SECOND-ORDER DUALITY FOR

Deo Brat Ojha, Int. J. Comp. Tech. Appl., Vol 2 (1), 103-111

ISSN : 2229-6093

SECOND-ORDER DUALITY FOR MULTIOBJECTIVE PROGRAMMING INVOLVING (  , ρ)-INVEXITY Deo Brat Ojha e-mail: [email protected] R.K.G.I.T., Ghaziabad,INDIA Abstract: The concepts of (  , ρ)-invexity have been given by Caristi, Ferrara and Stefanescu[1]. We consider a second-order dual model associated to a multiobjective programming problem involving support functions and a weak duality result is established under appropriate second-order (  , ρ)invexity conditions. AMS 2002 Subject Classification: 90C29, 90C30, 90C46. Key words: Higher-order (  ,ρ)-(pseudo/quasi)-convexity/invexity, multiobjective programming, second-order duality, duality theorem.

1. INTRODUCTION For nonlinear programming problems, a number of duals have been suggested among which the Wolfe dual [35,8] is well known. In 1981, Hanson[39], remarkably pointed out that the K-T sufficiency conditions lack invariance in that the applicability of the conditions for any given problems depends on the nature of the analytical formulation of the problem. Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for a minimum. This property was generalized by Craven(1981)[40], to a property, called K-invex, of a vector function in relation to a convex cone K .Also necessary conditions and sufficient conditions have been obtained for a function to be K-invex. This leads to a new second order sufficient condition for a constrained minimum.

appropriate pseudo-convexity/quasiconvexit assumptions. The study of second order duality is significant due to the computational advantage over first order duality as it provides tighter bounds for the value of the objective function when approximations are used [10,16,24].Mangasarian[12] considered a nonlinear programming problem and discussed second order duality under inclusion condition. Mond [14] was the first who present second order convexity. He also gave in [14] simpler conditions than Mangasarian using a generalized form of convexity. which was later called bonvexity by Bector and Chandra [2]. Further, Jeyakumar [37,30] and Yang [24] discussed also second order Mangasarian type dual formulation under ρ-convexity and generalized representation conditions respectively. In [20] Zhang and Mond established some duality theorems for secondorder duality in nonlinear programming under generalized second-order B-invexity, defined in their paper. In [14] it was shown that

While studying duality under generalized convexity, Mond and Weir [36] proposed a number of deferent duals for nonlinear programming problems with nonnegative variables and proved various duality theorems under

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second order duality can be useful from computational point of view, since one may obtain better lower bounds for the primal problem than otherwise. The case of some optimization problems that involve nset functions was studied by Preda [38]. Recently, Yang et al. [24] proposed four second-order dual models for nonlinear programming problems and established some duality results under generalized second-order F -convexity assumptions. For ( x, a, ( y, r ))  F ( x, a; y)  rd 2 ( x, a) , where F ( x, a;.) is sublinear on R n , the definition of (,  ) - invexity reduces to the definition of ( F ,  ) -convexity introduced by Preda[29], which in turn Jeyakumar[30] generalizes the concepts of F-convexity and  invexity[31]. The more recent literature, Xu[21], Ojha [27], Ojha and Mukherjee [22] for duality under generalized ( F ,  ) convexity, Mishra [23] and Yang et al.[24] for duality under second order F -convexity. Liang et al. [25] and Hachimi[26] for optimality criteria and (F , ,  , d ) duality involving convexity or generalized {F ,  ,  , d ) type functions. The ( F ,  ) -convexity was recently generalized to (,  ) invexity by Caristi , Ferrara and Stefanescu [32],and here we will use this concept to extend some theoretical results of multiobjective programming. Whenever the objective function and all active restriction functions satisfy simultaneously the same generalized invexity at a Kuhn-Tucker point which is an optimum condition, then all these functions should satisfy the usual invexity, too. This is not the case in

