Optimality Conditions and Second Order Duality for Nondifferentiable Multiobjective Continuous Programming Problems

Optimality Conditions and Second-Order Duality for

Nondifferentiable Multiobjective Continuous

Programming Problems

Iqbal Husain*, Vikas K. Jain*

Department of Mathematics, Jaypee University of Engineering and Technology, Guna (M.P.), India Email: *_{[email protected],} *_{[email protected]}

Received August 4, 2012; revised September 5, 2012; accepted September 18, 2012

ABSTRACT

Fritz John and Karush-Kuhn-Tucker type optimality conditions for a nondifferentiable multiobjective variational prob-lem are derived. As an application of Karush-Kuhn-Tucker type optimality conditions, Mond-weir type second-order nondifferentiable multiobjective dual variational problems is constructed. Various duality results for the pair of Mond-Weir type second-order dual variational problems are proved under second-order pseudoinvexity and sec-ond-order quasi-invexity. A pair of Mond-Weir type dual variational problems with natural boundary values is formu-lated to derive various duality results. Finally, it is pointed out that our results can be considered as dynamic generaliza-tions of their static counterparts existing in the literature.

Keywords: Nondifferentiable Multiobjective Programming; Second-Order Invexity; Second-Order Pseudoinvexity;

Second-Order Quasi-Invexity; Second-Order Duality; Nonlinear Multiobjective Programming

1. Introduction

Second-order duality in mathematical programming has been extensively investigated in the literature. In [1], Chen formulated second order dual for a constrained variational problem and established various duality re-sults under an involved invexity-like assumptions.

Sub-sequently, Husain et al. [2], have presented Mond-Weir

type second order duality for the problem of [1], by troducing continuous-time version of second-order in-vexity and generalized second-order inin-vexity. Husain and Masoodi [3] formulated a Wolfe type dual for a nondif-ferentiable variational problem and proved usual duality theorems under second-order pseudoinvexity condition while Husain and Srivastav [4] presented a Mond- Weir type dual to the problem of [2] to study duality under second-order pseudo-invexity and second-order quasi- invexity.

The purpose of this research is to present multiobjec-tive version of the nondifferentiable variational problems considered in [2,4] and study various duality in terms of efficient solutions. The relationship between these mul-tiobjective variational problems and their static counter-parts is established through problems with natural bound-ary values.

2. Definitions and Related Pre-Requisites

 

,

I a b _{be a real interval,} :_I n n

Let  R R R

:_{I R}n _Rn _Rm

   

and be twice continuously

differ-entiable functions. In order to consider 



t x t x t,

   

,



: n

where x IR is differentiable with derivativex,

denoted by x _and x the first order derivatives of 

with respect tox t

 

and x t

 

, respectively, that is,

1 2

1 2

, , , ,

, , ,

T n

T n

x x x

x x x

  

  

  

 

 _{  } 

 

  

 

 _{  } _







  

x

x





Further denote by xx, and x the Hessian

and

n n

m n Jacobian matrices respectively.

The symbols x, xx , xx and x have analogous

representations.

Designate by X the space of piecewise smooth

func-tions _{x I}: _Rn _{with the norm x x Dx}

 

 

 

 

d ,

t a

uDxx t 



u s s

,

where the differentiation operator D is given by

Thus d

dt D except at discontinuities.

required for the derivation of the duality results.

Definition 1. (Second-order Invex): If there exists a

vector function _



_{t x x}, ,



_Rn

0  

:_{I R} n_Rn_Rn _{and with} where

  a



t x x, ,



 at t  and

such that for a scalar function , the func-

t b

tional



, ,



d

I



_{I R} n_Rn_R

t x x t

 where : satisfies





 

 

 

 

d

d

T T

G t t

G t t

 

 

 

 

  

 

 

1

, , d , ,

2

, , , ,

T

I I

T

x x

I

t x x t t x t

t x D t x

x

x x

 

   

 

 _{ }

 



    

 _{ } _{ }

   







  

  

 

then



, ,



d

I

t x x t





 is second-order invex with respect to

 where ₂ 2 3

xx xx xx xx

G  D D D



, n



C I R

  the space of n-dimensional continuous    and

vector functions.

