Theoretical Mathematics & Applications, vol.3, no.1, 2013, 221-246
ISSN: 1792- 9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
On Vector valued Variational Problems
I. Husain 1 and Vikas K. Jain 2
Abstract
A Mond-Weir type second-order dual associated with a class of vector-valued
variational problem in the presence of equality and inequality constraints is
formulated. Under the generalized second-order invexity, various second-order
duality results are derived for this pair of multiobjective variational problems. A
mixed type second-order duality for this class of multiobjective variational
problems is also presented. Finally, the relationship between our second-order
duality results and those of multiobjective nonlinear programming problems is
incorporated through the pair of second-order dual multiobjective variational
problems with natural boundary values.
Mathematics Subject Classification: 90C30, 90C11, 90C20, 90C26
Keywords: Vector-valued variational problem, Second-order duality, Mixed type
second-order duality, Second-order generalized invexity, Multiobjective nonlinear
programming problem.
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: jainvikas13@yahoo.com
Article Info: Received : December 13, 2012. Revised : February 21, 2013
Published online : April 15, 2013
222
On Vector valued Variational Problems
1 Introduction
Second-order duality in mathematical programming has been extensively
studied. This type of duality enjoys computational advantage over the first order
duality because it offers tighter bounds. Following the approach of Mangasarian
[1], Chen [2] formulated a Wolfe type second-order dual associated with a class of
constrained variational problems. Later Husain et al [3] formulated a Mond-Weir
type dual to the variational problem considered in [2] to derive various duality
results under generalized invexity.
In the modeling of real-life problems, there can be more than one objective in
mathematical programming problems representing them. So the subject of
multiobjective programming has an important place in optimization theory.
Motivated with this faint glimpse of vast applications of multicriteria optimization
problems, Husain and Jain [4] have recently presented Wolfe type second-order
duality for multiobjective variational problems with equality and inequality
constraints under second-order pseudoinvexity.
In this research, we present Mond-Weir type duality for the class of
variational problems considered in [4] in order to relax the second-order invexity
requirements on the functions that constitute this pair of second-order dual
multiobjective variational problems. Further, in order to combine the Wolfe type
second-order duality results of [4] and Mond-Weir type second-order duality
results for multiobjective variational problems derived in this research, mixed type
second-order duality is presented and various second-order duality results are
obtained under second-order generalized invexity. Finally, a relationship between
our duality results and those of nonlinear multiobjective programming problems is
briefly outlined through the pair of second-order multiobjective variational
problems with natural boundary values.
I. Husain and Vikas K. Jain
223
2 Pre-Requisites and statement of the problem
Let I = [ a, b ] be a real interval, φ : I × R n × R n → 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 derivatives of φ with respect to x ( t ) and x ( t ) ,
respectively, that is,
∂φ 
∂φ 
 ∂φ ∂φ
 ∂φ ∂φ
=
φx =
, 2 , ,
, φx  1 , 2 ,  ,

1
n 
∂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
 1
 ∂x
 ∂ψ 2
ψ x =  ∂x1

 

m
 ∂ψ
 ∂x1







m 
∂ψ 


∂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
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
224
On Vector valued Variational Problems
Consider the following constrained multiobjective variational problem:


(VEP): Minimize  ∫ f 1 (t , x, x )dt ,..., ∫ f p (t , x, x )dt 
I
I

subject to
=
x ( a ) α=
, x (b) β
g ( t , x, x ) < 0, t ∈ I
(1)
h ( t , x, x=
) 0, t ∈ I
(2)
where f i : I × R n × R n =
→ R, i ∈ K {1, 2,..., p}, g : I × R n × R n → R m ,
h : I × R n × R n → R l are continuously differentiable functions.
The following convention for equality and inequality will be used. If
α , β ∈ R n , then
α =β ⇔ α i =β i
i =1, 2,..., n
α >β ⇔ αi > βi
i=
1, 2,..., n
α ≥ β ⇔ α > β and α ≠ β
α >β ⇔ αi > βi
i=
1, 2,..., n
Definition 2.1 A feasible solution x is efficient for (VEP) if there is no feasible
x̂ for (VEP) such that
∫f
I
∫f
I
i
(t , xˆ , xˆ )dt < ∫ f i (t , x , x )dt ,
for some i ∈ {1, 2,..., p}
I
j
(t , xˆ , xˆ )dt <
∫f
j
(t , x , x )dt ,
for all j ∈ {1, 2,..., p}
I
In the case of maximization, the signs of above inequalities are reversed.
The optimality conditions for the problems (VEP), derived by Husain and Jain [4]
are reproduced in the following theorems (Theorem 2.1 and Theorem 2.2).
Theorem 2.1 (Karush-Kuhn-Tucker type necessary optimality conditions): If
x is an normal and optimal solution of (VEP) and hx (., x (.), x (.)) maps onto a
I. Husain and Vikas K. Jain
225
closed subspace of C ( I , R l ) , then there exist piecewise smooth y : I → R m ,
z : I → R l such that
f x (t , x (t ), x (t )) + y (t )T g x (t , x (t ), x (t )) + z (t )T hx (t , x (t ), x (t ))
=
D ( f x (t , x (t ), x (t )) + y (t )T g x (t , x (t ), x (t )) + z (t )T hx (t , x (t ), x (t )) ) , t ∈ I
y (t )T g (t , x (t ),=
x (t )) 0,
y (t )> 0,
t∈I
t∈I
Theorem 2.2 (Fritz John type necessary optimality conditions): Let x be an
efficient solution of (VEP).Then there exist λ i ∈ R, i ∈ K and piecewise smooth
functions y : I → R m , z : I → R l such that
∑λ ( f
p
i
i =1
i
x
− Df xi ) + ( y (t )T g x + z (t )T hx ) − D ( y (t )T g x + z (t )T hx ) =0, t ∈ I
y (t )T g ( t , x ,=
x ) 0, t ∈ I
( λ , y (t ) ) > 0, t ∈ I
( λ , y (t ), z (t ) ) ≠ 0, t ∈ I
We shall make use of the following definitions in the subsequent analysis:
Definition 2.2 The function
∫ φ (t ,.,.)dt
is second-order pseudoinvex if for all
I
β (t ) ∈ R n , there exist an η = η (t , x, u ) such that
T
 T

