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Non-differentiable symmetric duality for multiobjective programming with cone constraints
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European Journal of Operational Research 188 (2008) 652–661 www.elsevier.com/locate/ejor
Continuous Optimization
Non-differentiable symmetric duality for multiobjective programming with cone constraints Moon Hee Kim a, Do Sang Kim
q
b,*
a
b
Department of Multimedia Engineering, Tongmyong University, Pusan 608-711, Republic of Korea Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Republic of Korea Received 16 February 2006; accepted 7 May 2007 Available online 13 May 2007
Abstract Two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597] to the non-differentiable multiobjective symmetric dual problem. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Non-differentiable programming; Duality theorems; Cone constraints; Weakly efficient solutions
1. Introduction and preliminaries A nonlinear programming problem and its dual are said to be symmetric if the dual of the dual is the original problem. Symmetric duality in nonlinear programming in which the dual of the dual is the primal was first introduced by Dorn [2]. Danzig et al. [3] formulated a pair of symmetric dual nonlinear programs and established duality results for convex and concave functions with non-negative orthant as the cone. The same result was generalized by Bazaraa and Goode [1] to arbitrary cones. Mond and Weir [7] presented two pair of symmetric dual multiobjective programming problems for efficient solutions and obtained symmetric duality results concerning pseudoconvex and pseudoconcave functions. Nanda and Das [8] also studied the symmetric dual nonlinear programming problem for arbitrary cones assuming the functions to be pseudo-invex. Kim et al. [4] studied a pair of multiobjective symmetric dual programs for pseudo-invex functions and arbitrary cones. Suneja et al. [10] formulated a pair of symmetric dual multiobjective programs of Wolfe type over q *
This work was supported by the Brain Korea 21 Project in 2006. Corresponding author. Tel.: +82 51 620 6315; fax: +82 51 611 6356. E-mail address: [email protected] (D.S. Kim).
0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.05.005
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
653
arbitrary cones in which the objective function is optimized with respect to an arbitrary closed convex cone by assuming the function involved to be cone-convex. Very recently, Khurana [5] formulated a pair of differentiable multiobjective symmetric dual programs of Mond–Weir type over arbitrary cones in which the objective function is optimized with respect to an arbitrary closed convex cone by assuming the involved functions to be cone-pseudoinvex and strongly cone-pseudoinvex. In this paper, two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [5] to the non-differentiable multiobjective symmetric dual problem. First we consider the following multiobjective programming problem. ðP Þ K Minimize subject to
f ðxÞ; gðxÞ 2 Q;
x 2 C;
where f : Rn ! Rp ; g : Rn ! Rm and C Rn , K and Q are closed convex cone with nonempty interior in Rp and Rm , respectively. Let X 0 ¼ fx 2 C : gðxÞ 2 Qg be the set of all feasible solutions of (P). Definition 1.1. A point x 2 X 0 is a weakly efficient solution of (P) if there exist no other x 2 X 0 such that f ðxÞ f ðxÞ 2 int K: Definition 1.2. The positive polar cone K* of K is defined by K ¼ fz 2 Rp : xT z = 0 for all x 2 Kg: Definition 1.3 [10]. Let f : Rn ! Rp be a function. Then f is K-preinvex with respect to the g if there exists a function g : C C ! Rn such that for any x; y 2 Rn and a 2 ½0; 1, af ðxÞ þ ð1 aÞf ðyÞ f ðy þ agðx; yÞÞ 2 K: Remark 1.1. If f : Rn ! Rp is differentiable and K-preinvex with respect to g T T f ðyÞ rf ðyÞ gðx; yÞ 2 K. Moreover, 8k 2 K ; ðkT f ÞðxÞ ðkT f ÞðyÞ rðkT f ÞðyÞ gðx; yÞ = 0:
then
f ðxÞ
Definition 1.4 [5]. Let f : Rn ! Rp be a differentiable function. Then f is K-pseudoinvex with respect to g at u 2 Rn if for all x 2 Rn T
rf ðuÞ gðx; uÞ 62 int K ) ðf ðxÞ f ðuÞÞ 62 int K; where rf ðuÞ ¼ ðrf1 ðuÞ; rf2 ðuÞ; . . . ; rfp ðuÞÞT . Definition 1.5 [6]. Let C be a compact convex set in Rn . The support function sðx j CÞ of C is defined by sðxjCÞ :¼ maxfxT y : y 2 Cg: The support function sðx j CÞ, being convex and everywhere finite, has a subdifferential at every x in the sense of Rockafellar [9], that is, there exists z such that sðy j CÞ P sðx j CÞ þ zT ðy xÞ for all y 2C. The subdifferential of sðx j CÞ is given by osðxjCÞ :¼ fz 2 C : zT x ¼ sðxjCÞg: For any set S Rn , the normal cone to S at a point x 2 S is defined by N S ðxÞ :¼ fy 2 Rn : y T ðz xÞ 6 0 for all z 2 Sg: It is readily verified that for a compact convex set C, y is in NC(x) if and only if sðyjCÞ ¼ xT y, or equivalently, x is in the subdifferential of s at y.
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2. Wolfe type symmetric duality We now state the following pair of Wolfe type non-differentiable symmetric dual problems and establish weak, strong and converse duality theorems. f1 ðx; yÞ þ sðxjD1 Þ y T z1 y T
ðWPÞ K minimize
p X
ki ðry fi ðx; yÞ zi Þ; . . . ; fp ðx; yÞ
i¼1 T
þsðxjDp Þ y zp y
T
p X
! ki ðry fi ðx; yÞ zi Þ ;
i¼1
subject to
p X
ki ðry fi ðx; yÞ zi Þ 2 C 2 ;
ð1Þ
i¼1
zi 2 E i ;
i ¼ 1; . . . ; p;
ð2Þ
k 2 K ;
kT e ¼ 1;
ð3Þ
x 2 C1;
and ðWDÞ
K maximize
f1 ðu; vÞ sðvjE1 Þ þ uT x1 uT
p X
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ sðvjEp Þ
i¼1 T
þu xp u
T
p X
!