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multiobjective programming ; Ferrara and Stefanescu[28] showed that sufficiency Kuhn-Tucker condition can be proved under (,  ) -invexity, even if Hanson’s invexity is not satisfied, Puglisi[34].The interested reader may consult[1,3,4,5,6,7,9,11,13,15,17,18,19 ,33]. Therefore, the results of this paper are real extensions of the similar results known in the literature. In Section 2, we define the secondorder (  , ρ)-invexity . In Section 3 and 4 we consider a class of multiobjective programming problems and for the dual model we prove a weak duality result. In section 5, we are only giving outline of fractional multiobjective mathematical programming. Section 6 is conclusion and remark. 2.Notation and Preliminaries: Throughout the paper, following convention for vectors in R n will be followed: x y if and only if xi yi , i  1, 2,.,., n , x  y if and only if x y and x  y , x y if and only if xi  yi , i  1, 2,.,., n . The problem to be considered here is the following multiobjective nonlinear programming problem: Minimize f ( x) (P) Subject to g ( x)0, x  X (1) where f  ( f1 , f 2 ,.,., f k ) : X  Rk ,

g  ( g1 , g2 ,.,., gm ) : X  Rm are assumed to be twice differentiable function over X , an open subset of Rn .

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f : R p  R , and by 2 f (a) the Hessian matrix of f at a . For a real valued twice differentiable function  ( x, y) defined on an open set in R p  Rq , we denote by  x (a, b) the gradient vector of  with respect to x at (a, b) , and by  xx (a, b) the Hessian matrix with respect to x at (a, b) . Similarly, we may define  y (a, b) ,  xy (a, b) and  yy (a, b) .

We consider n p n q f : R  R , g : R  R ,are differential functions and X  Rn is an open set. Let X0 be the set of all feasible solutions of (P) that is, X 0  {x  X g ( x) 0} . We quote some definitions and also give some new ones.

For convenience, let us write the definitions of ,   -univexity from[1], Let  : X 0  R be a differentiable

Definition 2.1 A vector a  X 0 is said to be an efficient solution of problem (P) if there exit no x  X 0 such that

function

each

convex on every

fi ( x)  fi (a) , there exist at least one f ( a) f ( x) j j

 x, a   X 0  X 0

and

if for any z  R n for fixed w  R n ,there exist a vector p  R n and real number i , such that 1 { f ( z, w)  f ( z , w)  pT  zz f ( z , w) p}( z, z ;( z f ( z , w)  zz f ( z , w) p, )) 0 0 0 0 0 2

and f (a)  f ( x)  K  f ( x)  f (a)  . i i j j 

Rn  1 and   x, a,  0, r    0, for

r  R , h : X  Rn  R be  differentiable function. Definition 2.1: The twice differentiable function f i over X is said to be second order (  , ρ)-invex at z0 with respect to X ,

i 1, 2...... p satisfying

i 1, 2...... p suchthat

and

dimensional Euclidean Space R n 1 is represented as the ordered pair is a real ( z, r ) with z  Rn and r  R,  number and  is real valued function defined on n 1 X 0  X 0  R , suchthat   x, a,. is

0

for

( X  Rn ) , X  X , 0 0

a  X . An element of all (n+1)0

f (a)  f ( x)  Rp \{0} i.e., fi ( x)  fi (a) for all i {1,.,.,., p} , and for at least one j {1,.,.,., p} we have f i ( x)  f i (a ) . Definition 2.2 A point a  X is said to be a weak efficient solution of problem (VP) if there is no x  X such that f ( x)  f (a). Definition 2.3 A point a  X 0 is said to be a properly efficient solution of (VP) if it is efficient and there exist a positive constant K such that for each x  X 0 and

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

Denoting by WE(P), E(P) and PE(P) the sets of all weakly efficient, efficient and properly efficient solutions of (VP), we have

Definition 2.2: The twice differentiable function f i over X is said to be second order

PE(P)  E(P)  WE(P).

We denote by f (a) the gradient vector at a of a differentiable function

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(  , ρ)-incave at z0 with respect to X ,

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w  R n ,there exist a vector p  R n and real number i , such that

z  R n for fixed 1 { f ( z, w)  f ( z , w)  pT  zz f ( z , w) p}( z, z ;(  z f ( z , w)   zz f ( z , w) p,  )) 0 0 0 0 0 2 (  , ρ)-pseudoinvex at z0 with respect

and

if

for

any

to X , if for any z  R n for fixed w  R n ,there exist a vector p  R n and real number i , such that

Definition 2.3: The twice differentiable function f i over X is said to be second order

1 ( z, z ;( z f ( z , w)   zz f ( z , w) p,  ))0  { f ( z, w)  f ( z , w)  pT  zz f ( z , w) p}0 0 0 0 0 0 2 Definition 2.4: The twice differentiable function and b : Rn  Rn  R n  R if for any f i over X is said to be second order z  R n for fixed w  R n ,there exist a (  , ρ)-incave at z0 with respect to X , vector p  R n and real number i , such that