 dη 
T


+
+
η
φ
t
u
u
φ
t
u
u
η
A
β
t
(
,
,
)
(
,
,
)
(
)

dt > 0

u
u


∫I 
dt




1


⇒ ∫ φ (t , x, x )dt > ∫  φ (t , u , u ) − β T (t ) Aβ (t )  dt
2

I
I 
or equivalently,
226
On Vector valued Variational Problems

1
∫ φ (t , x, x )dt < ∫  φ (t , u, u ) − 2 β (t )
I
T
I

Aβ (t )  dt

T
 T

 dη 
T

⇒ ∫ η φu (t , u , u ) + 
 φu (t , u , u ) + η Aβ (t ) dt < 0
 dt 

I 

where
A=
f xxi − 2 Df xxi + D 2 f xxi − D 3 f xxi
Definition 2.3 The function
∫ φ (t ,.,.)dt
is said to second-order quasi-invex if for
I
all β (t ) ∈ R n , there exist an η = η (t , x, u ) such that

1
∫ φ (t , x, x )dt < ∫  φ (t , u, u ) − 2 β
I
I
T

(t ) Aβ (t )  dt

T


 dη 
T
⇒ ∫ η T φu (t , u , u ) + 
 φu (t , u , u ) + η Aβ (t ) dt < 0
 dt 

I 

The
function
∫ φ (t ,.,.)dt
is
said
to
strictly
pseudoinvex
if
for
I
β (t ) ∈ R n and x(t ) ≠ u (t ), t ∈ I , there exist an η = η (t , x, u ) such that
T
 T

 dη 
T

+
η
φ
(
t
,
u
,
u
)

 φu (t , u , u ) + η Aβ (t ) dt > 0
∫I  u
 dt 


⇒ ∫ φ (t , x, x )dt > ∫ φ (t , u , u )dt
I
I
or equivalently
∫ φ (t , x, x )dt < ∫ φ (t , u, u )dt
I
I
T
 T

 dη 
T


+
+
η
φ
φ
η
β
t
u
u
t
u
u
A
t
(
,
,
)
(
,
,
)
(
)

dt < 0

u
u


∫I 
 dt 


Remark 2.1 If φ does not depend explicitly on t then the above definitions
reduced to those given in [5].
all
I. Husain and Vikas K. Jain
227
3 Mond-Weir type second-order Duality
In this section, we propose the following Mond-Weir type second-order to
the problem(VEP):
(M-WED): Maximize
  1
1
1

 p
 
T
1
T
p
 ∫  f (t , u , u ) − β (t ) F β (t )  dt ,..., ∫  f (t , u , u ) − β (t ) F β (t )  dt 
2
2

 
I
I 
subject to
, u (b) β
=
u ( a ) α=
(3)
(
)
λ T fu + y (t )T gu + z (t )T hu − D λ T fu + y (t )T gu + z (t )T hu + B β (t ) =
0, t ∈ I
1

∫  y(t ) g ( t , u, u ) − 2 β (t )
T
T
I
1

∫  z (t ) h ( t , u, u ) − 2 β (t )
T

G β (t ) dt > 0

(5)

H β (t ) dt > 0

(6)
T
I
(4)
y (t ) > 0, t ∈ I
(7)
λ>0
(8)
where B= λ T F + G + H with
F i = fui u − 2 Dfui u + D 2 fui u − D3 fui u , t ∈ I , i ∈ K
(
T
(
T
G=
y ( t ) gu
)
u
(
− 2 D y ( t ) gu
T
)
u
(
+ D 2 y ( t ) gu
T
)
u
(
− D 3 y ( t ) gu
T
)
u
,t ∈ I
and
H=
z ( t ) hu
)
u
(
− 2 D z ( t ) hu
T
)
u
(
+ D 2 z ( t ) hu
T
)
u
(
− D 3 z ( t ) hu
T
)
u
,t ∈ I
are n × n symmetric matrices.
In the following analysis, we shall designate the sets of feasible solutions of the
problems (VEP) and (M-WED) by X and Y respectively.
228
On Vector valued Variational Problems
Theorem 3.1(Weak duality): Let x(t ) ∈ X and ( u (t ), λ , y (t ), z (t ), β (t ) ) ∈ Y such
that with respect to the same η
(A1):
∫ ( λ f ( t ,.,.) )dt is second-order pseudoinvex
T
I
(A2):
∫ ( y(t ) g ( t ,.,.) )dt is second-order quasi-invex and
T
I
(A3):
∫ ( z (t ) h ( t ,.,.) )dt is second-order quasi-invex,
T
I
then
∫f
I
r
1


(t , x, x )dt < ∫  f r (t , u , u ) − β (t )T F r β (t ) dt , for some r ∈ K
2

I 
(9)
and
∫f
I
i
1


(t , x, x )dt < ∫  f i (t , u, u ) − β (t )T F i β (t ) dt , i ∈ K r
2

I 
(10)
cannot hold.
Proof: Suppose, to the contrary, that (9) and (10) holds. Since λ > 0 , the above
inequalities give
∫λ
I
T
1


f (t , x, x )dt < ∫  λ T f (t , u , u ) − β (t )T λ T F β (t )  dt
2

I 
which because of second-order pseudoinvexity of ∫ λ T f (t ,.,.)dt , this implies
I
T
 T T

 dη 
T
T
T


η
λ
f
t
u
u
λ
f
t
u
u
η
λ
F
β
t
+
+
(
,
,
)
(
,
,
)
(
)
(
)
(
)
(
)

dt < 0

u
u


∫I 
 dt 


Using x(t ) ∈ X and ( u (t ), λ , y (t ), z (t ), x(t ) ) ∈ Y , we have
∫ y(t )
T
I
1


g (t , x, x )dt < ∫  y (t )T g (t , u , u ) − β (t )T G β (t )  dt
2

I 
By the hypothesis (A2), this implies
(11)
I. Husain and Vikas K. Jain
229
T
 T

 dη 
T
T
T
η
y
t
g
(
)
+
u)

