Second-order multiobjective symmetric duality with ... - Semantic Scholar
European Journal of Operational Research 205 (2010) 247–252
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Continuous Optimization
Second-order multiobjective symmetric duality with cone constraints T.R. Gulati a,*, Himani Saini a, S.K. Gupta b a b
Department of Mathematics, Indian Institute of Technology, Roorkee 247 667, India Department of Mathematics, Indian Institute of Technology, Patna 800 013, India
a r t i c l e
i n f o
Article history: Received 18 June 2008 Accepted 26 December 2009 Available online 13 January 2010 Keywords: Multiobjective symmetric duality g-bonvexity/g-pseudobonvexity Cones Efficient solutions Properly efficient solutions
a b s t r a c t In this paper, we formulate Wolfe and Mond–Weir type second-order multiobjective symmetric dual problems over arbitrary cones. Weak, strong and converse duality theorems are established under g-bonvexity/g-pseudobonvexity assumptions. This work also removes several omissions in definitions, models and proofs for Wolfe type problems studied in Mishra [9]. Moreover, self-duality theorems for these pairs are obtained assuming the function involved to be skew symmetric. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Mangasarian [8] introduced the concept of second-order duality for nonlinear problems. Its study is significant due to computational advantage over first-order duality as it provides tighter bounds for the value of the objective function when approximations are used [7,8,11]. Bector and Chandra [2] introduced the concept of bonvex functions. Mond [11] established second-order duality for nonlinear programs under second-order convexity assumptions. Later on, Bector and Chandra [1] formulated second-order symmetric dual programs in the spirit of Mond and Weir [12] and established appropriate duality results involving pseudobonvex functions. Recently, Yang et al. [16,18] studied second-order symmetric dual programs and established duality relations under F-convexity assumptions. Devi [4] formulated a pair of second-order symmetric dual nonlinear programming problems over arbitrary cones under gpseudobonvexity assumptions. Mishra [9] formulated a similar second-order model for multiobjective problems, proved a weak duality theorem and stated that the proofs of strong and converse duality theorems follow on the lines of corresponding theorems in Devi [4]. Since the models and proofs in [4] contain several errors (see [5]), the same have been carried over in Mishra [9], who even did not observe that the proofs of strong and converse duality theorems in [4] are different, while in symmetric duality the statement and the proof of the converse duality theorem go exactly as for the strong duality theorem.
* Corresponding author. Tel.: +91 9837106279. E-mail address: [email protected] (T.R. Gulati). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.12.024
This paper is organized as follows. In the next section we present some relevant preliminaries. In Section 3, we formulate a pair of Wolfe type second-order multiobjective symmetric dual problems with cone constraints and establish weak and strong duality theorems under g-bonvexity assumptions. These two sections also serve to remove several omissions in definitions, models and proofs in [9]. Section 4 contains duality relations for Mond-Weir type symmetric dual models under g-pseudobonvexity assumptions. Self-duality results for these pairs have been stated in Section 5. The last section contains an appendix. 2. Preliminaries Let C 1 and C 2 be closed convex cones with nonempty interiors in Rn and Rm , respectively. For i ¼ 1; 2; C i , called the polar cone of C i , is defined as follows :
C i ¼ fz : xT z5 0 for all x 2 C i g: Suppose that S1 # Rn and S2 # Rm are open sets such that C 1 C 2 S1 S2 . Definition 2.1. [14]. A twice differentiable function f : S1 S2 ! R is said to be g1 -bonvex in the first variable at u 2 S1 , if there exists a function g1 : S1 S1 ! Rn such that for x 2 S1 ; v 2 S2 ; r 2 Rn ,
1 f ðx; v Þ f ðu; v Þ=gT1 ðx; uÞ½rx f ðu; v Þ þ rxx f ðu; v Þr rT rxx f ðu; v Þr 2 and f ðx; yÞ is said to be g2 -bonvex in the second variable at v 2 S2 , if there exists a function g2 : S2 S2 ! Rm such that for u 2 S1 ; y 2 S2 ; p 2 Rm ,
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1 f ðu; yÞ f ðu; v Þ =gT2 ðy; v Þ½ry f ðu; v Þ þ ryy f ðu; v Þp pT ryy f ðu; v Þp: 2
Subject to
Definition 2.2. [14]. A twice differentiable function f : S1 S2 ! R is said to be g1 -pseudobonvex in the first variable at u 2 S1 , if there exists a function g1 : S1 S1 ! Rn such that for x 2 S1 ; v 2 S2 ; r 2 Rn ,
k > 0;
ry ðkT f Þðx; yÞ þ ryy ðkT f Þðx; yÞp 2 C 2 ; Dual (WD):
Maximize Gðu; v ; k; rÞ ¼ f ðu; v Þ ½uT rx ðkT f Þðu; v Þe
rx f ðu; v Þ þ rxx f ðu; v Þr =0
T 1 ðx; uÞ½
g
½uT rxx ðkT f Þðu; v Þre 1 r T ½rxx ðkT f Þðu; v Þre 2
1 ) f ðx; v Þ =f ðu; v Þ rT rxx f ðu; v Þr 2 and f ðx; yÞ is said to be g2 -pseudobonvex in the second variable at v 2 S2 , if there exists a function g2 : S2 S2 ! Rm such that for u 2 S1 ; y 2 S2 ; p 2 Rm ,
Subject to
gT2 ðy; v Þ½ry f ðu; v Þ þ ryy f ðu; v Þp =0
k > 0;
1 ) f ðu; yÞ = f ðu; v Þ pT ryy f ðu; v Þp: 2
ð3:1Þ
kT e ¼ 1;
rx ðkT f Þðu; v Þ rxx ðkT f Þðu; v Þr 2 C 1 ;
ð3:2Þ
kT e ¼ 1;
where f : Rn Rm ! Rk is a twice differentiable p 2 Rm ; r 2 Rn ; k 2 Rk and e ¼ ð1; . . . ; 1Þ 2 Rk .
function,
For r and p to be zero vectors, the above definitions reduce to that of gi -convex/gi -pseudoconvex (i = 1,2) functions. A general multiobjective programming problem can be expressed in the following form :
3.1. The duality results
x 2 S ¼ fx 2 X : gðxÞ50g;
We now establish the duality results for (WP) and (WD). It may be noted that these models contain an additional second-order term in the objective functions as compared to the models in [9]. Without these terms the duality relations proved in this section can not be obtained.
where X is an open subset of Rn and the functions f : X ! Rk and g : X ! Rm are differentiable on X. All the vectors shall be considered as column vectors.
Theorem 3.1 (Weak duality). Let ðx; y; k; pÞ be a feasible solution of the primal problem (WP) and ðu; v ; k; rÞ be a feasible solution of the dual problem (WD). Let
ðPÞ Minimize f ðxÞ ¼ ðf1 ðxÞ; f2 ðxÞ; . . . ; fk ðxÞÞ Subject to
Definition 2.3. A point x 2 S is said to be a weakly efficient solution of (P) if there exists no x 2 S such that f ðxÞ < f ð xÞ. Definition 2.4. A point x 2 S is said to be an efficient (or Pareto optimal) solution of (P) if there exists no x 2 S such that f ðxÞ 6 f ð xÞ. 2 S is said to be a properly efficient soluDefinition 2.5. A point x tion of (P) if it is efficient and if there exists a scalar M > 0 such that xÞ, we have for each i 2 1; 2; . . . ; k and x 2 S satisfying fi ðxÞ < fi ð fi ð xÞfi ðxÞ 5M , for some j such that f ðxÞ > f ð x Þ. j j f ðxÞf ðxÞ j
j
Remark 2.1. It may be noted that gi (i = 1,2) involved in the definitions of second-order invex/second-order pseudoinvex functions in [9] are taken from C i C i ! C i ði ¼ 1; 2Þ. Since every secondorder convex (bonvex) function is second-order invex (g1 -bonvex) with g1 ðx; uÞ ¼ x u, taking g1 : C 1 C 1 ! C 1 amounts to assuming that
x 2 C1;
u 2 C 1 ) g1 ðx; uÞ ¼ x u 2 C 1 ;
which is not true for a closed convex cone C 1 . In particular, if C 1 ¼ Rnþ , then x=0; u =0 does not imply x u =0. 3. Wolfe type second-order multiobjective symmetric duality We now consider the following pair of Wolfe type second-order multiobjective symmetric dual nonlinear programming problems over arbitrary cones: Primal (WP):
Minimize Fðx; y; k; pÞ ¼ f ðx; yÞ ½yT ry ðkT f Þðx; yÞe ½yT ryy ðkT f Þðx; yÞpe 1 pT ½ryy ðkT f Þðx; yÞpe 2
(i) (ii) (iii) (iv)
kT f ð:; v Þ be g1 -bonvex in the first variable at u, kT f ðx; :Þ be g2 -bonvex in the second variable at y, g1 ðx; uÞ þ u 2 C 1 , g2 ðv ; yÞ þ y 2 C 2 .
