Theoretical Mathematics & Applications, vol.3, no.1, 2013, 123-144
ISSN: 1792- 9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
Mixed Type Second-order Duality for a
Nondifferentiable Continuous
Programming Problem
I. Husain 1 and Santosh K. Shrivastav 2
Abstract
A mixed type second-order dual is formulated for a class of continuous
programming problems in which the integrand of the objective functional
contains square root of positive semi-definite quadratic form; hence it is
nondifferentiable.
Under
second-order
pseudoinvexity
and
second-order
quasi-invexity, various duality theorems are proved for this pair of dual
nondifferentiable continuous programming problems. A pair of dual continuous
programming problems with natural boundary values is constructed and it is
briefly indicated that the duality results for this pair of problems can be validated
analogously to those for the earlier models. Lastly, it is pointed out that our
duality results can be regarded as dynamic generalizations of those for a
nondifferentiable nonlinear programming problem, 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 : November 21, 2012. Revised : January 12, 2013
Published online : April 15, 2013
124
Mixed Type Second-order Duality…
Mathematics Subject Classification: 90C30; 90C11; 90C20; 90C26
Keywords: Continuous programming; Second-order invexity; Second-order
pseudoinvexity; Second-order quasi-invexity; Mixed type Second-order duality;
Nondifferentiable nonlinear programming
1 Introduction
A number of researchers have studied second-order duality in mathematical
programming. A second order dual to a Nonlinear programming problem was first
formulated by Mangasarian [1] .Subsequently Mond [2] established various
duality theorem under a condition which is called second-order convexity, which
is much simpler than that used by Mangasarian [1]. Mond and weir [3]
reformulated the second-order and higher-order duals to validate duality results. It
is found that second-order dual to a mathematical programming problem offers a
tighter bound and hence enjoys computational advantage over a first-order dual.
In the spirit of Mangasarian [1], Chen [4] formulated second-order dual for
a constrained variational problem and established various duality results under an
involved invexity-like assumption. Later, Husain et al. [5], studied Mond-Weir
type second-order duality for the problem of [4], by introducing continuous-time
version of second-order invexity and generalized second-order invexity, validated
usual duality results. Subsequently Husain and Masoodi [6] presented Wolf type
duality while Husain and Srivastava [7] formulated Mond-weir type dual for a
class of continuous programming containing square root of a quadratic form to
relax to the assumption of second-order pseudoinvexity and second-order
quasi-invexity.
In this paper, in order to combine dual formulations of [6] and [7], a mix
type dual to the non-differentiable continuous programming problem of Chandra
et al. [8] is constructed and a number of duality results are proved under
I. Husain and S.K. Shrivastav
125
appropriate generalized second-order invexity established. A relation between our
duality results and those of a nondifferentiable nonlinear programming problem is
pointed out through natural boundary value variational problems.
2 Related 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
n 
∂x 
∂x  .
 ∂x ∂x
 ∂x ∂x
T
T
Denote by φxx the Hessian matrix of φ and ψ x the m × n Jacobian matrices
 ∂ 2φ 
, i, j 1, 2,...n , ψ x the m × n Jacobian
respectively, =
that is φxx =
i
j 
 ∂x ∂x 
matrix
 ∂ψ 1
 1
 ∂x
 ∂ψ 2
ψ x =  ∂x1

 

m
 ∂ψ
 ∂x1








∂ψ m 


∂x n m×n .
∂ψ 1
∂ψ 1

∂x 2
∂x n
∂ψ 2
∂ψ 2

∂x 2
∂x n


∂ψ m
∂x 2
The symbols φx , φxx , φxx and ψ x have analogous representations.
Designate by X the space of piecewise smooth functions x : I → R n , with the
126
Mixed Type Second-order Duality…
x
norm=
x ∞ + Dx ∞ , where the differentiation operator D is given by
t
u =D x ⇔ x ( t ) =∫ u ( s ) ds ,
a
Thus
d
= D except at discontinuities.
dt
Now we incorporate the following definitions which are required in the
subsequent analysis.
Definition2.1 (Second-order Invex) If there exist a vector function
=
η η ( t , x, x ) ∈ R n where η : I × R n × R n → R n and with η = 0 at t = a and t = b,
such that for a scalar function φ ( t , x, x ) , the functional ∫ φ ( t , x, x ) dt where
I
φ : I × R n × R n → R satisfies
1

∫ φ ( t , x, x ) dt − ∫ {φ ( t , x , x ) − 2 β ( t ) G β ( t ) dt
T
I
I
{
}
T
≥ ∫ η T φx ( t , x , x ) + ( Dη ) φx ( t , x , x ) + η T G β ( t ) dt ,
I
then ∫ φ ( t , x, x ) dt is second-order invex with respect toη where
I
G=
φxx − 2 Dφxx + D 2φxx  − D 3φx x , and β ∈ C ( I , R n ) , the space of n -dimensional
continuous vector functions.
Definition 2.2 (Second- order Pseudoinvex) If the functional
∫ φ ( t , x, x ) dt
I
satisfies
∫ {η
T
}
φx + ( Dη ) φx + η T G β ( t ) dt ≥ 0
T
I
⇒

∫ φ ( t , x, x ) dt ≥ ∫ φ ( t , x , x ) − 2 β ( t )
I
I
1
T

G β ( t ) dt ,

I. Husain and S.K. Shrivastav
127
Then ∫ φ ( t , x, x ) dt is said to be second-order pseudoinvex with respect to η .
I
Definition 2.3 (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
T
I
{

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 [2] for static cases.
The following giving Schwartz inequality is also required for the validation of our
duality results.
Lemma 2.1 (Schwartz inequality) It states that
(
x (t ) B (t ) z (t ) ≤ x (t ) B (t ) x (t )
T
) ( z (t )
1/ 2
T
B (t ) z (t )
)
1/ 2
, t∈I
(1)
0 for some q ( t ) ∈ R .
with equality in (1) if and only if B ( t ) ( x ( t ) − q ( t ) z ( t ) ) =
Proposition 2.1 If (CP) attains an optimal solution at x= x ∈ X , then there exist
Lagrange multiplier τ ∈ R and piecewise smooth y : I → R+m , not both zero, and
ˆ : I → R n , satisfying for all t ∈ I .
also piecewise smooth ω
 ) + B( t ) ω
 ) − D ( τ fx ( t,x ,x ) + y( t )T gx=
ˆ + y( t )T g x ( t,x ,x
τ f x ( t,x ,x
) 0, t ∈ I
y ( t ) g ( t,x ,x=
) 0,t ∈ I
T
128
Mixed Type Second-order Duality…
=
x (t ) B (t ) ω(t )
T
( x (t )
T
B (t ) x (t )
)
1/ 2
,t ∈ I
ω( t )T B( t ) ω( t ) ≤ 1,t ∈ I
The Fritz John necessary optimality conditions given in Proposition7.2.1 become
the Karush-Kuhn-Tucker type optimality conditions if τ =1 .
For τ =1, it suffices that the following slater’s conditions [8] holds.
 t ) < 0 , For some v ∈ X and all t ∈ I .
g ( t,x ,x ) + g x ( t,x ,x ) v( t ) + g x ( t,x ,x ) v(
Consider the following class of non-differentiable continuous programming
problem studied in [8].
(CP ) :
Minimize
(
)

T B( t ) x( t ) 1 / 2  dt

f
t,x,x
+
x(
t
)
(
)


∫

I
Subject
x (a) =
α, x (b) =
β
g ( t ,x,x ) ≤ 0 ,t ∈ I
( 2)
( 3)
Where
(i) I = [ a,b ] is a real interval (ii) f : I × R n × R n → R, g : I × R n × R n → R m
are twice continuously differentiable function with respect to its argument
x(t ) and x (t ) .
(ii) x : I → R n is four times differentiable with respect to t and these derivatives
....
 
are defined by x,
x, 
x and x .
(iii) B (t) is a positive semi-definite n × n matrix with B (.) continuous on I.
The popularity of mathematical programming problems as (CP) seems to stem
from the fact that, even though the objective functions and /or constraint functions
are differentiable, a simple formulation of the dual may be given.
Non-differentiable mathematical programming deals with much more general
kinds of functions by using generalized sub-differentials and quasi differentials.
I. Husain and S.K. Shrivastav
129
Husain et al. [5] formulated the following Wolf type second-order dual and
established duality results for the pair of problems (CP) and (W-CD) under the
second-order pseudoinvexity of
∫ { f ( t,.,.) + u(.)
T
B( t )w( t ) + y( t )T g ( t,.,.)} dt with
I
respect to η .
(WCD) :
Maximize

∫  f ( t,u,u ) + u( t )
B( t ) w( t ) + y( t )T gu ( t,u,u ) −
T
I
1

β ( t )T L β ( t )  dt
2

Subject to
u ( a )= 0= u ( b ) ,
fu ( t,u,u ) + u( t )T B( t ) + y( t )T gu ( t,u,u ) − D ( fu + y( t )T gu ) + L β ( t ) =0 , t ∈ I ,
y( t ) ≥ 0 ,t ∈ I ,
w( t )T B( t )w( t ) ≤ 1,t ∈ I .
where
(
L=
fuu ( t , u , u ) + y ( t ) gu ( t , u , u )
T
(
)
u
(
)
− 2 D[ fuu ( t , u , u ) + y ( t ) gu ( t , u , u ) ]
)
T
(
u
)
3
+ D 2 [ fuu
  ( t , u , u ) + y ( t ) gu ( t , u , u ) ] − D [ fuu
 ( t , u , u ) + y ( t ) gu ( t , u , u ) ]
T
u
T
u
Recently, Husain and Srivastava [7] investigated Mond-Weir type duality by
(
)
constructing the following dual to (CP) under ∫ f ( t,u ( t ) ,u ( t ) ) + wT B ( t ) w ( t ) dt,
I
for all w( t ) ∈ R n is second-order pseudo-invex and
∫ y ( t ) g ( t,..,) dt is
T
I
second-order quasi-invex with respect to the same η .
(M − WCD) :
Maximize