 ( y (t ) gu ) + η G β (t ) dt < 0
∫I  (
 dt 


∫ z (t )
T
Also
I
(12)
h(t , x, x )dt < ∫ z (t )T h(t , u , u )dt , because (A3) implies
I
T
 T

 dη 
T
T
T
∫I η ( z (t ) hu ) +  dt  ( z (t ) hu ) + η H β (t ) dt < 0


(13)
Combining (11), (12) and (13), we have
T
 T T

 dη  T
T
T
0 > ∫ η ( λ fu + y (t ) gu + z (t ) hu ) +   ( λ fu + y (t )T gu + z (t )T hu ) + η T (λ T F + G + H ) β (t ) dt
 dt 

I

T
T
T
T
T
T
T
T
= ∫ η ( λ fu + y (t ) gu + z (t ) hu ) − D ( λ fu + y (t ) gu + z (t ) hu ) + (λ F + G + H ) β (t ) dt
I
t =b
t=a
which, because of η = =
0, at t a=
and t b, yields
+η T ( λ T fu + y (t )T gu + z (t )T hu )
∫η [( λ
T
T
fu + y (t )T gu + z (t )T hu ) − D ( λ T fu + y (t )T gu + z (t )T hu )
I
+ (λ T F + G + H ) β (t )]dt < 0
i.e.
∫η ( λ
T
I
T
fu + y (t )T gu + z (t )T hu ) − D ( λ T fu + y (t )T gu + z (t )T hu ) + B β (t ) dt < 0
This is contrary to the equality constraint of the dual problem (M-WED). Hence
the theorem fully follows.
Theorem 3.2 (Strong duality): Let x (t ) be normal and efficient solution of
(VEP). Then there exist λ ∈ R p , piecewise smooth functions y : I → R m and
z : I → R l such that
( x (t ), λ , y (t ), z (t ), β (t ) = 0 ) is feasible for (M-WED) and the
two objective functional are equal. Furthermore, if the hypotheses of Theorem 2.1
230
On Vector valued Variational Problems
hold for all feasible solutions of (VEP) and (M-WED),the ( x (t ), λ , y (t ), z (t ), β (t ) = 0 )
is an efficient for (M-WED).
Proof: Since x (t ) is normal and an efficient solution of the problem (VEP),
therefore, by Theorem 2.1, there exist λ ∈ R p , piecewise smooth functions
y : I → R m and z : I → R l such that
λ T f x + y (t )T g x (t , x , x ) + z (t )T hx (t , x , x )
− D ( λ T f x (t , x , x ) + y (t )T g x (t , x , x ) + z (t )T hx (t , x , x ) ) =
0, t ∈ I
y (t )T g (t , x , =
x ) 0, t ∈ I
(14)
(15)
p
λ > 0, ∑ λ i =
1
(16)
y (t )T > 0, t ∈ I
(17)
i =1
The relation (15) implies

∫  y (t )
T
I
1

g (t , x , x ) − β (t )T G β (t )  dt =
0
2

Since x ( t ) is feasible for (VEP), we have h ( t,x ,x=
) 0,t ∈ I and

∫  z (t )
I
Consequently,
T
1

h(t , x , x ) − β (t )T H β (t )  dt =
0
2

.
( x , λ , y (t ), z (t ), β (t ) = 0 ) is feasible for (M-WED).
Consider
 1

p
 ∫ f (t , x , x )dt ,..., ∫ f (t , x , x )dt 
I
I

  1
1
1

 p
 
T
T
p
1
=
 ∫  f (t , x , x ) − β (t ) F β (t ) dt ,..., ∫  f (t , x , x ) − β (t ) F β (t ) dt 
2
2

 
I 
I
yielding the equality of the objective functionals.
If
( x , λ , y (t ), z (t ), β (t ) = 0 ) is not efficient, there exists ( uˆ, λˆ, yˆ , zˆ, βˆ )
feasible for
I. Husain and Vikas K. Jain
231
(M-WED) such that

∫  f
k
I
1

(t , uˆ , uˆ ) − βˆ (t )T Fˆ k βˆ (t ) dt > ∫ f k (t , x , x )dt , for some k ∈ K
2

I
and

∫  f
r
I
1

(t , uˆ , uˆ ) − βˆ (t )T Fˆ r βˆ (t ) dt > ∫ f r (t , x , x )dt , r ≠ k .
2

I
These inequality with λˆ > 0 implies

∫ λˆ
I
T
1

f (t , uˆ , uˆ ) − βˆ (t )T (λˆT Fˆ ) βˆ (t ) dt > ∫ λˆT f (t , x , x )dt
2

I
which because of pseudoinvexity of
∫ λˆ
T
f (t ,.,.)dt yield for η (t , x , uˆ ) = η
I
T
 T T
 dη  ˆT