ki ðrx fi ðu; vÞ þ xi Þ ;
i¼1
subject to
p X
ki ðrx fi ðu; vÞ þ xi Þ 2 C 1 ;
ð4Þ
i¼1
xi 2 Di ;
k2K ; n
m
i ¼ 1; . . . ; p; T
k e ¼ 1;
ð5Þ
v 2 C2;
ð6Þ
p
where f : R R ! R be a twice differentiable functions of x and y, Di and Ei (1 5 i 5 p) are compact convex set in Rn and Rm , C1 and C2 be closed convex cones in Rn ; Rm with nonempty interiors, respectively. C 1 and C 2 are positive polar cones of C1 and C2, respectively, K is a closed convex cone in Rp such that int K 6¼ ;. Theorem 2.1 (Weak duality). Let ðx; y; k; zÞ be feasible for (WP) and let ðu; v; k; xÞ be feasible for (WD). Let f ð; yÞ þ ðÞT x be K-preinvex with respect to g1 for fixed y and f ðx; Þ þ ðÞT z be K-preinvex with respect to g2 for fixed x. If g1 ðx; uÞ þ u 2 C 1 and g2 ðv; yÞ þ y 2 C 2 , then ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 y T
p X
ki ðry fi ðx; yÞ zi Þ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp
i¼1
yT
p X
ki ðry fi ðx; yÞ zi ÞÞ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 uT
i¼1
sðvjEp Þ þ uT xp uT
p X
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ
i¼1 p X
ki ðrx fi ðu; vÞ þ xi ÞÞ 62 intK:
i¼1 T
Proof. Let ðx; y; k; zÞ be feasible for (WP) and let ðu; v; k; xÞ be feasible for (WD). Since f ð; yÞ þ ðÞ x is Kpreinvex with respect to g1 for fixed y = v, by Remark 1.1, p X i¼1
ki ðfi ðx; vÞ þ xT xi fi ðu; vÞ uT xi Þ = g1 ðx; uÞ
T
p X i¼1
ki ðrx fi ðu; vÞ þ xi Þ:
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
655
From (4) and g1 ðx; uÞ þ u 2 C 1 , ½g1 ðx; uÞ þ u
T
p X
ki ðrx fi ðu; vÞ þ xi Þ = 0:
i¼1
Hence p X
ki ðfi ðx; vÞ þ xT xi fi ðu; vÞ uT xi Þ = uT
i¼1
p X
ki ðrx fi ðu; vÞ þ xi Þ:
ð7Þ
i¼1 T
Since f ðx; Þ þ ðÞ z is K-preinvex with respect to g2 for fixed x, by Remark 1.1, p X
T
ki ðfi ðx; yÞ y T zi fi ðx; vÞ þ vT zi Þ = g2 ðv; yÞ
i¼1
p X
ki ðry fi ðx; yÞ zi Þ:
i¼1
From (1) and g2 ðv; yÞ þ y 2 C 2 , T
½g2 ðv; yÞ þ y
p X
ki ðry fi ðx; yÞ zi Þ 5 0:
i¼1
Hence p X
ki ðfi ðx; yÞ y T zi fi ðx; vÞ þ vT zi Þ = y T
i¼1
p X
ki ðry fi ðx; yÞ zi Þ:
ð8Þ
i¼1
From (7) and (8), we have p X
ki ðfi ðx; yÞ y T zi þ vT zi fi ðu; vÞ þ xT xi uT xi Þ = uT
i¼1
p X
ki ðrx fi ðu; vÞ þ xi Þ
i¼1
þ yT
p X
ki ðry fi ðx; yÞ zi Þ:
i¼1
Finally using xT xi 5 sðx j Di Þ and vT zi 5 sðv j Ei Þ, we obtain p X
ki ðfi ðx; yÞ y T zi þ sðvjEi Þ fi ðu; vÞ þ sðxjDi Þ uT xi Þ = uT
i¼1
p X
ki ðrx fi ðu; vÞ þ xi Þ
i¼1
þ yT
p X
ki ðry fi ðx; yÞ zi Þ
i¼1
and hence p X
ki ðfi ðx; yÞ þ sðxjDi Þ y T zi y T
i¼1
uT
p X
ki ðry fi ðx; yÞ zi ÞÞ
i¼1 p X
p X
ki ðfi ðu; vÞ sðvjEi Þ þ uT xi
i¼1
ki ðrx fi ðu; vÞ þ xi ÞÞ = 0:
ð9Þ
i¼1
Suppose to the contrary that ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 y T
p X
ki ðry fi ðx; yÞ zi Þ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp
i¼1
yT
p X
ki ðry fi ðx; yÞ zi ÞÞ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 uT
i¼1
sðvjEp Þ þ uT xp uT
p X i¼1
p X i¼1
ki ðrx fi ðu; vÞ þ xi ÞÞ 2 int K:
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ
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M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
Then by the constraint (3), k 2 K* and k 5 0, p X
ki ðfi ðx; yÞ þ sðxjDi Þ y T zi y T
i¼1
p X
ki ðry fi ðx; yÞ zi ÞÞ
p X
i¼1
uT
p X
ki ðfi ðu; vÞ sðvjEi Þ þ uT xi
i¼1
ki ðrx fi ðu; vÞ þ xi ÞÞ < 0;
i¼1
which contradicts (9). Hence the result holds.
h
Theorem 2.2 (Strong duality). Let f : Rn Rm ! Rp be a twice differentiable. Let ðx; y ; k; zÞ be a weakly efficient solution of (WP) and suppose that (I) ryy fi ðx; y Þ is positive definite for all i ¼ 1; . . . ; p. (II) fry fi ðx; y Þ zi ; i ¼ 1; 2; . . . ; pg is linearly independent. (III) K is a closed convex cone with Rpþ K. such that ðx; y ; is feasible for (WD), and the objective values of (WP) and (WD) are Then, there exists x k; xÞ equal. Furthermore, if the hypotheses of Theorem 2.1 are satisfied for all feasible solutions of (WP) and is a weakly efficient solution for (WD). (WD), then ðx; y ; k; xÞ Proof. Since ðx; y ; k; zÞ is a weakly efficient solution of (WP), by modifying the Fritz John optimality condition in [6], we can check that there exist a 2 K , b 2 C 2 , c 2 C 1 , d 2 K such that p X
i þ ai ½rx fi ðx; y Þ þ x
i¼1 p X
p X
ki ryx fi ðx; y Þðb ðaT eÞy Þ c ¼ 0;
ð10Þ
i¼1
ðai ðaT eÞ ki Þ½ry fi ðx; y Þ zi þ
i¼1
p X
ki ryy fi ðx; y Þðb ðaT eÞy Þ ¼ 0;
T
ðb ðaT eÞy Þ ðry fi ðx; y Þ zi Þ di ¼ 0; i ¼ 1; . . . ; p; ai y þ ðb ðaT eÞy Þ ki 2 N E ðzi Þ; i ¼ 1; . . . ; p;
ð12Þ ð13Þ
i
bT
ð11Þ
i¼1
p X
ki ½ry fi ðx; y Þ zi ¼ 0;
ð14Þ
i¼1
cT x ¼ 0; dT k ¼ 0;
ð15Þ ð16Þ Ti x x
i 2 Di ; x
¼ sðxjDi Þ;
i ¼ 1; . . . ; p;
ð17Þ
ða; b; c; dÞ 6¼ 0:
ð18Þ
Multiplying (11) by ðb ðaT eÞy Þ, p X
T ðai ðaT eÞ ki Þ½ry fi ðx; y Þ zi ðb ðaT eÞy Þ þ ðb ðaT eÞy Þ
i¼1
i¼1
Using the result in equality (12) and (16), we get p X
p X
ai di þ ðb ðaT eÞy Þ
i¼1
T
p X
ki ryy fi ðx; y Þðb ðaT eÞy Þ ¼ 0:
i¼1 T
Since a 2 K* and d 2 K; a d = 0 and hence ðb ðaT eÞy Þ
T
p X i¼1
ki ryy fi ðx; y Þðb ðaT eÞy Þ 5 0:
ki ryy fi ðx; y Þðb ðaT eÞy Þ ¼ 0:
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
Since
657
Pp x; y Þ is positive definite, then i¼1 ki ryy fi ð b ¼ ðaT eÞy :
ð19Þ
Pp
T x; y Þ zi ¼ 0: Since fry fi ðx; y Þ zi ; By (12), di ¼ 0; i ¼ 1; . . . ; p. From (11), i¼1 ðai ða eÞki Þ½ry fi ð T i ¼ 1; . . . ; pg is linearly independent, ai ¼ ða eÞki ; i ¼ 1; . . . ; p. If a = 0, then (19) implies that b = 0 and using (10), we get that c = 0, which contradicts (18). Hence ai 6¼ 0; i ¼ 1; . . . ; p, since a 2 K Rpþ , ai > 0. Hence by (19), y 2 C 2 . By (10), (19) and ai ¼ ðaT eÞ ki ; i ¼ 1; . . . ; p, p X i ¼ c 2 C 1 : ðaT eÞ ki ½rx fi ðx; y Þ þ x