∫  f ( t,u,u ) + u( t )
I
T
1

B( t )w( t ) − β ( t )T F β ( t )dt
2

130
Mixed Type Second-order Duality…
Subject to
u ( a )= 0= u ( b )
fu ( t,u,u ) + B( t ) w( t ) + y( t )T gu ( t,u,u ) −
(
)
0
− D fu ( t,u,u ) + y( t )T gu ( t,u,u ) + ( F + K ) β ( t ) =
1

∫  y ( t ) g ( t,u,u ) − 2 β ( t )
T
T
I

K β ( t )  dt ≥ 0 , t ∈ I

y ( t ) ≥ 0, t ∈ I
w ( t ) B ( t ) w ( t ) ≤ 1, t ∈ I
T
where
3
F ( t , u , u , u, 
u)=
fuu − 2 Dfuu + D 2 fuu
  − D fuu
 , t ∈ I
and
3
K ( t , u , u , u, 
u)=
y ( t ) guu − 2 D y ( t ) guu + D 2 y ( t ) guu
  − D y ( t ) guu
 , t ∈ I
T
T
T
In this paper, we propose a mixed type second-order dual (MixCD) to
(CP) and prove various duality theorems under the assumptions of second-order
pseudoinvexity and second-order quasi-invexity. We formulate a pair of
nondifferentiable continuous programming problem with natural boundary values
rather than fixed end points. Finally, it is also pointed out that our duality results
derived in this research can be regarded as the generalization of those of the
nonlinear programming problem.
3 Mixed Type second-order duality
In this section we formulate the following mixed type second-order dual
Mixed (CD) to (CP):
Mixed (CD):


1
T
T
Maximize ∫  f ( t,u,u ) + u ( t ) B( t )ω ( t ) + ∑ y i ( t ) g i ( t,u,u ) − β ( t ) H 0 β ( t ) dt
2

i∈I 0
I

I. Husain and S.K. Shrivastav
131
Subject to
=
u (a ) 0,=
u (b) 0
(4)
fu (t , u , u ) + B(t ) w)t ) + y (t )T gu (t , u , u )
(
)
− D fu (t , u , u ) + y (t )T gu (t , u , u ) + H β (t ) =∈
0, t I
(5)


1
i
i
T α



(
)
(
,
,
)
(
)
(
)
1, 2,3....r
y
t
g
t
u
u
t
G
t
dt ≥ 0. α =
β
β
−
∫ ∑

2
i
I
∈
I α

(6)
w(t )T B(t ) w(t ) ≤ 1, and y (t ) ≥ 0, t ∈ I
(7)
Where
=
( i ) I=
} ,α
α ⊆ M {1, 2 ,....m
( ii )
H 0 =−
f xx (t , u , u )
−
r
0 ,1, 2 ,...r with  Iα = M and Iα  I β = φ if α ≠ β .
α =0
∑ ( yi (t ) g i (t , u, u ) ) xx − 2 D( f xx (t , u, u )
i∈I 0
∑ ( yi (t ) g i (t , u, u ) ) xx ) + D 2 ( f xx  (t , u, u ) − ∑ ( yi (t ) g i x (t , u, u ) ) x )
i∈I 0
− D3 ( fuu
  (t , u , u ) −
( iii )
∑ ( yi (t ) g iu (t , u, u ) )u )
i∈I 0
i∈I 0
(
)u − 2D( fuu (t, u, u) − ( y(t )T gu (t, u, u) )u )
T
+ D 2 ( fuu
  (t , u , u ) − ( y (t ) gu (t , u , u ) ) )
u
T
− D3 ( fuu
 (t , u , u ) − ( y (t ) gu (t , u , u ) ) )
u
H=
fuu (t , u , u ) − y (t )T gu (t , u , u )
and


i
i


=
− 2 D ∑ y (t ) gu (t , u , u ) 
( iv ) G
∑
 i∈I

i∈Iα
u
u 
 α



+ D 2  ∑ y i (t ) gu i (t , u , u )  − D3  ∑ y i (t ) g i (t , u , u )
 i∈I

 i∈I
u 
 α
 α
α
(
y (t ) gu (t , u , u )
i
i
(
and
)
(
)
)
(

 ,α =
1, 2,..., r.

u u
)
132
(v)
Mixed Type Second-order Duality…
∑
G
i∈M − I 0
+D
(


− 2 D  ∑ y i (t ) gu i (t , u , u ) 
 i∈M − I

0
u
u 



i
i
3


y (t ) gu (t , u , u )
−D
y i (t ) g i (t , u , u )
∑

 i∈M − I
0
u 

)
y i (t ) gu i (t , u , u )

∑
 i∈M
− I0

2
(
(
)
)
(

 ,

u u
)
Theorem 3.1 (Weak Duality)
Let x be feasible for (CP) and
feasible
( u, y,w, β )
feasible for Mixed (CD). If for all
( x, u, y, w, β ) ,

∫  f (t ,.,.) + ∑ y (t ) g (t ,.,.) + (.)
i
i
T
i∈I 0
I

B(t ) w(t )  dt

is second-order pseudoinvex and
∑ ∫ y (t ) g (t ,.,.) dt ,α = 1,2,.....r
i
i
i ∈Iα I
is second-order quasi-invex with respect to the same η ,
then
infimum(CP) ≥ supremum Mixed(CD)
( u, y,w, β ) for (CP) and Mixed (CD)
Proof: By the feasibility of x and
respectively, we have
1
∫ ( ∑ y ( t ) g ( t , x, x ))dt ≤ ∫ ( ∑ y ( t ) g ( t , u, u ) − 2 β ( t )
i
I
i
i
i∈Iα
I
i
T
Gα β ( t )) dt ,
i∈Iα
α = 1, 2,..., r.
By second-order quasi-invexity of
∑ ∫ y i( t ) g i ( t , x, x ) dt , α = 1, 2,..., r ,
i∈Iα
I
this inequality yields
T
i
i
i
i
T α
∫ [(η ∑ y ( t ) g u ( t , u, u )) + ( Dη ) ( ∑ y ( t ) g u ( t , u, u )) + η G β ( t )]dt ≥ 0,
T
I
i∈Iα
i∈Iα
I. Husain and S.K. Shrivastav
133
α = 1, 2,..., r.
This implies
0 ≥ ∫ [(η T
I
∑
i∈M − I 0
y i ( t ) g i u ( t , u , u ))
+ ( Dη ) (
T
∑
i∈M − I 0
=

y i ( t ) g i u ( t , u , u )) + η T G β ( t )]dt




y i ( t ) g i u ( t,u,u )  − D  ∑ y i ( t ) g i u ( t,u,u )  + G β ( t )  dt

 i∈M − I

 i∈M − I0

I
0





t =b


T
i
i


+ η ∑ y ( t ) gu ( t,u,u )
 i∈I

 0
t=a
T
∫ η 
∑
(by integrating by parts)
Using η = 0, at t = a and t = b , we obtain,
∫η
T
∑
[(
i∈M − I 0
I
y i ( t ) g i u ( t , u , u )) − D (
∑
i∈M − I 0
y i ( t ) g i u ( t , u , u )) + G β ( t )]dt ≤ 0
Using (5), we have
0 ≤ ∫ η T [ fu ( t , u , u ) + B ( t ) w ( t ) + ∑ y i ( t ) gui ( t , u , u )
i∈I 0
I
− D( fu ( t , u , u ) +
=
∫ [η
T
∑ yi ( t ) gui ( t , u, u )) + H 0 β ( t )]dt
i∈I 0
( fu ( t , u , u ) + B ( t ) w ( t ) +
I
+ ( Dη ) ( fu +
T
∑ yi ( t ) gui ( t , u, u ))
i∈I 0
∑ yi ( t ) gui ( t , u, u )) + η T H 0 β ( t )]dt
i∈I 0
t =b
− η ( fu +
T
∑ y ( t ) ( t , u, u ))
i
i∈I 0
gui
t=a
(by integrating by parts)
From this, as earlier η = 0 at t = a and t = b , we get,
134
Mixed Type Second-order Duality…
∫ [η
T
( fu ( t , u , u ) + B ( t ) w ( t ) +
I
∑ yi ( t ) g i ( t , u, u ))
i∈I 0
∑ yi ( t ) gui ( t , u, u )) + η T H 0 β ( t )]dt ≥ 0
+ ( Dη ) ( fu ( t , u , u ) +
T
i∈I 0
which by second-order pseudo-invexity of
T
i
i
∫ ( f ( t , x, x ) + x B ( t ) w ( t ) + ∑ y ( t ) g ( t , x, x )) dt
i∈I 0
I
implies
T
i
i
∫ ( f ( t , x, x ) + x ( t ) B ( t ) w ( t ) + ∑ y ( t ) g ( t , x, x )) dt
i∈I 0
I
≥ ∫ [( f ( t , u , u ) + u ( t ) B ( t ) w ( t ) +
T
I
1
∑ yi ( t ) g i ( t , u, u )) − 2 β ( t )
T
H 0 β ( t )]dt
T
H 0 β ( t )]dt
i∈I 0
Thus from y ( t ) ≥ 0 and g ( t , x, x ) ≤ 0, t ∈ I . The above gives,
T
∫ ( f ( t , x, x ) + x ( t ) B ( t ) w ( t ) ) dt
I
≥ ∫ [( f ( t , u , u ) + u ( t ) B ( t ) w ( t ) +
1
T
I
∑ yi ( t ) g i ( t , u, u )) − 2 β ( t )
i∈I 0
Since w ( t ) B ( t ) w ( t ) ≤1, t ∈ I , the Schwartz inequality (1), this inequality
T
implies
T
∫ ( f ( t , x, x ) + ( x ( t ) B ( t ) x ( t ) )
1/2
)dt
I
≥ ∫ [ ( f ( t , u , u ) + u ( t ) B ( t ) w ( t ) +
T
I
1
∑ yi ( t ) g i ( t , u, u )) − 2 β ( t )
i∈I 0
yielding
infimum(CP) ≥ supremum Mixed (CD)
T
H 0 β ( t )]dt
I. Husain and S.K. Shrivastav
135
Theorem 3.2 (Strong Duality)
If x is a optimal solution of (CP) and normal [8],then there exist piecewise
smooth y : I → R m and w : I → R n such that
( x , y,w, β ( t ) = 0 ) is
feasible for
Mixed (CD), and the corresponding values of (CP) and Mix (CD) are equal.
If, for all feasible ( x ,u, y,w, β ) ,




T
i
i
∫  f ( t,x,x ) + ∑ y ( t ) g ( t,x,x ) + x ( t ) B ( t ) w ( t ) dt
I


i∈I 0
is second-order pseudo-invex and
∫ ( ∑ y ( t ) g ( t , x, x )) dt ,
i
I
i
α = 1, 2,...., r is
i∈Iα
second-order quasi-invex, then ( x , y,w, β ( t ) ) is an optimal solution of Mix (CD).
Proof: Since x is an optimal solution of (CP) and normal [8], then by
Proposition 2.1, there exist piecewise smooth y : I → R m and w : I → R n such
that
T
f x ( t , x , x ) + B ( t ) w ( t ) + y ( t ) g x ( t ,, x , x )
(
)
− D f x ( t , x , x ) + y ( t ) g x ( t ,, x , x ) =
0, t ∈ I
T
y (t ) g (t,=
x, x ) 0,
T
) B (t ) x (t ))
( x (t=
T
1/2
t∈I
x (t ) B (t ) w (t ) , t ∈ I
w (t ) B (t ) w (t ) ≤ 1 , t ∈ I
T
y ( t ) ≥ 0,
t∈I
This implies that ( x , y,w, β ( t ) = 0 ) is feasible for Mix (CD) and the
corresponding value of (CP) and Mix (CD) are equal. If


T
i
i


+
+
+
f
t
,
x
,
x
y
g
t
,
x
,
x
x
t
B
t
w
t
)
(
)
(
)
(
)
(
)
(

 dt
∑
∫I 
i∈I o

is second-order pseudo-invex, and
136
Mixed Type Second-order Duality…
∑ ∫ y ( t ) g ( t , x, x ) dt , α = 1, 2,...., r
T
i∈Iα I
is second-order quasi-invex with respect to the same η , then from weak duality
Theorem3.1
( x , y,w, β ( t ) )
is an optimal solution of Mix (CD).
Theorem 3.3 (Converse Duality) Let ( x, y, w, β ) be an optimal solution at which
( A1 ) for all α = 1, 2,3.....r , either


(a ) ∫ β (t )T  Gα + ∑ ( y i g i x ) x  β ( t ) dt > 0, and
i∈Iα
I


T 
(b) ∫ β ( t )  Gα +
I

∑( y g )
i
i
x x
i∈Iα

∫ β (t )  ∑ ( y g )
T
i
i∈Iα
I

 β ( t ) dt < 0, and

i
x

 dt ≥ 0, or



 i∈Iα

T
i i
∫ β (t )  ∑ ( y g x ) x  dt ≤ 0,
I
=
H 0j ,Gαj ,α 1=
, 2 ,...,r and j 1, 2 ,...n are linearly independent,
( A2 ) the vectors
where H 0j is the jth row of the matrix H 0 and Gαj is the jth row of the matrix
Gα and
( A3 ) the vectors
1, 2,3,..., r
∑ ( y (t ) g ) − D ∑ ( y (t ) g ), α =
i
i∈Iα
i
x
i
i∈Iα
i
x
are linearly independent. If, for all ( x, u , y, w, β ),

∫  f (t ,...) + (.)
T
I

B(t ) w(t ) + ∑ y i (t ) g i (t ,.,.) dt
i∈Iα

is second-order pseudoinvex and
∫ ∑ y (t ) g (t ,.,.)dt , α = 1, 2,...,r ,
i
i
I i∈Iα
is second-order quasi-invex, then x is an optimal solution of (CP).
Proof: Since ( x, y, u , β ) is an optimal solution of Mix (CD), (2), by Proposition
2.1, there τ 0 ∈ R, τ α ∈ R, α = 1, 2,..., r , and piecewise smooth
θ : I → R m , r : I → R m and p : I → R n such that
I. Husain and S.K. Shrivastav
(
137
)
(
)
0 = τ 0 [( f x + ∑ y i (t ) g xi + B(t ) w(t )) − D( f x + ∑ y i (t ) g i ) −
i∈I 0
i∈I 0
(
)
(
)
(
)
x
(
1
β (t )T H 0 β (t )
2
(
1
1
1
D β (t )T H 0 β (t ) − D 2 β (t )T H 0 β (t ) + D 3 β (t )T H 0 β (t )


x
x
2
2
2
1 4
− D β (t )T H 0 β (t ) ]

x
2
+ θ (t )T [( f xx + y (t )T g xx ) − D f xx + y (t )T g xx − D f xx + ( y (t )T g x ) x
+
(
(
)
)
(
(
)
x
)

x
)
)
− D D( f xx  + y (t )T g xx  + D 2 D( f xx + ( y (t )T g x ) x + ( H β (t )) x
(8)
− D( H β (t )) x + D 2 ( H β ) x − D 3 ( H β )x + D 4 ( H β )x ]
(
)
r
1
1
− ∑τ α { ∑ y i (t ) g i ) x − D( y i (t ) g i ) x − ( β (t )Gα β (t )) + D( β (t )Gα β (t )) x
2
2
=
α 1
i∈Iα
1
1
1
− D 2 ( β (t )Gα β (t )) x + D 3 ( β (t )Gα β (t )) − D 2 ( β (t )Gα β (t ))}, t ∈ I
4
2
3
1
τ 0 ( g i − β (t )T g xxi β (t )) − θ (t )T ( g xi + g xxi β (t )) + r i (t=
i ∈ I0
) 0,
2
1
τ 0 ( g i − β (t )T g xxi β (t )) − θ (t )( g xi + g xxi β (t )) + r i (t ) =0, i ∈ Iα , α =1, 2,..., r
2
τ 0 B(t ) x(t ) − θ (t ) B(t ) − 2 p(t ) ( B(t ) w(t ) ) =
0, t ∈ I
r
(τ 0 β − θ (t ) ) H 0 + ∑ (τ α β − θ (t ) )Gα
=
0
(9)
(10)
(11)
(12)
α =1
0, α =
1, 2,3,..., r
τ α ∫ {∑ ( y i (t ) g i ) − β (t )T Gα β (t )}dt =
(13)
p (t )T ( w(t )T B(t ) w(t ) − 1)= 0, t ∈ I
(14)
r (t )T y (=
t ) 0, t ∈ I
(15)
I
1
2
i∈Iα
(τ 0 ,τ1 ,τ 2 ,......τ r , r (t ), p(t ) ) ≥ 0, t ∈ I
(τ 0 ,τ1 ,τ 2 ,......τ r , r (t ),θ (t ), p(t ) ) ≠ 0,
(16)
t∈I
(17)
Because of assumption ( A2 ), (10) implies
τ α β (t ) − θ (t ) = 0, α = 0,1, 2,...r
(18)
Multiplying (10) by y (t ), t ∈ Iα , α =
1, 2,..., r , and summing over i, we have,
i
1
2
0
τ α { y i (t ) g i − β (t )( y i g i ) xx β (t )} + θ (t ){( y i g i ) x + ( y i g i ) xx β (t )} + y i (t )r i (t ) =
138
Mixed Type Second-order Duality…
1
2
τ α { ∑ y i (t ) g i − β (t )T
i∈Iα
+ θ (t ){ ∑ ( y i g i ) x +
i∈Iα
∑ ( yi g i ) xx β (t )}
i∈Iα
∑ ( yi g i ) xx β (t )} =0,
i∈Iα
1, 2,..., r
i ∈ Iα , α =
(19)
Using (18) in (19), we have
1
2
τ α { ∑ y i (t ) g i − β (t )T
i∈Iα
∑ ( yi g i ) xx β (t )}
i∈Iα
− τ α β (t ){ ∑ ( y i g i ) x +
i∈Iα
−τ α {∫ β (t )T
I
0, i ∈ Iα , α =
1, 2,..., r.
∑ ( yi g i ) xx β (t )} =
i∈Iα
∑ ( yi g i ) x dt + ∫ β (t )T ∑ ( yi g i ) xx β (t )dt}
i∈Iα
i∈Iα
I
1
0, α =
1, 2,..., r.
+ τ α ∫ [ ∑ ( y i g i ) − β (t )T ( ∑ ( y i g i )) xx ] β ( t ) dt =
2
i∈I
i∈I
α
I
(20)
(21)
α
gives
τ α [ ∫ β (t )( ∑ ( y i g i ) x )dt +
i∈Iα
I
+
τα
β (t )G
2 ∫
α
1
β (t )T ( ∑ ( y i g i ) x ) β (t ) dt ]
2 ∫I
i∈Iα
0
β (t )dt =
(22)
I
τ α ∫ β (t )T
I
∑ ( yi g i ) x dt +
i∈Iα
τα
2
∫ β (t )( ∑ ( y g ) xx + G
i i
I
α
) β (t )dt =
0
(23)
i∈Iα
=
2....r ,τ α 0 , then (18) implies θ (=
If=
for all α 0,1,
t ) 0, t ∈ I
From (10), we have r (=
t ) 0, t ∈ I and p (t ) = 0 from (11) and (14) .
) ) 0, t ∈ I
Thus (τ 0 ,τ 1 ,τ 2 ,......τ r , r (t ), θ (t ), p (t =
This gives a contradiction. Hence there exists an α ∈ {0,1, 2,.....r} such that
τ α > 0. If β (t ) ≠ 0, t ∈ I , thus (18) gives (τ α − τ α ) β (t ) =
0, α = 1, 2,..., r.
I. Husain and S.K. Shrivastav
139
This implies that τ α= τ α > 0, from (20), we have
2 ∫ β (t )T ( ∑ ( y i g i ) x )dt + ∫ β (t )T (Gα +
I
i∈Iα
I
0
∑ ( yi g i ) xx )β (t )dt =
i∈Iα
This contradicts ( A1 ). Hence β (=
t ) 0, t ∈ I .
Using (4) and β (=
t ) 0, t ∈ I , (8) gives
r
0,
∑ (τα −τ 0 ){ ∑ ( yi g ix ) − D ∑ ( yi g ix )} =
α=
1
i∈Iα
(24)
i∈Iα
Which by the linear independence of
{∑ ( y i g xi ) − D ∑ ( y i g xi )}, α =
1, 2,3,..., r
i∈Iα
i∈Iα
yields
τ α = τ α , for all α ∈ {0,1, 2,.....r}
Now (9) and (10) gives
τ 0 g i + ri (t ) = 0, t ∈ I , i ∈ I 0 gives g i (.) ≤ 0, i ∈ I 0
(25)
1, 2,3,...r.
τ α g i + r i (t ) = 0, t ∈ I , i ∈ Iα implies g i (.) ≤ 0, i ∈ Iα , α =
(26)
Gives g (.) ≤ 0 implies x is a feasible solution of (CP).
1, 2,.., r
Multiplying (25) by y i (t ), i ∈ I 0 and to y i (t ), i ∈ Iα , α =
From (11), we have
B (t ) x (t ) =
2 p (t )
τ0
B (t )ω (t )
Hence
1
1
x(t )T B(t ) w(t ) = ( x(t )T B (t ) x(t ) ) 2 ( w(t )T B (t ) w(t ) ) 2
( 27 )
If p (t ) > 0, t ∈ I , (14) implies w(t )T B (t ) w(t ) = 1 and (25) implies
=
x(t )T B(t ) w(t )
If p (=
t ) 0, t ∈ I , then (27) gives
( x(t )
T
1
B(t ) x(t ) ) 2 , t ∈ I
140
Mixed Type Second-order Duality…
B ( t ) x (=
t ) 0, t ∈ I ,
Thus in either Case, we have
( x(t )
=
x(t ) B(t ) w(t )
T
T
1
2
B(t ) x(t ) ) , t ∈ I
1