ˆ ) + η T (λˆT Fˆ ) βˆ (t ) dt < 0
ˆ
ˆ
ˆ
ˆ
(
,
,
)
(
,
,
+
f
t
u
u
f
t
u
u
η
λ
λ


u
u


∫I 
 dt 


(
where
)
(
)
(18)
Fˆ = F (t , uˆ , uˆ , uˆ , 
uˆ )
From the feasibility, we have
∫ yˆ (t )
1


g (t , x , x )dt < ∫  yˆ (t )T g (t , uˆ , uˆ ) − βˆ (t )T Gˆ βˆ (t )  dt
2

I 
(19)
∫ zˆ(t )
1


h(t , x , x )dt < ∫  zˆ (t )T h(t , uˆ , uˆ ) − βˆ (t )T Hˆ βˆ (t )  dt
2

I 
(20)
T
I
T
I
where
(t , uˆ , uˆ , uˆ , 
=
Gˆ G=
uˆ ), Hˆ H (t , uˆ , uˆ , uˆ , 
uˆ )
These, because second-order quasi-invexity of
∫ yˆ (t )
T
g (t ,.,.)dt and
I
∫ zˆ(t )
T
h(t ,.,.)dt with respect to the same η respectively, yield
I
T
 T

 dη 
T
T
T

∫I η yˆ (t ) gu (t , uˆ, uˆ ) +  dt  yˆ (t ) gu (t , uˆ, uˆ ) + η Gˆ βˆ (t ) dt < 0


(
)
(
)
and
T
 T

 dη 
T
T
T

∫I η zˆ(t ) hu (t , uˆ, uˆ ) +  dt  zˆ(t ) hu (t , uˆ, uˆ ) + η Hˆ βˆ (t ) dt < 0


(
)
(
)
232
On Vector valued Variational Problems
Combining (18), (19) and (20), we have
(
) (
)
0 > ∫ η T  λˆT fu + yˆ (t )T gu + zˆ (t )T hu − D λˆT fu + yˆ (t )T gu + zˆ (t )T hu + (λˆT Fˆ + Gˆ + Hˆ ) βˆ (t ) dt


I
As earlier, this will yield
∫η ( λˆ
T
I
T
(
)
)
fu + yˆ (t )T gu + zˆ (t )T hu − D λˆT fu + yˆ (t )T gu + zˆ (t )T hu + B βˆ (t ) dt < 0,

contradicting the feasibility of
( uˆ, λˆ, yˆ , zˆ, βˆ ) for (M-WED).
We now give a Mangasarian type [5] strict-converse duality theorem for the dual
(M-WED) to (VEP).
Theorem 3.3 (Strict-converse duality): Let x (t ) and ( u (t ), λ , y (t ), z (t ), β (t ) ) ∈ Y
be efficient solutions of (VEP) and (M-WED) respectively, such that

f (t , x , x )dt ∫ λ
∫ λ=

T
I
I
T
1

f (t , x , x ) − β (t )T (λ T F ) β (t ) dt
2

(21)
If with respect to the same η
(B1):
∫ ( λ f ( t ,.,.) )dt is second-order strictly pseudoinvex
T
I
( B2):
∫ ( y(t ) g ( t ,.,.) )dt is second-order quasi-invex,
T
I
and
(B3):
∫ ( z (t ) h ( t ,.,.) )dt is second-order quasi-invex,
T
I
then
x (t ) u (t ) , t ∈ I .
=
Proof: Suppose x ( t ) ≠ u ( t ) ,t ∈ I
By hypothesis (B1), (21) implies for η = η (t , x , u )
T
 T T

 dη 
T
T
T
η
λ
λ
η
λ
β
f
f
F
t
+
+
(
)
(
)
(
)
(
)

dt < 0

u
u


∫I 
dt




From feasibility of (VEP) and (M-WED), we have
(22)
I. Husain and Vikas K. Jain
233
1


g (t , x , x )dt < ∫  y (t )T g (t , u , u ) − β (t )T G β (t )  dt
2

I 
∫ y (t )
T
I
where G = G (t , u , u , u, 
u , y , 
y , 
y)
This by the hypothesis (B2) gives
T
 T

 dη 
T
T

 ) ) + η T G β (t ) dt < 0
η
y
t
g
t
u
u
y
t
g
t
u
u
(
)
(
,
,
)
+
(
)
(
,
,
(
)
(


u
u


∫I 
 dt 


(23)
Also we have
∫ z (t )
T
I
h(t , x , x )dt < ∫ z (t )T h(t , u , u )dt
I
by the hypothesis (B3), this implies
T
 T

 dη 
T
T

 ) ) + η T H β (t ) dt < 0
(
)
(
,
,
)
(
)
(
,
,
z
t
h
t
u
u
z
t
h
t
u
u
η
+
(
)
(


u
u


∫I 
 dt 


(24)
Combining (22), (23) and (24), we have
∫ {η ( λ
T
T
fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) )
I
 dη 
T
T
T
T
T



+
 ( λ fu (t , u , u ) + y (t ) gu (t , u , u ) + z (t ) hu (t , u , u ) ) + η (λ F + G + H ) β (t )}dt < 0
 dt 
This, as earlier analysis, implies
T
∫ {η ( λ
T
T
fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) )
I
− D ( λ T fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) ) + B β (t )}dt < 0
This contradicts the feasibility of ( u , λ , y , z , β ) for (M-WED). Hence
=
x (t ) u (t ) , t ∈ I .
234
On Vector valued Variational Problems
4 Mixed type second-order duality
Let=
M {1, 2,..., m
=
}, L {1, 2,..., l ), Iα ⊆ =
M , α 0,1, 2,..., r with Iα ∩ I β =
φ,
r
r
α =1
α =1
0,1, 2,..., r with Jα ∩ J β =
φ , α ≠ β ,  Jα =
M and Jα ⊆ L, α =
α ≠ β ,  Iα =
L.
In relation to the problem (VEP), consider the following multiobjective variational
problem:
(MVED): Maximize
( ∫ ( f 1 (t , u , u ) + ∑ y j (t ) g j (t , u , u ) +
j∈I 0
I
∑z
(t )h k (t , u , u ) −
k∈J 0
..., ∫ ( f p (t , u , u ) + ∑ y j (t ) g j (t , u , u ) +
j∈I 0
I
k
∑z
k
1
β (t )T A1 β (t ))dt ,
2
(t )h k (t , u , u ) −
k∈J 0
1
β (t )T A p β (t ))dt )
2
subject to
=
u (a ) α=
, u (b) β
(25)
λ T fu (t , u, u ) + y (t )T gu (t , u, u ) + z (t )T hu (t , u, u )
− D ( λ T fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) ) + BAβ (t ) =
0, t ∈ I

