Journal of Applied Mathematics & Bioinformatics, vol.3, no.2, 2013, 15-34
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2013
On Multiobjective Variational Problems with
equality and inequality constraints
I. Husain 1 and Vikas K. Jain 2
Abstract
Fritz John type optimality conditions for a multiobjective variational Problems
with equality and inequality constraints are derived. By an application of Karush
-Kuhn-Tucker type optimality conditions, a Wolfe type second-order dual to this
problem is formulated and various duality results are proved under generalized
second-order invexity assumptions. A pair of Wolfe type second-order dual
multiobjective with natural boundary values is also presented to investigate duality.
Finally, it is pointed out that our duality results established in this research can be
viewed as dynamic generalizations of those of static cases already existing in the
literature.
Mathematics Subject Classification: 90C30, 90C11, 90C20, 90C26
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 21, 2012. Revised : February 1, 2013
Published online : June 30, 2013
16
On Multiobjective Variational Problems
Keywords: Multiobjective Variational Problems, Wolfe Type Second-Order
Duality,
Second-Order
Generalized
invexity,
Multiobjective
Nonlinear
Programming Problems
1 Introduction
Second-order duality in mathematical programming has been widely
researched. As second-order dual to a constrained optimization problem gives a
tighter bound and hence enjoys computational advantage over the first-order dual
to the problem. Mangasarian [1] was the first to study second-order duality in
non-linear programming. Motivated with analysis of Mangasarian [1], Chen [2]
presented Wolfe type second-order dual to a class of constrained variational
problems under an involved invexity like conditions. Later Husain et al [3]
introduced second-order invexity and generalized invexity and presented a
Mond-Weir type second-order dual to the problem of [2] in order to relax implicit
invexity requirements to the generalized second-order invexity.
Multiobjective optimization (also known as multiobjective programming,
vector optimization, multicriteria optimization, multiattribute optimization, or
Pareto optimization) is an area of multiple criteria decision making. This deals
with optimization problems involving more than one objective functions to be
optimized simultaneously which appear more often than single objective
optimization problem to represent the models of real life problems.
Multiobjective optimization has applications in various fields of science that
includes engineering, economics and logistics where optimal decisions need to be
taken in the presence of trade-off between two or more conflicting objectives.
Minimizing weight while maximizing the strength of a particular component,
maximizing performance whilst minimizing fuel consumption and emission of
pollutants of a vehicle are examples of multiobjective optimization problems
I. Husain and Vikas K. Jain
17
involving two or three objectives respectively. In real-life problems, there can be
more than three objectives.
Motivated with the above cursory remarks relating to multiobjective
optimization problems in this paper we present a multiobjective version of the
variational problem considered by Chen [2] with equality and inequality
constraints which represent more realistic problems than those variational problem
with an inequality constraint only. For this multiobjective variational problem,
Fritz John type optimality conditions are obtained. Using Karush-Kuhn-Tucker
type optimality conditions, deduced from the Fritz John type optimality conditions
requiring a suitable regularity conditions. Wolfe type second-order multiobjective
variational to the variational problems is formulated and various duality theorems,
viz.
weak, strong, strict-converse and converse duality theorems are proved
under second-order pseudoinvexity and second-order strict pseudo-invexity. A
pair of Wolfe type second-order dual multiobjective variational problems with
natural boundary values is also constructed to study duality. Finally, it is indicated
that our duality theorems can be regarded as dynamic generalizations of the
second-order duality theorems is nonlinear programming, existing in the literature.
2 Pre-Requisites and expression of the problem
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 derivatives of φ with respect to
x ( t ) and x ( t ) , respectively, that is,
∂φ 
∂φ 
 ∂φ ∂φ
 ∂φ ∂φ
, 2 , ,
, φx  1 , 2 ,  ,
=
φx =

