ¼ K, we get ðk1 þ k2 Þðx3 u3 3u2 ðx uÞÞ P 0 and k1 ðv3 y 3 3y 2 ðv yÞÞ P 0;
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that is, ðk1 þ k2 Þðx3 þ 2u3 3u2 xÞ P 0 and k1 ðv3 þ 2y 3 3y 2 vÞ P 0; which on adding yield ðk1 þ k2 Þðx3 þ 2u3 Þ þ k1 ðv3 þ 2y 3 Þ P ðk1 þ k2 Þð3u2 xÞ þ k1 3y 2 v P 0; where the last inequality follows on account of feasibility conditions of the problems (SP) and (SD). Thus, we have ðk1 þ k2 Þðx3 u3 Þ þ ðk1 þ k2 Þð3u3 Þ þ k1 ðv3 y 3 Þ þ k1 ð3y 3 Þ P 0:
ð3Þ
We are to show that for any e ¼ ðe1 ; e2 Þ 2 int K with k1 e1 þ k2 e2 ¼ 1, ððu3 v3 ; u3 Þ 3ðk1 þ k2 Þu3 ðe1 ; e2 ÞÞ ððx3 y 3 ; x3 Þ þ 3k1 y 3 ðe1 ; e2 ÞÞ 62 int K: Let ððu3 v3 ; u3 Þ 3ðk1 þ k2 Þu3 ðe1 ; e2 ÞÞ ððx3 y 3 ; x3 Þ þ 3k1 y 3 ðe1 ; e2 ÞÞ 2 int K: Then for k ¼ ðk1 ; k2 Þ, we have k1 ðu3 v3 x3 þ y 3 3ðk1 þ k2 Þu3 e1 3k1 y 3 e1 Þ þ k2 ðu3 x3 3ðk1 þ k2 Þu3 e2 3k1 y 3 e2 Þ > 0; which gives that ðk1 þ k2 Þðu3 x3 Þ þ k1 ðy 3 v3 Þ 3ðk1 þ k2 Þu3 3k1 y 3 > 0 or ðk1 þ k2 Þðx3 u3 Þ þ k1 ðv3 y 3 Þ þ 3ðk1 þ k2 Þu3 þ 3k1 y 3 < 0; which is a contradiction to (3). In order to prove the strong duality theorem, we now obtain necessary optimality conditions for a point to be a weak minimum of (VP). Lemma 1. If x is a weak minimum of (VP), then 9a 2 K and b 2 Q not both zero such that T
T
ða T rf ðx Þ þ b T rgðx Þ Þðx x Þ P 0
8x 2 C
and b T gðx Þ ¼ 0: Proof. We shall show that the system F ðxÞ 2 intðK QÞ has no solution x 2 C; where T
T
F ðxÞ ¼ ðrf ðx Þ ðx x Þ; rgðx Þ ðx x Þ þ gðx ÞÞ: Let, if possible, there be a solution x 2 C. Then rf ðx ÞT ðx x Þ 2 int K
and
ðrgðx ÞT ðx x Þ þ gðx ÞÞ 2 int Q:
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Now, for 0 < a < 1, we have gðx þ aðx x ÞÞ ¼ gðx Þ þ rgðx ÞT aðx x Þ þ oðaÞ T
¼ aðgðx Þ þ rgðx Þ ðx x ÞÞ þ ð1 aÞgðx Þ þ oðaÞ 2 Q; where lim oðaÞ ¼ 0:
a!0þ
Also for 0 < a < 1, f ðx þ aðx x ÞÞ ¼ f ðx Þ þ rf ðx ÞT aðx x Þ þ oðaÞ ) f ðx þ aðx x ÞÞ f ðx Þ T
¼ rf ðx Þ aðx x Þ þ oðaÞ 2 int K; where lim oðaÞ ¼ 0:
a!0þ
This contradicts the fact that f assumes its weak minimum at x . Thus by the Alternative Theorem, 9a 2 K and b 2 Q not both zero such that T
T
a T rf ðx Þ ðx x Þ þ b T ðrgðx Þ ðx x Þ þ gðx ÞÞ P 0
for all x 2 C:
For x ¼ x , the above relation gives b T gðx Þ P 0: Since gðx Þ 2 Q and b 2 Q , we get b T gðx Þ 6 0: Thus b T gðx Þ ¼ 0: Therefore, we obtain T
T
T
ða rf ðx Þ þ b T rgðx Þ Þðx x Þ P 0 and b T gðx Þ ¼ 0:
T