j

k
∑ ∫  y
j∈Iα I
∑ ∫  z
k∈Jα I
(26)
1

(t ) g j (t , u , u ) − β (t )T G j β (t ) dt > 0 , α =
1, 2,..., r
2

(27)
1

(t )h k (t , u , u ) − β (t )T H k β (t ) dt > 0, , α =
1, 2,..., r
2

(28)
y (t ) > 0, t ∈ I
(29)
p
λ > 0, ∑ λ i =
1,, α =
1, 2,..., r
i =1
where
(30)
I. Husain and Vikas K. Jain
235
Ai =
fuui − 2 Dfuu + D 2 fuu  − D 3 fuu +
∑(y
j∈Iα
j
(t ) guj (t , u , u ) ) − 2 D ∑ ( y j (t ) guj (t , u , u ) )
+ D 2 ∑ ( y j (t ) guj (t , u , u ) ) − D 3 ∑ ( y j (t ) guj (t , u , u ) ) +
j∈Iα
j∈Iα
u
−2 D ∑ ( z (t )h (t , u , u ) ) + D
k
k
u
k∈Jα
∑ (z
2
u
k
k∈Jα
u
j∈Iα
u
∑ (z
k∈Jα
(t )h (t , u , u ) ) − D
k
u
3
u
k
(t )huk (t , u , u ) )
∑ (z
k∈Jα
u
u
k
(t )h (t , u , u ) ) ,
k
u
u
the matrix B is the same as defined in the previous section.
=
Gj
( y ( t )g ( t,u,u )) − 2D ( y ( t )g ( t,u,u ))
+ D ( y ( t )g ( t,u,u )) − D ( y ( t )g ( t,u,u ))
j
j
u
2
=
Hk
(z
k
j
j
(t )huk (t , u , u )
(
j
u
u
3
j
u
)
u
u
j
j
u
u
(
− 2 D z k (t )huk (t , u , u )
+ D 2 z k (t )huk (t , u , u )
)
u
(
u
,
)
u
)
− D3 z k (t )huk (t , u , u ) ,
u
We denote by Ω the set of feasible solutions of (MVED).
Theorem 4.1(Weak duality): Let x ∈ X and
( u, λ , y, z, β (t ) ) ∈ Ω
be efficient
solutions of (VEP) and (MVED) respectively, such that with respect to the sameη
 T

j
j
k
k
λ
f
t
,.,.
+
y
(
t
)
g
t
,.,.
+
z
(
t
)
h
t
,.,.
dt is
(
)
(
)
(
)


∑
∑
∫I 
j∈I 0
k∈J 0

(C1):
second-order
pseudoinvex
∫ ( y(t ) g ( t ,.,.) )dt is second-order quasi-invex, and
T
(C2):
I
∫ ( z (t ) h ( t ,.,.) )dt is second-order quasi-invex.
T
(C3):
I
Then
∫f
r
(t , x, x )dt
I
1


< ∫  f r (t , u , u ) + ∑ y j (t ) g j (t , u , u ) + ∑ z k (t )h k (t , u , u ) − β (t )T Ar β (t ) dt
2
j∈I 0
k∈J 0

I 

for some r ∈ K , and
236
On Vector valued Variational Problems
∫f
q
(t , x, x )dt
I
1


< ∫  f q (t , u , u ) + ∑ y j (t ) g j (t , u , u ) + ∑ z k (t )h k (t , u , u ) − β (t )T Aq β (t ) dt
2
j∈I 0
k ∈J 0

I 

cannot hold.
Proof: Suppose to the contrary that there is x feasible for (VEP) and
( u, λ , y, z, β (t ) ) feasible for (MVED) such that
∫f
r
(t , x, x )dt
I
1


< ∫  f r (t , u , u ) + ∑ y j (t ) g j (t , u , u ) + ∑ z k (t )h k (t , u , u ) − β (t )T Ar β (t ) dt
2
j∈I 0
k∈J 0

I 

for some r ∈ {1, 2,...,, p}
∫f
q
(t , x, x )dt
I


1
< ∫  f q (t , u , u ) + ∑ y j (t ) g j (t , u , u ) + ∑ z k (t )h k (t , u , u ) − β (t )T Aq β (t ) dt
2
j∈I 0
k∈J 0

I 

,q ≠ r
Multiplying these by λ > 0 , we have
∫λ
T
f (t , x, x )dt
I


1
< ∫ λ T f (t , u, u ) + ∑ y j (t ) g j (t , u, u ) + ∑ z k (t )h k (t , u, u ) − β (t )T (λ T A) β (t ) dt
2
j∈I 0
k∈J 0

I 

which by the hypothesis (C1) yields
∫ {η
T
(λ T f u +
I
∑ y j (t ) guj + ∑ z k (t )huk )
j∈I 0
 dη 
T
+
 (λ fu +
 dt 
T
k∈J 0
∑ y j (t ) guj + ∑ z k (t )huk ) + η T (λ T A)β (t )}dt < 0
j∈I 0
k∈J 0
for α = 1, 2,..., r
∑ ∫( y
j∈Iα I
j
(t )T g j (t , x, x ) )dt <