ð20Þ
i¼1
P i 2 C 1 . Thus ðx; y ; is feasible for (WD). ki ½rx fi ðx; y Þ þ x k; xÞ Since pi¼1 By multiplying both sides of Eq. (20) by x, hence from (15) we get xT
p X
i ¼ 0: ki ½rx fi ðx; y Þ þ x
i¼1
By (14) and the fact that b ¼ ðaT eÞy , ðaT eÞy T y T
p X
Pp x; y Þ zi ¼ 0: Since ai > 0, i¼1 ki ½ry fi ð
ki ½ry fi ðx; y Þ zi ¼ 0:
i¼1
But from (13), and b ¼ ðaT eÞy , ai y 2 N Ei ðzi Þ: Since ai > 0, y 2 N Ei ðzi Þ, we have y T zi ¼ sðy jEi Þ; i ¼ 1; . . . ; p: Thus (WP) and (WD) have equal objectives values. is a weakly efficient solution of (WD), otherwise there would exist a We shall now show that ðx; y ; k; xÞ feasible solution ðu; v; k; xÞ of (WD) such that 1 xT ðf1 ðx; y Þ sðy jE1 Þ þ xT x
p X
i Þ; . . . ; fp ðx; y Þ ki ðrx fi ðx; y Þ þ x
i¼1
p xT sðy jEp Þ þ xT x
p X
i ÞÞ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ki ðrx fi ðx; y Þ þ x
i¼1
uT
p X
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp uT
i¼1
p X
ki ðrx fi ðu; vÞ þ xi ÞÞ 2 int K:
i¼1
Pp P i ¼ 0 ¼ y T pi¼1 i ¼ sðxjDi Þ; i ¼ 1; . . . ; p, ki ½ry fi ðx; y Þ zi , y T zi ¼ sðy jEi Þ and xT x x; y Þ þ x Since x i¼1 ki ½rx fi ð it follows that T
ðf1 ðx; y Þ þ sðxjD1 Þ y T z1 y T
p X
ki ðry fi ðx; y Þ zi Þ; . . . ; fp ðx; y Þ þ sðxjDp Þ y T zp
i¼1
y T
p X
ki ðry fi ðx; y Þ zi ÞÞ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 uT
i¼1
sðvjEp Þ þ uT xp uT
p X
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ
i¼1 p X
ki ðrx fi ðu; vÞ þ xi ÞÞ 2 int K;
i¼1
is a feasible solution of (WD), ðx; y ; k; xÞ is a weakly efficient which contradicts weak duality. Since ðx; y ; k; xÞ solution of (WD). Hence the result holds. h We now state a converse duality theorem whose proof follows on the lines of Theorem 2.2. be a weakly Theorem 2.3 (Converse duality). Let f : Rn Rm ! Rp be a twice differentiable. Let ðu; v; k; xÞ efficient solution of (WD). If rxx fi ð u; vÞ is negative definite for all i ¼ 1; . . . ; p, frx fi ðu; vÞ þ xi ; i ¼ 1; 2; . . . ; pg is linearly independent and K is a closed convex cone with Rpþ K, then there exists z such that ðu; v; k; zÞ is feasible for (WP), and the objective values of (WP) and (WD) are equal. Also under the assumptions of Theorem 2.1, ðx; y ; k; zÞ is a weakly efficient solution for (WP).
658
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
3. Mond–Weir type symmetric duality Now we state the following pair of Mond–Weir type non-differentiable symmetric dual problems and establish weak, strong and converse duality theorems. ðMPÞ K minimize subject to
ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp Þ; p X ki ðry fi ðx; yÞ zi Þ 2 C 2 ; yT
i¼1 p X
ki ðry fi ðx; yÞ zi Þ = 0;
ð21Þ ð22Þ
i¼1
zi 2 E i ;
i ¼ 1; . . . ; p; T
k e ¼ 1;
k2K ;
ð23Þ
x 2 C1;
ð24Þ
and ðMDÞ
K maximize subject to
ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp Þ; p X ki ðrx fi ðu; vÞ þ xi Þ 2 C 1 ;
ð25Þ
i¼1
uT
p X
ki ðrx fi ðu; vÞ þ xi Þ 5 0;
ð26Þ
i¼1
xi 2 Di ;
k2K ; n
m
i ¼ 1; . . . ; p; T
k e ¼ 1;
ð27Þ
v 2 C2;
ð28Þ
p
where f : R R ! R be a twice differentiable functions of x and y, Di and Ei (1 5 i 5 p) are compact convex set in Rn and Rm , C1 and C2 be closed convex cones in Rn ; Rm with nonempty interiors, respectively. C 1 and C 2 are positive polar cones of C1 and C2, respectively, K is a closed convex cone in Rp such that intK 6¼ ;. Theorem 3.1 (Weak duality). Let ðx; y; k; zÞ be feasible for (MP) and let ðu; v; k; xÞ be feasible for (MD). Let f ð; yÞ þ ðÞT x be K-pseudoinvex with respect to g1 for fixed y and f ðx; Þ þ ðÞT z be K-pseudoinvex with respect to g2 for fixed x. If g1 ðx; uÞ þ uinC 1 and g2 ðv; yÞ þ y 2 C 2 , then ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp Þ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp Þ 62 intK: Proof. Let ðx; y; k; zÞ be feasible for (MP) and let ðu; v; k; xÞ be feasible for (MD). From (25) and g1 ðx; uÞ þ u 2 C 1 , ½g1 ðx; uÞ þ uT
p X
ki ðrx fi ðu; vÞ þ xi Þ = 0:
i¼1
From (26), g1 ðx; uÞ
T Pp i¼1 ki ðrx fi ðu; vÞ
þ xi Þ = 0: This gives
g1 ðx; uÞT ðrx f ðu; vÞ þ xÞ 62 int K; T
T
where rx f ðu; vÞ þ x ¼ ðrx f1 ðu; vÞ þ x1 ; . . . ; rx fp ðu; vÞ þ xp Þ . Since f ð; yÞ þ ðÞ x is K-pseudoinvex with respect to g1 for fixed y =v, ðf1 ðx; vÞ þ xT x1 f1 ðu; vÞ uT x1 ; . . . ; fp ðx; vÞ þ xT xp fp ðu; vÞ uT xp Þ 62 int K: Then k 2K* and k 5 0, p X i¼1
ki ½fi ðx; vÞ þ xT xi fi ðu; vÞ uT xi = 0:
ð29Þ
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
659
Similarly from (21) and g2 ðv; yÞ þ y 2 C 2 , we get T
½g2 ðv; yÞ þ y
p X
ki ðry fi ðx; yÞ zi Þ 5 0:
i¼1
From (22), g2 ðv; yÞ
T Pp i¼1 ki ðry fi ðx; yÞ
zi Þ 5 0: This gives
T
g2 ðv; yÞ ðry f ðx; yÞ zÞ 62 int K; T
T
where ry f ðv; yÞ z ¼ ðry f1 ðv; yÞ z1 ; . . . ; ry fp ðv; yÞ zp Þ . As f ðx; Þ þ ðÞ z be K-pseudoinvex with respect to g2 for fixed x, ðf1 ðx; yÞ y T z1 f1 ðx; vÞ þ vT z1 ; . . . ; fp ðx; yÞ y T zp fp ðx; vÞ þ vT zp Þ 62 int K: Then k 2 K* and k 5 0, p X
ki ½fi ðx; yÞ y T zi fi ðx; vÞ þ vT zi = 0:
ð30Þ
i¼1
From (29) and (30), p X
ki ½fi ðx; yÞ þ xT xi y T zi fi ðu; vÞ þ vT zi uT xi = 0:
i¼1
Finally using xT xi 5 sðx j Di Þ and vT zi 5 sðv j Ei Þ, p X
ki ½fi ðx; yÞ þ sðxjDi Þ y T zi fi ðu; vÞ þ sðvjEi Þ uT xi = 0:
ð31Þ
i¼1
Suppose to the contrary that ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp Þ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp Þ 2 int K: Then k 2 K* and k 5 0, p X
ki ½fi ðx; yÞ þ sðxjDi Þ y T zi fi ðu; vÞ þ sðvjEi Þ uT xi < 0;
i¼1
which contradicts (31). Hence the result holds.
