Journal of Applied Mathematics & Bioinformatics, vol.3, no.2, 2013, 65-86
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2013
On Second-Order Fritz John Type Duality for
Variational Problems
I. Husain 1 and Santosh K. Shrivastav 2
Abstract
Second-order dual to a variational problem is formulated. This dual uses the Fritz
John type necessary optimality conditions instead of the Karush-Kuhn-Tucker
type necessary optimality conditions and thus, does not require a constraint
qualification. Weak, strong, Mangasarian type strict-converse, and Huard type
converse duality theorems between primal and dual problems are established
under appropriate generalized second-order invexity conditions. A pair of
second-order dual variational problems with natural boundary conditions is
constructed, and it is briefly indicated that duality results for this pair can be
validated analogously to those for the earlier models dealt with in this research.
Finally, it is pointed out that our results can be viewed as the dynamic
generalizations of those for nonlinear programming problems, already treated in
the literature.
1
2
Department of Mathematics, Jaypee University of Engineering and Technology,
Guna, M.P, India: e-mail: ihusain11@yahoo.com
Department of Mathematics, Jaypee University of Engineering and Technology,
Guna, M.P, India: e-mail: santosh_00147@rediffmail.com
Article Info: Received : December 19, 2012. Revised : March 1, 2013
Published online : June 30, 2013
66
On Second-Order Fritz John Type Duality for Variational Problems
Mathematics Subject Classification: 90C30, 90C11, 90C20, 90C26
Keywords: Variational Problem, Generalized Second-order invexity, Natural
boundary values, Nonlinear programming
1 Introduction
Second-order duality in mathematical programming problem has been widely
investigated in the recent past. Mangasarian [1] formulated second-order dual to
nonlinear programming problem and gave duality result under assumptions that
are involved and rather difficult to verify, Subsequently Mond [2] established the
duality results for non-linear programming problem under simpler assumption.
Subsequently many researchers investigated second-order duality under invexity
and generalized invexity conditions. Duality for continuous programming problem
has been studied by a number of researcher researches. Mond and Hanson [3]
were the first to consider a class of constraint variational problems and study first
order-duality for such problem. Motivated with the results of [3], a number of
duality theorems appeared in the literature.
Chen [4] was the first to identify second-order dual formulated for a
constrained variational problem and established various duality results under an
involved invexity like assumptions. Recently, Husain et al [5] have presented
Mond-Weir type duality for the problem of [6] and by introducing
continuous-time version of second-order invexity and generalized second-order
invexity, validated various duality results. Earlier Weir and Mond studied duality
for nonlinear programming problem using Fritz John type optimality conditions
instead of Karush-Kuhn-Tucker optimality conditions and thus their duality results
do not require a constraint qualification.
In this research we study Fritz type second-order duality using Fritz John
type optimality conditions and validate various duality theorems under the
I. Husain and Santosh K. Shrivastav
67
assumption of second-order pseudoinvexity and second-order quasi-invexity. A
pair of Fritz John type second-order dual variational problems with natural
boundary condition by ignoring fixed point condition is formulated and a close
relationship between our duality results and those of Husain et al [7] is briefly
outlined.
2 Pre-Requisites
Let
I = [ a, b ]
be
a
real
interval,
φ : I × Rn × Rn → R
and
ψ : I × R n × R n → R m be twice continuously differentiable functions. In order to
consider φ ( t , x ( t ) , x ( t ) ) , where x : I → R n is differentiable with derivative x ,
denoted by φx and φx , the first order of φ with respect to x ( t ) and x ( t ) ,
respectively, that is,
∂φ 
∂φ 
 ∂φ ∂φ
 ∂φ ∂φ
=
φx =
, 2 , ,
, φx  1 , 2 , ,

n 
1
∂x 
∂x n 
 ∂x ∂x
 ∂x ∂x
T
T
Denote by φxx the Hessian matrix of φ , and ψ x the m × n Jacobian matrix
 ∂ 2φ 
, i, j 1, 2,..., n,
respectively, that is, with respect to x ( t=
) , that is φxx =
i
j 
 ∂x ∂x 
ψ x the m × n Jacobian matrix
 ∂ψ 1 ∂ψ 1
∂ψ 1 
 n 
 1
∂x 2
∂x
 ∂x

2
2
 ∂ψ
∂ψ
∂ψ 2 

ψ x =  ∂x1 ∂x 2
∂x n 


 
 


m
∂ψ m
∂ψ m 
 ∂ψ


∂x 2
∂x n m×n .
 ∂x1
The symbols φx , φxx , φxx and ψ x have analogous representations.
68
On Second-Order Fritz John Type Duality for Variational Problems
Designate by X, the space of piecewise smooth functions x : I → R n , with the
x
norm =
x
∞
+ Dx ∞ , where the differentiation operator D is given by
t
u =D x ⇔ x ( t ) =∫ u ( s ) ds ,
a
d
= D except at discontinuities.
dt
Thus
We incorporate the following definitions which are required in the subsequent
analysis.
Definition2.1 (Second-order Pseudoinvex) If the functional
∫ φ ( t , x, x ) dt
I
satisfies
∫ {η φ + ( Dη )
T
T
x
}
φx + η T G β ( t ) dt ≥ 0
I
1
T