Then 1 f ðx;yÞ ½yT ry ðkT f Þðx;yÞe ½yT ryy ðkT f Þðx; yÞpe pT ½ryy ðkT f Þðx;yÞpe 2 1 T T T T if ðu; v Þ ½u rx ðk f Þðu; v Þe ½u rxx ðk f Þðu; v Þre r T ½rxx ðkT f Þðu; v Þre: 2
Proof. By g1 -bonvexity ðkT f Þðx; :Þ, we have
of
ðkT f Þð:; v Þ
and
g2 -bonvexity of
ðkT f Þðx; v Þ ðkT f Þðu; v Þ =gT1 ðx; uÞ½rx ðkT f Þðu; v Þ 1 þ rxx ðkT f Þðu; v Þr r T rxx ðkT f Þðu; v Þr; 2 ð3:3Þ ðkT f Þðx; yÞ ðkT f Þðx; v Þ = gT2 ðv ; yÞ½ry ðkT f Þðx; yÞ 1 þ ryy ðkT f Þðx; yÞp þ pT ryy ðkT f Þðx; yÞp: 2 ð3:4Þ Adding inequalities (3.3) and (3.4), we obtain
ðkT f Þðx; yÞ ðkT f Þðu; v Þ =gT1 ðx; uÞfrx ðkT f Þðu; v Þ þ rxx ðkT f Þðu; v Þrg 1 r T rxx ðkT f Þðu; v Þr gT2 ðv ; yÞfry ðkT f Þðx; yÞ 2 1 þ ryy ðkT f Þðx; yÞpg þ pT ryy ðkT f Þðx; yÞp: 2 From the dual constraint (3.2) and hypothesis (iii), we get
ð3:5Þ
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ða ða T eÞkÞ þ ryy ðkT f Þðx; y T eÞy Þðb Þ ry f ðx; yÞða
½g1 ðx; uÞ þ uT frx ðkT f Þðu; v Þ þ rxx ðkT f Þðu; v Þrg =0
T eÞryy ðkT f Þðx; y Þp þ ry fryy ðkT f Þðx; y Þp g ða 1 ða T eÞ y þ p b ¼ 0; 2
or
gT1 ðx; uÞfrx ðkT f Þðu; v Þ þ rxx ðkT f Þðu; v Þrg T
T
= u frx ðk f Þðu; v Þ þ rxx ðk f Þðu; v Þrg: T
ð3:6Þ
Similarly, the primal constraint (3.1) and hypothesis (iv) yield
gT2 ðv ; yÞfry ðkT f Þðx; yÞ þ ryy ðkT f Þðx; yÞpg =yT fry ðkT f Þðx; yÞ þ ryy ðkT f Þðx; yÞpg:
ð3:7Þ
(
T 1 ða T eÞ y þ p b 2 ) T 1 T eÞ y Þp ; . . . ; b ða þ p ryy f1 ðx; y ryy fk ðx; yÞp ¼ 0; 2
T y f ðx; yÞðb
r
ð3:10Þ
T eÞy þl Þ x e þ ða
ð3:11Þ
Using inequalities (3.6) and (3.7) in (3.5), we have
ðkT f Þðx; yÞ ðkT f Þðu; v Þ T
1 r T rxx ðkT f Þðu; v Þr þ yT fry ðkT f Þðx; yÞ 2 1 þ ryy ðkT f Þðx; yÞpg þ pT ryy ðkT f Þðx; yÞp: 2
ð3:8Þ
ð3:12Þ
T ðry ðkT f Þðx; y Þ þ ryy ðkT f Þðx; y Þp Þ ¼ 0; b
ð3:13Þ
T k ¼ 0; x
ð3:14Þ
T
T
Þ=0; ; x ða
T
f ðx; yÞ ½y ry ðk f Þðx; yÞe ½y ryy ðk f Þðx; yÞpe
6 f ðu; v Þ ½uT rx ðkT f Þðu; v Þe ½uT rxx ðkT f Þðu; v Þre 1 ½r T rxx ðkT f Þðx; yÞre: 2
which implies (3.9). Eqs. (3.10) and (3.11) are obtained similarly. =0, Eq. (3.14) yields x ¼ 0. Since k > 0 and x Using hypothesis (i) in (3.12), we get
ðkT f Þðx; yÞ ½yT ry ðkT f Þðx; yÞ ½yT ryy ðkT f Þðx; yÞp 1 ½pT ryy ðkT f Þðx; yÞp 2
¼a T eðy þp Þ: b
ð3:16Þ
¼ 0. Therefore (3.11) yields l ¼ 0, then (3.16) implies b ¼ 0. If a x ;l –0 or a P 0 or ; b; Þ ¼ 0, contradicting (3.15). Hence a Thus ða
< ðkT f Þðu; v Þ ½uT rx ðkT f Þðu; v Þ ½uT rxx ðkT f Þðu; v Þr 1 ½r T rxx ðkT f Þðx; yÞr; 2
a T e > 0:
ð3:17Þ
Using (3.16) and (3.17) in (3.10), we obtain
h
; Þ be a weakly efficient Theorem 3.2 (Strong duality). Let ð x; y k; p solution of (WP). Fix k ¼ k in (WD). Assume that
ryy ððkT f Þðx; yÞÞ is nonsingular, Þ; . . . ; ry fk ð Þ are linearly independent, x; y x; y the vectors ry f1 ð Þp Þp –0, –0 implies ry ðryy ð kT f Þð x; y p Þp Þp R Span fry f1 ð Þ; . . . ; ry fk ð kT f Þð x; y x; y x; the vector ry ðryy ð Þg nf0g. y
; Then ð x; y k; r ¼ 0Þ is feasible for (WD) and the objective function values of (WP) and (WD) are equal. Furthermore, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of the primal and ; dual problems, then ð x; y k; r ¼ 0Þ is a properly efficient solution of (WD). ; Þ is a weakly efficient solution of (WP), by Proof. Since ð x; y k; p the Fritz John optimality condition [3,13], there exist 2 Rk such that 2 R and x a 2 Rk ; b 2 C 2 ; l
ða T f Þðx; y T eÞy Þ þ rxy ðkT f Þðx; y Þðb Þ rx ða 1 ða T eÞ y Þp g b þ p ¼ 0; þ rx fryy ðkT f Þðx; y 2
ð3:15Þ
for all x 2 Rn ;
Since k > 0 and kT e ¼ 1, the above inequality becomes
(i) (ii) (iii) (iv)
l Þ–0:
ða T f Þðx; y T eÞy Þ þ rxy ðkT f Þðx; y Þðb Þ ðx xÞT rx ða 1 ða T eÞ y Þp g b þ p þrx fryy ðkT f Þðx; y =0 2
1 ½pT ryy ðkT f Þðx; yÞpe 2
which is a contradiction to (3.8). Thus the result holds.
2 C 2 ; ða x ; b; ; and b
It may be noted that like Eq. (4) in [13] the main Fritz John necessary optimality condition is a lengthy inequality involving all the terms in Eqs. (3.9)–(3.11) leading to
Now suppose that T
T eÞðy þp ÞÞ ¼ 0; ryy ðkT f Þðx; yÞðb ða T
= u frx ðk f Þðu; v Þ þ rxx ðk f Þðu; v Þrg T
ð3:9Þ
1 2
ða T eÞkÞ þ ry fryy ðkT f Þðx; y T eÞ p ¼ 0; Þp gða ry f ðx; yÞða or
ry fryy ðkT f Þðx; yÞpgp ¼
2 ða T eÞkÞ: Þða ½r f ðx; y T eÞ y ða
ð3:18Þ
¼ 0. Suppose p – 0, then hypothesis (iii) We now claim that p implies
ry ðryy ðkT f Þðx; yÞpÞp–0; which along with (3.18) contradicts the hypothesis (iv). Hence Þ; . . . ; ry fk ðx; y Þg is linearly ¼ 0. Since the set of vectors {ry f1 ð x; y p independent, Eq. (3.18) implies
a ¼ ða T eÞk:
ð3:19Þ
Using Eqs. (3.16) and (3.19) in (3.9), we have
T f Þðx; y T eÞ ¼ 0 2 C 1 : Þ ¼ rx ða Þ=ða rx ðkT f Þðx; y ; Hence ðx; y k; r ¼ 0Þ is feasible for (WD). From (3.13), (3.16) and (3.17), we get
T ry ðkT f Þðx; y Þ ¼ 0: y Also, from (3.20),
ð3:20Þ
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xT rx ðkT f Þðx; y Þ ¼ 0:
k X
Therefore, we have ; ¼ 0Þ ¼ Gð ; Fð x; y k; p x; y k; r ¼ 0Þ, i.e., the two objectives are equal. Finally, similar to the proof of Theorem 2 in [6], one can show ; that ð x; y k; r ¼ 0Þ is a properly efficient solution of (WD).