h
Theorem 3.2 (Strong duality). Let f : Rn Rm ! Rp be a twice differentiable. Let ðx; y ; k; zÞ be a weakly efficient solution of (MP) and suppose that (I) ryy fi ðx; y Þ is positive definite for all i ¼ 1; . . . ; p. (II) fry fi ðx; y Þ zi ; i ¼ 1; 2; . . . ; pg is linearly independent. (III) K is a closed convex cone with Rpþ K. such that ðx; y ; is feasible for (MD), and the objective values of (MP) and (MD) are Then, there exists x k; xÞ equal. Furthermore, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of (MP) and is a weakly efficient solution for (MD). (MD), then ðx; y ; k; xÞ Proof. Since ðx; y ; k; zÞ is a weakly efficient solution of (MP), by modifying the Fritz John optimality condition in [6], there exist a 2 K*, b 2 C 2 , c 2 Rþ ; d 2 C 1 , f 2 K such that
660
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661 p X
i þ ai ½rx fi ðx; y Þ þ x
i¼1
p X
ki ryx fi ðx; y Þðb cy Þ d ¼ 0;
p p X X ki ryy fi ðx; y Þðb cy Þ ¼ 0; ðai c ki Þ½ry fi ðx; y Þ zi þ i¼1
ð33Þ
i¼1 T
ðb cy Þ ðry fi ðx; y Þ zi Þ fi ¼ 0; ai y þ ðb cy Þ ki 2 N Ei ðzi Þ; bT
ð32Þ
i¼1
p X
i ¼ 1; . . . ; p;
ð34Þ
i ¼ 1; . . . ; p;
ð35Þ
ki ½ry fi ðx; y Þ zi ¼ 0;
ð36Þ
i¼1
cy T
p X
ki ½ry fi ðx; y Þ zi ¼ 0;
ð37Þ
i¼1
dT x ¼ 0;
ð38Þ
k ¼ 0; fT
ð39Þ
i 2 x
Ti x Di ; x
¼ sðxjDi Þ;
i ¼ 1; . . . ; p;
ð40Þ
ða; b; c; d; fÞ 6¼ 0:
ð41Þ
Multiplying (33) by ðb cy Þ, p X
T ðai c ki Þ½ry fi ðx; y Þ zi ðb cy Þ þ ðb cy Þ
p X
i¼1
ki ryy fi ðx; y Þðb cy Þ ¼ 0:
i¼1
Using the result in equality (34) and (39), we get p X
ai fi þ ðb cy Þ
i¼1
T
p X
ki ryy fi ðx; y Þðb cy Þ ¼ 0:
i¼1
Since a 2 K* and f 2 K; aT f = 0 and hence ðb cy ÞT
p X
ki ryy fi ðx; y Þðb cy Þ 5 0:
i¼1
Since
Pp x; y Þ is positive definite, then i¼1 ki ryy fi ð b ¼ cy :
Pp
ð42Þ
By (34), fi ¼ 0; i ¼ 1; . . . ; p. From (33), i¼1 ðai cki Þ½ry fi ðx; y Þ zi ¼ 0: Since fry fi ðx; y Þ zi ; i ¼ 1; . . . ; pg is linearly independent, ai ¼ c ki , i ¼ 1; . . . ; p. If c = 0, then a = 0, (42) implies that b = 0 and using (32), d = 0, which contradicts (41). Thus c > 0. Hence by (42), y 2 C 2 . By (32), and the fact that b ¼ cy and ai ¼ c ki ; i ¼ 1; . . . ; p, c
p X
i ¼ d 2 C 1 : ki ½rx fi ðx; y Þ þ x
i¼1
Pp i 2 C 1 . By multiplying both sides of equation by x, hence from (38) we get ki ½rx fi ðx; y Þ þ x Since c > 0, c i¼1 P p i ¼ 0. Thus ðx; y ; is feasible for (MD). xT i¼1 ki ½rx fi ðx; y Þ þ x k; xÞ By (35) and the fact that b ¼ cy , ai y 2 N Ei ðzi Þ; i ¼ 1; . . . ; p: Since a 2 K Rpþ , ai > 0, and hence y 2 N Ei ðzi Þ, so that y T zi ¼ sðy jEi Þ; i ¼ 1; . . . ; p: Thus (MP) and (MD) have equal objectives values. is a weakly efficient solution for (MD), otherwise there would exist a feasible solution Clearly, ðx; y ; k; xÞ ðu; v; k; xÞ of (MD) such that 1 ; . . . ; fp ðx; y Þ sðy jEp Þ þ xT x p Þ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ ðf1 ðx; y Þ sðy jE1 Þ þ xT x sðvjEp Þ þ uT xp Þ 2 int K:
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
661
i ¼ sðxjDi Þ; i ¼ 1; . . . ; p, it follows that Since y T zi ¼ sðy jEi Þ and xT x ðf1 ðx; y Þ þ sðxjD1 Þ y T z1 ; . . . ; fp ðx; y Þ þ sðxjDp Þ y T zp Þ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp Þ 2 int K; is a feasible solution of (MD), ðx; y ; k; xÞ is a weakly effiwhich contradicts by weak duality. Since ðx; y ; k; xÞ cient solution of (MD). Hence the result holds. h We now state a converse duality theorem whose proof follows on the lines of Theorem 3.2. be a weakly Theorem 3.3 (Converse duality). Let f : Rn Rm ! Rp be a twice differentiable. Let ðu; v; k; xÞ i ; i ¼ 1; 2; . . . ; pg efficient solution of (MD). If rxx fi ð u; vÞ is negative definite for all i ¼ 1; . . . ; p, frx fi ðu; vÞ þ x is linearly independent and K is a closed convex cone with Rpþ K, then there exists z such that ðu; v; k; zÞ is feasible for (MP), and the objective values of (MP) and (MD) are equal. Also under the assumptions of Theorem 3.1, ðx; y ; k; zÞ is a weakly efficient solution for (MP). Remark 3.1. When Di = Ei = {0}, then the support functions and inner products in the problems (MP) and (MD) in the draft are disappeared, and hence (MP) and (MD) in the draft collapse to (P) and (D) in the paper of S. Khurana (EJOR, Vol. 165, 2005, pp. 592–597), respectively. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M.S. Bazaraa, J.J. Goode, On symmetric duality in nonlinear programming, Operations Research 21 (1) (1973) 1–9. W.S. Dorn, A symmetric dual theorem for quadratic programs, Journal of Operations Research Society of Japan 2 (1960) 93–97. G.B. Dantzig, E. Eisenberg, R.W. Cottle, Symmetric dual nonlinear programs, Pacific Journal of Mathematics 15 (1965) 809–812. D.S. Kim, Y.B. Yun, W.J. Lee, Multiobjective symmetric duality with cone constraints, European Journal of Operational Research 107 (1998) 686–691. S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597. B. Mond, M. Schechter, Non-differentiable symmetric duality, Bulletin of the Australian Mathematical Society 53 (1996) 177–188. B. Mond, T. Weir, Generalized concavity and duality, in: S. Schaible, W.T. Ziemba (Eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981, pp. 263–280. S. Nanda, L.S. Das, Pseudo-invexity and duality in nonlinear programming, European Journal of Operational Research 88 (1996) 572–577. R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970. S.K. Suneja, S. Aggarwal, S. Davar, Multiobjective symmetric duality involving cones, European Journal of Operational Research 141 (2002) 471–479.
