A note on higher-order symmetric duality

Applied Mathematics and Computation 222 (2013) 553–558

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A note on higher-order symmetric duality T.R. Gulati, Khushboo Verma ⇑ Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247 667, India

a r t i c l e

i n f o

Keywords: Wolfe and Mond–Weir type higher-order dual Higher-order invexity Symmetric duality Efficient solution

a b s t r a c t In this paper, a pair of Wolfe type higher-order symmetric dual multiobjective programs is formulated. Strong and converse duality theorems are established under invexity assumptions. Duality relations for Mond–Weir type dual models have also been obtained under pseudoinvexity assumptions. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Mangasarian [14] introduced the concept of second and higher-order duality in nonlinear problems. This motivated several authors [2,6,8–10,13] in this field. One practical advantage of second and higher order duality is that it provides tighter bounds for the value of the objective function of the primal problem when approximations are used because there are more parameters involved. Arana et al. [4] have discussed Kuhn–Tucker and Fitz–John type necessary and sufficient conditions and duality for a multiobjective problem. Mond and Zhang [15] and Yang et al. [17] have discussed multiobjective higher-order symmetric duality under invexity assumptions. Chen [6] studied higher-order symmetric duality for multiobjective nondifferentiable programming problems by introducing higher-order F-convexity. Recently, Nahak and Padhan [16] have presented higher order symmetric dual programs for multiobjective problems. In the literature strong and converse duality theorems have been established assuming conditions on known quantities. How for Wolfe ever, in strong and converse duality theorems in [16] an assumption involves the unknown lagrange multiplier a  and c  for Mond–Weir type symmetric duals. In this note we establish type symmetric duals, and the lagrange multipliers a these results under the assumptions on the lines of [1,5,7–9,11,12] involving known quantities. 2. Preliminaries Consider the following multiobjective programming problem:

ðPÞ Minimize FðxÞ ¼ fF 1 ðxÞ; F 2 ðxÞ; . . . ; F k ðxÞg;

x2X

where F : Rn ! Rk and X # Rn . The following convention for vector inequalities will be used: If a; b 2 Rn , then a=b () ai =bi ; i ¼ 1; 2; . . . ; n; a=b () a=b and a – b; a > b () ai > bi ; i ¼ 1; 2; . . . ; n. Definition 2.1. A point  x 2 X is said to be an efficient solution of (P) if there exists no x 2 X such that FðxÞ 6 Fð xÞ. Definition 2.2. A function F : Rn ! Rk is said to be higher-order invex at u 2 Rn with respect to g : Rn  Rn ! Rn and h : Rn  Rn ! R, if for all ðx; pÞ 2 Rn  Rn and e 2 Rk ⇑ Corresponding author. E-mail addresses: [email protected] (T.R. Gulati), [email protected] (K. Verma). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.07.064

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    FðxÞ  FðuÞ = gT ðx; uÞ rx FðuÞ þ ðrp hðu; pÞÞe þ hðu; pÞ  pT rp hðu; pÞ e:

Definition 2.3. A function F : Rn ! Rk is said to be higher-order pseudoinvex at u 2 Rn with respect to g : Rn  Rn ! Rn and h : Rn  Rn ! R, if for all ðx; pÞ 2 Rn  Rn and e 2 Rk

gT ðx; uÞ½rx FðuÞ þ ðrp hðu; pÞÞe = 0 ) FðxÞ  FðuÞ  ½hðu; pÞ þ pT rp hðu; pÞe = 0: Let f : Rn  Rm # Rk ; g : Rn  Rm  Rn # R e ¼ ð1; 1; . . . ; 1ÞT 2 Rk .

and

h : Rn  Rm  Rm # R

be

differentiable

functions,

p 2 Rm ; r 2 Rn

and

3. Wolfe type symmetric duality We now establish duality theorems for the following pair of Wolfe type higher order multiobjective problems. Primal problem (WHP)