i¼1
uT
ki ðrx fi ðu; v Þ rxx fi ðu; v Þr i Þ 2 C 1 ;
k X
ki ðrx fi ðu; v Þ þ rxx fi ðu; v Þri Þ50;
i¼1
k > 0;
Remark 3.1. It may be noted that in the statement of the strong duality theorem, Mishra [9] has assumed ryyy ðkT f Þðx; yÞ to be negative definite, which is meaningless. Since ðkT f Þ is a scalar function, so ryyy ðkT f Þðx; yÞ is not a matrix. Moreover, in the secondorder symmetric dual pair Mishra [9] has taken two constraints
where F i ðx; y; pÞ ¼ fi ðx; yÞ 12 pTi ryy fi ðx; yÞpi ,
ry ðkT f Þðx; yÞ 2 C 2 and ryy ðkT f Þðx; yÞp 2 C 2
Also, in this section, p ¼ ðp1 ; p2 ; . . . ; pk Þ and r ¼ ðr 1 ; r 2 ; . . . ; rk Þ. We state the duality relations between these problems.
instead of the single constraint (3.1) in (WP), while in the proof of the strong duality the above two constraints have been taken together in gðzÞ ¼ ry ðkT f Þðx; yÞ þ ryy ðkT f Þðx; yÞp for writing the Fritz John necessary optimality conditions. Remark 3.2. Assumption (iii), which is equivalent to ry ðryy ðkT f Þðx; yÞpÞp ¼ 0 implies p ¼ 0, has been taken as in [5,16,18]. This assumption holds [11], if (iii)0 one of the matrices @ Þ; i ¼ 1; . . . ; m is positive or negative definite. Hence ðkT f Þyy ð x; y @yi assumption (iii) in Theorem 3.2 may be replaced by ðiiiÞ0 . A converse duality theorem may be merely stated as its proof would run analogous to that of Theorem 3.2. ; v ; k; r Þ be a weakly efficient Theorem 3.3 (Converse duality). Let ðu solution of (WD). Fix k ¼ k in (WP). Assume that (i) (ii) (iii) (iv)
; v ÞÞ is nonsingular, rxx ððkT f Þðu ; v Þ; . . . ; rx fk ðu ; v Þ are linearly independent, the vectors rx f1 ðu ; v Þr Þr –0, r–0 implies rx ðrxx ð kT f Þðu ; v Þr Þr R Span frx f1 ðu ; v Þ; . . . ; kT f Þðu the vector rx ðrxx ð ; v Þg n f0g. rx fk ðu
¼ 0Þ is feasible for (WP) and the objective function val; v ; k; p Then ðu ues of (WP) and (WD) are equal. Furthermore, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of the primal and ¼ 0Þ is a properly efficient solution of ; v ; k; p dual problems, then ðu (WP).
4. Mond–Weir type second-order multiobjective symmetric duality In this section we consider the following pair of Mond-Weir type second-order multiobjective symmetric dual nonlinear programming problems over arbitrary cones : Primal (MP):
Minimize Fðx; y; pÞ ¼ ðF 1 ðx; y; pÞ; F 2 ðx; y; pÞ; . . . ; F k ðx; y; pÞÞ Subject to k X
ki ðry fi ðx; yÞ þ ryy fi ðx; yÞpi Þ 2 C 2 ;
i¼1
yT
k X
ki ðry fi ðx; yÞ þ ryy fi ðx; yÞpi Þ=0;
i¼1
k > 0; Dual (MD):
Maximize Gðu; v ; rÞ ¼ ðG1 ðu; v ; rÞ; G2 ðu; v ; rÞ; . . . ; Gk ðu; v ; rÞÞ Subject to
1 Gi ðu; v ; rÞ ¼ fi ðu; v Þ r Ti rxx fi ðu; v Þr i ; 2 ki 2 R; pi 2 Rm ; r i 2 Rn ; i ¼ 1; 2; . . . ; k:
Theorem 4.1 (Weak duality). Let ðx; y; k; pÞ and ðu; v ; k; rÞ be feasible solutions of (MP) and (MD) respectively. Let either of the following conditions hold : (a) for i ¼ 1; 2; . . . ; k; f i be g1 -bonvex in the first variable at u and fi be g2 -bonvex in the second variable at y, or Pk (b) i¼1 ki fi be g1 -pseudobonvex in the first variable at u and P ki¼1 ki fi be g2 -pseudobonvex in the second variable at y. Also, let
g1 ðx; uÞ þ u 2 C 1 ; g2 ðv ; yÞ þ y 2 C 2 : Then
Fðx; y; pÞi Gðu; v ; rÞ: Proof. Follows on the lines of Suneja et al. [14]. h Theorem 4.2 (Strong duality). Let f : Rn Rm ! Rk be thrice differ; Þ be a weakly efficient solution of (MP); Fix k ¼ entiable. Let ð x; y k; p k in (MD) and suppose that (a) either the Hessian matrix ryy fi is positive definite for each Pk T ry fi = 0 or the Hessian matrix ki p i ¼ 1; 2; . . . ; k; and i¼1
i
ryy fi is negative definite for each i ¼ 1; 2; . . . ; k, and Pk T i¼1 ki pi ry fi 5 0, and 1 ; ry f2 þ ryy f2 p 2 ; . . . ; ry fk þ (b) the vectors fry f1 þ ryy f1 p ryy fk pk g are linearly independent, Þ; i ¼ 1; 2; . . . ; k. Then ð ; x; y x; y k; r ¼ 0Þ is feasible for where fi ¼ fi ð (MD) and objective function values of (MP) and (MD) are equal. Furthermore, if the hypotheses of Theorem 4.1 are satisfied for all feasible ; solutions of (MP) and (MD), then ð x; y k; r ¼ 0Þ is a properly efficient solution for (MD). Proof. Follows on the lines of Yang et al. [17] after writing the necessary optimality conditions as in Theorem 3.2. h Theorem 4.3 (Converse duality). Let f : Rn Rm ! Rk be thrice dif; v ; k; r Þ be a weakly efficient solution of (MD). Fix ferentiable. Let ðu k¼ k in (MP) and suppose that (a) either the Hessian matrix rxx fi is positive definite for each Pk T i ¼ 1; 2; . . . ; k, and i¼1 ki r i rx fi = 0 or the Hessian matrix rxx fi is negative definite for each i ¼ 1; 2; . . . ; k, and Pk T i¼1 ki r i rx fi 5 0, and (b) the vectors frx f1 þ rxx f1r1 ; rx f2 þ rxx f2r 2 ; . . . ; rx fk þ rxx fkrk g are linearly independent,
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; v ; v ¼ 0Þ is feasible for Þ; i ¼ 1; 2; . . . ; k. Then ðu ; where fi ¼ fi ðu k; p (MP) and objective function values of (MP) and (MD) are equal. Furthermore, if the hypotheses of Theorem 4.1 are satisfied for all feasible ¼ 0Þ is a properly efficient ; v ; k; p solutions of (MP) and (MD), then ðu solution for (MP).