⇒ ∫ φ ( t , x, x ) dt ≥ ∫ φ ( t , x , x ) − β ( t ) G β ( t )  dt ,
2

I
I 
or

∫ φ ( t , x, x ) dt ≤ ∫ φ ( t , x , x ) − 2 β ( t )
I
1
I
{
T

G β ( t )  dt

}
⇒ ∫ η T φx + ( Dη ) φx + η T G β ( t ) dt ≤ 0
T
I
then
∫ φ ( t , x, x ) dt
is said to be second-order pseudoinvex with respect to η .
I
Where G =
φxx − 2 D φxx + D 2φxx  , and β ∈ C ( I , R n ) , the space of n -dimensional
continuous vector functions.
Definition 2.2 (Strictly-pseudoinvex) If the functional
∫ φ ( t , x, x ) dt
I
satisfies
I. Husain and Santosh K. Shrivastav
∫ {η φ + ( Dη )
}
φx + η T G β ( t ) dt ≥ 0
T
T
69
x
I

∫ φ ( t , x, x ) dt > ∫ φ ( t , x , x ) − 2 β ( t )
⇒
I
then
∫ φ ( t , x, x ) dt
1
T
I

G β ( t )  dt ,

is said to be second-order strictly- pseudoinvex with respect to
I
η.
Definition2.3
(Semi-strictly
∫y
g : I × Rn × Rn → Rm ,
T
pseudoinvex)
If
( SGFr CD )
and
(t ) g dt will be semi-strictly pseudoinvex with respect
I
to η , if
∫y
T
(t ) g dt
is strictly pseudoinvex for all
y (t ) ≥ 0, t ∈ I ,
I
y (t ) ≠ 0, t ∈ I .
Definition2.4. (Second- order Quasi-invex) If the functional
∫ φ ( t , x, x ) dt
I
satisfies
∫ φ ( t , x, x ) dt ≤ ∫ {φ ( t , x , x ) − 2 β ( t )
1
I
I
T
{

G β ( t ) dt

}
⇒ ∫ η T φx + ( Dη ) φx + η T G ( t ) β ( t ) dt ≤ 0,
T
I
Then
∫ φ ( t , x, x ) dt
is said to be second-order quasi-invex with respect to η .
I
Remark 2.1 If ϕ does not depend explicitly on t, then the above definitions
reduce to those given in [8] for static cases.
Consider the following class of nondifferentiable continuous programming
problem studied in [9]:
(CP0):
Minimize
∫ { f ( t , x ( t ) , x ( t ) )}dt
I
70
On Second-Order Fritz John Type Duality for Variational Problems
Subject to
x ( a )= 0= x ( b ) ,
g ( t , x ( t ) , x ( t ) ) ≤ 0,
t∈I
h ( t , x ( t ) ,=
x ( t ) ) 0,
t∈I
where, (i) f, g and h are twice differentiable functions from I × R n × R n into
R, R m and R k respectively, and
(ii) B ( t ) is a positive semi definite n × n matrix with B ( ⋅) continuous on I .
The following proposition gives the Fritz John type optimality conditions which
are derived in [9].
Proposition 2.1 (Fritz-John Conditions) If x is an optimal solution of (CP0 )
and hx (., x(.), x (.) ) maps on the closed subspace of C ( I , R n ), then there exist
Lagrange multipliers r ∈ R , and piecewise smooth y : I → R m such that
T
T
r f x ( t , x ( t ) , x ( t ) ) + y ( t ) g x ( t , x ( t ) , x ( t ) ) + µ ( t ) hx ( t , x ( t ) , x ( t ) )
T
T
D  rf x ( t , x ( t ) , x ( t ) ) + y ( t ) g x ( t , x ( t ) , x ( t ) ) + µ ( t ) hx ( t , x ( t ) , x ( t ) )  t ∈ I ,
=


T
y (t ) g (t, x (t ) , =
x ( t ) ) 0,
t∈I
(r , y (t )) ≥ 0, t ∈ I
(r , y (t )) ≠ 0, t ∈ I
If r = 1, then x is called a normal solution and the above conditions reduce to
Karush-Kuhn-Tucker conditions.
Ignoring the equality constraint in (CP0 ) , consider the following variational
problem:
(CP ) :
Minimize
∫ f (t , x, x )dt
I
Subject to
x(a )= 0= x(b)
g (t , x, x ) ≤ 0
(1)
(2)
I. Husain and Santosh K. Shrivastav
71
Chen [4] formulated the following Wolf type dual (CD1 ) to (CP) :
(CD1 ) :