1
n 
∂x 
∂x n 
 ∂x ∂x
 ∂x ∂x
T
T
18
On Multiobjective Variational Problems
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
The symbols φx , φxx , φxx







m 
∂ψ 


∂x n m×n .
∂ψ 1
∂ψ 1
 n
∂x 2
∂x
2
∂ψ
∂ψ 2

∂x 2
∂x n


∂ψ m
∂x 2
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
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
h ( t , x, x=
) 0, t ∈ I
(1)
(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 ,
I. Husain and Vikas K. Jain
19
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
I
∫f
j
(t , xˆ , xˆ )dt < ∫ f i (t , x , x )dt ,
for some i ∈ {1, 2,..., p}
I
(t , xˆ , xˆ )dt <
I
∫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. We need
the following lemma due to Chankong and Haimes [4] for the proof of strong
duality result.
Lemma 2.1 If x is efficient solution of (VEP) if and only if x solves Pk ( x ) , for
all K , defined as
Pk ( x ) :
Minimize
∫f
k
(t , x, x )dt
I
subject to
=
x ( a ) α=
, x (b) β
g ( t , x, x ) < 0,
t∈I
h ( t=
, x, x ) 0,
t∈I
f j (t , xˆ , xˆ ) < f j (t , x , x ) , t ∈ I for all j ∈ {1, 2,..., p}, j ≠ k
We incorporate the following definitions which are required in the subsequent
analysis.
20
On Multiobjective Variational Problems
Definitions 2.2 The function
∫ φ (t , x, x )dt
where φ : I × R n × R n → R , is said to be
I
second order-pseudo invex with respect to η = η (t , x, u ) if
T
 T

 dη 
T
∫I η φu (t , u, u ) +  dt  φu (t , u, u ) + η Aβ (t ) dt > 0


1


⇒ ∫ φ (t , x, x )dt > ∫  φ (t , u , u ) − β T (t ) Aβ (t )  dt
2

I
I 
where A =
φxx (t , x, x ) − 2 Dφxx (t , x, x ) + D 2φxx  (t , x, x ) − D 3φxx (t , x, x ) , t ∈ I .
The functional
∫ φ (t , x, x )dt is said to strictly pseudoinvex with respect to
η if
I
T
 T

 dη 
T


+
+
η
φ
t
u
u
φ
t
u
u
η
A
β
t
(
,
,
)
(
,
,
)
(
)