Definition 6.1. The function f ð; yÞ is said to be second-order Fconvex in the first variable at u 2 S1 , if there exists a sublinear functional F : S1 S1 Rn ! R such that for x 2 S1 ; r 2 Rn ,
Remark 4.1. The duality results of Sections 3 and 4 also hold if the constraints x 2 C 1 and v 2 C 2 are included in the primal and dual models respectively.
and f ðx; Þ is said to be second-order G-concave in the second variable at v 2 S2 , if there exists a sublinear functional G : S2 S2 Rm ! R such that for y 2 S2 ; p 2 Rm ,
5. Self-duality
1 f ðx; yÞ f ðx; v Þ þ pT ryy f ðx; v Þp 5 Gðy; v ; ry f ðx; v Þ þ ryy f ðx; v ÞpÞ: 2
A mathematical problem is said to be self-dual if it is formally identical with its dual, that is, if the dual can be recast in the form of primal, the new problem so obtained is the same as the primal. In general Wolfe and Mond-Weir type primal and dual are not selfdual without an additional restriction on f. Self-duality follows from symmetric duality (see [15]) if m ¼ n; C 1 ¼ C 2 and the vector function f is skew-symmetric, i.e., f ðy; xÞ ¼ f ðx; yÞ or fi ðy; xÞ ¼ fi ðx; yÞ; i ¼ 1; 2; . . . ; k. Theorem 5.1 (Self-duality). Let m ¼ n; C 1 ¼ C 2 and f be skew symmetric. Then (WP) is self-dual. Furthermore, if (WP) and (WD) ; Þ is a joint properly efficient solution, are dual problems and ð x; y k; p ; Þ and the common optimal value of the objective then so is ðy x; k; p functions is 0. Theorem 5.2 (Self-duality). Let m ¼ n; C 1 ¼ C 2 and f be skew symmetric. Then (MP) is self-dual. Furthermore, if (MP) and (MD) are dual ; Þ is a joint properly efficient solution, then so is problems and ð x; y k; p ; Þ and Fð ; p Þ ¼ Gðy ; Þ ¼ 0. ðy x; k; p x; y x; p
1 f ðx; yÞ f ðu; yÞ þ r T rxx f ðu; yÞr = Fðx; u; rx f ðu; yÞ þ rxx f ðu; yÞrÞ; 2
In [10], vectors p 2 Rm and r 2 Rn have been taken as variables in the primal and dual problems, while in the definitions and proofs, they have been treated as functions of x and y. In the weak duality theorem, Mishra [10] assumed conditions like
Fðx; u; n1 þ n2 Þ þ uT n1 þ uT n2 P 0 for n1 2 Rnþ ; n2 2 Rnþ ; and used it in the proof taking n1 ¼ rx f ðu; v Þ 2 Rnþ and n2 ¼ rxx f ðu; v Þr 2 Rnþ . The values of n1 and n2 as taken above are not in Rnþ : However, in view of the dual constraint (6.4), n1 þ n2 is in Rnþ . Similar mistakes occur throughout the paper. Theorem 6.1 (Weak duality). Let ðx; y; pÞ be feasible for the primal problem (WP1) and ðu; v ; rÞ be feasible for the dual problem (WD1). Let (i) (ii) (iii) (iv)
f ð; v Þ be second-order F-convex in the first variable at u, f ðx; Þ be second-order G-concave in the second variable at y, Fðx; u; nÞ þ uT n=0; for all n 2 Rnþ and Gðv ; y; gÞ þ yT g=0; for all g 2 Rm þ.
Then
Acknowledgements
Fðx; y; pÞ=Gðu; v ; rÞ: The authors are thankful to the reviewers for their comments and suggestions, in particular, for the addition of the appendix. The second author is also thankful to the University Grants Commission, New Delhi (India) for providing financial support during this work.
Proof. By second-order F-convexity of f ð; v Þ and G-concavity of f ðx; Þ, we have
1 f ðx; v Þ f ðu; v Þ þ rT rxx f ðu; v Þr = Fðx; u; rx f ðu; v Þ þ rxx f ðu; v ÞrÞ; 2
Appendix A This appendix has been added as per suggestion of a reviewer to Þ to be negative definite for x; y correct the mistake of assuming ryyy f ð a scalar function f ðx; yÞ by Mishra [10, Theorem 2 ]. The second-order Wolfe type symmetric dual programs considered in [10] are :
and
1 f ðx; yÞ f ðx; v Þ pT ryy f ðx; yÞp = Gðv ; y; ry f ðx; yÞ 2 ryy f ðx; yÞpÞ: Adding these inequalities, we get
Primal (WP1):
1 1 f ðx; yÞ f ðu; v Þ þ r T rxx f ðu; v Þr pT ryy f ðx; yÞp = 2 2 Fðx; u; rx f ðu; v Þ þ rxx f ðu; v ÞrÞ þ Gðv ; y; ry f ðx; yÞ ryy f ðx; yÞpÞ:
Minimize Fðx; y; pÞ ¼ f ðx; yÞ yT ry f ðx; yÞ yT ryy f ðx; yÞp 1 pT ryy f ðx; yÞp 2 Subject to ry f ðx; yÞ þ ryy f ðx; yÞp 5 0;
ð6:2Þ
ð6:6Þ
x=0;
ð6:3Þ
Now, by the dual constraint (6.4), n ¼ rx f ðu; v Þ þ rxx f ðu; v Þr 2 Rnþ and so from the hypothesis (iii), we have
ð6:1Þ
Dual (WD1):
Maximize Gðu; v ; rÞ ¼ f ðu; v Þ uT rx f ðu; v Þ uT rxx f ðu; v Þr 1 rT rxx f ðu; v Þr 2 Subject to rx f ðu; v Þ þ rxx f ðu; v Þr = 0; ð6:4Þ
v =0;
ð6:5Þ
where f : S1 S2 ! R is a thrice differentiable function, p 2 Rm ; r 2 Rn and S1 # Rn ; S2 # Rm are open sets. Duality for these models was first studied in 1974 by Mond [11].
Fðx; u; rx f ðu; v Þ þ rxx f ðu; v ÞrÞ þ uT ½rx f ðu; v Þ þ rxx f ðu; v Þr =0; or
Fðx; u; rx f ðu; v Þ þ rxx f ðu; v ÞrÞ = uT ½rx f ðu; v Þ þ rxx f ðu; v Þr: Similarly, g ¼ ry f ðx; yÞ ryy f ðx; yÞp 2 sis (iv) yields
ð6:7Þ Rm þ
and therefore hypothe-
Gðv ; y; ry f ðx; yÞ ryy f ðx; yÞpÞ=yT ½ry f ðx; yÞ þ ryy f ðx; yÞp:
ð6:8Þ
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Inequalities (6.6)–(6.8) give
1 1 f ðx; yÞ f ðu; v Þ þ rT rxx f ðu; v Þr pT ryy f ðx; yÞp 2 2 = uT ½rx f ðu; v Þ þ rxx f ðu; v Þr þ yT ½ry f ðx; yÞ þ ryy f ðx; yÞp; or
1 f ðx; yÞ yT ½ry f ðx; yÞ þ ryy f ðx; yÞp pT ryy f ðx; yÞp 2 1 T =f ðu; v Þ u ½rx f ðu; v Þ þ rxx f ðu; v Þr r T rxx f ðu; v Þr; 2 or
Fðx; y; pÞ = Gðu; v ; rÞ: Once the weak duality theorem for (WP1) and (WD1) is established, the strong duality follows as in Mond [11]. Below we state the strong duality theorem. Its proof is exactly same as given for Theorem 6 in [11]. Also, in view of Remark 3.2, the proof can be obtained on the lines of Theorem 2 in Yang et al. [16]. h ; p Þ be a local optimal soluTheorem 6.2 (Strong duality). Let ð x; y tion for (WP1). If Þ is nonsingular, and x; y (i) ryy f ð Þ; i ¼ 1; . . . ; m is positive or negax; y (ii) one of the matrices @y@ fyy ð i ¼ 0; ð ; r ¼ 0Þ is feasible for the dual tive definite, then p x; y problem (WD1) and the objective function values of (WP1) and (WD1) are equal. Furthermore, if the hypotheses of Theorem 6.1 are satisfied for all feasible solutions of the problems ; p ¼ 0Þ and ð ; r ¼ 0Þ are global (WP1) and (WD1), then ð x; y x; y optimal solutions for (WP1) and (WD1) respectively.
References [1] C.R. Bector, S. Chandra, Generalized bonvex functions and second-order duality in mathematical programming, Research Report 85-2. The University of Manitoba, Winnipeg, Manitoba January 1985.