1
∫ ( f (t , u, u ) + y(t ) g (t , u, u ) ) − 2 β (t )Gβ (t )  dt
Maximize
I
Subject to
=
u (a ) 0,=
u (b) 0
fu (t , u , u ) + y (t )T gu (t , u , u ) − D ( fu (t , u , u ) + y (t )T gu (t , x, x ) ) + G β (t ) =
0, t ∈ I
y (t ) ≥ 0, t ∈ I ,
where
G=
fuu (t , u , u ) + y (t )T g xu (t , u , u ) − 2 D  fuu (t , u , u ) + ( y (t )T gu (t , u , u ) )  
u

+ D 2 ( ( f x (t , u , u ) + y (t )T gu )u ) − D 3 ( fu (t , u , u ) + y (t )T gu )  ,
u
and β (t ) ∈ R n , t ∈ I
Using the invexity-like assumptions on the functions that constitute the
primal problem, Chen [4] derived second-order, strong and converse duality
results for the above pair of problem (CP ) as (CD1 ) . Recently in order to relax
invexity requirement on the function, further, Husain et al [5] formulated the
following Mond-Weir type second-order dual to (CP) which is given below and
established various duality theorems:
( M − WCD) :
1
∫ ( f (t , u, u ) − 2 β (t )
Maximize
T
F β (t )) dt
I
Subject to
=
u (a ) 0,=
u (b) 0
fu (t , u , u ) + y (t )T gu (t , u , u ) − D ( fu (t , u , u ) + y (t )T gu (t , u , u ) )
+ ( F + H ) β (t ) = 0, t ∈ I