Definition 2. (Second-order Pseudoinvex): If the func-

tional



, ,



d

I

t x x t









satisfies

 

 











 

 

d 0

d

T

t

t G t t

 



 

 

 



, ,



d

I

t x x t







1

, , d , ,

2

T

T T

x x

I

I I

D G t

t x x t t x x

     

 

 

  









 

then is said to be second-order pseudo-

invex with respect to .

Definition 3. (Second-order strict-pseudoinvex): If

the functional



, ,



d

I

t x x t





 satisfies

 

 













 

 

d 0

d

T

t

t G t t

 



 _ 

 

 

1

, , d , ,

2

T

T T

x x

I

I I

D G t

t x x t t x x

     

 

 

 









 

then



, ,



d

I

t x x t





 is said to be second-order pseudo-

invex with respect to .

Definition 4. (Second-order Quasi-invex): If the func-

tional



, ,



d

I

t x x t





 satisfies









 

 

d

d 0

T

t G t t

t

  





 

 





1

, , d , ,

2

I I

T

T T

x x

I

t x x t t x x

D G t

 

     



  



  









 

then



, ,



d

I

t x x t





 is said to be second-order quasi-in-

vex with respect to .

 does not depend explicitly on t, then

the above definitions reduce to those for static cases.

Remark 1. If

The following inequality will also be required in the forthcoming analysis of the research:

Lemma: 1 (Schwartz inequality): It states that

     



     





     



2

1 2

1

T T T

x t B t z t  x t B t x t z t B t z t

       

with equality in (1) if B t x t q t z t 0 for

some q t

 

R t I,  .

, _Rn

 

Throughout the analysis of this research, the following conventions for the inequalities will be used:



1_, 2_{, ,} n



If with α    and



_{ }1_, 2_{, ,}n





β 









, 1, 2, ,

and

, 1, 2, , .

i i

i i

i n

i n

   

     

   

 

  

   





 



   







 

   



, then

3. Statement of the Problem and Necessary

Optimality Conditions

Consider the following nondifferentiable Multiobjective variational problem:

(VCP): Minimize

   







 

   



1

1 1 2

1 2

, , d , ,

, , d

T I

T

p p

I

f t x t x t x t B t x t t f t x t x t x t B t x t t

  

   

 _ _





 

 

 _{ }

 

  











subject to

 

,

 

x a  x b  (1)



,

   

,



0,

g t x t x t  t I



, n



(2)

x C I R



where 1) _{C I R}, n



_{denote the space of piecewise}

smooth functions x with norm x  x  Dx_



 , where

differentiation operator D already defined.



2) : , 1, 2, , ,

:

i n n

n n m

f I R R R i K p

g I R R R

    

  







are assumed to be continuously differentiable functions, and 3) for each _{t I i K} , _{ } 1, 2, ,_{ p B t}, i

 

n n

is an

 positive semi definite (symmetric) matrix, with

 

.

B continuous on I.

In this section we will derive Fritz John and Ka-rush-Kuhn-Tucker type necessary optimality conditions for (VCP).

Definition: A point xX is said to be efficient











 

   



 

   





1

1 2

2

d

d , , ,

, ,

T

r r

I

T

r r

I

f t x x x t B t x t t

f t x x t B



  

 _{  }

 

 











 _{t t t}

x x

 

 _

 

 

 

 





1, 2, , p



 





for some r K  and









 

 

 

   





1

1 2 2 _d

d , , ,

, ,

T

i i

I

T

i i

I

f t

x

x x x t B t x t t

f t x x t B



  



  

 

 











 _ t x t _ t

 

 

 

 

 

 

.

K r

  

for r

The following result which is a recast of a result of Chankong and Haimes [5] giving a linkage between an efficient solution of (VCP) and an optimal solution of p-single objective variational problem:

r K

Proposition 1. (Chankong and Haimes [5]): A point

 

x t X is an efficient solution of (VCP) if and only if

 

x t is an optimal solution of

 

Pr for each rK.

 

Pr : Minimize

 





 



r _{, ,}





 

T r

   



12 _d

I

f t x t x t  x t B



_  t x t _ t

 



, , 0,

subject to

 

,

 

,



x a  x b  g t x x  t I







 

   













 

   



1 2

, ,

, ,

T

i i

I

T

i i

I

f t x x t B t x t

f t x x x t B t x t

1 2

d

d , r

t

t K

x





 

 

 



 









 i



  

 

 

for obtaining the optimal conditions for (VCP) we will

use the optimal conditions obtained by Chandra et al. [6]

for a single-objective variational problem which does not

contain integral inequality constraints of Pr .

The validity of the following proposition is quite

es-sential in obtaining the optimality conditions for (VCP),

Proposition 2. If x t( )X is an efficient solution of

(VCP), then x t( ) is an optimal solution of the

follow-ing problem

 

Pˆr for each rK.



_Pˆ



r

   





: Minimize



_t,



 

_{t x t} 12 _dt

 

 

 

 , 

T

r r

I

f x x  x t B





subject to

 

,

 

,



, ,



0,

x a  x b  g t x x  t I







 

   









 

   



1 2

1 2

, , d

, , d , ,

T

i i

I

T

i i

r I

f t x x x t B t x t t

f t x x x t B t x t t i K t I

 _ 

 

 

 

 _  _  

 









Proof: Let x( )t be an efficient solution of (VCP).

Suppose that x( )t is not optimal solution of



Pˆ



.

r K r

, for

some _{ Then there exists an x t}0

 

_{X such that}



, ,



0,

g t x x  t I







 

   









 

   



1 2

0 0 0 0

1 2

, , d

, , d , ,

T

i i

I

T

i i

r I

f t x x x t B t x t t

f t x x x t B t x t t i K t I

 _ 

 

 

 

 _  _  

 



















 

   



(3) and











 

   



1

0 0 0 0 2

1 2

, , d

, , d ,

T

r r

I

T

r r

I

f t x x x t B t x t t f t x x x t B t x t t

 



 

 

 

 

 _  _

 









r

i K The inequality (3) for







 

   













 

   



1

0 0 0 0 2

1 2

, , d

, , d

T

i i

I

T

i i

I

f t x x x t B t x t t f t x x x t B t x t t

 



 

 

 

 



 

 

 











(4)

The inequalities (3) along with (4) contradicts the fact that x( )t is an efficient solution of (VCP).

Hence x( )t

 

Pˆr

.

r K

is an optimal solution of , for some



Theorem 1. (Fritz John Type necessary optimality condition): Let x( )t be an efficient solution of (VCP).

Then there exist _i_{R i K}, _{ and piecewise smooth}

functions _y:Ι_Rm

and zi:ΙR i Kn,  , such that

   









   

   











   



 



   



1

, ,

, , ( ) , ,

, , 0,

p

i i i i

x i

i T

x x

T x

f t x t x t B t z t Df t x t x t y t g t x t x t Dy t g t x t x t t I







 

  





 

 



(5)



 

T ,

   

,



0,

y t g t x t x t  t I

(6)

 

T i

   

i



 

T i

   



12_,

x t B t z t  x t B t x t i K

(7)

 

T

   

1,

i _{t B}i i

z t z t  iK

(8)



,y t

 



0,t I

(9)

Proof: Since x( )t is an efficient solution of (VCP),

for each and hence in particular 1 . So by the results of [6] there exist

.

rK

 

Pˆ





,

R i K

 

i

 and piecewise

smooth functions _y:Ι_Rm

and zi:ΙR i Kn,  such that

   









   

   









 



   



 



   



1

, ,

, 0

, ,

i i

x p

T i

x i i

x T

x

f t x t x t B t Df t x t x t y t g Dy t g t x t x t t









 









 



, , , ,

i

z t t x t x t

I

 

 

T



,

   

, t



0,t I

y t g t x t x

 

T i

   

i



 

T i

   



12_{, ,}

x t B t z t  x t B t x t i K t I

 

T

   

1, ,

i i i

z t B t z t  i K t I 

 



_{ }1_, 2_{, , }k_{,y t}



0,t I .

The above conditions yield the relations (5) to (9).

Theorem 2 (Kuhn-Tucker type necessary optimal-ity condition):

Let x t( )

r k be an efficient solution of (VCP) and let for _Pˆ



each , the conditions of



r satisfy Slaters or

Robinson condition [6] at x t( ). Then there exist _Rp

and piecewise smooth functions _{y I}: _Rm _and

:

i

z IRn,_{iK such that}

   









   

   









 



   



 



   





1

, ,

, ,

x

T



, ,

, ,

0,

i i i

i i

i x

T

f t x t x t B t z t

Df t x t x t y t g t x t x t

D y t g t x t x t t I





 

 









 



p



(10)

 

T



,

   

,



0,

y t g t x t x t  t I (11)

 

T i

   

i



 

T i

   



12_,

x t B t z t  x t B t x t

1, 2, , ,

i  p t I

(12)

 

T

   

i _{t B}i i

z t z t 1,tI (13)

 

0,

y t  t I (14)

0

  (15)

Proof: Since x t( ) is an efficient solution of (VCP)

by Proposition 2, x t( ) is an optimal solution of

 

P

r K r

for each r k . Since for each , the contradicts of

 

Pr , satisfy Slaters or Robinson conditions [6] at x t( )



2, ,

, by Kuhn-Tucker necessary condition of [6], for each

, there exist



p

1,

r K  _vi_{R i K,}



_



r r and

piecewise smooth function j

 

r K

 t R i,  such that

























1

, ( ), ( ) ( ) ( ) , ( ), ( )

, ( ), ( ) ( ) ( ) , ( ), ( )

( ) , ( ), ( ) 0, , ( ), ( )

r

r r r r

x x

i i i i i

r x x

i K

j T j m r x

j j

x

f t x t x t B t z t Df t x t x t v f t x t x t B t z t Df t x t x t

t g t x t x t

t I Dg t x t x t







 

  

 

 

 _{ } 

 

 











 

 





 



   



1

, , 0,

m T

j j

r j

t g t x t x t t I





 





 

T i

   

i



 

T i

   



12_{, ,}

x t B t z t  x t B t x t i K t I 

 

0, ,



1, 2, ,



j

r t t I j m

    

0,

i

r r

v  i K

r K _{we obtain}

Summing over





























1 2

1

1 2

1

1 2

, ( ), ( ) ( ) ( )

...

, ( ), ( )

( ) ( ) ... ( ) , ( ), ( )

0,

( ) ( ) ... ( ) , ( ), ( )

i i i

p x

i i i p

i i

x

j j j j

p x m

j j j j

j

p x

f t x t x t B t z t

v v v

Df t x t x t

t t t g t x t x t

D t t t g t x t x t

t I

  

  





  

 

   _{ }

_ 

 

    

 _{ }

 

   

 

 









 







 

 

 



1j t 2j t pj t g t x t x t



j



,

   

,



0,

t I

    





_ 

1

i

v i K.

where _r  for each 

These can be written as,













 



 



1

1

, ( ), ( ) ( ) ( )

, ( ), ( )

( ) , ( ), ( )

0, , ( ),

i i i

p x i

i i

x j j

m x

j j j

x

f t x t x t B t z t v

Df t x t x t

t g t x t x t

D t g t x t x t

 





  

 

 

_ 

 

  

 

 

 

 







 









 

   



1

, , 0,

m j

j _{t g t x t x t}j _{t I}





 





where 1 _r 0

r

i i

r K

v  



v 

and

 

 





1

0, , 1, 2, ,

m

j j

r j

t t t I j m

 







   

Setting

 

 





1

, and , 1,2, ,

j i

i i

p i

i i

y t

v

i K t j m

v

 





   



we get

   





   



   







 



   





 



   





0,

i _{t z t}i

x t

t I

  



1

1

, ,

, ,

, ,

, ,

i i x i x m

i i x i

i x p

i

i

f t x t x t B

Df t x t x t

t g t x t

D t g y

y t x t x t



 

 









 





 



   

t



0,t I

1

, ,

m j

j _{t g t x t x}j

y







That is

   





   



   







 



   



 



   







0,

i

t z t

t

1

, ,

, ,

, ,

, ,

i i i

x i x T

T p

i

f t x t x t B Df t x t x t

t g t x t x t D t g t x t x t y

y I























  

 

T



,

   

,x t



0,t I

y t g t x t 

 

T i

   

i



 

T i

   



12_,

x t B t z t  x t B t x t i K

 

T

   

1,

i i i

z t B t z t  i K

 

_



1

 

_, 2

 

_{t ,},_ym

 

_t



_0,_{t I}

y t y t y



1_, 2_{, ,} p



      0.



  

   

4. Mond-Weir Type Second Order Duality

In this section, we present the following Mond-Weir type second-order dual to (VCP) and validate duality results:

(M-WD): Maximize

 

 



  

   

 

2

1 , ,

2

I

T T

p p

I

 

1 _{, ,} 1 1 1 _{d , ,}

d

T T

p

f t x x x t B t z t t H t t

f t x x x t B t z t t



H t t

 

 

  _{ } 

  

 



 



 

    



 

 

,

 

x a x b





subject to

 

 

 

 





 

0,

i i i x z t Dfx H t

t t I



  

  



  



(16)

 

1

n

i i i i

f B t







 

T

 

T

x x

y t g Dy t g G

  _

(17)

 

 

1

, , d 0

2

T T

I

y t g t x x  t G t t

 _ 

 

 



 

 

 

   

0, 0, 1,

,

T i i i

y t z t B t z t

i K t I

 

 

 

(18)

(19)

where

















2 3

2

T T

x _x x _x

T T

x _x x _x

G y g D y g

D y g D y g

 

 



 _{  }





2 3

2 , ,

1, 2, , .

i i i i i xx xx xx xx

and

H f Df D f D f t I

i K p

    

 

  



We denote by CP and CD the sets of feasible solutions

to (VCP) and (M-WD) respectively.

Theorem 3. (Weak Duality): Assume that

 

(A1) x t CP and



_x

     

_{, ,} 1 _{, ,} p

   

_,



D

t y t z t  z t  t C

   

,.,. .T

 

d

i i i i I

.

(A2)

 

 _f t  B z t _ t

   

T ,.,. d

I

y t g t t



is second-order

pseudoinvex.

(A3) is second-order quasi-invex.

Then

   







 

   





  

   

 

 

1 2

, , d

, ,

1 _{d , for some}

2

T

r r

I

T

r r r

I

T r



f t x t x t x t B t x t t f t x x x t B t z t

t H t t r K

 

 

 

 

 

 

 

 

 







 (20)

and

   







 

   





  

   

 

 

1 2

, , d

, ,

1

d , 2

T

i r

I

T

i i i

I

T i

r



f t x t x t x t B t x t t f t x x x t B t z t

t H t t i K

 

 

 

 

 

 _





 















0,

(21)

cannot hold.

Proof. Suppose to the contrary, that (20) and (21)

hold.







 

   





  

   

 

 

1 1

, ,

, ,

1 _d

2

p

T i i

i I

T

i i i

p

T i i I

1

2 _d

i

i

f t x x x t B

f t x x x t B

t H t t





 

 



 _



 









 



t x t t

t z t

 

 

 

 

Since

 

   



 

   



1 2_{, ,}

T i i T i

x t B t z t x t B t x t t I



we have

 







 

   





  

   

 

 

1

1

, ,

, ,

1 _d

2

p

T i i

i I p

T

i i i

i I

T i

1

2 _d

i

i

f t x x t x t B

f t x x x t B t z

t H t t





 





 

 _ 

 











t x t t

t

 

 

 

 

Now, by the constraints (2), (18) and (19), we have

  



  

T





 

t GT 

 

t dt



 

d 0

G t _ t

, , d

1 , ,

2

T

I I

y t g t x x t

y t g t x x

 _

 











This by (A3), yields

 









 



 







, ,

, ,

T T

x I

T T T

x

y t g t x x

D y t g t x x



 

 

 



 



By integration by parts, we have

 







 



 





 

0 T T T T

x x

I T

x

y t g D y t g

t b y t g

t a

 



 _{ }



 









 _{ }T_{G t} d_t



 



T





 

T



T T

x x

I

y t g D y t g

  







 

0  _{ }T_{G t} d_t

 

 

 



_Hi_{ t}



d_t0

 

0 T i d

This, by using (4) gives

 

1

p

i T i i i i

x x

I i

f B z t Df

 



 







p

(22)

   





1

i T i i i T i

x x

i I

f B t z t Df

  



 













 

H t t

 



 

 

1

1

( ) ( )

T i i i

p  _f _{B t z t}  _D

0 d

( )

T _i

x x

i

T i i I

p T i i

x i

f t H t

t a f

t b



 

 





 

 



_ 

 

 



 







   





 

 

1

0

d

p

i T i i i

x

i I

T i T i

x

f B t z t

D f H t t

 

  







  _

 

 





By hypothesis (A1), it implies





 

 







  

   



 

 

1

1

, , ( ) d

, , 1

d 2

p

T

i i i i

i p

T

i i i i

i

T I

i I

f t x x x t B t z t t f t x x x t B t z t

t H t t

 

 











 _













 





Using

 

   



 

   



1 2_,

T i i T i

x t B t z t  x t B t x t t I in the above, we have







 

   





  

   



 

 

1 2 1

1

, , d

, , 1

d 2

p

T

i i i

i p

T

i i i i

i

T I

I

i

f t x x x t B t x t t f t x x x t B t z t

t H t t





 





 



 

 

 





 _













 



This contradicts (20) and (21). Hence the result.

 

Theorem 4 (Strong duality): Let x t be normal

and is an efficient solution of (VP). Then there exist

p

R

 _{, a piecewise smooth function y I}: _Rm _such

that



_{x t}

   

_{,y t , , z t}1

 

_{, , z t}p

   

_{, t 0}



_is

feasi-ble for (M-WD) and the two objective functions are equal. Furthermore, if the hypotheses of Theorem 3 hold for all feasible solutions of (VCP) and (M-WD) ,then



_{x t y t( ), ( ), , ( ), , z t}1 _{ z t}p_{( ), ( ) t}



_{is an efficient solution}

of (M-WD).

 

Proof: Since x t is normal and an efficient solution

of (VP), by Proposition 2, there exist _Rp

: m and

piecewise smooth y IR

: , 1, 2, ,

i m

w I R i  n





   

and satisfying







 







 





1

, , , ,

0,

p

i i i i i

x x

i

T T

x x

f t x x B t z t Df t x x

y t g D y t g t I





 

   



 

 

 

(23)



, ,



0,

T

y t g t x x  t I ₍₂₄₎

 

T i

   

i



 

T i

   



12_,

x t B t z t  x t B t x t t I

 

T

   

1,

i i i

z t B t z t t I



(25)

(26)

 

0,y t 0,t I.

   (27)



From (24) along with  t



0,t I

  



, we have

 

 

1

, , d 0

2

T T

I

y t g t x x  t G t t

 _  _

 

 

Hence

   

 

   



 t , t 0





 



1

, , , , , p

x t y t z t  z

satisfies the constraints of (M-WD) and







 

   





  

   

 

1 2

, , d

1 , ,

2

T

i i

I

T

i i i

I

f t x x x t B t x t t

 

d

T i

f t x x x t B t z t t

 



 

 

 



 _  









 

  H  t _ t



 

That is, the two objective functionals have the same value.

Suppose that



_{x t z t}

   

_, 1 _{, , z t y}p

 

_, t ,

 

t _0



     

   



_{,ˆ ,ˆ}1 _{, ,ˆ}p _, ˆ



_,

is not the efficient solution of (M-WD). Then there exists

ˆ _D

x t y t z t  z t  t C such that

1 1 1

1 1 1

1

1

ˆ ˆ

ˆ ˆ ˆ ˆ

( , , ) ( ) ( ) ( ) ( ) ( ) d ,...,

2

1 ˆ ˆ

ˆ ˆ ˆ ˆ

( , , ) ( ) ( ) ( ) ( ) ( ) d

2 1

ˆ ˆ

( , , ) ( ) ( ) ( ) ( ) ( ) d ,...,

2 ˆ

( , , )

T T

I

p T p p T p

I

T T

I p

f t x x x t B t z t 1 t H t t f t x x x t B t z t t H t t f t x x x t B t z t t H t t f t x x x

 

 

 

 

 

 

 _{ } 

 _{ }

  _{ }  

 

 _{ } 

 

 _{ } 

 

 

















 

1 ˆ

( ) ( ) ( ) ( ) ( ) d

2

T p p T p I

t B t z t  t H  t t

 

 

 

  _  

 

 _{ } 







 

ˆ _{t 0}

 

As , we have









1 1 1

1

1 1 2

1 1

1

ˆ ˆ ˆ ˆ

( , , ) ( ) ( ) ( ) ( )

2 1

ˆ ˆ ˆ ˆ

( , , ) ( ) ( ) ( ) ( )

2

( , , ) ( ) ( ) ( ) d ,..

( , , ) ( ) ( ) ( )

T T

I

p T p p I

T T

I

p T T

f t x x x t B t z t ˆ t H1ˆ( ) d ,...,

ˆ _{ˆ( ) d}

.,

T p

t t f t x x x t B t z t t

f t x x x t B t x t t f t x x x t B t x t

 

 

 



 

 



















H t t

 

 

   

 

 _{ } 

 

   

 

 _{ } 

 

2 _d

I

t

 

 

 



 

 

 

 

 

 

 

This contradicts the conclusion of Theorem 3. Hence



_{x t y t( ), ( ), , ( ), z t}1 _,_{z t}p( ), ( )_{ t}



_{is an efficient solu-}

tion of (M-WD).

Theorem 5 (Converse duality):

(A1): Assume that



x, , , , , y z1 zp,



, 1, 2, ,

is an effi-cient solution of (M-WD)

(A2): The vectors



H G t I i Kij, j,  , 



i j n

 

are linearly independent where Hj the th row of is

i j

H and Gj is the jth row of G,

(A3)





   





, , , ,

i i

x

i i

x

f t x x B t Df t x x H  t



  





 

_{, ,}

i

z t

t I i K 

t I



 

d 0

x  t t

 

are linearly independent and

(A4) for either

a)

 

T





 

T x



I

t G y t g





and ( )T T 0

x I

t y t g

 



 





 

T x





 

d 0

x I

t G y t g t t

 

or b)



 



 



and ( )T T 0

x I

t y t g







Then x is feasible for (VCP) and the two objective

functionals have the same value. Also, if Theorem 3 holds for all feasible solutions of (CP) and (M-WD), the

x is an efficient solution of (VCP).



Proof: Since x, , , , , y z1 zp,



is an efficient

solution of (M-WD), there exist _Rp_{, R}p

r R

and

:_{I R}n

  _, i:_I

 , and piecewise smooth  R

1, 2, ,

i n

,

 _and :_I n

  R such that the following

Fritz John optimality conditions (Theorem 1)









































2

3 4

2 3 4

1

( ) ( ) ( ) ( ) 2

1 _{( ) ( )} 1 _{( ) ( )}

2 2

1 1

( ) ( ) ( ) ( )

2 2

( ) ( )

( ) ( ) ( )

i i i i T i

x x _x

i T i T i

x x

T i T i

x x

i i i

x x

i i i

i _{x x x}

f B t z t Df t H t

D t H t D t H t

D t H t D t H t

H H t D H t

D H t D H t D H t

G G

 

    

   

 

  





 _{  } 

 

 

_{ } 

 

 

_{ } 

 

 

 

  



 





 

 



  























 

 



















2 1

3 4

2

3 4

( )

( ) ( ) ( )

( ) ( )

1

( ) ( ) ( ) ( )

2

1 _{( ) ( )} 1 _{( ) ( )}

2 2

1 1

( ) ( ) ( ) ( )

2 2

p i

x x x

x x

T T T i

x x _x

T i T i

x x

T i T i

x x

t

t D G t D G t

D G t D G t

y t g D y t g t H t

D t H t D t H t

D t H t D t H t



 

 

 

    

   



 

 

 

 

 _{ } 

 

_{ } 

 

 _{ } 

 

 

 

 _  _

 

  



 



 

 

 

 

0,t I

 



 

   

 

(28)





0,

,

T i i i i i

x

t f B t z t H t t I i K

     

 

 



 



(29)

 

 

 

1 2

0, , 1, 2, ,

T j j j T j

x xx xx

j

t g g t g t g t

t t I j m

    



 

  _  _

 

    

   

 

=1 =1

0 , , 1, 2, ,

p p

T

i i i i

i i

H t t H G G t

t I j m

       

(30)

 _  _ 

 

 







     

 

2

     

0,

.

T

i T_{x t B t}i _{t B t}i i _{t B t z t}i i

i K

  

(31)

  



 

(32)

 

 

1 _{d 0}

2

T T

I

y t g t G t t

 _    _ 

 

 

T

t y



 

t 0,t I



,i1, 2, , n

0

T

 



_1_{, ,}n



_0



1 2

, , , , ,_{  }n _0

 



0

p

t  t G

 

(34) i K from (32) and i

   

t z ti T B t z ti

   

i i

 

t ,

t I

   



   

1 0

i i T i i

t z t B t z t

   ₍₃₅₎

(36)

 

, , t , ,

   

 

 



 , t , ,  t 

(37)

(38)

From (31), we have

 

 







 

1

i i i

i

H   t   t 



 



(39)

This, by the hypothesis (A2) gives

 

0,

i _t i _{t t I}

      (40)

and

 

t  

 

t 0,t I





  (41)

Using (40), (41) and (17), we have

































2 3

4

2

3 4

1 _{( ) ( )} 1 _{( )}

2 2

1 1

( ) ( ) ( )

2 2

1

( ) ( )

2

( ) ( )

( ) (

T i T i

x

i T i

T i x

i i

x x

i i

i _x

t H t D t H

D t H t D

D t H t

H t D H t D

D H t D H

   

  

 

 

 



 

  



 

 



































( ) ( ) ( ) ( )

( )

( )

( )

i i i i i i i

x x

x T i

x x

i x

f B t z t Df H t

t

t H t

H t

  

 



   

 

 

 

 

 

 

 

 









 

























( ) ( )

( ) 0,

( )

x

x x

i x

t

H t

H t

 



 

 



 

 

 

 

 

 

 

 

 

 _{ }

 

 

 

 

 



0.

2 1

3 4

2

3 4

)

( ) ( )

( ) ( )

1

( ) ( )

2

1 _{( ) ( )} 1 _{( )}

2 2

1 1

( ) ( ) ( )

2 2

p

x i

x x

x x

T i x

T i T i

T i T

x

t G

G t D G t D G t

D G t D G t

t H t

D t H t D t

D t H t D t

t

 

 

 

   

 







  

 



  

 

 





 



I



(42)

Let   

 

t 0

 

0

i _t

   t I

 

t

 

 

t 0

Then (41) gives and (40) gives

,

Using 0 and _{ }i , (42) implies

 







i

 

i i

 



0

x z t Dfx H  t 

0

i i

   0

i i

 _ _fi_{B t}i



This, because of (A3) yields

(43)

The relation (43) with   _{gives  }i 0

Since 0, (36) gives  0.

I

 The relation (30)

   

yields

 

t 0,t we have _i _{t B t z t}i i

 

_0 _,

 , i K from (35). These yield _i

 

_t