1 { f ( z, w)  f ( z , w)  pT  zz f ( z , w) p}0  ( z, z ;(  z f ( z , w)   zz f ( z , w) p,  ))0 0 0 0 0 0 2 Lemma 2.1: (i) If f (., w) is (  , ρ)-invex and g (., w) is (  , ρ)-incave, then p(., w) is (  , ρ)-pseudoinvex. (ii) If f (., w) is (  , ρ)-inccave and g (., w) is (  , ρ)-invex, then p(., w) is (  , ρ)-pseudoincave.

Dual (SMD) Maximize  (u, v, p )  1 1  { f (u, v)  uT  f (u, v)  uT  f (u, v) p  p T  f (u, v) p } z zz 1 2 1 zz 1 Subject to  f (u, v)   f (u, v) p 0 z zz 1 v0 ;

3. Second order Wolfe type symmetric duality : We consider the following higherorder Mond-Weir type pair and prove a weak duality theorem.

Theorem 3.1:

Let f (., w) be second-order  0 univex at u and f ( z,.) be second1  { f ( z, w)  wT w f ( z, w)  wT ww f ( z, w) p  pT ww f ( z, w) p} order 1 -incave at w and for all 2 ( z, w) feasible for (SMP) and all (u, v) feasible for (SMD) . Subject to w f ( z, w)  ww f ( z, w) p0 Primal (SMP) Minimize  (z, w, p) 

z0 ; 4

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(I) F ( z, u;    )  uT   uT  0,  ,   Rn 0 1 2 1 2 1 2

(II) F1 (v, w;1  2 )  wT1  wT2 0,1 ,2  Rn Then inf(SMP)  Sup(SMD) .

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 ( z, u;(   ,  ))  uT   uT  , 0 1 2 1 2 and 1 (v, w;(1  2 ,  ))  wT1  wT2 . Thereafter, we get 1 1 f ( z, w)  pT  ww f ( z, w) p  f ( z, v)  pT  zz f (u, v) p 2 2

Then inf(SMP)  Sup(SMD) . Proof: Let b : Rn  Rn  Rn  Rnn  R

1   f (u, v) 2   z

zz

The second strong duality theorem can be developed on the lines of [6] in the view of the above theorem.

f (u, v) .

Then  ( z, u;(   ,  ))  uT   uT  0, 0 1 2 1 2

Which by the second order (  0 , ρ)invexity of f (., w) at u yields, 1 f ( z, v)  f (u, v)  pT  zz f (u, v) p ( z, u;(   ,  )) 0 1 2 2 -----------(1) Let, 1  w f ( z, w),2  ww f ( z, w) p

4. Mond-Weir type symmetric duality in second-order: We consider the following second order Mond-Weir type pair and prove a weak duality theorem. Primal (SMP) Minimize  (z, w, p)  1  { f ( z, w)  wT  w f ( z, w)  wT  ww f ( z, w) p  pT  ww f ( z, w) p} 2

Subject to w f ( z, w)  ww f ( z, w) p0

1 (v, w;(1  2 ,  ))  wT1  wT2 0 , which by the second order ( 1 , ρ)incavity of f ( z,.) at w gives 1 f ( z, w)  f ( z, v)  pT ww f ( z, w) p1 (v, w;(1  2 ,  )) 2 ---------- (2)

wT w f ( z, w)  wT ww f ( z, w) p0 , z0 ; Dual (SMD) Maximize  (u, v, p )  1 1  { f (u, v)  uT  f (u, v)  uT  f (u, v) p  p T  f (u, v) p } z zz 1 2 1 zz 1

combining (1) and (2) , 1 1 { f ( z, w)  pT  ww f ( z, w) p}  { f ( z, v)  pT  zz f (u, v) p} 2 2  ( z, u;(   ,  ))  1 (v, w;(1  2 ,  )) 0 1 2 ------------(3)

Subject to  f (u, v)   f (u, v) p 0 z zz 1 T T u  f (u, v)  u  f (u, v) p 0 z zz 1 v0 ;

Using the hypothesis of the theorem ,

Theorem 4.1: Let f (., w) be second order  0 pseudoinvex at u and f ( z,.) be

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second order 1 -pseudoincave at w and for all ( z, w) feasible for (SMP) and all (u, v) feasible for (SMD) .