[2] C.R. Bector, S. Chandra, Generalized bonvexity and higher order duality for fractional programming, Opsearch 24 (1987) 143–154. [3] B.D. Craven, Lagrangian conditions and quasi duality, Bulletin of the Australian Mathematical Society 16 (1977) 325–339. [4] G. Devi, Symmetric duality for nonlinear-programming problem involving gbonvex functions, European Journal of Operational Research 104 (1998) 615– 621. [5] T.R. Gulati, S.K. Gupta, I. Ahmad, Second-order symmetric duality with cone constraints, Journal of Computational and Applied Mathematics 220 (2008) 347–354. [6] T.R. Gulati, I. Husain, A. Ahmed, Multiobjective symmetric duality with invexity, Bulletin of the Australian Mathematical Society 56 (1997) 25–36. [7] M.A. Hanson, Second-order invexity and duality in mathematical programming, Opsearch 30 (4) (1993) 313–320. [8] O.L. Mangasarian, Second and higher-order duality in nonlinear programming, Journal of Mathematical Analysis and Applications 51 (1975) 607–620. [9] S.K. Mishra, Multiobjective second-order symmetric duality with cone constraints, European Journal of Operational Research 126 (2000) 675–682. [10] S.K. Mishra, Second order symmetric duality in mathematical programming with F-convexity, European Journal of Operational Research 127 (2000) 507– 518. [11] B. Mond, Second order duality for nonlinear programs, Opsearch 11 (1974) 90– 99. [12] 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. [13] S.K. Suneja, S. Aggarwal, S. Davar, Multiobjective symmetric duality involving cones, European Journal of Operational Research 141 (2002) 471–479. [14] S.K. Suneja, C.S. Lalitha, S. Khurana, Second order symmetric duality in multiobjective programming, European Journal of Operational Research 144 (2003) 492–500. [15] T. Weir, B. Mond, Symmetric and self duality in multiple objective programming, Asia-Pacific Journal of Operational Research 5 (1988) 124–133. [16] X.M. Yang, X.Q. Yang, K.L. Teo, Nondifferentiable second order symmetric duality in mathematical programming with F-convexity, European Journal of Operational Research 144 (2003) 554–559. [17] X.M. Yang, X.Q. Yang, K.L. Teo, S.H. Hou, Second order symmetric duality in non-differentiable multiobjective programming with F-convexity, European Journal of Operational Research 164 (2005) 406–416. [18] X.M. Yang, X.Q. Yang, K.L. Teo, S.H. Hou, Multiobjective second-order symmetric duality with F-convexity, European Journal of Operational Research 165 (2005) 585–591.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Continuous Optimization
Second-order multiobjective symmetric duality with cone constraints T.R. Gulati a,*, Himani Saini a, S.K. Gupta b a b
Department of Mathematics, Indian Institute of Technology, Roorkee 247 667, India Department of Mathematics, Indian Institute of Technology, Patna 800 013, India
a r t i c l e
i n f o
Article history: Received 18 June 2008 Accepted 26 December 2009 Available online 13 January 2010 Keywords: Multiobjective symmetric duality g-bonvexity/g-pseudobonvexity Cones Efficient solutions Properly efficient solutions
a b s t r a c t In this paper, we formulate Wolfe and Mond–Weir type second-order multiobjective symmetric dual problems over arbitrary cones. Weak, strong and converse duality theorems are established under g-bonvexity/g-pseudobonvexity assumptions. This work also removes several omissions in definitions, models and proofs for Wolfe type problems studied in Mishra [9]. Moreover, self-duality theorems for these pairs are obtained assuming the function involved to be skew symmetric. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Mangasarian [8] introduced the concept of second-order duality for nonlinear problems. Its study is significant due to computational advantage over first-order duality as it provides tighter bounds for the value of the objective function when approximations are used [7,8,11]. Bector and Chandra [2] introduced the concept of bonvex functions. Mond [11] established second-order duality for nonlinear programs under second-order convexity assumptions. Later on, Bector and Chandra [1] formulated second-order symmetric dual programs in the spirit of Mond and Weir [12] and established appropriate duality results involving pseudobonvex functions. Recently, Yang et al. [16,18] studied second-order symmetric dual programs and established duality relations under F-convexity assumptions. Devi [4] formulated a pair of second-order symmetric dual nonlinear programming problems over arbitrary cones under gpseudobonvexity assumptions. Mishra [9] formulated a similar second-order model for multiobjective problems, proved a weak duality theorem and stated that the proofs of strong and converse duality theorems follow on the lines of corresponding theorems in Devi [4]. Since the models and proofs in [4] contain several errors (see [5]), the same have been carried over in Mishra [9], who even did not observe that the proofs of strong and converse duality theorems in [4] are different, while in symmetric duality the statement and the proof of the converse duality theorem go exactly as for the strong duality theorem.
* Corresponding author. Tel.: +91 9837106279. E-mail address: [email protected] (T.R. Gulati). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.12.024
This paper is organized as follows. In the next section we present some relevant preliminaries. In Section 3, we formulate a pair of Wolfe type second-order multiobjective symmetric dual problems with cone constraints and establish weak and strong duality theorems under g-bonvexity assumptions. These two sections also serve to remove several omissions in definitions, models and proofs in [9]. Section 4 contains duality relations for Mond-Weir type symmetric dual models under g-pseudobonvexity assumptions. Self-duality results for these pairs have been stated in Section 5. The last section contains an appendix. 2. Preliminaries Let C 1 and C 2 be closed convex cones with nonempty interiors in Rn and Rm , respectively. For i ¼ 1; 2; C i , called the polar cone of C i , is defined as follows :
C i ¼ fz : xT z5 0 for all x 2 C i g: Suppose that S1 # Rn and S2 # Rm are open sets such that C 1 C 2 S1 S2 . Definition 2.1. [14]. A twice differentiable function f : S1 S2 ! R is said to be g1 -bonvex in the first variable at u 2 S1 , if there exists a function g1 : S1 S1 ! Rn such that for x 2 S1 ; v 2 S2 ; r 2 Rn ,
1 f ðx; v Þ f ðu; v Þ=gT1 ðx; uÞ½rx f ðu; v Þ þ rxx f ðu; v Þr rT rxx f ðu; v Þr 2 and f ðx; yÞ is said to be g2 -bonvex in the second variable at v 2 S2 , if there exists a function g2 : S2 S2 ! Rm such that for u 2 S1 ; y 2 S2 ; p 2 Rm ,
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1 f ðu; yÞ f ðu; v Þ =gT2 ðy; v Þ½ry f ðu; v Þ þ ryy f ðu; v Þp pT ryy f ðu; v Þp: 2
Subject to
Definition 2.2. [14]. A twice differentiable function f : S1 S2 ! R is said to be g1 -pseudobonvex in the first variable at u 2 S1 , if there exists a function g1 : S1 S1 ! Rn such that for x 2 S1 ; v 2 S2 ; r 2 Rn ,
k > 0;
ry ðkT f Þðx; yÞ þ ryy ðkT f Þðx; yÞp 2 C 2 ; Dual (WD):
Maximize Gðu; v ; k; rÞ ¼ f ðu; v Þ ½uT rx ðkT f Þðu; v Þe
rx f ðu; v Þ þ rxx f ðu; v Þr =0
T 1 ðx; uÞ½
g
½uT rxx ðkT f Þðu; v Þre 1 r T ½rxx ðkT f Þðu; v Þre 2
1 ) f ðx; v Þ =f ðu; v Þ rT rxx f ðu; v Þr 2 and f ðx; yÞ is said to be g2 -pseudobonvex in the second variable at v 2 S2 , if there exists a function g2 : S2 S2 ! Rm such that for u 2 S1 ; y 2 S2 ; p 2 Rm ,
Subject to
gT2 ðy; v Þ½ry f ðu; v Þ þ ryy f ðu; v Þp =0
k > 0;
1 ) f ðu; yÞ = f ðu; v Þ pT ryy f ðu; v Þp: 2
ð3:1Þ
kT e ¼ 1;
rx ðkT f Þðu; v Þ rxx ðkT f Þðu; v Þr 2 C 1 ;
ð3:2Þ
kT e ¼ 1;
where f : Rn Rm ! Rk is a twice differentiable p 2 Rm ; r 2 Rn ; k 2 Rk and e ¼ ð1; . . . ; 1Þ 2 Rk .
function,
For r and p to be zero vectors, the above definitions reduce to that of gi -convex/gi -pseudoconvex (i = 1,2) functions. A general multiobjective programming problem can be expressed in the following form :
3.1. The duality results
x 2 S ¼ fx 2 X : gðxÞ50g;
We now establish the duality results for (WP) and (WD). It may be noted that these models contain an additional second-order term in the objective functions as compared to the models in [9]. Without these terms the duality relations proved in this section can not be obtained.
where X is an open subset of Rn and the functions f : X ! Rk and g : X ! Rm are differentiable on X. All the vectors shall be considered as column vectors.