European Journal of Operational Research 188 (2008) 652–661 www.elsevier.com/locate/ejor
Continuous Optimization
Non-differentiable symmetric duality for multiobjective programming with cone constraints Moon Hee Kim a, Do Sang Kim
q
b,*
a
b
Department of Multimedia Engineering, Tongmyong University, Pusan 608-711, Republic of Korea Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Republic of Korea Received 16 February 2006; accepted 7 May 2007 Available online 13 May 2007
Abstract Two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597] to the non-differentiable multiobjective symmetric dual problem. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Non-differentiable programming; Duality theorems; Cone constraints; Weakly efficient solutions
1. Introduction and preliminaries A nonlinear programming problem and its dual are said to be symmetric if the dual of the dual is the original problem. Symmetric duality in nonlinear programming in which the dual of the dual is the primal was first introduced by Dorn [2]. Danzig et al. [3] formulated a pair of symmetric dual nonlinear programs and established duality results for convex and concave functions with non-negative orthant as the cone. The same result was generalized by Bazaraa and Goode [1] to arbitrary cones. Mond and Weir [7] presented two pair of symmetric dual multiobjective programming problems for efficient solutions and obtained symmetric duality results concerning pseudoconvex and pseudoconcave functions. Nanda and Das [8] also studied the symmetric dual nonlinear programming problem for arbitrary cones assuming the functions to be pseudo-invex. Kim et al. [4] studied a pair of multiobjective symmetric dual programs for pseudo-invex functions and arbitrary cones. Suneja et al. [10] formulated a pair of symmetric dual multiobjective programs of Wolfe type over q *
This work was supported by the Brain Korea 21 Project in 2006. Corresponding author. Tel.: +82 51 620 6315; fax: +82 51 611 6356. E-mail address: [email protected] (D.S. Kim).
0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.05.005
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
653
arbitrary cones in which the objective function is optimized with respect to an arbitrary closed convex cone by assuming the function involved to be cone-convex. Very recently, Khurana [5] formulated a pair of differentiable multiobjective symmetric dual programs of Mond–Weir type over arbitrary cones in which the objective function is optimized with respect to an arbitrary closed convex cone by assuming the involved functions to be cone-pseudoinvex and strongly cone-pseudoinvex. In this paper, two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [5] to the non-differentiable multiobjective symmetric dual problem. First we consider the following multiobjective programming problem. ðP Þ K Minimize subject to
f ðxÞ; gðxÞ 2 Q;
x 2 C;
where f : Rn ! Rp ; g : Rn ! Rm and C Rn , K and Q are closed convex cone with nonempty interior in Rp and Rm , respectively. Let X 0 ¼ fx 2 C : gðxÞ 2 Qg be the set of all feasible solutions of (P). Definition 1.1. A point x 2 X 0 is a weakly efficient solution of (P) if there exist no other x 2 X 0 such that f ðxÞ f ðxÞ 2 int K: Definition 1.2. The positive polar cone K* of K is defined by K ¼ fz 2 Rp : xT z = 0 for all x 2 Kg: Definition 1.3 [10]. Let f : Rn ! Rp be a function. Then f is K-preinvex with respect to the g if there exists a function g : C C ! Rn such that for any x; y 2 Rn and a 2 ½0; 1, af ðxÞ þ ð1 aÞf ðyÞ f ðy þ agðx; yÞÞ 2 K: Remark 1.1. If f : Rn ! Rp is differentiable and K-preinvex with respect to g T T f ðyÞ rf ðyÞ gðx; yÞ 2 K. Moreover, 8k 2 K ; ðkT f ÞðxÞ ðkT f ÞðyÞ rðkT f ÞðyÞ gðx; yÞ = 0:
then
f ðxÞ
Definition 1.4 [5]. Let f : Rn ! Rp be a differentiable function. Then f is K-pseudoinvex with respect to g at u 2 Rn if for all x 2 Rn T
rf ðuÞ gðx; uÞ 62 int K ) ðf ðxÞ f ðuÞÞ 62 int K; where rf ðuÞ ¼ ðrf1 ðuÞ; rf2 ðuÞ; . . . ; rfp ðuÞÞT . Definition 1.5 [6]. Let C be a compact convex set in Rn . The support function sðx j CÞ of C is defined by sðxjCÞ :¼ maxfxT y : y 2 Cg: The support function sðx j CÞ, being convex and everywhere finite, has a subdifferential at every x in the sense of Rockafellar [9], that is, there exists z such that sðy j CÞ P sðx j CÞ þ zT ðy xÞ for all y 2C. The subdifferential of sðx j CÞ is given by osðxjCÞ :¼ fz 2 C : zT x ¼ sðxjCÞg: For any set S Rn , the normal cone to S at a point x 2 S is defined by N S ðxÞ :¼ fy 2 Rn : y T ðz xÞ 6 0 for all z 2 Sg: It is readily verified that for a compact convex set C, y is in NC(x) if and only if sðyjCÞ ¼ xT y, or equivalently, x is in the subdifferential of s at y.
654
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
2. Wolfe type symmetric duality We now state the following pair of Wolfe type non-differentiable symmetric dual problems and establish weak, strong and converse duality theorems. f1 ðx; yÞ þ sðxjD1 Þ y T z1 y T
ðWPÞ K minimize
p X
ki ðry fi ðx; yÞ zi Þ; . . . ; fp ðx; yÞ
i¼1 T
þsðxjDp Þ y zp y
T
p X
! ki ðry fi ðx; yÞ zi Þ ;
i¼1
subject to
p X
ki ðry fi ðx; yÞ zi Þ 2 C 2 ;
ð1Þ
i¼1
zi 2 E i ;
i ¼ 1; . . . ; p;
ð2Þ
k 2 K ;
kT e ¼ 1;
ð3Þ
x 2 C1;
and ðWDÞ
K maximize
f1 ðu; vÞ sðvjE1 Þ þ uT x1 uT
p X
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ sðvjEp Þ
i¼1 T
þu xp u
T
p X
!