Minimize ðLðx; y; pÞ ¼ f ðx; yÞ þ ½hðx; y; pÞ  pT rp hðx; y; pÞe  yT ½ry ðkT f Þðx; yÞ þ rp hðx; y; pÞe

ð1Þ

subject to

ry ðkT f Þðx; yÞ þ rp hðx; y; pÞ50;

ð2Þ

k > 0; kT e ¼ 1:

ð3Þ

Dual problem (WHD) Maximize Mðu; v ; rÞ ¼ f ðu; v Þ þ ½gðu; v ; rÞ  rT rr gðu; v ; rÞe  uT ½rx f ðu; v Þ þ rr gðu; v ; rÞe

ð4Þ

subject to

rx ðkT f Þðu; v ÞÞ þ rr gðu; v ; rÞ=0; k > 0;

kT e ¼ 1:

ð5Þ ð6Þ

k will be denoted by ðWHDÞk . Any problem, say (WHD), in which k is fixed to be  Theorem 3.1 [16] Weak duality. Let ðx; y; k; pÞ be feasible for the primal problem (WHP) and ðu; v ; k; rÞ be feasible for the dual problem (WHD). Let (i) (ii) (iii) (iv)

f ð:; v Þ be higher-order invex at u with respect to g1 and gðu; v ; rÞ, f ðx; :Þ be higher-order invex at y with respect to g2 and hðx; y; pÞ, g1 ðx; uÞ þ u=0 and g2 ðv ; yÞ þ y=0.

Then

Lðx; y; pÞiMðu; v ; rÞ: ;  Þ be an efficient solution of (WHP). Suppose that k; p x; y Theorem 3.2 (Strong duality). Let ð (i) (ii) (iii) (iv) (v)

rpp hðx; y; pÞ is nonsingular, Þ; i ¼ 1; . . . ; kg is linearly independent, the set fry fi ð x; y  R spanfry fi ð Þ; i ¼ 1; . . . ; kg n f0g x; y rp hðx; y; pÞ þ ry hðx; y; pÞ  ryy ðkT f Þðx; yÞp ¼0)p  ¼ 0 and rp hðx; y; pÞ þ ry hðx; y; pÞ  ryy ðkT f Þðx; yÞp ; 0Þ ¼ gð ; 0Þ; rx hð ; 0Þ ¼ rr gð ; 0Þ; ry hð ; 0Þ ¼ rp hð ; 0Þ. hð x; y x; y x; y x; y x; y x; y

Then ;  k; r ¼ 0Þ is feasible for (WHD) and (I) ð x; y  ; p Þ ¼ Mð ; r Þ. (II) Lðx; y x; y ;  Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of ðWHPÞk and ðWHDÞk , then ð x; y k; r ¼ 0Þ is an efficient solution of ðWHDÞk .

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Remark 3.1. In the above theorem assumptions (iii) and (iv) are on the lines of [1,7–9]. These replace the assumption

ry f ðx; yÞða  ðaT eÞÞ  ½rp hðx; y; pÞ þ ry hðx; y; pÞðaT eÞ þ ryy ðkT f Þðx; yÞðaT eÞp ¼ 0 ) p ¼ 0; involving the unknown Lagrange multiplier a in Theorem 3.2 in [16]. ;  Þ is an efficient solution for (WHP), by the Fritz–John necessary optimality conditions [3,14], there exist k; p Proof. Since ð x; y

a 2 Rk ; b 2 Rm ; x 2 Rk and l 2 R, such that the following conditions are satisfied: rx ðaT f Þðx; yÞ þ rx hðx; y; pÞðaT eÞ þ rxy ðkT f Þðx; yÞðb  ðaT eÞyÞ þ rpx hðx; y; pÞðb  ðaT eÞðp þ yÞÞ ¼ 0;