∫  y(t )
I
T
1

g (t , u , u ) − β (t ) H β (t ) dt ≥ 0
2

y (t ) ≥ 0, t ∈ I
where
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On Second-Order Fritz John Type Duality for Variational Problems
F=
fuu (t , u , u ) − 2 Dfuu (t , u , u ) + D 2 fuu  (t , u , u ) − D 3 fuu (t , u , u ), t ∈ I
and
=
H y (t )T guu (t , u , u ) − 2 D ( y (t )T guu (t , u , u ) )
+ D 2 ( y (t )T guu  (t , u , u ) )  − D 3 ( y (t )T gu ) 
u
u
Husain et al [5] establish weak duality theorem under the assumption that
∫ f (t ,.,.)dt
is second-order pseudoinvex and
I
∫ y(t )
T
g (t ,.,.)dt is second-order
I
quasi-invex with respect to η . They proved strong duality for the
Mond-weir
type
dual
continuous
programming
problem,
pair of
using
the
Karush-Kuhn-Tucker type necessary conditions at the optimal for the primal
(CP ) and hence regularity condition was needed at the optimal point of the
problem (CP ) .
In this research a second-order dual and a generalized dual to (CP ) are
proposed and establish duality theorems using Fritz John necessary conditions at
the optimal point for the primal (CP ) . Thus the requirement for a constraint
qualification or regularity conditions is eliminated.
3 Fritz John Type second-order duality
We formulate the following Fritz John type second-order dual to problem
(CP):
( SFr CD ) :
Maximize
1
∫ { f (t , u, u ) − 2β (t )
T
F (t ) β (t )}dt
I
Subject to
u (a )= 0= u (b)
r fu + y (t )T gu − D(r fu + y (t )T gu ) + (rF + H ) β (t ) = 0, t ∈ I
(3)
I. Husain and Santosh K. Shrivastav
∫ ( y(t )
T
I
73
1
g (t , u , u ) − β (t )T H β (t )) dt ≥ 0
2
(4)
(r , y (t )) ≥ 0, t ∈ I
(r , y (t )) ≠ 0, t ∈ I
(5)
Theorem 3.1 (Weak Duality) Let
( r , u (t ), y (t ), β (t ) )
be feasible to
x∈ X
( SFr CD )
be feasible to (CP)
. If
∫ f dt
and
be second-order
I
pseudoinvex and ∫ y ( t ) g dt is second-order semi-strictly pseudoinvex with
T
I
respect to the same η .
Then
inf(CP) ≥ Sup ( SFr CD )
Proof: Let x be feasible for (CP ) and
( r , u (t ), y (t ), β (t ) )
1
of ( SFr CD) .
∫ f (t , x, x )dt < ∫ ( f (t , u, u ) − 2 β ( t )
F β ( t )) dt .
This, because of second-order pseudoinvexity of
∫ f (t ,.,.)dt
Then, suppose
I
T
I
with respect to η
I
yields,
∫ {η
T
fu + ( Dη )T fu + η T F β (t )}dt < 0
I
This gives
∫
r{η T fu + ( Dη )T fu + η T F β (t )}dt ≤ 0
(6)
I
with strict inequality in (6) if r > 0.
From the constraint of ( CP ) and ( SFr CD ) , we have
∫ y(t )
T
I
1
T
g (t , x, x ) dt ≤ ∫ ( y (t )T g (t , u , u ) − β ( t ) H β ( t )) dt .
2
I
This, because of second-order semi-strict pseudoinvexity of
∫ y(t )
T
I
g (t , x, x ) dt ,
74
On Second-Order Fritz John Type Duality for Variational Problems
with respect to η implies,
∫ {η
T
( y (t )T gu ) + ( Dη )T ( y (t ) gu )T + η T H β (t )} dt ≤ 0
(7)
I
with strict inequality in (7) if some y i (t ) > 0, t ∈ I , i ∈{1, 2,3, 4,..., m}.
Combining (6) and (7), we obtain
0 > ∫ η T (rfu + y (t )T gu ) + ( Dη )T ( rfu + y (t ) gu ) + η T ( rF + H ) β (t )  dt ,
I
0 > ∫ η T (rfu + y (t )T gu ) − D ( rfu + y (t ) gu ) + ( rF + H ) β (t )  dt
I
+η
T
( rf
u
+ y (t ) gu )
t =b
T
,
t=a
(by integrating by parts)
=
∫η (rf
u
+ y (t )T gu ) − D ( rfu + y (t ) gu ) + ( rF + H ) β (t )  dt ,
I
η
(U sin g =
0,=
t a and=
t b)
That is,
∫η
T
[(rfu + y (t )T gu ) − D ( rfu + y (t ) gu ) + ( rF + H ) β (t )] dt < 0,
I
contradicting the equality constraint (3) of ( SFr CD ) .
Hence
1
∫ f (t , x, x )dt ≥ ∫ ( f (t , u, u ) − 2 β ( t )
I
T
F β ( t )) dt ,
I
giving
inf ( CP ) ≥ Sup ( SFr CD ) .
Theorem 3.2 (Strong duality) If x is optimal to (CP), then there exist r ∈ R
and piecewise smooth y : I → R m such that (r , x , y ) is feasible for ( SFr CD)
and the corresponding values of (CP) and ( SFr CD) are equal. If also,
I. Husain and Santosh K. Shrivastav
∫ f ( t ,.., ) dt
75
is second-order pseudoinvex and
I
∫ y(t )
T
g (t ,.,.) dt is semi-strictly
I
pseudoinvex. Then x and (r , x , y ), are optimal solution of (CP) and ( SFr CD) ,
respectively.
Proof: Since x is optimal to (CP) by Proposition 2.1, there exist r ∈ R and
piecewise smooth y : I → R m such that
(r f x + y (t )T g x ) − D(r f x + y (t )T g x ) =
0, t ∈ I
y (t )T g (t , x , =
x ) 0, t ∈ I
(r , y (t )) ≥ 0, t ∈ I
(r , y (t )) ≠ 0, t ∈ I
This implies that ( x , r , y , β=
(t ) 0), t ∈ I is feasible for ( SFr CD) . Equality of
objective functionals is obvious. In view of second-order pseudoinvexity of
∫ f (t ,.,.) dt
and second-order semi-strict pseudoinvexity of
I
∫ y ( t ) g ( t ,.., ) dt
T
I
with respect to η , optimality follows by Theorem 3.1.
4 Generalized Second-order Fritz John Duality
In this section, we generalized the dual (SFrCD).
Let M = {1, 2,3,..., m}, Iα ⊂ M , α =
0,1, 2,3,..., k with Iα  I β = φ , (α ≠ β )
k
and
 Iα = M .
Let
K = {0,1, 2,3,..., k} and N ⊂ K .
α =0
The following is the generalized second-order Fritz John dual to (CP):
( SGFr CD) :
Minimize
∫ f ( t , u, u ) dt
I
Subject to
u (a )= 0= u (b)
(8)
76
On Second-Order Fritz John Type Duality for Variational Problems