Theorem 3.1 (Weak duality). Let ðx; y; k; pÞ be a feasible solution of the primal problem (WP) and ðu; v ; k; rÞ be a feasible solution of the dual problem (WD). Let
ðPÞ Minimize f ðxÞ ¼ ðf1 ðxÞ; f2 ðxÞ; . . . ; fk ðxÞÞ Subject to
Definition 2.3. A point x 2 S is said to be a weakly efficient solution of (P) if there exists no x 2 S such that f ðxÞ < f ð xÞ. Definition 2.4. A point x 2 S is said to be an efficient (or Pareto optimal) solution of (P) if there exists no x 2 S such that f ðxÞ 6 f ð xÞ. 2 S is said to be a properly efficient soluDefinition 2.5. A point x tion of (P) if it is efficient and if there exists a scalar M > 0 such that xÞ, we have for each i 2 1; 2; . . . ; k and x 2 S satisfying fi ðxÞ < fi ð fi ð xÞfi ðxÞ 5M , for some j such that f ðxÞ > f ð x Þ. j j f ðxÞf ðxÞ j
j
Remark 2.1. It may be noted that gi (i = 1,2) involved in the definitions of second-order invex/second-order pseudoinvex functions in [9] are taken from C i C i ! C i ði ¼ 1; 2Þ. Since every secondorder convex (bonvex) function is second-order invex (g1 -bonvex) with g1 ðx; uÞ ¼ x u, taking g1 : C 1 C 1 ! C 1 amounts to assuming that
x 2 C1;
u 2 C 1 ) g1 ðx; uÞ ¼ x u 2 C 1 ;
which is not true for a closed convex cone C 1 . In particular, if C 1 ¼ Rnþ , then x=0; u =0 does not imply x u =0. 3. Wolfe type second-order multiobjective symmetric duality We now consider the following pair of Wolfe type second-order multiobjective symmetric dual nonlinear programming problems over arbitrary cones: Primal (WP):
Minimize Fðx; y; k; pÞ ¼ f ðx; yÞ ½yT ry ðkT f Þðx; yÞe ½yT ryy ðkT f Þðx; yÞpe 1 pT ½ryy ðkT f Þðx; yÞpe 2
(i) (ii) (iii) (iv)
kT f ð:; v Þ be g1 -bonvex in the first variable at u, kT f ðx; :Þ be g2 -bonvex in the second variable at y, g1 ðx; uÞ þ u 2 C 1 , g2 ðv ; yÞ þ y 2 C 2 .
Then 1 f ðx;yÞ ½yT ry ðkT f Þðx;yÞe ½yT ryy ðkT f Þðx; yÞpe pT ½ryy ðkT f Þðx;yÞpe 2 1 T T T T if ðu; v Þ ½u rx ðk f Þðu; v Þe ½u rxx ðk f Þðu; v Þre r T ½rxx ðkT f Þðu; v Þre: 2
Proof. By g1 -bonvexity ðkT f Þðx; :Þ, we have
of
ðkT f Þð:; v Þ
and
g2 -bonvexity of
ðkT f Þðx; v Þ ðkT f Þðu; v Þ =gT1 ðx; uÞ½rx ðkT f Þðu; v Þ 1 þ rxx ðkT f Þðu; v Þr r T rxx ðkT f Þðu; v Þr; 2 ð3:3Þ ðkT f Þðx; yÞ ðkT f Þðx; v Þ = gT2 ðv ; yÞ½ry ðkT f Þðx; yÞ 1 þ ryy ðkT f Þðx; yÞp þ pT ryy ðkT f Þðx; yÞp: 2 ð3:4Þ Adding inequalities (3.3) and (3.4), we obtain
ðkT f Þðx; yÞ ðkT f Þðu; v Þ =gT1 ðx; uÞfrx ðkT f Þðu; v Þ þ rxx ðkT f Þðu; v Þrg 1 r T rxx ðkT f Þðu; v Þr gT2 ðv ; yÞfry ðkT f Þðx; yÞ 2 1 þ ryy ðkT f Þðx; yÞpg þ pT ryy ðkT f Þðx; yÞp: 2 From the dual constraint (3.2) and hypothesis (iii), we get
ð3:5Þ
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ða ða T eÞkÞ þ ryy ðkT f Þðx; y T eÞy Þðb Þ ry f ðx; yÞða
½g1 ðx; uÞ þ uT frx ðkT f Þðu; v Þ þ rxx ðkT f Þðu; v Þrg =0
T eÞryy ðkT f Þðx; y Þp þ ry fryy ðkT f Þðx; y Þp g ða 1 ða T eÞ y þ p b ¼ 0; 2
or
gT1 ðx; uÞfrx ðkT f Þðu; v Þ þ rxx ðkT f Þðu; v Þrg T
T
= u frx ðk f Þðu; v Þ þ rxx ðk f Þðu; v Þrg: T
ð3:6Þ
Similarly, the primal constraint (3.1) and hypothesis (iv) yield
gT2 ðv ; yÞfry ðkT f Þðx; yÞ þ ryy ðkT f Þðx; yÞpg =yT fry ðkT f Þðx; yÞ þ ryy ðkT f Þðx; yÞpg:
ð3:7Þ
(
T 1 ða T eÞ y þ p b 2 ) T 1 T eÞ y Þp ; . . . ; b ða þ p ryy f1 ðx; y ryy fk ðx; yÞp ¼ 0; 2
T y f ðx; yÞðb
r
ð3:10Þ
T eÞy þl Þ x e þ ða
ð3:11Þ
Using inequalities (3.6) and (3.7) in (3.5), we have
ðkT f Þðx; yÞ ðkT f Þðu; v Þ T
1 r T rxx ðkT f Þðu; v Þr þ yT fry ðkT f Þðx; yÞ 2 1 þ ryy ðkT f Þðx; yÞpg þ pT ryy ðkT f Þðx; yÞp: 2
ð3:8Þ
ð3:12Þ
T ðry ðkT f Þðx; y Þ þ ryy ðkT f Þðx; y Þp Þ ¼ 0; b
ð3:13Þ
T k ¼ 0; x
ð3:14Þ
T
T
Þ=0; ; x ða
T
f ðx; yÞ ½y ry ðk f Þðx; yÞe ½y ryy ðk f Þðx; yÞpe
6 f ðu; v Þ ½uT rx ðkT f Þðu; v Þe ½uT rxx ðkT f Þðu; v Þre 1 ½r T rxx ðkT f Þðx; yÞre: 2
which implies (3.9). Eqs. (3.10) and (3.11) are obtained similarly. =0, Eq. (3.14) yields x ¼ 0. Since k > 0 and x Using hypothesis (i) in (3.12), we get
ðkT f Þðx; yÞ ½yT ry ðkT f Þðx; yÞ ½yT ryy ðkT f Þðx; yÞp 1 ½pT ryy ðkT f Þðx; yÞp 2
¼a T eðy þp Þ: b
ð3:16Þ
¼ 0. Therefore (3.11) yields l ¼ 0, then (3.16) implies b ¼ 0. If a x ;l –0 or a P 0 or ; b; Þ ¼ 0, contradicting (3.15). Hence a Thus ða
< ðkT f Þðu; v Þ ½uT rx ðkT f Þðu; v Þ ½uT rxx ðkT f Þðu; v Þr 1 ½r T rxx ðkT f Þðx; yÞr; 2
a T e > 0:
ð3:17Þ
Using (3.16) and (3.17) in (3.10), we obtain
h
; Þ be a weakly efficient Theorem 3.2 (Strong duality). Let ð x; y k; p solution of (WP). Fix k ¼ k in (WD). Assume that
ryy ððkT f Þðx; yÞÞ is nonsingular, Þ; . . . ; ry fk ð Þ are linearly independent, x; y x; y the vectors ry f1 ð Þp Þp –0, –0 implies ry ðryy ð kT f Þð x; y p Þp Þp R Span fry f1 ð Þ; . . . ; ry fk ð kT f Þð x; y x; y x; the vector ry ðryy ð Þg nf0g. y
; Then ð x; y k; r ¼ 0Þ is feasible for (WD) and the objective function values of (WP) and (WD) are equal. Furthermore, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of the primal and ; dual problems, then ð x; y k; r ¼ 0Þ is a properly efficient solution of (WD). ; Þ is a weakly efficient solution of (WP), by Proof. Since ð x; y k; p the Fritz John optimality condition [3,13], there exist 2 Rk such that 2 R and x a 2 Rk ; b 2 C 2 ; l
ða T f Þðx; y T eÞy Þ þ rxy ðkT f Þðx; y Þðb Þ rx ða 1 ða T eÞ y Þp g b þ p ¼ 0; þ rx fryy ðkT f Þðx; y 2
ð3:15Þ
for all x 2 Rn ;
Since k > 0 and kT e ¼ 1, the above inequality becomes
(i) (ii) (iii) (iv)
l Þ–0:
ða T f Þðx; y T eÞy Þ þ rxy ðkT f Þðx; y Þðb Þ ðx xÞT rx ða 1 ða T eÞ y Þp g b þ p þrx fryy ðkT f Þðx; y =0 2
1 ½pT ryy ðkT f Þðx; yÞpe 2
which is a contradiction to (3.8). Thus the result holds.