ð7Þ

ry f ðx; yÞða  ðaT eÞkÞ þ ry hðx; y; pÞðaT eÞ þ ryy ðkT f Þðx; yÞðb  ðaT eÞyÞ  rp hðx; y; pÞðaT eÞ ; p Þðb  ðaT eÞðp þy ÞÞ ¼ 0; þ rpy hðx; y

ð8Þ

rpp hðx; y; pÞðb  ðaT eÞðp þ yÞÞ ¼ 0;

ð9Þ

ÞðaT eÞ þ bT ry f ðx; y Þ  x þ le ¼ 0; yT ry f ðx; y

ð10Þ

Þ þ rp hðx; y ; p ÞÞ ¼ 0; bT ðry ðkT f Þðx; y

ð11Þ

xT k ¼ 0;

ð12Þ

ða; b; xÞ  0; ða; b; x; lÞ – 0;

ð13Þ

Since  k > 0 and x = 0, Eq. (12) implies

x ¼ 0:

ð14Þ

Using hypothesis (i) in Eq. (9) we get

þy Þ; b ¼ ðaT eÞðp

ð15Þ

Now, we claim that a  0. Indeed, if a ¼ 0, then (15) gives b ¼ 0 and so Eqs. (10) and (14) imply ða; b; x; lÞ ¼ 0, a contradiction to (13). Thus a  0 or

aT e > 0:

l ¼ 0. Therefore ð16Þ

Eqs. (15) and (16), reduce Eq. (8) to

rp hðx; y; pÞ þ ry hðx; y; pÞ  ryy ðkT f Þðx; yÞp ¼

ða  ðaT eÞkÞ Þg: fry f ðx; y aT e

Using hypothesis (iii) and (iv), the above relation gives

 ¼ 0: p

ð17Þ

Therefore in view of Eq. (15) and Assumption (v), Eq. (8) gives

Þða  ðaT eÞkÞ ¼ 0: ðry f Þðx; y Since the set fry fi ; i ¼ 1; . . . ; kg is linearly independent, the above equation implies

a ¼ ðaT eÞk:

ð18Þ

Using (15)–(18) in (7), we get

rx ðkT f Þðx; yÞ þ rr gðx; y; 0Þ ¼ 0:

ð19Þ

  ; h; k; r ¼ 0Þ is a feasible solution for the dual problem (WHD). Also, from Eqs. (11), (15), (16) and (17), we get Thus ð x; y

T ½ry ðkT f Þðx; y Þ þ rp hðx; y ; 0Þ ¼ 0: y

ð20Þ

Hypothesis (v) and Eqs. (19) and (20) leads to the equality of the two objective function values. Using weak duality, it can be   ; h;  ¼ 0Þ is an efficient solution of ðWHDÞk . h k; p easily shown that ð x; y ; v ;  k; r Þ be an efficient solution of (WHD). Suppose that Theorem 3.3 (Converse Duality). Let ðu ; v  ; r Þ is nonsingular, (i) rrr gðu ; v  Þ; i ¼ 1; . . . ; kgis linearly independent, (ii) the set fru g i ðu

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; v  ; r Þ þ ru hðu ; v  ; r Þ  ruu ð ; v  Þr R spanfru g i ðu ; v  Þ; i ¼ 1; . . . ; kg n f0g (iii) rr hðu kT gÞðu ; v  ; gÞ þ ru gðu ; v  ; r Þ  ruu ð ; v  Þr ¼ 0 ) r ¼ 0 and (iv) rr gðu kT gÞðu ; v  ; 0Þ ¼ gðu ; v  ; 0Þ; rx hðu ; v  ; 0Þ ¼ rr gðu ; v  ; 0Þ; ry hðu ; v  ; 0Þ ¼ rp hðu ; v  ; 0Þ. (v) hðu Then ; v  ¼ 0Þ is feasible for (WHP) and ;  k; p (I) ðu  ; p Þ ¼ Mðu ; v  ; rÞ. (II) Lðu; v ; v ;   ¼ 0Þ is an Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of ðWHPÞk and ðWHDÞk , then ðu k; p efficient solution of ðWHPÞk . Proof. Follows on the lines of Theorem 3.2. h 4. Mond–Weir type symmetric duality We now establish duality theorems for the following pair of Mond–Weir type higher order multiobjective problems. Primal problem (MWHP) Minimize