(r f x + y (t )T gu ) − D(r fu + y (t )T gu ) + (r F + H ) β (t ) =∈
0, t I
∑ ∫ y(t ) g (t , u, u )dt ≥ 0,
α=
0,1, 2,3,..., k ,
(9)
(10)
i∈Iα I
(r , y i (t ), i ∈ Iα , α ∈ N ) ≠ 0, t ∈ I .
(r , y (t )) ≥ 0,
∫ f ( t , u, u ) dt
Theorem 4.1 (Weak duality) If
(11)
is pseudoinvex
I
∑α ∫ y ( t ) g ( t ,.., )dt , α ∈ N
i
i∈I
i
second-order semi-strictly pseudoinvex,
I
∑ ∫ y (t ) g (t ,..,..)dt , α ∈ K \ N
i
i
second-order quasi-invex with respect to the
i∈Iα I
same η .
Then
inf ( CP ) ≥ sup ( SGFr CD ) .
Proof: Let x be feasible to (CP) and (u , r , y , β ) be feasible for ( SGFr CD ) .
Suppose
1
∫ f ( t , x, x ) dt < ∫ ( f ( t , u, u ) − 2 β ( t )
I
T
F β ( t )) dt.
I
This, by second-order pseudoinvexity of
∫ f ( t , u, u ) dt ,
I
∫ {η
T
fu + ( Dη )T fu + η T F β (t )}dt < 0
I
Thus
∫ r{η
T
fu + ( Dη )T fu + η T F β (t )}dt ≤ 0
(12)
I
with strict inequality in (12) if
r > 0,
∑ ∫ y (t ) g (t , x, x )dt ≤ ∑ ∫ y (t ) g (t , u, u )dt ,
i
i∈Iα I
i
i
i
α∈N
i∈Iα I
By second-order semi-strict pseudoinvexity of
∑ ∫ y (t ) g dt , α ∈ N ,
i
i∈Iα I
i
I. Husain and Santosh K. Shrivastav
∑ ∫ {η ( y (t ) g ) + ( Dη )
T
i
i
T
u
i∈Iα I
77
}
( y i gu ) + η T H i β (t ) dt ≤ 0, α ∈ N ,
(13)
where
Hi =
( yi (t ) gui ) − 2D ( yi (t ) gui )  − D 2 ( yi (t ) gui )  − D3 ( yi (t ) gui )
u
u
u
u
with strict inequality in (5), if y i (t ) > 0, t ∈ I , i ∈ Iα , α ∈ N .
Also
∑ ∫ y (t ) g (t ,.,..,.)dt ≤ ∑ ∫ y (t ) g (t , u, u )dt , α ∈ K \ N
i
i
i
i∈Iα I
i
i∈Iα I
∑ ∫ y (t ) g (t ,.,..,.)dt , α ∈ K \ N , this
i
By second-order quasi-invexity condition
i
i∈Iα I
implies that
∑ ∫ {η
i∈Iα
T
( y i (t ) g i ) + ( Dη )T ( y i (t ) g i u ) + η T H i β (t )} dt ≤ 0, α ∈ K \ N
(14)
I
Combining (12), (13) and (14) we have
∫ {η
T
(r fu + y (t )T gu ) + ( Dη )T (r fu + y (t )T gu ) + η T (rF + H ) β (t )} dt < 0
I
This, as earlier, yields
∫ {η ( r f
T
u
+ y (t )T gu ) − D ( r fu + y (t )T gu ) + ( rF + H ) β (t )} dt < 0 .
I
This contradicts the dual constraint of (SGFrCD). Hence the conclusion follows.
Theorem 4.2 (Strong duality) If x is an optimal solution of (CP) then there
r ∈ R and piecewise smooth y : I → R m such that
exist
( r ,x , y , β ( t ) = 0 )
is
feasible for (SGFrCD) and the corresponding value of functions of (CP) and
(SGFrCD) are equal. If, also,
∫ f (t ,.,.)dt
is second-order pseudoinvex,
I
∑ ∫ y (t ) g dt , α ∈ N
i
i
is
second-order
semi-strictly
pseudo-invex
and
i∈Iα I
∑ ∫ y (t ) g dt , α ∈ K \ N
i
i
is second-order quasi-invex with respect to the same η ,
i∈Iα I
then x and (r , x , y , β ), are optimal solution of (CD) and (SFCD) respectively.
r
78
On Second-Order Fritz John Type Duality for Variational Problems
Proof: Since x is an optimal solution of (CP), by Proposition 2.1, there exist
r ∈ R and piecewise smooth y : I → R m such that
(r f x + yT (t ) g x ) − D(r f x + y (t )T g x ) =
0, t ∈ I
y T (t ) g (t , x,=
x ) 0, t ∈ I
(r , y (t )) ≥ 0, t ∈ I
(r , y (t )) ≠ 0, t ∈ I
y (t )T g (t , x, x ) =∑ y i (t ) g i (t , x, x ) +
∑
i∈ Iα
i∈ Iα
α∈N
α ∈K / N
y i (t ) g i (t , x, x ) =0,
implies
dt
∑ ∫ y (t ) g (t , x, x )=
i
i
0,=
α 0,1, 2,..., k .
i∈Iα I
∑
y i (t ) g i (t , x,=
x ) 0, α ∈ K \ N .
i∈Iα
α ∈K / N
This implies that
( x , r , y , β (t ) = 0 )
is feasible for
( SGFr CD ) . Equality follows
since the objective functionals are the same in the formulations.
Optimality of
( x , r , y , β (t ) = 0 )
for
( SGFr CD ) follows
by Theorem 4.1. The
following is the Mangasarian type [10] strict converse duality.
Theorem 4.3 (Strict Converse Duality) Assume that
(i) ∫ f ( t , x, x ) dt is second-order strict pseudo-invex,
I
second-order
semi-strictly
pseudoinvex
and
∑ ∫ y (t ) g dt ,
i
i
α ∈ N are
i∈Iα
∑ ∫ y (t ) g dt , α ∈ K \ N
i
i
are
i∈Iα
second-order quasi-invex with respect to the same η .
(ii) (CP) has an optimal solution x . If
(u , r , y , β (t ))
( SGFr CD ) ,
Then u = x i.e., u is an optimal solution of (CP).
is an optimal solution of
I. Husain and Santosh K. Shrivastav
79
Proof: We assume that u ( t ) ≠ x ( t ) and exhibit a contradiction. Since x is
optimal to (CP), by Theorem 3.5 there exists (r , y ) with piecewise r ∈ R and
piecewise smooth y : I → R m such that (r , x , y , β (t )) is optimal to ( SGFr CD).
(u , r , y , β )
Since
is an optimal solution of ( SGFr CD) , it implies that
∫ f ( t , x , x ) dt = ∫ f ( t , u , u ) dt.
I
I
This, by the second-order strict pseudoinvexity of
∫ f (.) dt , gives
I
∫ (η
T
fu + ( Dη )T fu + η F β (t ) ) dt < 0.
I
Multiplying this by r , we have
∫ r (η
T
fu + ( Dη )T fu + η F β (t ) ) dt ≤ 0
(15)
I
with strict inequality (15) in the above with r > 0 .
Also
∑ ∫ y (t )g (t , x , x )dt ≤ ∑ ∫ y (t )g (t , u , u )dt ,
i
i∈Iα I
i
i
i
α=
0,1, 2,3,..., k .
i∈Iα I
This, because of second-order semi-strict pseudoinvexity of
∑ ∫ y (t )g (t ,...,..)dt ,
i
i
α ∈ N.
i∈Iα I
We have
η T ( y i (t ) gu i (t , u , u ) ) + ( Dη )T ( y i (t ) gui (t , u , u ) ) 
0,1, 2,..., k
 dt ≤ 0, α =
∑∫ T i
i∈Iα I  +η H (t ) β (t )