Theorem 2 (Strong duality). Let ðx; k; y Þ be a weak minimum for (SP). Fix k ¼ k in (SD). If ryy ðk f Þðx; yÞ is negative definite and the set fry fi ðx; yÞ : i ¼ 1; 2; . . . ; pg is linearly independent, then ðx; k; yÞ is feasible for (SD), and the objective values of (SP) and (SD) are equal. Furthermore, under the assumptions of Theorem 1, ðx; k; yÞ is a weak maximum of (SD). Proof. Since ðx; k; yÞ is a weak minimum for (SP), from Lemma 1, there exist r0 2 K ; r 2 ðC2 Þ ¼ C2 ; ðr0 ; rÞ 6¼ 0 such that for each ðx; yÞ 2 C1 C2 and k 2 K , T
T
T
½r0T rx f ðx; yÞ þ ðr ðr0T eÞyÞ ryx ðk f Þðx; yÞðx xÞ T
þ ½ðr0 ðr0T eÞkÞT ry f ðx; yÞT þ ðr ðr0T eÞyÞT ryy ðk f Þðx; yÞðy yÞ þ ½ðr ðr0T eÞyÞT ry f ðx; yÞðk kÞ P 0
ð4Þ
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477
and T
rT ry ðk f Þðx; yÞ ¼ 0:
ð5Þ
We claim that r0 6¼ 0. Letting x ¼ x 2 C1 and k ¼ k 2 K in inequality (4), we get for each y 2 C2 , T
T
T
T
½ðr0 ðr0T eÞkÞ ry f ðx; yÞ þ ðr ðr0T eÞyÞ ryy ðk f Þðx; yÞðy yÞ P 0:
ð6Þ
If r0 ¼ 0 2 K , putting y ¼ r þ y 2 C2 , we get T
rT ryy ðk f Þðx; yÞr P 0:
ð7Þ
T
Since the matrix ryy ðk f Þ ðx; yÞ is negative definite, we get r ¼ 0. This is not possible since ðr0 ; rÞ 6¼ 0. Thus r0 6¼ 0. Now substituting x ¼ x and y ¼ y in inequality (4), we get T
½ðr ðr0T eÞyÞ ry f ðx; yÞðk kÞ P 0
for each k 2 K :
ð8Þ
This implies that T
ðr ðr0T eÞyÞ ry f ðx; yÞ ¼ 0; which gives that T
ry f ðx; yÞ ðr ðr0T eÞyÞ ¼ 0:
ð9Þ
Replacing y by r=ðr0T eÞ 2 C2 in inequality (6) and using Eq. (9), we obtain T
T
ðr ðr0T eÞyÞ ryy ðk f Þðx; yÞðr ðr0T eÞyÞ P 0: T
As the matrix ryy ðk f Þðx; yÞ is negative definite, we get r ¼ ðr0T eÞy:
ð10Þ
Using the result in inequality (6), we get T
T
ðr0 ðr0T eÞkÞ ry f ðx; yÞ ¼ 0:
ð11Þ
Now as fry fi ðx; yÞ; i ¼ 1; 2; . . . :; pg is linearly independent, we have r0 ¼ ðr0T eÞk:
ð12Þ
Using Eqs. (10) and (12), we obtain from inequality (4) T
rx ðk f Þðx; yÞT ðx xÞ P 0
for each x 2 C1 :
ð13Þ
Since for each x 2 C1 , x 2 C1 , x þ x 2 C1 as C1 is a closed convex cone, inequality (13) gives T
T
xT rx ðk f Þðx; yÞ P 0 ) rx ðk f Þðx; yÞ 2 C1 : Thus ðx; k; yÞ is feasible for (SD). Putting x ¼ 0 and x ¼ x in inequalities (13) and (14) respectively, we get T
xT rx ðk f Þðx; yÞ ¼ 0 and substituting Eq. (10) in Eq. (5), we get T
y T ry ðk f Þðx; yÞ ¼ 0:
ð14Þ
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Thus (SP) and (SD) have equal objective values at ðx; k; yÞ. We shall now show that ðx; k; yÞ is a weak maximum for (SD), otherwise there would exist a feasible solution ðu; k; vÞ such that T
T
ðf ðu; vÞ ½uT rx ðk f Þðu; vÞeÞ ðf ðx; yÞ ½xT rx ðk f Þðx; yÞeÞ 2 int K: As T
T
xT rx ðk f Þðx; yÞ ¼ y T ry ðk f Þðx; yÞ; we get T
T