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The second order strong duality theorem can be developed on the lines

 ( z, u;(   ,  ))  uT   uT  0,  ,   Rn 0 1 2 1 2 1 2

(II) 1 (v, w;(1  2 ,  ))  wT1  wT2 0,1,2  Rn

of [6] in the view of the above theorem. 5. Second order dual fractional programming In this section, we extend above to the second order complex fractional symmetric dual pair (SFP) and (SFD) as follows :

Then inf(SMP)  Sup(SMD) .

Proof: Let 1   f (u, v) 2   f (u, v) . z zz Then  ( z, u;(   ,  ))  uT   uT  0, 0 1 2 1 2

Primal (HFP) 1  { f ( z, w)  wT  w f ( z, w)  wT  ww f ( z, w) p  pT  ww f ( z , w) p} 2 Minimize  1  {g ( z, w)  wT  w g ( z, w)  wT  ww g ( z , w) p  pT  ww g ( z , w) p} 2

Which by the second order  0 pseudoinvexity of f (., w) at u yields, 1 f ( z, v)  f (u, v)  pT  zz f (u, v) p0 2 -----------(3) Let, 1  w f ( z, w),2  ww f ( z, w) p

Subject to G( z, w, p){w f ( z, w) ww f ( z, w) p}  H ( z, w, p){w g ( z, w) ww g ( z, w) p}0

wT {G( z, w, p1 ) w f ( z, w)  H ( z, w, p1 ) w g ( z, w)}  wT {G( z, w, p1 ) ww f ( z, w) p1  H ( z, w, p1 ) ww g ( z, w) p1 0 z0 ; where 1 G( z, w, p1 )   {g ( z, w)  wT w g ( z, w)  wT ww g ( z, w) p1  p1T ww g ( z, w) p1} 2 and 1 H ( z, w, p)   { f ( z, w)  wT w f ( z, w)  wT ww f ( z, w) p  pT ww f ( z, w) p} 2

1 (v, w;(1  2 ,  ))  wT1  wT2 0 , which by the second order 1 pseudoincavity of f ( z,.) at w gives 1 f ( z, w)  f ( z, v)  pT  ww f ( z, w) p0 2 ---------- (4) combining (1) and (2) , and using property of function  and b 1 1 { f ( z, w)  pT ww f ( z, w) p}  { f ( z, v)  pT  zz f (u, v) p} 2 2

Dual (HMD) 1  { f (u, v)  uT  f (u, v)  uT  f (u, v) p  p T  f (u, v) p } z zz 1 2 1 zz 1 Maximize 1  {g (u, v)  uT  g (u, v)  uT  g (u, v) p  p T  g (u, v) p } z zz 1 2 1 zz 1 ; G(u, v, p1 ){ f (u, v)  f (u, v) p }  H ((u, v, p1){ g (u, v)  g (u, v) p }0 z zz 1 z zz 1

That i.e., inf(SMP)  Sup(SMD) . 

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uT {G(u, v, p1 ){ f (u, v)  H (u, v, p1 ) g (u, v)}} z z T u {G(u, v, p1 ){ f (u, v) p  H (u, v, p1 ) g (u, v) p }}0 zz 1 zz 1 , v0 ;

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fractional programming, Opsearch 24 (1987) 143–154. [3] S. Chandra, B.D. Craven, B. Mond, Generalized concavity and duality with a

where 1 G(u, v, p1 )   {g (u, v)  uT  g (u, v)  uT  g (u, v) p  p T  g (u, v) p } z zz 1 2 1 zz 1 and 1 H (u, v, p3 ) { f (u, v)  uT  f (u, v)  uT  f (u, v) p  p T  f (u, v) p } z zz 1 2 1 zz 1

It is assumed that Re G  0 and the feasible Re H  0 throughout regions defined by the primal (SFP) and the dual problem (SFD). Lemma 2.1 above can be extended to the higher order and also next theorem on the line of Mishra et.al. [23]. 6. Particular cases and remark in applications (i) If we take, then this is an earlier work by Mishra et al. [6]. On the same condition if it is single objective and f is real and differential then this is an earlier work [7] . (ii) If we take, the function f to be real and differentiable and F0  F1 , then this is an earlier work by Gulati and Ahmad [15]. 7. Acknowledgement Author is grateful to anonymous referee for their valuable suggestion.

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