i
with strict inequality in the above if y ( t ) > 0, i ∈ Iα , α ∈ N .
Combining (15) and (16) together with
∫ {η
T
(r fu + y (t )T gu ) + ( Dη )T (r fu + y (t )T gu ) + η T (r F + H ) β (t )}dt < 0
I
which, as earlier analysis, gives,
∫η
T
(rfu + y (t )T gu ) − D(r fu + y (t )T gu ) + η T (r F + H ) β (t )  dt < 0
I
contradicting the feasibility of (r , u , y , β (t )) of ( SFr CD). Hence x = u .
(16)
80
On Second-Order Fritz John Type Duality for Variational Problems
5 Converse Duality
In this section, we shall establish Huard type [10] converse duality for the
pair of dual problems (CP) and (SFrCD) considered in earlier section.
Theorem 5.1 (Converse Duality) Let (r , x , y , β ) be an optimal solution of
( SFr CD ) .
If for each t ∈ I ,
( A1 ) the vectors {Fi , H i , i = 1, 2,3,..., n} are linearly independent where Fi and
H i are the i th row of the matrices F and H respectively, and
( A2 ) y ( t )
T
( A3 )
g x − D y ( t ) g x + H β ( t ) ≠ 0, t ∈ I
T
(a ) ∫ β (t )T ( H + ( y (t )T g x ) x ) β (t ) dt > 0 and
∫ β (t )
T
( y (t )T g x )dt ≥ 0
or
(b)
∫ β ( H + ( y(t )
T
∫ β (t )
g x ) x ) β (t )dt < 0 and
T
I
( y (t )T g x )dt ≤ 0.
I
Then x (t ) is feasible for (CP ) and the two objective functional have the same
value. Also, if, Theorem 3.1 holds for all feasible solutions of (CP ) and
( SFr CD ) ,
then x is an optimal solution of (CP ) .
Proof: Since
( x, r , y , β )
is an optimal solution of (SFrCD), by Proposition 2.1,
there exist, τ ∈ R, ω ∈ R, ξ ∈ R, piecewise smooth θ : I → R m and η : I → R n
such that
1
1
T
T