ki ðrx fi ðu; vÞ þ xi Þ ;
i¼1
subject to
p X
ki ðrx fi ðu; vÞ þ xi Þ 2 C 1 ;
ð4Þ
i¼1
xi 2 Di ;
k2K ; n
m
i ¼ 1; . . . ; p; T
k e ¼ 1;
ð5Þ
v 2 C2;
ð6Þ
p
where f : R R ! R be a twice differentiable functions of x and y, Di and Ei (1 5 i 5 p) are compact convex set in Rn and Rm , C1 and C2 be closed convex cones in Rn ; Rm with nonempty interiors, respectively. C 1 and C 2 are positive polar cones of C1 and C2, respectively, K is a closed convex cone in Rp such that int K 6¼ ;. Theorem 2.1 (Weak duality). Let ðx; y; k; zÞ be feasible for (WP) and let ðu; v; k; xÞ be feasible for (WD). Let f ð; yÞ þ ðÞT x be K-preinvex with respect to g1 for fixed y and f ðx; Þ þ ðÞT z be K-preinvex with respect to g2 for fixed x. If g1 ðx; uÞ þ u 2 C 1 and g2 ðv; yÞ þ y 2 C 2 , then ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 y T
p X
ki ðry fi ðx; yÞ zi Þ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp
i¼1
yT
p X
ki ðry fi ðx; yÞ zi ÞÞ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 uT
i¼1
sðvjEp Þ þ uT xp uT
p X
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ
i¼1 p X
ki ðrx fi ðu; vÞ þ xi ÞÞ 62 intK:
i¼1 T
Proof. Let ðx; y; k; zÞ be feasible for (WP) and let ðu; v; k; xÞ be feasible for (WD). Since f ð; yÞ þ ðÞ x is Kpreinvex with respect to g1 for fixed y = v, by Remark 1.1, p X i¼1
ki ðfi ðx; vÞ þ xT xi fi ðu; vÞ uT xi Þ = g1 ðx; uÞ
T
p X i¼1
ki ðrx fi ðu; vÞ þ xi Þ:
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
655
From (4) and g1 ðx; uÞ þ u 2 C 1 , ½g1 ðx; uÞ þ u
T
p X
ki ðrx fi ðu; vÞ þ xi Þ = 0:
i¼1
Hence p X
ki ðfi ðx; vÞ þ xT xi fi ðu; vÞ uT xi Þ = uT
i¼1
p X
ki ðrx fi ðu; vÞ þ xi Þ:
ð7Þ
i¼1 T
Since f ðx; Þ þ ðÞ z is K-preinvex with respect to g2 for fixed x, by Remark 1.1, p X
T
ki ðfi ðx; yÞ y T zi fi ðx; vÞ þ vT zi Þ = g2 ðv; yÞ
i¼1
p X
ki ðry fi ðx; yÞ zi Þ:
i¼1
From (1) and g2 ðv; yÞ þ y 2 C 2 , T
½g2 ðv; yÞ þ y
p X
ki ðry fi ðx; yÞ zi Þ 5 0:
i¼1
Hence p X
ki ðfi ðx; yÞ y T zi fi ðx; vÞ þ vT zi Þ = y T
i¼1
p X
ki ðry fi ðx; yÞ zi Þ:
ð8Þ
i¼1
From (7) and (8), we have p X
ki ðfi ðx; yÞ y T zi þ vT zi fi ðu; vÞ þ xT xi uT xi Þ = uT
i¼1
p X
ki ðrx fi ðu; vÞ þ xi Þ
i¼1
þ yT
p X
ki ðry fi ðx; yÞ zi Þ:
i¼1
Finally using xT xi 5 sðx j Di Þ and vT zi 5 sðv j Ei Þ, we obtain p X
ki ðfi ðx; yÞ y T zi þ sðvjEi Þ fi ðu; vÞ þ sðxjDi Þ uT xi Þ = uT
i¼1
p X
ki ðrx fi ðu; vÞ þ xi Þ
i¼1
þ yT
p X
ki ðry fi ðx; yÞ zi Þ
i¼1
and hence p X
ki ðfi ðx; yÞ þ sðxjDi Þ y T zi y T
i¼1
uT
p X
ki ðry fi ðx; yÞ zi ÞÞ
i¼1 p X
p X
ki ðfi ðu; vÞ sðvjEi Þ þ uT xi
i¼1
ki ðrx fi ðu; vÞ þ xi ÞÞ = 0:
ð9Þ
i¼1
Suppose to the contrary that ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 y T
p X
ki ðry fi ðx; yÞ zi Þ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp
i¼1
yT
p X
ki ðry fi ðx; yÞ zi ÞÞ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 uT
i¼1
sðvjEp Þ þ uT xp uT
p X i¼1
p X i¼1
ki ðrx fi ðu; vÞ þ xi ÞÞ 2 int K:
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ
656
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
Then by the constraint (3), k 2 K* and k 5 0, p X
ki ðfi ðx; yÞ þ sðxjDi Þ y T zi y T
i¼1
p X
ki ðry fi ðx; yÞ zi ÞÞ
p X
i¼1
uT
p X
ki ðfi ðu; vÞ sðvjEi Þ þ uT xi
i¼1
ki ðrx fi ðu; vÞ þ xi ÞÞ < 0;
i¼1
which contradicts (9). Hence the result holds.
h
Theorem 2.2 (Strong duality). Let f : Rn Rm ! Rp be a twice differentiable. Let ðx; y ; k; zÞ be a weakly efficient solution of (WP) and suppose that (I) ryy fi ðx; y Þ is positive definite for all i ¼ 1; . . . ; p. (II) fry fi ðx; y Þ zi ; i ¼ 1; 2; . . . ; pg is linearly independent. (III) K is a closed convex cone with Rpþ K. such that ðx; y ; is feasible for (WD), and the objective values of (WP) and (WD) are Then, there exists x k; xÞ equal. Furthermore, if the hypotheses of Theorem 2.1 are satisfied for all feasible solutions of (WP) and is a weakly efficient solution for (WD). (WD), then ðx; y ; k; xÞ Proof. Since ðx; y ; k; zÞ is a weakly efficient solution of (WP), by modifying the Fritz John optimality condition in [6], we can check that there exist a 2 K , b 2 C 2 , c 2 C 1 , d 2 K such that p X
i þ ai ½rx fi ðx; y Þ þ x
i¼1 p X
p X
ki ryx fi ðx; y Þðb ðaT eÞy Þ c ¼ 0;
ð10Þ
i¼1
ðai ðaT eÞ ki Þ½ry fi ðx; y Þ zi þ
i¼1
p X
ki ryy fi ðx; y Þðb ðaT eÞy Þ ¼ 0;
T
ðb ðaT eÞy Þ ðry fi ðx; y Þ zi Þ di ¼ 0; i ¼ 1; . . . ; p; ai y þ ðb ðaT eÞy Þ ki 2 N E ðzi Þ; i ¼ 1; . . . ; p;
ð12Þ ð13Þ
i
bT
ð11Þ
i¼1
p X
ki ½ry fi ðx; y Þ zi ¼ 0;
ð14Þ
i¼1
cT x ¼ 0; dT k ¼ 0;
ð15Þ ð16Þ Ti x x
i 2 Di ; x
¼ sðxjDi Þ;
i ¼ 1; . . . ; p;
ð17Þ
ða; b; c; dÞ 6¼ 0:
ð18Þ
Multiplying (11) by ðb ðaT eÞy Þ, p X
T ðai ðaT eÞ ki Þ½ry fi ðx; y Þ zi ðb ðaT eÞy Þ þ ðb ðaT eÞy Þ
i¼1
i¼1
Using the result in equality (12) and (16), we get p X
p X
ai di þ ðb ðaT eÞy Þ
i¼1
T
p X
ki ryy fi ðx; y Þðb ðaT eÞy Þ ¼ 0:
i¼1 T
Since a 2 K* and d 2 K; a d = 0 and hence ðb ðaT eÞy Þ
T
p X i¼1
ki ryy fi ðx; y Þðb ðaT eÞy Þ 5 0:
ki ryy fi ðx; y Þðb ðaT eÞy Þ ¼ 0:
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
Since
657
Pp x; y Þ is positive definite, then i¼1 ki ryy fi ð b ¼ ðaT eÞy :
ð19Þ
Pp
T x; y Þ zi ¼ 0: Since fry fi ðx; y Þ zi ; By (12), di ¼ 0; i ¼ 1; . . . ; p. From (11), i¼1 ðai ða eÞki Þ½ry fi ð T i ¼ 1; . . . ; pg is linearly independent, ai ¼ ða eÞki ; i ¼ 1; . . . ; p. If a = 0, then (19) implies that b = 0 and using (10), we get that c = 0, which contradicts (18). Hence ai 6¼ 0; i ¼ 1; . . . ; p, since a 2 K Rpþ , ai > 0. Hence by (19), y 2 C 2 . By (10), (19) and ai ¼ ðaT eÞ ki ; i ¼ 1; . . . ; p, p X i ¼ c 2 C 1 : ðaT eÞ ki ½rx fi ðx; y Þ þ x