Rðx; y; pÞ ¼ f ðx; yÞ þ ½hðx; y; pÞ  pT rp hðx; y; pÞe

ð21Þ

subject to

ry ðkT f Þðx; yÞ þ rp hðx; y; pÞ 5 0;

ð22Þ

yT ½ry ðkT f Þðx; yÞ þ rp hðx; y; pÞ = 0;

ð23Þ

k > 0;

kT e ¼ 1:

ð24Þ

Dual problem (MWHD) Maximize

Sðu; v ; rÞ ¼ f ðu; v Þ þ ½gðu; v ; rÞ  r T rr gðu; v ; rÞe

ð25Þ

subject to

rx f ðu; v Þ þ rr gðu; v ; rÞ = 0;

ð26Þ

uT ½rx ðkT f Þðu; v Þ þ rr gðu; v ; rÞ 5 0;

ð27Þ

k > 0;

kT e ¼ 1;

ð28Þ

Any problem, say (MWHD), in which k is fixed to be  k will be denoted by ðMWHDÞk . Theorem 4.1 ( [16] Weak duality). Let ðx; y; k; pÞ be feasible for the primal problem (MWHP) and ðu; v ; k; rÞ be feasible for the dual problem (MWHD). Let (i) f ð:; v Þ be higher-order pseudoivexinvex at u with respect to g1 and gðu; v ; rÞ, (ii) f ðx; :Þ be higher-order pseudoinvex at y with respect to g2 and hðx; y; pÞ, (iii) g1 ðx; uÞ þ u = 0 and (iv) g2 ðv ; yÞ þ y = 0. Then

Rðx; y; pÞiSðu; v ; rÞ: ;  Þ be an efficient solution of (MWHP). Suppose that Theorem 4.2 (Strong duality). Let ð k; p x; y (i) (ii) (iii) (iv) (v)

rpp hðx; y; pÞ is nonsingular, Þ; i ¼ 1; . . . ; kg is linearly independent, the set fry fi ð x; y ry ðkT f Þðx; yÞ þ rp hðx; y; pÞ – 0, Þ þ rp hð ; p Þ ¼ 0 ) p  ¼ 0 and pT ½ry ð kT f Þð x; y x; y   ; 0Þ; rx hð ; 0Þ ¼ rr gð ; 0Þ ¼ 0; ry hð ; 0Þ ¼ rp hð ; 0Þ ¼ 0. hðx; y; 0Þ ¼ gð x; y x; y x; y x; y x; y

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Then ; y ;  k; r ¼ 0Þ is feasible for (MWHD) and (I) ðx ; y ; p Þ ¼ Sð ; r Þ. (II) Rðx x; y ;  Also, if the hypotheses of Theorem 4.1 are satisfied for all feasible solutions of ðMWHPÞk and ðMWHDÞk , then ð k; r ¼ 0Þ x; y is an efficient solution of ðMWHDÞk . Remark 4.1. In the above theorem assumption (iii) and (iv) are on the lines of [5,11,12]. These replace the assumption

ry f ðx; yÞða  ckÞ  crp hðx; y; pÞ þ ½ry hðx; y; pÞ þ ryy ðkT f Þðx; yÞpðaT eÞ ¼ 0 ) p ¼ 0; involving the unknown lagrange multipliers a and c in Theorem 4.2 in [16]. ; y ;  Þ is an efficient solution for (MWHP), by the Fritz–John necessary optimality conditions [3,14], there Proof. Since ðx k; p k exist a 2 R ; b 2 Rm ; x 2 Rk and c; l 2 R, such that the following conditions are satisfied:

rx ðaT f Þðx; yÞ þ rx hðx; y; pÞðaT eÞ þ rxy ðkT f Þðx; yÞðb  cyÞ þ rpx hðx; y; pÞðb  cy  ðaT eÞpÞ ¼ 0;