( f x − Df x ) − 2 β ( t ) F β ( t ) x + 2 D β ( t ) F β ( t ) x

τ

 − 1 D 2 β ( t )T F β ( t ) + 1 D3 β ( t )T F β ( t ) − 1 D 4 β ( t )T F β ( t ) .... 



x 2
x 2
x 
 2
(
(
+θ (t )
T
)
)
(
) + ( y (t ) g )
)
(
)
(
)
(
)
(
)
T
T
T


− D y ( t ) g x − D  rf xx

 + y ( t ) g x
x
 r ( f xx − Df xx
x
x
x  



+
 + ( ( rF + H ) β ( t ) ) x − D ( ( rF + H ) β ( t ) ) x


 + D 2 ( rF + H ) β ( t )  − D3 ( ( rF + H ) β ( t ) ) + D 4 ( ( rF + H ) β ( t ) ).... 
x

x
x


I. Husain and Santosh K. Shrivastav
((
81
)
)
)) (
) (
)
(
1
1
T
T


T
T
 y( t ) g x − D y( t ) g x − 2 β ( t ) H β ( t ) x + 2 D β ( t ) H β ( t ) x 
+ω 
0 (17 )
=
 − 1 D 2 β ( t )T H β ( t ) + 1 D3 β ( t )T H β ( t ) − 1 D 4 β ( t )T H β ( t ) .... 
 2


2
x 2
x
x 

)
(
θ (t )
T
(g
x
+ g xx
(
1
β ( t ) ) + ω{g − ( β ( t )
2
T
( r θ −τ β ) F + (θ (t ) − ω β ( t ) ) H =0,
θ (t )
T
ξ
( f x − D f x + F β ) +=
ω ∫ { y (t )T g −
I
)
(
)
g xx β ( t ) } + η ( t ) = 0, t ∈ I
(18)
t∈I
(19)
0 , t∈I
(20)
1
0, t ∈ I
( β (t )T H β (t ) )} dt =
2
(21)
t ) 0, t ∈ I
η ( t ) y (=
(22)
ξ r = 0,
(23)
(τ , ξ , ω ,η (t ) ) ≥ 0, t ∈ I
(24)
T
(τ , ξ ,θ (t ), ω ,η (t ) ) ≠ 0, t ∈ I
(25)
By the hypothesis ( A1 ) , (19) yields
r θ (t ) −τ β (t ) =
0, t ∈ I
(26)
θ (t ) − ω β (t ) =
0, t ∈ I
(27)
Using equality constraint of (SFrCD) along with (26), (27) in (17), we have
(τ − r ω )  y (t )T g x − D y (t )T g x + H β ( t )
(
(
)
(
)
1
1
1
T
T
T
β (t ) F β (t ) + D β (t ) F β (t ) − D2 β (t ) F β (t )

x
x
2
2
2
1
1
T
T
+ D 3 β ( t ) F β ( t ) − D 4 β ( t ) F β ( t ) .... }

x
x
2
2
+ rτ {−
(

x
)
(
)
)
+ rθ ( t ) {( ( rF + H ) β ( t ) ) x − D ( ( rF + H ) β ( t ) ) x + D 2 ( rF + H ) β ( t ) x
T
− D 3 ( ( rF + H ) β ( t ) )x + D 4 ( ( rF + H ) β ( t ) )....x }
(
(
)
(
)
)
1
1
1
T
T
T
β (t ) H β (t ) + D β (t ) H β (t ) − D2 β (t ) H β (t )
x
x
x
2
2
2
1
1
T
T
+ D 3 β ( t ) H β ( t ) − D 4 β ( t ) H β ( t ) .... } =
0
(28)