∑ ∫  y
j∈Iα I
j
1

(t )T g j (t , u , u ) − β (t )T G j β (t ) dt ,
2

(31)
I. Husain and Vikas K. Jain
237
which by the hypothesis (C2) gives
T
 T j T j

 dη 
j
T
j
T
j


η
y
t
g
t
u
u
y
t
g
t
u
u
η
G
β
t
+
+
(
)
(
,
,
)
(
)
(
,
,
)
(
)
(
)
(
)

dt < 0,
∑

u
u


∫
 dt 
j∈Iα I 

α = 1, 2,..., r
and so
T
 T j T j

 dη 
j
T
j
T
j


+
+
η
η
β
y
t
g
t
u
u
y
t
g
t
u
u
G
t
dt < 0
(
)
(
,
,
)
(
)
(
,
,
)
(
)
)
)
(
(



u
u


∫I 
 dt 


∑
r
j∈
 Iα
α =1
implies
T
 T

 dη 
T
T
T


η
η
β
y
t
g
t
u
u
y
t
g
t
u
u
G
t
+
+
(
)
(
,
,
)
(
)
(
,
,
)
(
)
(
)
(
)

dt < 0

u
u


∫I 
 dt 


(32)
Also for α = 1, 2,..., r ,
∑ ∫(z
k
k∈Jα I
(t )T h k (t , x, x ) )dt <

∑ ∫  z
k
k∈Jα I
1

(t )T h k (t , u , u ) − β (t )T H k β (t ) dt ,
2

By the hypothesis (C3), this yields
T
 T k T k

 dη  k T k
T
k


η
z
t
h
t
u
u
z
t
h
t
u
u
η
H
β
t
+
+
(
)
(
,
,
)
(
)
(
,
,
)
(
)
(
)
(
)

dt < 0,
∑

u
u


∫
 dt 
k∈Jα I 

α = 1, 2,..., r
and so
∑
r
k∈
 Jα
T
 T k T k

 dη 
k
T k
T
k


η
z
(
t
)
h
(
t
,
u
,
u
)
+
z
(
t
)
h
(
t
,
u
,
u
)
+
η
H
β
(
t
)
dt < 0
(
)
(
)



u
u


∫I 
 dt 


α =1
implies
T
 T

 dη 
T
T
T


η
z
t
h
t
u
u
z
t
h
t
u
u
η
H
β
t
+
+
(
)
(
,
,
)
(
)
(
,
,
)
(
)
(
)
(
)

dt < 0

u
u


∫I 
dt




Combining (31), (32) and (33), we have
(33)
238
On Vector valued Variational Problems
 dη  T
T
T
T
T
T
T
∫I [η ( λ fu + y(t ) gu + z(t ) hu ) +  dt  ( λ fu + y(t ) gu + z(t ) hu )
T
+ η T (λ T A + G + H ) β (t )]dt < 0
0 > ∫ η T ( λ T fu + y (t )T gu + z (t )T hu ) − D ( λ T fu + y (t )T gu + z (t )T hu ) + B β (t ) dt
I
+ η T ( λ T f u + y (t )T gu + z (t )T hu )
t =b
t=a
t =b
which,=
by using η η=
(t , x, u )
0 implies
t=a
∫η ( λ
T
I
T
fu + y (t )T gu + z (t )T hu ) − D ( λ T fu + y (t )T gu + z (t )T hu ) + Bβ (t ) dt < 0,
contradicting to the feasibility of
( u, λ , y, z, β (t ) ) for (MVED).
Hence the conclusion of the theorem is true under stated hypotheses.
Theorem 4.2 (Strong Duality): Let x be normal and efficient. Then there exist
λ ∈ Rp
and piecewise smooth functions y : I → R m and z : I → R l such that
( x (t ), λ , y (t ), z (t ), β (t ) = 0 ) is feasible for (MVED) and the objective values of
(VEP) and (MVED) are equal. If also hypothesis of Theorem 4.1 hold, then
( x (t ), λ , y (t ), z (t ), β (t ) = 0 ) is efficient for (MVED).
Proof:
By Theorem 2.1, there exist λ ∈ R p , piecewise smooth functions
y : I → R m and z : I → R l such that
0, t ∈ I
λ T f x + y (t )T g x + z (t )T hx − D ( λ T f x + y (t )T g x + z (t )T hx ) =
y (t )T g (t , x , x ) = 0
z (t )T h(t , x , x ) = 0
p
λ > 0, ∑ λ i =
1
i =1
y (t )T > 0, t ∈ I
I. Husain and Vikas K. Jain
239
From the above relation, we have

T

T
∫  y (t )
I
1

0
g (t , x , x ) − β (t )T G β (t )  dt =
2

and
∫  z (t )
I
1

h(t , x , x ) − β (t )T G β (t )  dt =
0
2

( x , λ , y , z , β = 0) for (MVED).
consequently
Consider
∫f
i
(t , x , x )dt
I
 i

 ) + ∑ y j (t ) g j (t , x , x ) + ∑ z k (t )h k (t , x , x ) − 1 β (t )T Ai β (t ) dt
(
,
,
f
t
x
x
=

∫I 
2
j∈I 0
k∈J 0

for i ∈ K .
This implies the objective values of (VEP) and (MVED) are equal.
( x (t ), λ , y (t ), z (t ), β (t ) = 0 ) is not efficient solution for (MVED), then there
If
exist feasible ( u , λ , y , z , β ) for (MVED) such that

∫  f
r
(t , u , u ) + ∑ y j (t ) g j (t , u , u ) +
j∈I 0
I
∑z
k
k∈J 0
1

(t )h k (t , u , u ) − β (t )T Ar β (t ) dt
2



> ∫  f r (t , x , x ) + ∑ y j (t ) g j (t , x , x ) + ∑ z k (t )h k (t , x , x ) dt ,
j∈I 0
k∈J 0

I 

for some r ∈ K

∫  f
I
q
(t , u , u ) + ∑ y j (t ) g j (t , u , u ) +
j∈I 0
∑z
k∈J 0
k

1
(t )h k (t , u , u ) − β (t )T Aq β (t ) dt
2



> ∫  f q (t , x , x ) + ∑ y j (t ) g j (t , x , x ) + ∑ z k (t )h k (t , x , x ) dt , q ≠ r.
j∈I 0
k∈J 0

I 

Multiplying by λ (> 0) , these imply
240

∫ λ
On Vector valued Variational Problems
T
f (t , u , u ) + ∑ y j (t ) g j (t , u , u ) +
j∈I 0
I
∑z
k∈J 0
k