ð29Þ

ry f ðx; yÞða  ckÞ þ ry hðx; y; pÞðaT eÞ þ ryy ðkT f Þðx; yÞðb  cyÞ  crp hðx; y; pÞ þ rpy hðx; y; pÞðb  cy  ðaT eÞpÞ ¼ 0;

ð30Þ

rpp hðx; y; pÞðb  cy  ðaT eÞpÞ ¼ 0;

ð31Þ

ry f ðx; yÞðb  cyÞ  x þ le ¼ 0;

ð32Þ

Þ þ rp hðx; y ; p ÞÞ ¼ 0; bT ðry ðkT f Þðx; y

ð33Þ

cyT ðry ðkT f Þðx; yÞ þ rp hðx; y; pÞÞ ¼ 0;

ð34Þ

xT k ¼ 0;

ð35Þ

ða; b; c; xÞ  0; ða; b; c; x; lÞ – 0;

ð36Þ

k > 0 and x=0, Eq. (35) implies Since 

x ¼ 0:

ð37Þ

Using hypothesis (i) in (31), we get

 þ ðaT eÞp : b ¼ cy

ð38Þ

Now, we claim that a  0. Indeed, if a ¼ 0, then (38) yields

; b ¼ cy then Eqs. (32), (37) and (38) imply

ð39Þ

l ¼ 0. Also from Eqs. (39) and (30), we obtain

c½ry ðkT f Þðx; yÞ þ rp hðx; y; pÞ ¼ 0; which along with hypothesis (iii) gives c ¼ 0 and so from (39), b ¼ 0. Therefore ða; b; c; x; lÞ ¼ 0, a contradiction to Eq. (7). Thus a  0 or

aT e > 0:

ð40Þ

Adding Eqs. (33) and (34), we get

Þ½ry ðkT f Þðx; y Þ þ rp hðx; y ; p Þ ¼ 0; ðb  cy

ð41Þ

Further, using (38) and (40) and hypothesis (iv) in (41), we have

 ¼ 0; p

ð42Þ

Therefore in view of Eqs. (39), (42) and hypothesis (v), Eq. (30) gives

ry f ðx; yÞða  ckÞ ¼ 0: Since the set fry fi ; i ¼ 1; . . . ; kg is linearly independent, the above equation implies

a ¼ ck:

ð43Þ

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As  k > 0 and a  0,

c > 0:

ð44Þ

This together with Eqs. (38), (42), (43), hypothesis (v) and Eq. (29) give

rx ðkT f Þðx; yÞ ¼ 0;

ð45Þ

xT ½rx ðkT f Þðx; y Þ ¼ 0:

ð46Þ

and

;  x; y k; r ¼ 0Þ is a feasible solution for the dual problem (MWHD). Also, hypothesis (v) leads to the equality of the two Thus ð ;  k; r ¼ 0Þ is an efficient solution of objective function values. Using weak duality it can be easily shown that ð x; y ðMWHDÞk . h ; v ;  k; rÞ be an efficient solution of (MWHD). Suppose that Theorem 4.3 (Converse Duality). Let ðu (i) (ii) (iii) (iv) (v)

; v  ; rÞ is nonsingular, rpp hðu ; v  Þ; i ¼ 1; . . . ; kg is linearly independent, the set fru g i ðu ; v  Þ þ rr gðu ; v ; p Þ – 0, ru ðkT gÞðu ; v ; v  Þ þ rr gðu  ; rÞ ¼ 0 ) r ¼ 0 and rT ½ru ð kT gÞðu ; v  ; 0Þ ¼ gðu ; v  ; 0Þ; rx hðu ; v  ; 0Þ ¼ rr gðu ; v  ; 0Þ ¼ 0; ry hðu ; v  ; 0Þ ¼ rp hðu ; v  ; 0Þ ¼ 0. hðu