x
x
2
2
+ rω{−
(
)
(
)
82
On Second-Order Fritz John Type Duality for Variational Problems
( A2 ) , the equality constraint of the problem (SFrCD), implies that
In view of
r ≠ 0. Hence r > 0.
t ) 0, t ∈ I and from (26) implies
Let ω = 0, (27) implies θ ( =
τ
r
β = 0, i.e., τ β = 0,
for r > 0.
Consequently from (28), we have
τ  y (t )T g x − D y (t )T g x + H β ( t )  =
0, t ∈ I ,
which because of hypothesis (A2), yields τ = 0. From (20), we have ξ = 0 . Hence
ω)
(τ ,θ (t ), ξ ,η (t ),=
0, t ∈ I , ensuing a contradiction to Fritz John condition to
(25), thus ω > 0 .
Multiplying (18) by y (t), then (26), (27) and (21) along with ω > 0 , we get
∫ {2β (t )
T
( y (t )T g x ) + β (t )T ( H + ( yg x ) x ) β (t )}dt =
0
I
∫ {β (t )
T
I
But
( H + ( yg x ) x ) β (t )} dt =
−2 ∫ β (t )T ( y (t )T g x )dt ≤ 0
I
∫ β (t ) ( H + ( y(t )
T
T
g x ) x ) β (t )dt > 0.
I
Hence β (=
t ) 0, t ∈ I . From equation (27), θ (=
t ) 0, t ∈ I and from equation (18)
η (t )
≤ 0, t ∈ I , implying the feasibility of x for (CP) and the objective
ω
−
g=
functions of (CP) and (SFrCD) are equal there. Under the stated second-order
generalized invexity condition, by Theorem 4.1, x is an optimal solution for
(CP).
6 Natural Boundary Values
In this section, we formulate a pair of dual variational problems with natural
boundary values rather than fixed end points.
I. Husain and Santosh K. Shrivastav
(CP0 )
83
∫ f ( t , x, x ) dt
Minimize
I
Subject to
g ( t , x, x ) ≤ 0,
( SFr CD0 ) Maximize
t∈I.
1
∫ { f (t , u, u ) − 2 β (t )
T
(29)
F (t ) β (t )} dt
I
Subject to
r fu + y (t )T gu − D (r fu + y (t )T gu ) + (rF + H ) β (t ) = 0, t ∈ I
∫ { y(t )
T
I
1
g (t , u , u ) − β (t ) T H β (t )}dt ≥ 0, t ∈ I
2
(30)
(31)
( r , y (t )) ≥ 0, t ∈ I
(32)
(r , y (t )) ≠ 0, t ∈ I
(33)
r fu ( t , u , u ) + y ( t ) gu ( t , u , u )
T
r fu ( t , u , u ) + y ( t ) gu ( t , u , u )
t =a
T
t =b
=
0, t ∈ I
(34)
=
0, t ∈ I
(35)
We shall not repeat the proofs of Theorems 3.1-5.1, as these follow exactly
on the lines of the analysis of the preceding section with slight modifications.
7 Nonlinear Programming Problems
If all functions in the problems (CP0) and (SFrCD0) are independent of t
1 , then these problems will reduce to following pair of Fritz John dual
and b − a =
nonlinear programming problems, studied by Husain et.al [7].
(NP):
Minimize
f ( x)
Subject to
g ( x ) ≤ 0.
84
On Second-Order Fritz John Type Duality for Variational Problems
( SFr D0 )
Maximize
1
f (u ) − β T Fˆ β
2
Subject to
r fu + yT gu + (r Fˆ + Hˆ ) β =
0
1
yT g − β T Hˆ β ≥ 0
2
(r , y ) ≥ 0, (r , y ) ≠ 0,
where
Fˆ =
f uu ( u ) =
∇ 2 f ( u ) , and Hˆ =
∇ 2 ( yT g ) .
( y(t )T g x ) =
x
( SGFr CD0 ) :
1


Maximize  f (u ) − β T Fˆ β 
2


Subject to
rfu (u ) + yT gu (u ) − ∇ 2 ( rf (u ) + yT g (u ) ) β = 0

1
i
( r, y ) ≥ 0
( r, y , i ∈ I
α

0,1, 2,..., k
∑  y g (u ) − 2 β Hˆ β  ≥ 0, α =
i
i
i∈Iα
i
i
, α ∈ N ) ≠ 0.
It is pointed out that the duality results between (CP0 ) and ( SFr D0 ) , and
between (CP0 ) and ( SGFr CD0 ) are immediate consequences of the preceding
extensive analysis of this research. If β = 0 , the dual problems ( SFr CD) and
( SGFr CD) reduce to the problem to the following nonlinear programming
problems whose duality is extensively discussed by Weir and Mond [11]:
(NP):
Minimize
f ( x)
Subject to
g ( x ) ≤ 0.
I. Husain and Santosh K. Shrivastav
( SFr D)
85
Maximize
f (u )
Subject to
r fu + yT gu =
0,
yT g ≥ 0,
(r , y ) ≥ 0,
(r , y ) ≠ 0.
( SGFr CD ) :
( f (u ) )
Maximize
Subject to
rfu (u ) + yT gu (u ) =
0,
∑ ( y g (u ) ) ≥ 0,
i
i
α=
0,1, 2,..., k ,
i∈Iα
( r , y ) ≥ 0,
( r , y , i ∈ I ,α ∈ N ) ≠ 0.
i
i
α
References
[1] O.L. Mangasarian, Second and higher order duality in nonlinear
programming, J. Math. Anal. Appl., 51, (1979), 605-620.
[2] B. Mond, Second order duality in non-linear programming, Opsearch, 11,
(1974), 90-99.
[3] Mond and M.A. Hanson, Duality for variational problems, J. Math. Anal.
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[4] X.Chen, Second order duality for the variational problems, J. Math. Anal.
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[5] I. Husain, A. Ahmed and Mashoob Masoodi, Second order duality for
variational
problems,
European
Mathematics, 2(2), (2009), 278- 295.
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of
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and
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86
On Second-Order Fritz John Type Duality for Variational Problems
[6] B. Mond and T. Weir, Generalized convexity and higher order duality, J.
Math. Anal. App., 46, (1974), 169-174.
[7] I. Husain, N.G. Rueda and Z. Jabeen, Fritz John Second-Order Duality for
Nonlinear Programming, Applied Mathematics Letters, 14, (2001), 513-518.
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