2 C 2 ; ða x ; b; ; and b
It may be noted that like Eq. (4) in [13] the main Fritz John necessary optimality condition is a lengthy inequality involving all the terms in Eqs. (3.9)–(3.11) leading to
Now suppose that T
T eÞðy þp ÞÞ ¼ 0; ryy ðkT f Þðx; yÞðb ða T
= u frx ðk f Þðu; v Þ þ rxx ðk f Þðu; v Þrg T
ð3:9Þ
1 2
ða T eÞkÞ þ ry fryy ðkT f Þðx; y T eÞ p ¼ 0; Þp gða ry f ðx; yÞða or
ry fryy ðkT f Þðx; yÞpgp ¼
2 ða T eÞkÞ: Þða ½r f ðx; y T eÞ y ða
ð3:18Þ
¼ 0. Suppose p – 0, then hypothesis (iii) We now claim that p implies
ry ðryy ðkT f Þðx; yÞpÞp–0; which along with (3.18) contradicts the hypothesis (iv). Hence Þ; . . . ; ry fk ðx; y Þg is linearly ¼ 0. Since the set of vectors {ry f1 ð x; y p independent, Eq. (3.18) implies
a ¼ ða T eÞk:
ð3:19Þ
Using Eqs. (3.16) and (3.19) in (3.9), we have
T f Þðx; y T eÞ ¼ 0 2 C 1 : Þ ¼ rx ða Þ=ða rx ðkT f Þðx; y ; Hence ðx; y k; r ¼ 0Þ is feasible for (WD). From (3.13), (3.16) and (3.17), we get
T ry ðkT f Þðx; y Þ ¼ 0: y Also, from (3.20),
ð3:20Þ
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xT rx ðkT f Þðx; y Þ ¼ 0:
k X
Therefore, we have ; ¼ 0Þ ¼ Gð ; Fð x; y k; p x; y k; r ¼ 0Þ, i.e., the two objectives are equal. Finally, similar to the proof of Theorem 2 in [6], one can show ; that ð x; y k; r ¼ 0Þ is a properly efficient solution of (WD).
i¼1
uT
ki ðrx fi ðu; v Þ rxx fi ðu; v Þr i Þ 2 C 1 ;
k X
ki ðrx fi ðu; v Þ þ rxx fi ðu; v Þri Þ50;
i¼1
k > 0;
Remark 3.1. It may be noted that in the statement of the strong duality theorem, Mishra [9] has assumed ryyy ðkT f Þðx; yÞ to be negative definite, which is meaningless. Since ðkT f Þ is a scalar function, so ryyy ðkT f Þðx; yÞ is not a matrix. Moreover, in the secondorder symmetric dual pair Mishra [9] has taken two constraints
where F i ðx; y; pÞ ¼ fi ðx; yÞ 12 pTi ryy fi ðx; yÞpi ,
ry ðkT f Þðx; yÞ 2 C 2 and ryy ðkT f Þðx; yÞp 2 C 2
Also, in this section, p ¼ ðp1 ; p2 ; . . . ; pk Þ and r ¼ ðr 1 ; r 2 ; . . . ; rk Þ. We state the duality relations between these problems.
instead of the single constraint (3.1) in (WP), while in the proof of the strong duality the above two constraints have been taken together in gðzÞ ¼ ry ðkT f Þðx; yÞ þ ryy ðkT f Þðx; yÞp for writing the Fritz John necessary optimality conditions. Remark 3.2. Assumption (iii), which is equivalent to ry ðryy ðkT f Þðx; yÞpÞp ¼ 0 implies p ¼ 0, has been taken as in [5,16,18]. This assumption holds [11], if (iii)0 one of the matrices @ Þ; i ¼ 1; . . . ; m is positive or negative definite. Hence ðkT f Þyy ð x; y @yi assumption (iii) in Theorem 3.2 may be replaced by ðiiiÞ0 . A converse duality theorem may be merely stated as its proof would run analogous to that of Theorem 3.2. ; v ; k; r Þ be a weakly efficient Theorem 3.3 (Converse duality). Let ðu solution of (WD). Fix k ¼ k in (WP). Assume that (i) (ii) (iii) (iv)
; v ÞÞ is nonsingular, rxx ððkT f Þðu ; v Þ; . . . ; rx fk ðu ; v Þ are linearly independent, the vectors rx f1 ðu ; v Þr Þr –0, r–0 implies rx ðrxx ð kT f Þðu ; v Þr Þr R Span frx f1 ðu ; v Þ; . . . ; kT f Þðu the vector rx ðrxx ð ; v Þg n f0g. rx fk ðu
¼ 0Þ is feasible for (WP) and the objective function val; v ; k; p Then ðu ues of (WP) and (WD) are equal. Furthermore, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of the primal and ¼ 0Þ is a properly efficient solution of ; v ; k; p dual problems, then ðu (WP).
4. Mond–Weir type second-order multiobjective symmetric duality In this section we consider the following pair of Mond-Weir type second-order multiobjective symmetric dual nonlinear programming problems over arbitrary cones : Primal (MP):
Minimize Fðx; y; pÞ ¼ ðF 1 ðx; y; pÞ; F 2 ðx; y; pÞ; . . . ; F k ðx; y; pÞÞ Subject to k X
ki ðry fi ðx; yÞ þ ryy fi ðx; yÞpi Þ 2 C 2 ;
i¼1
yT
k X
ki ðry fi ðx; yÞ þ ryy fi ðx; yÞpi Þ=0;
i¼1
k > 0; Dual (MD):
Maximize Gðu; v ; rÞ ¼ ðG1 ðu; v ; rÞ; G2 ðu; v ; rÞ; . . . ; Gk ðu; v ; rÞÞ Subject to
1 Gi ðu; v ; rÞ ¼ fi ðu; v Þ r Ti rxx fi ðu; v Þr i ; 2 ki 2 R; pi 2 Rm ; r i 2 Rn ; i ¼ 1; 2; . . . ; k:
Theorem 4.1 (Weak duality). Let ðx; y; k; pÞ and ðu; v ; k; rÞ be feasible solutions of (MP) and (MD) respectively. Let either of the following conditions hold : (a) for i ¼ 1; 2; . . . ; k; f i be g1 -bonvex in the first variable at u and fi be g2 -bonvex in the second variable at y, or Pk (b) i¼1 ki fi be g1 -pseudobonvex in the first variable at u and P ki¼1 ki fi be g2 -pseudobonvex in the second variable at y. Also, let
g1 ðx; uÞ þ u 2 C 1 ; g2 ðv ; yÞ þ y 2 C 2 : Then
Fðx; y; pÞi Gðu; v ; rÞ: Proof. Follows on the lines of Suneja et al. [14]. h Theorem 4.2 (Strong duality). Let f : Rn Rm ! Rk be thrice differ; Þ be a weakly efficient solution of (MP); Fix k ¼ entiable. Let ð x; y k; p k in (MD) and suppose that (a) either the Hessian matrix ryy fi is positive definite for each Pk T ry fi = 0 or the Hessian matrix ki p i ¼ 1; 2; . . . ; k; and i¼1
i
ryy fi is negative definite for each i ¼ 1; 2; . . . ; k, and Pk T i¼1 ki pi ry fi 5 0, and 1 ; ry f2 þ ryy f2 p 2 ; . . . ; ry fk þ (b) the vectors fry f1 þ ryy f1 p ryy fk pk g are linearly independent, Þ; i ¼ 1; 2; . . . ; k. Then ð ; x; y x; y k; r ¼ 0Þ is feasible for where fi ¼ fi ð (MD) and objective function values of (MP) and (MD) are equal. Furthermore, if the hypotheses of Theorem 4.1 are satisfied for all feasible ; solutions of (MP) and (MD), then ð x; y k; r ¼ 0Þ is a properly efficient solution for (MD). Proof. Follows on the lines of Yang et al. [17] after writing the necessary optimality conditions as in Theorem 3.2. h Theorem 4.3 (Converse duality). Let f : Rn Rm ! Rk be thrice dif; v ; k; r Þ be a weakly efficient solution of (MD). Fix ferentiable. Let ðu k¼ k in (MP) and suppose that (a) either the Hessian matrix rxx fi is positive definite for each Pk T i ¼ 1; 2; . . . ; k, and i¼1 ki r i rx fi = 0 or the Hessian matrix rxx fi is negative definite for each i ¼ 1; 2; . . . ; k, and Pk T i¼1 ki r i rx fi 5 0, and (b) the vectors frx f1 þ rxx f1r1 ; rx f2 þ rxx f2r 2 ; . . . ; rx fk þ rxx fkrk g are linearly independent,
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; v ; v ¼ 0Þ is feasible for Þ; i ¼ 1; 2; . . . ; k. Then ðu ; where fi ¼ fi ðu k; p (MP) and objective function values of (MP) and (MD) are equal. Furthermore, if the hypotheses of Theorem 4.1 are satisfied for all feasible ¼ 0Þ is a properly efficient ; v ; k; p solutions of (MP) and (MD), then ðu solution for (MP).