ð20Þ
i¼1
P i 2 C 1 . Thus ðx; y ; is feasible for (WD). ki ½rx fi ðx; y Þ þ x k; xÞ Since pi¼1 By multiplying both sides of Eq. (20) by x, hence from (15) we get xT
p X
i ¼ 0: ki ½rx fi ðx; y Þ þ x
i¼1
By (14) and the fact that b ¼ ðaT eÞy , ðaT eÞy T y T
p X
Pp x; y Þ zi ¼ 0: Since ai > 0, i¼1 ki ½ry fi ð
ki ½ry fi ðx; y Þ zi ¼ 0:
i¼1
But from (13), and b ¼ ðaT eÞy , ai y 2 N Ei ðzi Þ: Since ai > 0, y 2 N Ei ðzi Þ, we have y T zi ¼ sðy jEi Þ; i ¼ 1; . . . ; p: Thus (WP) and (WD) have equal objectives values. is a weakly efficient solution of (WD), otherwise there would exist a We shall now show that ðx; y ; k; xÞ feasible solution ðu; v; k; xÞ of (WD) such that 1 xT ðf1 ðx; y Þ sðy jE1 Þ þ xT x
p X
i Þ; . . . ; fp ðx; y Þ ki ðrx fi ðx; y Þ þ x
i¼1
p xT sðy jEp Þ þ xT x
p X
i ÞÞ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ki ðrx fi ðx; y Þ þ x
i¼1
uT
p X
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp uT
i¼1
p X
ki ðrx fi ðu; vÞ þ xi ÞÞ 2 int K:
i¼1
Pp P i ¼ 0 ¼ y T pi¼1 i ¼ sðxjDi Þ; i ¼ 1; . . . ; p, ki ½ry fi ðx; y Þ zi , y T zi ¼ sðy jEi Þ and xT x x; y Þ þ x Since x i¼1 ki ½rx fi ð it follows that T
ðf1 ðx; y Þ þ sðxjD1 Þ y T z1 y T
p X
ki ðry fi ðx; y Þ zi Þ; . . . ; fp ðx; y Þ þ sðxjDp Þ y T zp
i¼1
y T
p X
ki ðry fi ðx; y Þ zi ÞÞ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 uT
i¼1
sðvjEp Þ þ uT xp uT
p X
ki ðrx fi ðu; vÞ þ xi Þ; . . . ; fp ðu; vÞ
i¼1 p X
ki ðrx fi ðu; vÞ þ xi ÞÞ 2 int K;
i¼1
is a feasible solution of (WD), ðx; y ; k; xÞ is a weakly efficient which contradicts weak duality. Since ðx; y ; k; xÞ solution of (WD). Hence the result holds. h We now state a converse duality theorem whose proof follows on the lines of Theorem 2.2. be a weakly Theorem 2.3 (Converse duality). Let f : Rn Rm ! Rp be a twice differentiable. Let ðu; v; k; xÞ efficient solution of (WD). If rxx fi ð u; vÞ is negative definite for all i ¼ 1; . . . ; p, frx fi ðu; vÞ þ xi ; i ¼ 1; 2; . . . ; pg is linearly independent and K is a closed convex cone with Rpþ K, then there exists z such that ðu; v; k; zÞ is feasible for (WP), and the objective values of (WP) and (WD) are equal. Also under the assumptions of Theorem 2.1, ðx; y ; k; zÞ is a weakly efficient solution for (WP).
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3. Mond–Weir type symmetric duality Now we state the following pair of Mond–Weir type non-differentiable symmetric dual problems and establish weak, strong and converse duality theorems. ðMPÞ K minimize subject to
ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp Þ; p X ki ðry fi ðx; yÞ zi Þ 2 C 2 ; yT
i¼1 p X
ki ðry fi ðx; yÞ zi Þ = 0;
ð21Þ ð22Þ
i¼1
zi 2 E i ;
i ¼ 1; . . . ; p; T
k e ¼ 1;
k2K ;
ð23Þ
x 2 C1;
ð24Þ
and ðMDÞ
K maximize subject to
ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp Þ; p X ki ðrx fi ðu; vÞ þ xi Þ 2 C 1 ;
ð25Þ
i¼1
uT
p X
ki ðrx fi ðu; vÞ þ xi Þ 5 0;
ð26Þ
i¼1
xi 2 Di ;
k2K ; n
m
i ¼ 1; . . . ; p; T
k e ¼ 1;
ð27Þ
v 2 C2;
ð28Þ
p
where f : R R ! R be a twice differentiable functions of x and y, Di and Ei (1 5 i 5 p) are compact convex set in Rn and Rm , C1 and C2 be closed convex cones in Rn ; Rm with nonempty interiors, respectively. C 1 and C 2 are positive polar cones of C1 and C2, respectively, K is a closed convex cone in Rp such that intK 6¼ ;. Theorem 3.1 (Weak duality). Let ðx; y; k; zÞ be feasible for (MP) and let ðu; v; k; xÞ be feasible for (MD). Let f ð; yÞ þ ðÞT x be K-pseudoinvex with respect to g1 for fixed y and f ðx; Þ þ ðÞT z be K-pseudoinvex with respect to g2 for fixed x. If g1 ðx; uÞ þ uinC 1 and g2 ðv; yÞ þ y 2 C 2 , then ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp Þ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp Þ 62 intK: Proof. Let ðx; y; k; zÞ be feasible for (MP) and let ðu; v; k; xÞ be feasible for (MD). From (25) and g1 ðx; uÞ þ u 2 C 1 , ½g1 ðx; uÞ þ uT
p X
ki ðrx fi ðu; vÞ þ xi Þ = 0:
i¼1
From (26), g1 ðx; uÞ
T Pp i¼1 ki ðrx fi ðu; vÞ
þ xi Þ = 0: This gives
g1 ðx; uÞT ðrx f ðu; vÞ þ xÞ 62 int K; T
T
where rx f ðu; vÞ þ x ¼ ðrx f1 ðu; vÞ þ x1 ; . . . ; rx fp ðu; vÞ þ xp Þ . Since f ð; yÞ þ ðÞ x is K-pseudoinvex with respect to g1 for fixed y =v, ðf1 ðx; vÞ þ xT x1 f1 ðu; vÞ uT x1 ; . . . ; fp ðx; vÞ þ xT xp fp ðu; vÞ uT xp Þ 62 int K: Then k 2K* and k 5 0, p X i¼1
ki ½fi ðx; vÞ þ xT xi fi ðu; vÞ uT xi = 0:
ð29Þ
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
659
Similarly from (21) and g2 ðv; yÞ þ y 2 C 2 , we get T
½g2 ðv; yÞ þ y
p X
ki ðry fi ðx; yÞ zi Þ 5 0:
i¼1
From (22), g2 ðv; yÞ
T Pp i¼1 ki ðry fi ðx; yÞ
zi Þ 5 0: This gives
T
g2 ðv; yÞ ðry f ðx; yÞ zÞ 62 int K; T
T
where ry f ðv; yÞ z ¼ ðry f1 ðv; yÞ z1 ; . . . ; ry fp ðv; yÞ zp Þ . As f ðx; Þ þ ðÞ z be K-pseudoinvex with respect to g2 for fixed x, ðf1 ðx; yÞ y T z1 f1 ðx; vÞ þ vT z1 ; . . . ; fp ðx; yÞ y T zp fp ðx; vÞ þ vT zp Þ 62 int K: Then k 2 K* and k 5 0, p X
ki ½fi ðx; yÞ y T zi fi ðx; vÞ þ vT zi = 0:
ð30Þ
i¼1
From (29) and (30), p X
ki ½fi ðx; yÞ þ xT xi y T zi fi ðu; vÞ þ vT zi uT xi = 0:
i¼1
Finally using xT xi 5 sðx j Di Þ and vT zi 5 sðv j Ei Þ, p X
ki ½fi ðx; yÞ þ sðxjDi Þ y T zi fi ðu; vÞ þ sðvjEi Þ uT xi = 0:
ð31Þ
i¼1
Suppose to the contrary that ðf1 ðx; yÞ þ sðxjD1 Þ y T z1 ; . . . ; fp ðx; yÞ þ sðxjDp Þ y T zp Þ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp Þ 2 int K: Then k 2 K* and k 5 0, p X
ki ½fi ðx; yÞ þ sðxjDi Þ y T zi fi ðu; vÞ þ sðvjEi Þ uT xi < 0;
i¼1
which contradicts (31). Hence the result holds.
h
Theorem 3.2 (Strong duality). Let f : Rn Rm ! Rp be a twice differentiable. Let ðx; y ; k; zÞ be a weakly efficient solution of (MP) and suppose that (I) ryy fi ðx; y Þ is positive definite for all i ¼ 1; . . . ; p. (II) fry fi ðx; y Þ zi ; i ¼ 1; 2; . . . ; pg is linearly independent. (III) K is a closed convex cone with Rpþ K. such that ðx; y ; is feasible for (MD), and the objective values of (MP) and (MD) are Then, there exists x k; xÞ equal. Furthermore, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of (MP) and is a weakly efficient solution for (MD). (MD), then ðx; y ; k; xÞ Proof. Since ðx; y ; k; zÞ is a weakly efficient solution of (MP), by modifying the Fritz John optimality condition in [6], there exist a 2 K*, b 2 C 2 , c 2 Rþ ; d 2 C 1 , f 2 K such that
660
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661 p X
i þ ai ½rx fi ðx; y Þ þ x
i¼1
p X
ki ryx fi ðx; y Þðb cy Þ d ¼ 0;
p p X X ki ryy fi ðx; y Þðb cy Þ ¼ 0; ðai c ki Þ½ry fi ðx; y Þ zi þ i¼1
ð33Þ
i¼1 T
ðb cy Þ ðry fi ðx; y Þ zi Þ fi ¼ 0; ai y þ ðb cy Þ ki 2 N Ei ðzi Þ; bT
ð32Þ
i¼1
p X
i ¼ 1; . . . ; p;
ð34Þ
i ¼ 1; . . . ; p;
ð35Þ
ki ½ry fi ðx; y Þ zi ¼ 0;
ð36Þ
i¼1
cy T
p X
ki ½ry fi ðx; y Þ zi ¼ 0;
ð37Þ
i¼1
dT x ¼ 0;
ð38Þ
k ¼ 0; fT
ð39Þ
i 2 x
Ti x Di ; x
¼ sðxjDi Þ;
i ¼ 1; . . . ; p;
ð40Þ
ða; b; c; d; fÞ 6¼ 0:
ð41Þ
Multiplying (33) by ðb cy Þ, p X
T ðai c ki Þ½ry fi ðx; y Þ zi ðb cy Þ þ ðb cy Þ
p X
i¼1
ki ryy fi ðx; y Þðb cy Þ ¼ 0:
i¼1
Using the result in equality (34) and (39), we get p X
ai fi þ ðb cy Þ
i¼1
T
p X
ki ryy fi ðx; y Þðb cy Þ ¼ 0:
i¼1
Since a 2 K* and f 2 K; aT f = 0 and hence ðb cy ÞT
p X
ki ryy fi ðx; y Þðb cy Þ 5 0:
i¼1
Since
Pp x; y Þ is positive definite, then i¼1 ki ryy fi ð b ¼ cy :
Pp
ð42Þ
By (34), fi ¼ 0; i ¼ 1; . . . ; p. From (33), i¼1 ðai cki Þ½ry fi ðx; y Þ zi ¼ 0: Since fry fi ðx; y Þ zi ; i ¼ 1; . . . ; pg is linearly independent, ai ¼ c ki , i ¼ 1; . . . ; p. If c = 0, then a = 0, (42) implies that b = 0 and using (32), d = 0, which contradicts (41). Thus c > 0. Hence by (42), y 2 C 2 . By (32), and the fact that b ¼ cy and ai ¼ c ki ; i ¼ 1; . . . ; p, c
p X
i ¼ d 2 C 1 : ki ½rx fi ðx; y Þ þ x
i¼1
Pp i 2 C 1 . By multiplying both sides of equation by x, hence from (38) we get ki ½rx fi ðx; y Þ þ x Since c > 0, c i¼1 P p i ¼ 0. Thus ðx; y ; is feasible for (MD). xT i¼1 ki ½rx fi ðx; y Þ þ x k; xÞ By (35) and the fact that b ¼ cy , ai y 2 N Ei ðzi Þ; i ¼ 1; . . . ; p: Since a 2 K Rpþ , ai > 0, and hence y 2 N Ei ðzi Þ, so that y T zi ¼ sðy jEi Þ; i ¼ 1; . . . ; p: Thus (MP) and (MD) have equal objectives values. is a weakly efficient solution for (MD), otherwise there would exist a feasible solution Clearly, ðx; y ; k; xÞ ðu; v; k; xÞ of (MD) such that 1 ; . . . ; fp ðx; y Þ sðy jEp Þ þ xT x p Þ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ ðf1 ðx; y Þ sðy jE1 Þ þ xT x sðvjEp Þ þ uT xp Þ 2 int K:
M.H. Kim, D.S. Kim / European Journal of Operational Research 188 (2008) 652–661
661
i ¼ sðxjDi Þ; i ¼ 1; . . . ; p, it follows that Since y T zi ¼ sðy jEi Þ and xT x ðf1 ðx; y Þ þ sðxjD1 Þ y T z1 ; . . . ; fp ðx; y Þ þ sðxjDp Þ y T zp Þ ðf1 ðu; vÞ sðvjE1 Þ þ uT x1 ; . . . ; fp ðu; vÞ sðvjEp Þ þ uT xp Þ 2 int K; is a feasible solution of (MD), ðx; y ; k; xÞ is a weakly effiwhich contradicts by weak duality. Since ðx; y ; k; xÞ cient solution of (MD). Hence the result holds. h We now state a converse duality theorem whose proof follows on the lines of Theorem 3.2. be a weakly Theorem 3.3 (Converse duality). Let f : Rn Rm ! Rp be a twice differentiable. Let ðu; v; k; xÞ i ; i ¼ 1; 2; . . . ; pg efficient solution of (MD). If rxx fi ð u; vÞ is negative definite for all i ¼ 1; . . . ; p, frx fi ðu; vÞ þ x is linearly independent and K is a closed convex cone with Rpþ K, then there exists z such that ðu; v; k; zÞ is feasible for (MP), and the objective values of (MP) and (MD) are equal. Also under the assumptions of Theorem 3.1, ðx; y ; k; zÞ is a weakly efficient solution for (MP). Remark 3.1. When Di = Ei = {0}, then the support functions and inner products in the problems (MP) and (MD) in the draft are disappeared, and hence (MP) and (MD) in the draft collapse to (P) and (D) in the paper of S. Khurana (EJOR, Vol. 165, 2005, pp. 592–597), respectively. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M.S. Bazaraa, J.J. Goode, On symmetric duality in nonlinear programming, Operations Research 21 (1) (1973) 1–9. W.S. Dorn, A symmetric dual theorem for quadratic programs, Journal of Operations Research Society of Japan 2 (1960) 93–97. G.B. Dantzig, E. Eisenberg, R.W. Cottle, Symmetric dual nonlinear programs, Pacific Journal of Mathematics 15 (1965) 809–812. D.S. Kim, Y.B. Yun, W.J. Lee, Multiobjective symmetric duality with cone constraints, European Journal of Operational Research 107 (1998) 686–691. S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597. B. Mond, M. Schechter, Non-differentiable symmetric duality, Bulletin of the Australian Mathematical Society 53 (1996) 177–188. B. Mond, T. Weir, Generalized concavity and duality, in: S. Schaible, W.T. Ziemba (Eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981, pp. 263–280. S. Nanda, L.S. Das, Pseudo-invexity and duality in nonlinear programming, European Journal of Operational Research 88 (1996) 572–577. R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970. S.K. Suneja, S. Aggarwal, S. Davar, Multiobjective symmetric duality involving cones, European Journal of Operational Research 141 (2002) 471–479.