Then ; v ;   ¼ 0Þ is feasible for (MWHP) and (I) ðu k; p ; v Þ ¼ Sðu ; v ; p  ; r Þ. (II) Rðu ; v ;  Also, if the hypotheses of Theorem 4.1 are satisfied for all feasible solutions of ðMWHPÞk and ðMWHDÞk , then ðu k; r ¼ 0Þ is an efficient solution for ðMWHPÞk . Proof. Follows on the lines of Theorem 4.2. h Acknowledgements The authors are thankful to a reviewer for his suggestions. The second author is also thankful to the MHRD, Government of India, for providing financial support. References [1] I. Ahmad, Z. Husain, On multiobjective second-order symmetric duality with cone constraints, Eur. J. Oper. Res. 204 (2010) 402–409. [2] I. Ahmad, Z. Husain, Sarita Sharma, Higher-order duality in non-differentiable multiobjective programming, Numer. Func. Anal. Opt. 28 (2007) 989– 1002. [3] B.D. Craven, Lagrangian conditions and quasiduality, Bull. Austral. Math. Soc. 16 (1977) 587–592. [4] M. Arana, A. Rufin, R. Osuna, G. Ruiz, ’Pseudoinvexity, optimality conditions and efficiency in multiobjective problems; duality, Nonlinear Anal. TMA 68 (2008) 24–34. [5] S. Chandra, V. Kumar, A note on pseudo-invexity and symmetric duality, Eur. J. Oper. Res. 105 (1998) 626–629. [6] X. Chen, Higher order symmetric duality in non-differentiable multiobjective programming problems, J. Math. Anal. Appl. 290 (2004) 423–435. [7] T.R. Gulati, Geeta, On some symmetric dual models in multiobjective programming, Appl. Math. Comput. 215 (2009) 380–383. [8] S.K. Gupta, N. Kailey, Multiobjective second-order mixed symmetric duality with a square root term, Appl. Math. Comput. 218 (2012) 7602–7613. [9] S.K. Gupta, N. Kailey, A note on multiobjective second-order symmetric duality, Eur. J. Oper. Res. 201 (2010) 649–651. [10] S.K. Gupta, N. Kailey, M.K. Sharma, Higher-order ðF; a; q; dÞ-convexity and symmetric duality in multiobjective programming, Comput. Math. Appl. 60 (2010) 2373–2381. [11] S.H. Hou, X.M. Yang, On second-order symmetric duality in nondifferentiable programming, J. Math. Anal. Appl. 255 (2001) 491–498. [12] Mohamed Abd El-Hady Kassem, Multiobjective nonlinear second order symmetric duality with ðK; FÞ-pseudoconvexity, Appl. Math. Comput. 219 (2012) 2142–2148. [13] D.S. Kim, H.S. Kang, Y.J. Lee, Y.Y. Seo, Higher-order duality in multiobjective programming with cone constraints, Optimization 59 (1) (2010) 29–43. [14] O.L. Mangasarian, Second and higher-order duality in nonlinear programming, J. Math. Anal. Appl. 51 (1975) 607–620. [15] B. Mond, J. Zhang, Higher-order invexity and duality in mathematical programming, in: J. P. Crouzeix, et al. (Eds.), Generalized Convexity, Generalized Monotonicity: Recent Results, Kluwer Academic, Dordrecht, 1998, pp. 357–372. [16] C. Nahak, S.K. Padhan, Higher-order symmetric duality in multiobjective programming problems under higher-order invexity, Appl. Math. Comput. 218 (2011) 1705–1712. [17] X.M. Yang, X.Q. Yang, K.L. Teo, Higher-order symmetric duality in multiobjective mathematical programming with invexity, J. Ind. Manag. Optim. 4 (2008) 335–391.