dt > 0

u
u


∫I 
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
∫I  u
 dt 


Remark 2.1 If φ does not depend explicitly on t then the above definitions
reduce to those given in [5].
3 Necessary optimality conditions
In order to obtain the Fritz John type necessary optimality conditions, we
state the following variational problem treated by Chandra et al [6]:
(P): Minimize
∫ f (t , x, x )dt
I
I. Husain and Vikas K. Jain
21
subject to
=
x ( a ) α=
, x (b) β
g ( t , x, x ) < 0, t ∈ I
h ( t , x=
, x ) 0, t ∈ I
where f : I × R n × R n → R and g and h are the same as earlier.
The problem (P) may be written as
Minimize
F(x)
subject to
G ( x) ∈ S ,
H ( x) = 0 ,
where G : X → C ( I , R m ) giving G ( x)(t ) = g (t , x, x ) for all x ∈ X and t ∈ I and
C ( I , R m ) denotes the space of continuous function from an interval I into R m , S
is the convex cone of function in C ( I , R m ) where components are non-negative.
The function H : X → C ( I , R l ) is defined by
=
H ( x)(t ) h(t , x, x ) for x ∈ X , t ∈ I .
The following theorem gives the Fritz John type necessary optimality conditions
for the single objective variational problems derived in [6].
Theorem 3.1 (Fritz John optimality conditions) If x is an optimal solution of
(P) and hx (., x(.), x (.)) maps onto a closed subspace of C ( I , R l ) , then there exist
Lagrange multipliers τ ∈ R and piecewise smooth y : I → R m , z : I → R l such
that
τ 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 (τ 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
22
On Multiobjective Variational Problems
y (t )T g (t , x (t ),=
x (t )) 0,
t∈I
(τ , y (t ) ) > 0,
(τ , y (t ), z (t ) ) ≠ 0,
t∈I
t∈I
Remark 3.1 The above Fritz John necessary optimality conditions become
Karush-Kuhn-Tucker John optimality condition if τ = 1 (then x may be called
normal). It is sufficient for τ = 1 that the Zowe’s form of Slater condition [6] is
assumed, that is, there exist q ∈ X ,
G ( x ) + G′( x )q ∈ int S
H ′( x )q = 0
The Karush-Kuhn-Tucker type necessary optimality conditions for (P) can
explicitly be given in the following theorem:
Theorem 3.2 (Karush-Kuhn-Tucker type necessary optimality conditions) If
x is an normal and optimal solution of (VEP) and hx (., x(.), x (.)) maps onto a
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
In this section, we will establish the following theorem giving the Fritz John type
necessary optimality conditions for (VEP):
Theorem 3.3 (Fritz John type necessary optimality conditions) Let x be an
efficient solution of (VEP). Then there exist λ i ∈ R, i ∈ K and piecewise
I. Husain and Vikas K. Jain
23
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
Proof: Since x is an efficient solution of (VEP), by Lemma 2.1, x is an
optimal
( Pk )
for each p ∈ K and in particular of
( P1 ) .
Therefore, by Theorem
3.1, there exist λ i ∈ R, i ∈ K
(λ
T
f x − Dλ T f x ) + ( 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
which validates the theorem.
4 Wolfe type Duality
In this section, we present the following Wolfe type second- order dual to the
problem (VEP) and prove various duality theorems
(WVED):
1
Maximize ( ∫ ( f 1 (t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x ) − β (t )T H 1β (t ))dt ,
2
I
1
..., ∫ ( f p (t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x ) − β (t )T H p β (t ))dt )
2
I
subject to
=
x ( a ) α=
, x (b) β
(3)
24
On Multiobjective Variational Problems
0
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 ) + H β (t ) =
(4)
y (t ) > 0
(5)
λ > 0, λ T e= 1, where e= (1,1,...,1) ∈ R n
(6)
where
Hi =
f xxi − 2 Df xxi + D 2 f xxi − D 3 f xxi + ( y (t )T g x + z (t )T hx ) − 2 D ( y (t )T g x + z (t )T hx ) 
+D
2
( y(t )
T
g x + z (t ) hx )  − D
T
x
3
( y(t )
T
x
x
g x + z (t ) hx ) 
T
x
H=
λ T f xx − 2 Dλ T f xx + D 2 λ T f xx  − D 3λ T f xx + ( y (t )T g x + z (t )T hx )
− 2 D ( y (t ) g x + z (t ) hx )  + D
T
T
x
2
( y(t )
T
g x + z (t ) hx )  − D
T
x
3
x
( y(t )
T
g x + z (t )T hx )  ,
x
Ai =
f xxi − 2 Df xxi + D 2 f xxi − D 3 f xxi .
and
Theorem 4.1 (Weak Duality) Assume that for all feasible x for (VEP) and all
feasible
( u, y, z, λ , β ) for (WVED).
∫{ f
i
(t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x )}dt
I
is second-order pseudoinvex with respect to η . Then the following cannot hold:
∫f
I
i
1


(t , x, x )dt < ∫  f i (t , u , u ) + y (t )T g (t , u , u ) + z (t )T h(t , u , u ) − β (t )T H i β (t ) dt
2

I 
for some i ∈ {1, 2,..., p} .
∫f
I
j
(7)
1


(t , x, x )dt < ∫  f j (t , u , u ) + y (t )T g (t , u , u ) + z (t )T h(t , u , u ) − β (t )T H j β (t ) dt
2

I 
for all j ∈ {1, 2,..., p} .
(8)
Proof: Suppose contrary to the result that (7) and (8) hold. Since x is feasible for
(VEP) and
( u, y, z , λ , β )
y (t ) > 0 , (7) and (8) that
is feasible for (WVED), it follows from g ( t , x, x ) < 0,
I. Husain and Vikas K. Jain
∫{ f
i
25
(t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x )}dt
I
1


< ∫  f i (t , u , u ) + y (t )T g (t , u , u ) + z (t )T h(t , u , u ) − β (t )T H i β (t ) dt
2

I 
for some i ∈ {1, 2,..., p} .
∫{ f
j
(9)
(t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x )}dt
I
1


< ∫  f j (t , u , u ) + y (t )T g (t , u , u ) + z (t )T h(t , u , u ) − β (t )T H j β (t ) dt
2

I 
for all j ∈ {1, 2,..., p} , with j ