On Continuous Programming with Support Functions

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On Continuous Programming with Support Functions

Iqbal Husain1, Santosh K. Shrivastav1, Abdul Raoof Shah2 1

Department of Mathematics, Jaypee University of Engineering and Technology, Guna, India 2

Department of Statistics, University of Kashmir, Srinagar, India Email: [email protected]

Received June 12, 2013; revised July 12, 2013; accetped July 19, 2013

Copyright © 2013 Iqbal Husain etal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

A second-order Mond-Weir type dual problem is formulated for a class of continuous programming problems in which both objective and constraint functions contain support functions; hence it is nondifferentiable. Under second-order strict pseudoinvexity, second-order pseudoinvexity and second-order quasi-invexity assumptions on functionals, weak, strong, strict converse and converse duality theorems are established for this pair of dual continuous programming problems. Special cases are deduced and a pair of dual continuous problems with natural boundary values is constructed. A close relationship between the duality results of our problems and those of the corresponding (static) nonlinear pro-gramming problem with support functions is briefly outlined.

Keywords: Continuous Programming; Second-Order Generalized Invexity; Second-Order Duality; Nonlinear Programming; Support Functions; Natural Boundary Values

1. Introduction

Chen [1] was the first to identify second-order dual for-mulated for a constrained variational problem and estab-lished various duality results under an involved invexity- like assumptions. Husain etal. [2] have presented Mond- Weir type second-order duality for the problem of [1] and by introducing continuous-time version of second- order invexity and generalized second-order invexity, vali-dated various duality results. Subsequently, for a class of nondifferentiable continuous programming problems, Husain and Masoodi [3] studied Wolfe type second-order duality while Husain and Srivastava [4] investigated Mond- Weir type second-order duality. Recently, in the spirit of Mangasarian [5], Husain and Masoodi [6] studied Wolfe type second-order duality for a continuous programming problem having support functions appearing in the inte-grand of the functional as well as in the constraint func-tions under second-order invexity and second-order pseu-doinvexity conditions. They also incorporated a pair of second-order dual variational problems with natural bound-ary values rather than fixed end points and indicated their close relationship with those of corresponding (static) second-order duality results established for nonlinear pro-gramming problem with support functions, considered by Husain etal. [7]. The popularity of this type of nondif-ferentiable continuous programming problems seems to

originate from the fact that, even though the objective function and/or constraint functions are non-smooth, a simple representation of the dual problem may be written. The theory of non-smooth mathematical programming deals with more general type of functions by means of generalized subdifferentials. However, square root of posi-tive semi-definite quadratic form and support functions are amongst few cases of the nondifferentiable functions for which one can write down the subdifferentials explic-itly.

(2)

2. Pre-Requisites

Let I 

 

a b, be a real interval :_I__Rn__Rn __R

and

:IRnRn Rm

  be twice continuously

differenti-able functions. In order to consider 



t x t,

   

,x t



, where _{x I}: __Rn

is differentiable with derivative x

denoted by x and x the first order derivative of 

with respect to x t

 

and x t

 

respectively, that is,

T T

1, 2, , , 1, 2, ,

x n x n

x x x x x x

     

 _   _  _  

     

         .

 



Denote by xx the n n Hessian matrix of  and

x

 the Jacobian matrix respectively, that is,

with respect to

mn

 

x t , that is,

2



, , 1, 2, ,

xx i j i j n

x x

 _ _ 

 

  

 

x

 the m n _Jacobian

matrix.

1 1 1

1 2

2 2 2

1 2

n

n x

m m m

n m n

x x x

  



  



   

 _ _ _ 

 

   



 _  

 

  

 _ _ _ 

 



   



    

The symbols   _x, _xx , _xx and_x

: n

have analogous rep-resentations. Designate by X the space of piecewise

smooth functions x IR , 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.

We incorporate the following definitions which needed in the subsequent analysis:

Definition1. (Second-Order Invex):

If there exists a vector function  



t x x, ,



Rn

where : n n n

I R R R

    and with 0



n n

at t = a

and t= b such that for a scalar function the

functional where



t x x, ,





, ,



d

I

t x x t





 :IR R R satisfies









   





 





 





T T

1

, , d , , d

2

, , , , d .

I I

x x

I

t x x t t x x p t Gp t t

t x x D t x x Gp t t

 

    

 

   

 

  





 

Then



, ,



d is second-order invex with respect

I

t x x t







to  where G_xx2D_xx_D2_xx_D3_xx,



, n



the space of -dimensional continuous

vector functions.



, n

I R

pC

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

tional



, ,



d

I

t x x t





 satisfies

 

   













 

T

T T _d ₀

1

, , d , , d .

2

x x

I

T

I I

D Gp t p t t

t x x t t x x p t Gp t t

    

 

  

 

  _  _

 





 

Then



, ,



d

I

t x x t





 is said to be second-order pseu-

doinvex with respect to .

Definition 3. (Second-Order Strictly Pseudoinvex):

If the functional



, ,



d

I

t x x t





 satisfies

 













 

T

T T

T

d 0

1

, , d , , d ,

2

x x

I

I I

D Gp t t

    

 

  

 

    

 





 

then



, ,



d

I

t x x t





 is said to be second-order strictly

pseudoinvex with respect to .

Definition 4. (Second-Order Quasi-invex):

If the functional



, ,



d

I

t x x t





 satisfies









 

   





T T

1

, , d , , d

2

d 0.

I I

x x

I

D G t p t t

 

    

 

 _  _

 

   





 

Then



, ,



d

I

t x x t





 is said to be second-order quasi-

invex with respect to .

Consider the following nondifferentiable continuous programming problem with support functions treated by Husain and Jabeen [8]:

(CP): Minimize





, ,





 





d

I

f t x x S x t K t





subject to

 

0





,

x a  x b (1)



, ,





 



0, 1, 2, , , ,

j j

g t x x S x t C  j  m tI (2) where f and g are continuously differentiable and each





, 1, 2, ,

j

C j  m is a compact convex set in Rn.

Husain and Jabeen [8] derived the following optimal-ity condition for (CP):

Lemma 1. (Fritz-John Necessary Optimality Con-ditions):

If the problem (CP) attains a minimum at x x X

then there exist rR_{and piecewise smooth functions}

: m

y IR with

 



1

 

2

 



, , , m

(3)

: n

z IR and wj:IRn, j1, 2,, ,m such that





 





 





 





, ,

, , ,

j j

t g t x x

t x x t I



 

 

 

1 T

, ,

m

j

x x

j

x x

r f t x x z t y w t

D rf t x x y t g



    

  

  



 

 





 

T

 

1

, , 0,

m

j j j

j

y t g t x x x t w t t I



 _ _

 



 _

   

T



 



,

x t z t S x t K tI

 

T

 



 



, 1, 2, , ,

j j

x t w t S x t C j  m tI

 

, j

 

j, 1, 2, , ,

z t K w t C j  m tI

 



r y t,



0, tI

 



r y t,



0, tI

The minimum x t

 

of (CP) may be described as

normal if r 1,_{so that the Fritz John optimality} tions reduce to Karush-Kuhn-Tucker optimality condi-tions. It suffices for r 1 that Slater’s [8] condition holds at x t

 

.

Now we review some well known facts about a sup-port function for easy reference.

Let K_{be a compact set in} , then the support

func-tion of _{is defined by}

n

R K

 







     

T



max : , .

S x t K  x t v t v t K tI

A support function, being convex everywhere finite, has a subdifferential in the sense of convex analysis i.e., there exist _{z t}

 

__{R t}n, __I such that

 







 





   



T

 

.

S y t K S x t K  y t x t z t

From [9], the subdifferential of S x t K



 



is given by

 







     T



 





, such that .

S x t K z t K t I x t z t S x t K

    

For any set , the normal cone to at a point

is defined by

n

R

  

 

x t  

 







 

n

     





0,

 



N x t  y t R y t z t x t  z t  



It can be verified that for a compact convex set C,

if and only if

 

C



 

y t N x t

 





   

T

, .

S y t C x t y t tI

3. Mond-Weir Type Second-Order Duality

In this section, we present the following problem as the Mond-Weir type dual to (CP) and validate usual duality theorems:

(M-WCD):



    

T 1

 

T

 

Maximize , , d

2

I

f t u u u t z t p t FP t t

 _ _ 

 

 





subject to

 

0



u a  u b



(3)

 



 



 







  

1

T

0

m

j j j

u u

j

u u

f z t y t g w t

D f y t g F G p t



  

   



  

(4)

 



 

T

 



T

   

1

d 2

m

j j j

j I

y t g u t  t p t Gp t t



 

0

  

 









(5)

 

, j

 

j, , 1, 2,

z t K w t C tI j ,m (6)

 

0,

y t  tI (7) where

1) p t

 

R tn, I

2) F f_uu2Df_uuD f2 _uu  D f3 _uu,tI

 

3) T



T



 



T





 

T



2 3

2

uu u

u

uu uu

G y t g D y t g

D y t g D y t g

 

 



  

Theorem 1. (Weak Duality): Let x t

 

X be fea-sible solution of (CP) and

       

 

   



_{u t} _,_{y t} _,_{z t} _,_{w t}1 _,_w2 _t _,__,_wm _t _,_{p t}



be feasible for (M-WCD). Assume that for all feasible

         

 



1 2



, , , , , , , m

x t u t y t z t w t w t  w t

and with respect to vector function  



t x u, ,



,

1)



     

,.,. .T



d

I

f t  z t



t is second-order pseudo-

invex and

2)



     

T T

 



1

,.,. . d

m

j j j

j I

y t g t w t t









is second-or-

der quasi-invex. Then,

 



inf CP sup M-WCD



.

Proof: Since x t

 

is feasible for (CP) and

       

 

   



1 2



, , , , , , m ,

u t y t z t w t w t  w t p t

is feasible of (M-WCD), we have

 





 



 



  

 



 

1

T

1

T

, , d

, ,

1

d 2

m

j j j

j I m

j j j

j I

y t g t x x S x C t

y t g t u u u t w t

p t Gp t t

 



 

 _  _

 





 _











(4)

Using x t

 

Twj

 

t S x t



 

/Cj



,tI j, 1, 2,,m,

we have,

 



  

 



 



  

 



 

T 1

T

, , d

, ,

1

d . 2

m

j j j

j I

m

j j j

j I

y t g t x x x t w t t

p t Gp t t





 

 _  _

 





 _











By the second-order quasi-invexity of

 



   

T

 



1

,.,. . d ,

m

j j j

j I

y t g t w t t







for wj

 

t Rn, j1, 2,,m with respect to , from this we have,

 



 



 



 



 



T

1

T T T

d 0

m

j j j

u j

I

u

y t g w t

D y t g Gp t t



 



  

 _

  

  



 





  .

By integrating by parts, we have

 



 



 





 



 



 



T

1

T T

d

0.

m

j j j

u j

I

u

y t g w t

D y t g Gp t t

t b

t y t g

t a



 

 

 

 _

  

  



 



 











Using 0, at ta and tb, this yields,

 



 



 





 



T 1

T

d 0

m

j j j

u j

I

u

y t g w t

D y t g Gp t t







 



 





 



Using equality constraint (4), we have

 





 





 



T

T T

0

d

0, by integrating by parts

u u

I

u u

T u

f z t Df Fp t



f z t D f Fp t t

t b

f

t a



  



   

   



 





  

As earlier, this becomes

 





 



_T T _T



d 0

u u

I

     





This, because of second-order pseudoinvexity of

     



T



 

,.,. . d , n,

I

f t  z t t z t R t



I with respect to

,

 gives



    







    

 

T

T T

, , d

1

, , d .

2

I

f t x x x t z t t

f t u u u t z t p t Fp t t



 

 _   _

 





Since x t

   

Tz t S x t



 

K



,tI, we have







 









    

T

 

T

 

, , d

1

, , d ,

2

I

f t x x S x t K t

f t u u u t z t p t Fp t t



 

 _   _

 





implying,

 



inf CP sup M-WCD



.

Theorem 2. (Strong Duality):

If x t

 

X be an optimal solution of (CP) and is

normal, then there exist piecewise smooth functions

: m,

y IR : n and

z IR j: n

w IR such that

       

   



1 t ,p t 0



, , , , m

x t y t z t w t w is a

feasi-ble solution of (CD) and the two objective values are equal. Furthermore, if the hypothesis of Theorem1 holds,

then



       

1 , , m

   

,



, , ,

x t y t z t w t  w t p t is an

op-timal solution of (M-WCD).

Proof: From Lemma 1 there exist piecewise smooth

functions y I: Rm, : n

z IR and j: n

w IR



j1, 2,,m



such that





 





 







 







1



, , , ,

, , , , 0,

m

j j j

x x

j

T

x x

f t x x z t y t g t x x w t

D f t x x y t g t x x t I



  

   



 

 





 

T

 



1

, , 0,

m

j j j

j

y t g t x x x t w t t I



  





   

T



 



/ ,

x t z t S x t K tI

 

T

 



 



/ , 1, 2, , ,

j j

x t w t S x t C j  m tI

 

, j

 

j, 1, 2, , ,

z t K w t C j  m tI

 

0,

y t  tI

The above relations imply that

       

   



1



, , , , , m , 0

x t y t z t w t  w t p t  is feasible

for (M-WCD). Also







 











   

 

T

 

, , d

1

, , d .

2

I

f t x x S x t K t

f t x x x t z t p t Fp t t



 

 _   _

 





This shows the equality of objective functions of the problem. Hence the optimality of

       

   



1



, , , , , m ,

(5)

follows from weak duality theorem (Theo

me that

(C is second-or

pseu is

second-order quasi-invex with respect to the same rem1).

Theorem 3 (Strict Converse Duality): Assu

1):



     



T



,.,. . d der strictly

I

f t  z t t

doinvex and

 

1 m j I y t 







   

T

 



,.,. .

j j j

g t w t

 



 

 

.

(C2): x t

 

is an optimal solution for (CP), If

 



, , ,z ,wm,p t is optimal solution of (M-



1 2

, ,

u y w w

WCD), then

 

, I.

Proof: We

u t x t t

assume that u t

 

x t

 

and show that a contradiction follows. Since x t

 

is a

The

n optimal solution

of (CP), it follows from orem 2, there exist

: m,

y IR z I: Rn and



,m



suc

: ,

j n

w IR j1, 2, h that

       

 

   



x t ,y t ,z t ,w t1 ,w2 t ,,wm t ,p t 0



is optimal solution of (M-WCD). Since

       

1 2

 

   

,y t ,z t ,w t ,w t ,,wm t ,p t

is an optimal solution of (M-WCD), it follows that



u t







   









   

T



 T

, , d

1

, , d

2

I

f t x x x t z t t

f t u u u t z t p t Fp t t



 

 _   _

 





This, because of the second-order strict pseudoinvexity

of for all

(8)

From the constraint of (CP) and (M-WCD), we have

     



T





 

__Rn

gives

,.,. . d

I

f t  z t t z t

 





 



T T T



_d ₀

u u

I

     





 







 



 



  

 



 

1 T 1 T

, , d

, ,

1

d 2

m

j j j

 

j I

m

j j j

j I

y t g t x x S x t C t

p t Gp t t

        _     _ 









  Using 

 

T

 



 



, , 1, 2, ,

j j

,

x t w t S x t C tI j  m

from this, we have





  

 



 



  

 



 

T 1 T 1 T

, , d

, , 1 d 2 m j j  _j j I m

j j j

j I

y t g t x x x t w t t

p t Gp t t

          _     _ 









 

This, because of (C1) we have

 





 



 







 





T 1

T T T

, ,

d 0

m

j j j

u j

I

u

y t g t u u w t

D y g Gp t t

        _          





  (9)

Combining (8) and (9), we have

 



 



 



 





 

 



 



 







  



 







T 0 m j

f z t y t g



  





   





1

T T T

T T T T d . j j u u j I u u m

j j j

u u

w t

D f y t g F G p

f z t y t g w t

D f y t g F G p t t

t b

f y t g

t a                               



      Using 1 u u j I     



d t t

at ta andtb, this implies

, 0,



 



 



 



T





  



d 0

m

T j j j

u u

j I

u u

f z t y t g w t

D f y t g F G p t t

     _ _ _         





  

contradicting the equality constraint of (M-WCD), hence

 

,

u t x t tI

(Converse Duality):

(H1):

.

Theorem 4. Assume that

       

 

   



1 2



, , , , , , m ,

x t y t z t w t w t  w t p t

mal solution of (M-WCD). is an opti

(H2): The vectors



F G ii, i, 1, 2,,m



i

are linear

independent where F and _Gi

are the ith row of F and G respectively, and

(H3):

 



 





 

T



   

j j j

x x

y t g w t D y t g G t p t 0,

t I

  



 



and

(H4): either

 



 



 

T

d 0

xx I

p t T Gy t g p t t



d p t





T

 



 



1

d 0

m

j j j

x j

I

y t g w t t



 





an

 

T



 

T



 

d 0

xx I

p t Gy t g p t t



or

and

 

j I p t 

 



 



T 1 d 0 m

j j j

x

y t g w t t





 

x t

Then is feasible for (CP) and the two objective

ionals have the same value. Also, if Theorem1 holds feasible solution of (CP) and (M-WCD), then funct

for all

 

(6)

Since

Proof:

       

1

   

, , , , , m ,





   

 

j



 

j j

C

t y t x t N w t

  



(14)

x t y t z t w t  w t p t

solution of (M-WCD), by results of Sc

s R, R

is an optimal hester [10], there

 



 

T

 



1

 

T

 

0 2

j j j

i I I

y t g x t w t p t Gp t





 _ _ _

 









(15)

exist   and piecewise sm

: n

ooth function

I R

  and :IRm such that following Fritz

John optimality conditions are satisfied:

 







 



T

   

t y t 0,tI (16)

 







 



 



 





 



 



 





 

....

T T

3 4

T T

2 2

1 1

2 2

x x

x

xx xx x

t

D t D p t Fp t

t f Df y t g D y





 

 

 _   

 





 











 



 











  





  







  







T ₂ T

T

T T

T 2

2 3

x

x x

xx x xx x

x x

xx x _x

x

x x

D p t Fp t D p t Fp

p t Fp

t g

D f y t g D D f y t g

D Df y t g F G p t

D F G p t D F G p t

D F G



 



   

   

   

 





   



 



 

 







  



 







 



T

1 2

x x

f z t Df p t Fp t

 _    

 



 , t , 



0,tI (17)

1 1

   



 , t , t , 



0,tI (18) Using the hypothesis (H2) in (12), we have

 

t p t

 

0,t

   I (19)

 

t p t

 

0,t

   I (20)

Using (4), (19) and (20) in (10), we have (see (21) below)

Let  0, then (20) gives (19)

im-plies

 

t 0,t ,

  I

 

p t

 0,tI.

Consequently from (21), we have

 





 



 





 



1 T

, ,

0,

m

j j j

x j

x

y t g t x x w t

D y t g Gp t t I







 



  





 







 



 







 



 





....

4 T 1

T T

2 3

T 4

1 1

2 2

1 1

2 2

1

0, 2

x x

m

j j j

x x

j

x x

x

p t D F G p t

y t g w D y t g

p t Gp t D p t Gp t

D p t Gp t D p t Gp t

D p t Gp t







  _



 _  



 



 









 

(10)



By the hypothesis (H3), this implies  0.

The relation (11) implies 

 

t 0,tI.

Hence



 ,

   

t , t ,



0 contradicting the Fritz John condition, Hence  0, & 0.

 

Pre-multiplying (11) by and Using (16), we

have

j

y t

 



 



 





 

T

T 1 T

2 0,

j j j

x xx

j j j

xx

j

t g w t g p t

g x t w t p t g p t

t t I







   

 

 

 _   _

 

  

 



 





 



 



 



 



 



 

T T

1

T

1

T T

1

0. 2

m

j j j

xx j

m

j j j

j

xx

t y t g w t y t g p t

y t g x t w t

p t y t g p t



 



 

 

 

 



  





 _





(11)

I

 (12)

 



 t p t



F





 

t p t

 



G0,t

   

k



 



x t t N z t

   (13)

Integrating and then using (15), we have





 



 





 



 



 





 





 



 







 



  



  





  





  





  







  



....



 

T

2

 





 





 



 

....

T T ₂ T

1

T T

3 4 2

T T

3 4 2

T 3

2

m

T

j j j

x x

j

T

x x x

x x

x x _x _x _x

y t g w t D y t g G p t Fp t D p t Fp t D p t Fp t

D p t Fp t D p t Fp t t p t D F G p t D F G p t

D F G p t D F G p t t D p t Gp t D p t Gp t

D p t

 

 

 

 _    _ _  

 



  _    

     







 



 _ _

 







 



....

T 4

0,

x x

Gp t D p t Gp t _ tI



(21)

p t 

F G

 



p t Gp

 _

 

(7)

 



 





 



 



 



 





T T

1

T T T

d

d 0 2

m

j j j

x xx

j I

xx I

t y t g w t y t g p t

p t Gp t p t y t g p t t



 

 

 

 

 

 

 _  _

 





t



Putting 

 

t  p

 

t we have

 



 



 



 



 

T

1

T T

d

d 0 2

m

j j j

x j

I

xx I

p t y t g w t t

p t G y t g p t t





 

 _  _

 

 







ives

d

  

of the hyp yields,

g

 



 



 



 



 

T

1

T T

2

d 0

m

j j j

x j

I

xx I

p t y t g w t t

p t G y t g p t t







 



 





This, in view othesis (H4)

 

0,

p t  tI we have x t

 

N_k



z t

 



and

 

j



 



, 1, 2, ,

j

C .

x t N w t j  m

These respectively imply x t

   

Tz t S



 

t K



and

x

 

T

 



 



, 1, 2, ,

j j

x t w t S x t C j  m.

Multiplying the relation (11) by _yj

 

_t

and using (16) ith

along w p t

 

0,tI, we have

 



 

T

 



1

0,

m

j j j

j

y t g x t w t t I



  



and also j



, ,





 

j



0,

g t x x S x t C  tI im lying the

feasibility of

p

 

x t for (CP).

Finally,







 











   

 

T

 

, d

1

d 2

I

f t x K t

T

, ,

,x S x t

I

f t x x t





_

_  _{x t} _{z t} _ _{p t} _Fp _ _t

 



 

By Theorem 1, it implies that x t

 

i P).

4. Special Cases

,

s an optimal so-lution of (C

Let for tI, A t

 

,Bj

 

t ,j1, 2,,m

us on .

be positive

semidefinite matrices and continuo I

Then



     





 



1 2 T

, ,

x t A t x t S x t K tI where

         



T



1,

K  A t z t z t A t z t  tI

 

   





1 2



 



T

, 1, 2, , ,

j j _.

x t B t x t S x t C j  m tI

Replacing S x t



 

K



by



     



1 2 T

x t A t x t and

 



j



,

S x t C

 

   





1 2

T

, 1, 2, , ,

j

.

x t B t x t j  m tI

We have the following problems:













     





d

1 2 T

2

CP : Minimize , ,

I

f t x x  x t A t x t





subject to

t

 

0

 

,

x a  x b







 

   



1 2 T

, , 0,

, 1, 2, , .

j j

g t x x x t B t x t

t I j m

 

 





(M-WCD2):



      



 

T

Maximize , ,

1

d 2

f

I

t u u u t A t z

p t FP t t





 _







subject to

t

 

0



,

u a  u b







   

 

m

j





j

 





  











  

1

T

, ,

, , , , 0,

u

j j

j

u u

f t u u A t z t

y t g t u u B t w t

D f t u u y t g t u u F G p t





 _u 

by

t I

    



   





 



 

   



   

T

1

T

1

d 0 2

m

j j j j

j I

y t g u t B t t

p t Gp t t

 



 





 _ 







     

T

1,

z t A t z t  tI

     

1, , 1, 2, ,

j j j

w t B t w t  tI j  m

If



 

   



1 2 T _j

, 1, 2, ,

x t B t x t j  m are suppressed

from the constraints of (CP2), we have the following

problem studied for duality by Husain and Srivastava [4].











     



1 2 T

3

CP : Minimize , , d

I

f t x x x t A t x t t

 _ 

 

 





subject to

 

0

 

x a  x b



, ,



0, .

g t x x  tI

(8)



      



 

T

Maximize , ,

1

d 2

I

f t u u u t A t z t

p t FP t t





 _







 

0

 

u a  u b





     







  











  

T

, , , ,

0,

u u

f t u u A t z t y t g t u u

D f t u u y t g t u u

F G p t t I

 

 

   

 

  

T



1

 

T

 

, , d 0,

2

I

y t g t u u p t Gp t t t I

 _ _

 

 



  

     

T

1,

z t A t z t  tI

 

0, .

y t  tI

dual variational problems with natural boundary values rather than fixed end points.

5. Problems with Natural Boundary Values

In this section, we formulate a pair of nondifferentiable



CP : Minimize0







, ,





 





d

I

f t x x S x t K t





subject to x a

 

 0 x

 

b ,



, ,





 



0, , 1, 2 ,

j j

g t x x S x t C  tI j  m







    



     

0

T

- :

Maximize , ,

1

d 2

I

M WCD

f t u u u t z t

p t F t p t t





 _







subject to



 

0

 

u a  u b



  

 





 



  











  

T

1 T

, , , ,

0,

m

j j j

u u

j

u u

f t u u z t y t g t u u w t

D f t u u y t g t u u

F G p t t I



  

 

   



 



 





 



  

 

T 1

T

, ,

1

d 0,

2

m

j j j

j I

p t Gp t t t I





 





 _ 











 

0,

y t  tI

 

, j

 

j, 1, 2, , ,

z t K w t C j  m tI



, ,



0 at and ,

u

f_ t u u  ta tb

 

T





, , at and .

u

g t u u 0, t a t b

y t     

6. Nonlinear Programming Problems

If all functions in the problems (CP0) and (M-WCD0) are

independent of t, then these problems will reduce to the

following nonlinear programming problems ied by

Husain etal. [7]. (CP1):Minimize

stud

 

f x S x K

subject to

 





0, 1, 2, , .

j j

g x S x C  j  m





 

T T

1

1 CD : Maximize

2

f u u z p Fp

subject to

 



 







1

0

m

j j j

u u

j

f u z y g u w F G p



 



   



T



T

1

0, 2

m

j j j

j

y g u w p Gp



 

  

 







, j j, 1, 2,

zK w C j , ,m

where F fuu

 

u and

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[1] X. H. Chen, “Second Order Du y for the Variational Problems,” Journal of Mathematical Analysis and Appli-cations, Vol. 286, No. 1, 2003, pp. 261-270.

http://dx.doi.org/10.1016/S0022-247X(03)00481-5

 

.

uu

Gy gT u

alit

[2] I. Husain, A. Ahmed and M. Masoodi, “Second Order Duality for Variational Problems,” European Journal of Pure and Applied Mathematics, Vol. 2, No. 2 009, pp. 278-295.

for a ming Pro-

f Operations Research, Vol. 2, No. 3,

dx.doi.org/10.4236/ajor.2012.23035

, 2 [3] I. Husain and M. Masoodi, “Second-Order Duality

Class of Nondifferentiable Continuous Program

blems,” European Journal of Pure & Applied Mathemat- ics, Vol. 5, No. 3, 2012, pp. 390-400.

[4] I. Husain and S. K. Shrivastav, “On Second-Order Dual- ity in Nondifferentiable Continuous Programming,” Ame- rican Journal o

2012, pp. 289-295.

http://

[5] O. L. Mangasarian, “Second and Higher Ord Duality in Nonlinear Programming,” Journal of Mathematical Ana- lysis and Applications, Vol. 51, No. 3, 1979, pp. 605-620.

dx.doi.org/10.1016/0022-247X(75)90111-0

er

http://

[6] I. Husain and M. Masoodi “Second-Order Duality for Continuous Programming Containing support Functions,” Applied Mathematics, Vol. 1, No. 6, 2010, pp. 534-541.

http://dx.doi.org/10.4236/am.2010.16071

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[8] I. Husain and Z. Jabeen, “Continuous Programming Con-taining Support Functions,” Journal of Applied Mathe-tics & Infor ol. 26, No. 1-2 008, pp. 75-106. [9] B. Mond and M. Schechter, “Nondifferentiable

Symmet-ric Duality,” B Mathematical ciety, Vol. 53,

ma matics, V , 2

ulletin of the Australian 1996, pp. 177-188.

So-

http://dx.doi.org/10.1017/S0004972700016890

[10] M. Schechter, “More on Subgradient Duality,” Journal of Mathematical Analysis and Applications, Vol. 71, No. 1, 1979, pp. 251-262.