1
(t )h k (t , u , u ) − β (t )T (λ T A) β (t ) dt
2



> ∫ λ T f (t , x , x ) + ∑ y j (t ) g j (t , x , x ) + ∑ z k (t )h k (t , x , x ) dt
j∈I 0
k∈J 0

I 

which because of pseudoinvexity of

∫  λ f ( t ,.,.) + ∑ y
T
j
(t ) g j ( t ,.,.) +
j∈I 0
I
∑z
k
k∈J 0

(t )h k ( t ,.,.) dt implies

 dη 
y (t ) g + ∑ z (t )h ) +   (λ T fu + ∑ y j (t ) guj + ∑ z k (t )huk )
∫I [η (λ fu + ∑
 dt 
j∈I 0
k∈J 0
j∈I 0
k ∈J 0
1
− β (t )T (λ T A) β (t )]dt < 0
2
T
T
T
j
j
u
k
k
u
As earlier, it implies
 T
  T

T
j
j
k
k
j
j
k
k
+
+
−
+
+
η
λ
f
y
t
g
z
t
h
D
λ
f
y
t
g
z
t
h
[
(
)
(
)
(
)
(
)



∑
∑
∑
∑



u
u
u
u
u 
∫I  u j∈I0
k∈J 0
j∈I 0
k∈J 0
 

T
+ (λ A) β (t ) ]dt < 0
Also for α = 1, 2 ,...,r , we have
∑ ∫( y
j
∑ ∫(z
k
j∈Iα I
k∈Jα I

j

k
(t )T g j (t , x , x ) )dt <
∑ ∫  y
(t )T h k (t , x , x ) )dt <
∑ ∫  z
j∈Iα I
k∈Jα I
1

(t )T g j (t , u , u ) − β (t )T G j β (t ) dt
2

1

(t )T h k (t , u , u ) − β (t )T H k β (t ) dt
2

This, because of the second-order quasi-invexity of
∑ ∫( y
j∈Iα I
and
∑ ∫(z
k∈Jα I
k
j
(t )T g j ( t ,.,.) )dt
(t )T h k ( t ,.,.) )dt , for α = 1, 2,..., r gives
T
 T j T j

 dη 
j
T
j
j


η ( y (t ) gu (t , u , u ) ) + 
∑
 ( y (t ) gu (t , u , u ) ) + G β (t ) dt < 0,
∫
 dt 
j∈Iα I 


α = 1, 2,..., r and
(34)
I. Husain and Vikas K. Jain
241
T
 T k T k

 dη 

z
t
h
t
u
u
η
+
(
)
(
,
,
)
)  dt  ( z k (t )T huk (t , u , u ) ) + H k β (t )dt < 0,
 (
∑
u
∫
k∈Jα I 


α = 1, 2,..., r .
These imply








T
j
j
j
j
j
−
+
η
y
(
t
)
g
D
y
(
t
)
g
G
β
(
t
)

dt < 0


∑r (
∑r
u )
u )
∫I  ∑r (



j∈ Iα
j∈ Iα
 j∈ Iα 


=
 α1

 α 1 =α 1=
(35)
and








T
k
k
k
k
k
−
+
η
z
(
t
)
h
D
z
(
t
)
h
H
β
(
t
)

dt < 0


(
)
(
)
∑
∑
∑

u
u
∫I  r
r
r



k ∈ J α
k ∈ J α
 k ∈ J α

=
 α1

 α 1 =α 1=

(36)
Combining (34), (35) and (36), we have
∫η ( λ
T
I
T
fu + y (t )T gu + z (t )T hu ) − D ( λ T fu + y (t )T gu + z (t )T hu ) + B β (t ) dt < 0
contradicting the feasibility of
,
( u , λ , y , z , β ) for (MVED).Hence the efficiency of
( x , λ , y , z , β = 0 ) for (MVED) follows.
The following theorem gives Mangasarian type [5] strict-converse duality for
(VEP) and (MVED):
Theorem 4.3(Strict converse duality): Let x and
( u , λ , y , z , β (t ) ) ∈ Ω
be
efficient solutions of (VEP) and (MVED) respectively, such that
∫λ
T
f (t , x , x )dt
I
1
 T

y j (t ) g j (t , u , u ) + ∑ z k (t )h k (t , u , u ) − β (t )T (λ T A) β (t ) dt
=
∫I λ f (t, u , u ) + ∑
2
j∈I 0
k∈J 0

If with respect to the same η
(37)
242
On Vector valued Variational Problems

∫  λ f ( t ,.,.) + ∑ y
(D1):
T
j
(t ) g j ( t ,.,.) +
j∈I 0
I
∑z
k∈J 0
k

(t )h k ( t ,.,.) dt is

second-order
strictly pseudoinvex
(D2):
∑ ∫( y
j
(t )T g j ( t ,.,.) )dt , α = 1, 2,..., r is second-order quasi-invex, and
∑ ∫(z
k
(t )T h k ( t ,.,.) )dt , α = 1, 2,..., r is second-order quasi-invex,
j∈Iα I
(D3):
k∈Jα I
then
=
x (t ) u (t ), t ∈ I
Proof: Suppose x (t ) ≠ u (t ), t ∈ I . By the hypothesis (D1), (37) implies for
η = η (t , x , u )
 dη 
y (t ) g + ∑ z (t )h ) +   (λ T fu + ∑ y j (t ) guj + ∑ z k (t )huk )
∫I [η (λ fu + ∑
 dt 
j∈I 0
k∈J 0
j∈I 0
k ∈J 0
T
T
T
j
j
u
k
k
u
+ η T (λ T A) β (t )]dt < 0
(38)
As earlier, we have
∫[ 
I
r
j∈
 Iα
 dη 
(η ( y (t ) g ) + 