ðf ðu; vÞ ½uT rx ðk f Þðu; vÞeÞ ðf ðx; yÞ ½y T ry ðk f Þðx; yÞeÞ 2 int K; which contradicts the weak duality theorem. T
Theorem 3 (Converse duality). Let ðu; k; vÞ be a weak maximum of (SD). Fix k ¼ k in (SP). If rxx ðk f Þðu; vÞ is positive definite and the set frx fi ðu; vÞ; i ¼ 1; . . . ; pg is linearly independent, then ðu; k; vÞ is feasible for (SP), and the objective values of (SP) and (SD) are equal. Also under the assumptions of Theorem 1, ðu; k; vÞ is a weak minimum for (SP). Proof. It follows on the lines of Theorem 2.
4. Self-duality An optimization problem is said to be self-dual if, when the dual is written in the form of the primal, the new problem so obtained is the same as the primal. We now assume that m ¼ n, f ðx; yÞ ¼ f ðy; xÞ, that is, f is skew-symmetric and C1 ¼ C2 . Rewriting the dual problem (SD) as a minimization problem: ðSD0 Þ
K minimize subject to
f ðu; vÞ þ ½uT rx ðkT f Þðu; vÞe ðu; vÞ 2 C1 C2 ; rx ðkT f Þðu; vÞ 2 C1 ; k 2 K ;
e 2 int K;
kT e ¼ 1:
Since rx f ðu; vÞ ¼ ry f ðv; uÞ, the problem ðSD0 Þ reduces to K minimize
f ðv; uÞ ½uT ry ðkT f Þðv; uÞe
subject to
ry ðkT f Þðv; uÞ 2 C1 ; k 2 K ;
e 2 int K;
kT e ¼ 1:
This shows that ðSD0 Þ is formally identical to (SP), that is, the objective and constraint functions are identical. Therefore, this problem is self-dual. Consequently, the feasibility of ðx; k; yÞ for (SP) implies the feasibility of ðy; k; xÞ for (SD) and conversely. Theorem 4 (Self-duality). Assume that (SP) is self-dual and that the conditions of Theorem 1 are satisfied. If T ðx; k; yÞ is a weak minimum for (SP), and ryy ðk f Þðx; yÞ is negative definite and the set fry fi ðx; yÞ : i ¼ 1; 2; . . . ; pg is linearly independent, then ðy; k; xÞ is a weak minimum and weak maximum, respectively, for both (SP) and (SD), and the common optimal value is 0.
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Proof. By Theorem 2, ðx; k; yÞ is a weak maximum for (SD), and the optimal values of (SP) and (SD) are equal to f ðx; yÞ. Using self-duality ðy; k; xÞ is feasible for both (SP) and (SD), and using Theorems 1 and 2, we get that it is optimal for both the problems. Now as f is skew-symmetric, we have f ðx; yÞ ¼ f ðy; xÞ: Hence f ðx; yÞ ¼ f ðy; xÞ ¼ f ðx; yÞ; and so f ðx; yÞ ¼ f ðy; xÞ ¼ 0:
5. Conclusion A pair of symmetric dual programs have been formulated by considering the optimization with respect to an arbitrary cone under the assumption of cone-convexity. The results may be further generalized by relaxing the condition of cone-convexity to cone-pseudoconvexity.
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