T
This gives ∫  f (t , x, x ) + ( x(t ) B(t ) x(t ) ) 2 dt

I 


1
T
0
i i


=
+
+
−
f
t
,
x
,
x
x
t
B
t
w
t
y
g
t
,
x
,
x
t
H
t
β
β
(
)
(
)
(
)
(
)
(
)
(
)
(
)

 dt
∑
∫I 
2
i∈I 0

implies x is optimal to (CP).
4 Special Cases
If I 0 = M , then Mix (CD) becomes (WD) and from Theorem 3.1-3.3, it
follows that it is a second-order dual to (CP), if
∫ { f ( t,x,x ) + y ( t ) g ( t,x,x ) + x ( t ) B ( t ) w ( t )} dt
T
T
is a second-order pseudoinvex.
I
If I 0 is empty set and Iα = M for some α ∈{1, 2,..., r} , then Mix (CD) reduces
to (M-WD) which is a second-order dual to (CP).
If
∫ { f ( t,x,x ) + x ( t ) B ( t ) w ( t )} dt
T
I
is
second-order
pseudoinvex
and
∫ y ( t ) g ( t,x,x ) dt,
T
is
second-order
I
quasi-invex.
If B (=
t ) 0 ,t ∈ I , the dual problem Mixed (CD) will reduce to the Mixed type dual
treated by Husain and Bilal [9].
I. Husain and S.K. Shrivastav
141
5 Natural Boundary Values
In this section, we formulate a pair of nondifferentiable mixed type dual
variational problems with natural boundary values rather than fixed end points.
(CP0):
Minimize

T
∫I  f ( t , x, x ) + x ( t ) B ( t ) x ( t )

(
)
1
2

 dt

Subject to
g ( t , x, x ) ≤ 0,
t∈I
(MixCD0):
Maximize
1
∫ ( f ( t , u ( t ) , u ( t ) ) + u ( t ) B ( t ) w ( t ) + ∑ y ( t ) g ( t , u, u ) − 2 β (t )
T
i
i
T
H 0 β (t ))dt
i∈I 0
I
Subject to
( f (t, u, u ) + B (t ) w (t ) + y (t ) g (t, u, u ))
− D ( f ( t , u , u ) + y ( t ) g ( t , u , u ) ) + H β ( t ) =
0,
T
u
u
T
u
u
1
∫ ( ∑ y (t ) g (t , u, u ) − 2 β (t )
i
I
i
i∈Iα
T
1, 2,3, , r
Gα β (t ))dt ≥ 0, α =
w ( t ) B ( t ) w ( t ) ≤ 1, y ( t ) ≥ 0, t ∈ I
∑ y ( t ) g ( t , u, u ) = 0,
i
i∈Iα
i
u
and
fu ( t , u , u ) + ∑ y i ( t ) gui ( t , u , u ) =
0, at t =
a and t =
b.
i∈I 0
t∈I
142
Mixed Type Second-order Duality…
6 Nondifferentiable Nonlinear Programming Problems
If all the functions of the problem (CP0) and Mix (CD0) are independent of t
as b − a =
1. Then these problem will reduces to the following dual problem to
(CP0) formulated by Zhang and Mond [10] :
(CP1):
f ( x ) + ( xT B x )
1/ 2
Minimize
g ( x ) ≤ 0,
Subject to
Mix (CD1):

1
Maximize f ( u ) + ∑ y i g i ( u ) + u T B w − β T H 0 β
2
i∈I 0
Subject to

fu ( u ) + B w + yT gu ( u ) + yT gu ( u ) + H β =
0
1
∑ y g (u ) − 2 β
i
i∈I 0
i
T
Gˆ β ≥ 0
wT B w ≤ 1, y ≥ 0
where


Hˆ 0 =
fuu ( u ) −  ∑ y i g i  and
 i∈I0
uu


i i
Gˆ =
1, 2,..., r .
∑ y g  , α =
∈
i
I
 α
uu
0
In Theorem 3.3, the symbols H ,H 0j ,Gα and Gαj will respectively become as


H 0 =−
f xx ( x )  ∑ y i g i ( x ) 
 i∈I0
 xx


=
∇ 2  f ( x ) − ∑ y i g i ( x ) 
i∈I 0


 2

i i
H 0j =
∇  f ( x ) − ∑ y g ( x )   ,
i∈I 0
 
  j
 2





i i
i i
∇ 2  ∑ y i g i ( x )  and Gαj =
Gα =
 ∇  ∑ y g ( x )  
 ∑ y g ( x )  =
  i∈Iα
 i∈Iα
 xx
 i∈Iα

  j
For the problems (CP0) and Mix (CD0) will be stated as the following
theorem (Theorem 6.1) established by Yang and Zhang [11].Theorem 6.1 is
Theorem 2 of [11] and the rectified version of [10].
I. Husain and S.K. Shrivastav
Theorem 6.1 Let
143
( x, y,w, β ) be an optimal solution of Mix (CD0) at which
( A1 ) for all α = 1, 2,..,r. either
2
i i
(a) the n × n Hessian matrix ∇ ∑ y g ( x ) is positive definite, and
i∈Iα
β T ∇ ∑ yi g i ( x ) ≥ 0
i∈Iα
or
(b) the n × n Hessian matrix ∇ 2 ∑ y i g i ( x ) is negative definite and
i∈Iα
β T ∇ ∑ y i g i ( x ) ≤ 0,
i∈Iα
( A2 ) The vectors
 2
  

i i
∇  f ( x ) − ∑ y g ( x )   ,  ∇ 2  ∑ y i g i ( x )   ,
i∈I 0
 
  j   i∈Iα
  j
α = 1, 2,..,r. j = 1, 2,...,n are linearly independent, where it is the jth row of the
matrix H 0 and
Gα .
1, 2,.., r} are linearly independent.
( A3 ) the vectors {∇ ∑ y i g i ( x), α =
i∈Iα
( x,u, y,w, p ) ,
If, for all feasible
pseudoconvex and
f ( .) + ∑ y i g i ( .) + ( .) B w is second-order
∑ y g ( .),α = 1, 2,..,r
i
i
T
i∈Iα
is second-order quasiconcave with
i∈Iα
respect to the same η , then x is an optimal solution to (CP0).
References
[1] O.L. Mangasarian, Second and higher order duality in nonlinear
programming, Journal of Mathematical Analysis and Applications, 51,
(1979), 605-620.
144
Mixed Type Second-order Duality…
[2] B. Mond, Second order duality in nonlinear programming, Opsearch, 11,
(1974), 90-99.
[3] B. Mond and T. Weir, Generalized Convexity and High Order-Duality,
Journal of Mathematical Analysis and Applications, 46(1), 1974), 169-174.
[4] X. Chen, Second-Order Duality for Variational Problems, Journal of
Mathematical Analysis and Applications, 286, (2003), 261-270.
[5] I. Husain, A. Ahmed and Mashoob Masoodi, Second-Order Duality for
Variational Problems, European Journal of Pure and Applied Mathematics,
2(2), (2009), 278-295.
[6] Iqbal Husain and Mashoob Masoodi, Second-Order Duality for a Class of
Nondifferentiable Continuous Programming Problems, European Journal of
Pure and Applied Mathematics, 5(3), (2012), 390-400.
[7] I. Husain and Santosh K. Shrivastav, On Second-Order Duality in
Nondifferentiable
Continuous
Programming,
American
Journal
of
Operations Research, 2(3), (2012), 289-295.
[8] S. Chandra, B.D. Craven and I. Husain, A class of nondifferentiable
continuous programming problems, J. Math. Anal. Appl., 107, (1985),
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