 dt 
T
T
j
j
u
(y
j
(t ) guj ) + η T G j β (t ))
α =1

+
r
k∈
 Jα
 dη 
(η ( z (t )h ) + 

 dt 
T
T
k
k
u
(z
k
(t )huk ) + η T H k β (t ))]dt < 0
(39)
α =1
implying
T
 T

 dη 
T
T
T
T
T
η
y
t
g
z
t
h
y
t
g
z
t
h
η
G
H
β
t
dt < 0
+
+
+
+
+
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)




u
u
u
u


∫I 
 dt 


Combining (38) and (39) and then integrating by parts with η = 0, at t = a and
t = b , we have
∫η ( λ
T
I
T
fu + y (t )T gu + z (t )T hu ) − D ( λ T fu + y (t )T gu + z (t )T hu ) + B β (t ) dt < 0,
contradicting the feasibility of ( u , λ , y , z , β ) for (MVED). Hence
=
x (t ) u (t ), t ∈ I .
I. Husain and Vikas K. Jain
243
5 Natural boundary values
It is possible to extend the duality theorems validated in the previous two
sections to the corresponding multiobjective variational problem with natural
boundary values rather than fixed end points. The proofs of the duality theorems
for these pairs of dual problems can be proved analogously to the proofs of the
theorems of the preceding sections except that some slight modifications are
needed.


(VEP)N: Minimize  ∫ f 1 (t , x, x )dt ,..., ∫ f p (t , x, x )dt 
I
I

subject to
g ( t , x, x ) < 0, t ∈ I
h ( t , x, x=
) 0, t ∈ I


(M-WED)N: Maximize  ∫ ( f 1 (t , u , u ) ) dt ,..., ∫ ( f p (t , u , u ) ) dt 
I
I

subject to
(
)
λ T fu + y (t )T gu + z (t )T hu − D λ T fu + y (t )T gu + z (t )T hu + Bβ (t ) =
0
1

∫  y(t ) g ( t , u, u ) − 2 β (t )

G β (t ) dt > 0

1

∫  z (t ) h ( t , u, u ) − 2 β (t )

H β (t ) dt > 0

T
T
I
T
I
T
λ > 0,
y (t ) > 0, t ∈ I
λ T fu + y (t )T gu + z (t )T hu =
0, at t = a and t = b .
244
On Vector valued Variational Problems
(MVED)N: Maximize
( ∫ ( f 1 (t , u , u ) + ∑ y j (t ) g j (t , u , u ) +
j∈I 0
I
∑z
k∈J 0
..., ∫ ( f p (t , u , u ) + ∑ y j (t ) g j (t , u , u ) +
j∈I 0
I
k
1
(t )h k (t , u , u ) − β (t )T A1 β (t ))dt ,
2
∑z
k∈J 0
k
1
(t )h k (t , u , u ) − β (t )T A p β (t ))dt )
2
subject to
(λ
T
fu + y (t )T gu + z (t )T hu ) − D ( λ T fu + y (t )T gu + z (t )T hu ) + B β (t ) =∈
0, t I

∑ ∫  y
j∈Iα I

j
1

(t ) g j (t , u , u ) − β (t )T G j β (t ) dt > 0, α =
1, 2,..., r
2

∑ ∫  z
k∈Jα I
k
1

(t )h k (t , u , u ) − β (t )T H k β (t ) dt > 0, α =
1, 2,..., r
2

p
λ > 0, ∑ λ i =
1
i =1
y (t ) > 0, t ∈ I
λ T fu + y (t )T gu + z (t )T hu =
0, at t = a and t = b .
6 Multiobjective non-linear programming problems
If all the function in the problems are independent of t, then we have
(VEP)0: Minimize
(f
1
( x),..., f p ( x) )
subject to
g ( x) < 0
h ( x) = 0
1
1


(M-WED)0: Maximize  f 1 (u ) − β (t )T ∇ 2 f 1 (u ),..., f p (u ) − β (t )T ∇ 2 f p (u ) 
2
2


I. Husain and Vikas K. Jain
245
subject to
0
λ T fu + yT gu + zT hu =
yT g ( u ) −
1 T 2 T
β ∇ y g (u ) β > 0
2
zT h ( u ) −
1 T 2 T
β ∇ z h (u ) β > 0
2
λ > 0, y > 0 .
(MVED)0: Maximize
1
( f 1 (u ) + ∑ y j g j (u ) + ∑ z k h k (u ) − β T A1β ,...,
2
j∈I 0
k∈J 0
1
f p (u ) + ∑ y j g j (u ) + ∑ z k h k (u ) − β T A p β )
2
j∈I 0
k∈J 0
subject to
λ T fu + yT gu + zT hu =
0

j

k
∑  y g
j
j∈Iα
1

(u ) − β T ∇ 2 y j g j (u ) β  > 0, α =
1, 2,..., r
2

1
∑  z h (u ) − 2 β
k
k∈Jα
T

1, 2,..., r
∇ 2 z k h k (u ) β  > 0, α =

p
λ > 0, ∑ λ i =
1
i =1
y (t ) > 0 .
where
Ai =
fuui + ∑ y j guuj (u ) +
j∈Iα
∑z
k∈Jα
k
huuk (u ), i ∈ K
These problems (M-WED)0 and (MVED)0 are not explicitly reported in the
literature. However, if β = 0 , these reduce to the problems treated by Weir [6].
246
On Vector valued Variational Problems
References
[1] O.L.
Mangasarian,
Second-and
higher–order
duality
in
nonlinear
programming, J. Math. Anal. Appl., 51, (1979), 607-620.
[2] X.Chen, Second order duality for the variational problems, J. Math. Anal.
Appl., 286, (2003), 261-270.
[3] 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.
[4] I. Husain and Vikas K. Jain, On Multiobjective Variational Problems with
equality and inequality constraints, Journal of Applied Mathematics and
Bioinformatics, 3(2), (2013), 45-64.
[5] O.L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.
[6] T. Weir, Proper Efficiency and Duality for Vector Valued Optimization
Problems, J. Austral. Math. Soc., (Series A), 43, (1987), 21-34.