Definition 6.1. The function f ð; yÞ is said to be second-order Fconvex in the first variable at u 2 S1 , if there exists a sublinear functional F : S1 S1 Rn ! R such that for x 2 S1 ; r 2 Rn ,
Remark 4.1. The duality results of Sections 3 and 4 also hold if the constraints x 2 C 1 and v 2 C 2 are included in the primal and dual models respectively.
and f ðx; Þ is said to be second-order G-concave in the second variable at v 2 S2 , if there exists a sublinear functional G : S2 S2 Rm ! R such that for y 2 S2 ; p 2 Rm ,
5. Self-duality
1 f ðx; yÞ f ðx; v Þ þ pT ryy f ðx; v Þp 5 Gðy; v ; ry f ðx; v Þ þ ryy f ðx; v ÞpÞ: 2
A mathematical problem is said to be self-dual if it is formally identical with its dual, that is, if the dual can be recast in the form of primal, the new problem so obtained is the same as the primal. In general Wolfe and Mond-Weir type primal and dual are not selfdual without an additional restriction on f. Self-duality follows from symmetric duality (see [15]) if m ¼ n; C 1 ¼ C 2 and the vector function f is skew-symmetric, i.e., f ðy; xÞ ¼ f ðx; yÞ or fi ðy; xÞ ¼ fi ðx; yÞ; i ¼ 1; 2; . . . ; k. Theorem 5.1 (Self-duality). Let m ¼ n; C 1 ¼ C 2 and f be skew symmetric. Then (WP) is self-dual. Furthermore, if (WP) and (WD) ; Þ is a joint properly efficient solution, are dual problems and ð x; y k; p ; Þ and the common optimal value of the objective then so is ðy x; k; p functions is 0. Theorem 5.2 (Self-duality). Let m ¼ n; C 1 ¼ C 2 and f be skew symmetric. Then (MP) is self-dual. Furthermore, if (MP) and (MD) are dual ; Þ is a joint properly efficient solution, then so is problems and ð x; y k; p ; Þ and Fð ; p Þ ¼ Gðy ; Þ ¼ 0. ðy x; k; p x; y x; p
1 f ðx; yÞ f ðu; yÞ þ r T rxx f ðu; yÞr = Fðx; u; rx f ðu; yÞ þ rxx f ðu; yÞrÞ; 2
In [10], vectors p 2 Rm and r 2 Rn have been taken as variables in the primal and dual problems, while in the definitions and proofs, they have been treated as functions of x and y. In the weak duality theorem, Mishra [10] assumed conditions like
Fðx; u; n1 þ n2 Þ þ uT n1 þ uT n2 P 0 for n1 2 Rnþ ; n2 2 Rnþ ; and used it in the proof taking n1 ¼ rx f ðu; v Þ 2 Rnþ and n2 ¼ rxx f ðu; v Þr 2 Rnþ . The values of n1 and n2 as taken above are not in Rnþ : However, in view of the dual constraint (6.4), n1 þ n2 is in Rnþ . Similar mistakes occur throughout the paper. Theorem 6.1 (Weak duality). Let ðx; y; pÞ be feasible for the primal problem (WP1) and ðu; v ; rÞ be feasible for the dual problem (WD1). Let (i) (ii) (iii) (iv)
f ð; v Þ be second-order F-convex in the first variable at u, f ðx; Þ be second-order G-concave in the second variable at y, Fðx; u; nÞ þ uT n=0; for all n 2 Rnþ and Gðv ; y; gÞ þ yT g=0; for all g 2 Rm þ.
Then
Acknowledgements
Fðx; y; pÞ=Gðu; v ; rÞ: The authors are thankful to the reviewers for their comments and suggestions, in particular, for the addition of the appendix. The second author is also thankful to the University Grants Commission, New Delhi (India) for providing financial support during this work.
Proof. By second-order F-convexity of f ð; v Þ and G-concavity of f ðx; Þ, we have
1 f ðx; v Þ f ðu; v Þ þ rT rxx f ðu; v Þr = Fðx; u; rx f ðu; v Þ þ rxx f ðu; v ÞrÞ; 2
Appendix A This appendix has been added as per suggestion of a reviewer to Þ to be negative definite for x; y correct the mistake of assuming ryyy f ð a scalar function f ðx; yÞ by Mishra [10, Theorem 2 ]. The second-order Wolfe type symmetric dual programs considered in [10] are :
and
1 f ðx; yÞ f ðx; v Þ pT ryy f ðx; yÞp = Gðv ; y; ry f ðx; yÞ 2 ryy f ðx; yÞpÞ: Adding these inequalities, we get
Primal (WP1):
1 1 f ðx; yÞ f ðu; v Þ þ r T rxx f ðu; v Þr pT ryy f ðx; yÞp = 2 2 Fðx; u; rx f ðu; v Þ þ rxx f ðu; v ÞrÞ þ Gðv ; y; ry f ðx; yÞ ryy f ðx; yÞpÞ:
Minimize Fðx; y; pÞ ¼ f ðx; yÞ yT ry f ðx; yÞ yT ryy f ðx; yÞp 1 pT ryy f ðx; yÞp 2 Subject to ry f ðx; yÞ þ ryy f ðx; yÞp 5 0;
ð6:2Þ
ð6:6Þ
x=0;
ð6:3Þ
Now, by the dual constraint (6.4), n ¼ rx f ðu; v Þ þ rxx f ðu; v Þr 2 Rnþ and so from the hypothesis (iii), we have
ð6:1Þ
Dual (WD1):
Maximize Gðu; v ; rÞ ¼ f ðu; v Þ uT rx f ðu; v Þ uT rxx f ðu; v Þr 1 rT rxx f ðu; v Þr 2 Subject to rx f ðu; v Þ þ rxx f ðu; v Þr = 0; ð6:4Þ
v =0;
ð6:5Þ
where f : S1 S2 ! R is a thrice differentiable function, p 2 Rm ; r 2 Rn and S1 # Rn ; S2 # Rm are open sets. Duality for these models was first studied in 1974 by Mond [11].
Fðx; u; rx f ðu; v Þ þ rxx f ðu; v ÞrÞ þ uT ½rx f ðu; v Þ þ rxx f ðu; v Þr =0; or
Fðx; u; rx f ðu; v Þ þ rxx f ðu; v ÞrÞ = uT ½rx f ðu; v Þ þ rxx f ðu; v Þr: Similarly, g ¼ ry f ðx; yÞ ryy f ðx; yÞp 2 sis (iv) yields
ð6:7Þ Rm þ
and therefore hypothe-
Gðv ; y; ry f ðx; yÞ ryy f ðx; yÞpÞ=yT ½ry f ðx; yÞ þ ryy f ðx; yÞp:
ð6:8Þ
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Inequalities (6.6)–(6.8) give
1 1 f ðx; yÞ f ðu; v Þ þ rT rxx f ðu; v Þr pT ryy f ðx; yÞp 2 2 = uT ½rx f ðu; v Þ þ rxx f ðu; v Þr þ yT ½ry f ðx; yÞ þ ryy f ðx; yÞp; or
1 f ðx; yÞ yT ½ry f ðx; yÞ þ ryy f ðx; yÞp pT ryy f ðx; yÞp 2 1 T =f ðu; v Þ u ½rx f ðu; v Þ þ rxx f ðu; v Þr r T rxx f ðu; v Þr; 2 or
Fðx; y; pÞ = Gðu; v ; rÞ: Once the weak duality theorem for (WP1) and (WD1) is established, the strong duality follows as in Mond [11]. Below we state the strong duality theorem. Its proof is exactly same as given for Theorem 6 in [11]. Also, in view of Remark 3.2, the proof can be obtained on the lines of Theorem 2 in Yang et al. [16]. h ; p Þ be a local optimal soluTheorem 6.2 (Strong duality). Let ð x; y tion for (WP1). If Þ is nonsingular, and x; y (i) ryy f ð Þ; i ¼ 1; . . . ; m is positive or negax; y (ii) one of the matrices @y@ fyy ð i ¼ 0; ð ; r ¼ 0Þ is feasible for the dual tive definite, then p x; y problem (WD1) and the objective function values of (WP1) and (WD1) are equal. Furthermore, if the hypotheses of Theorem 6.1 are satisfied for all feasible solutions of the problems ; p ¼ 0Þ and ð ; r ¼ 0Þ are global (WP1) and (WD1), then ð x; y x; y optimal solutions for (WP1) and (WD1) respectively.
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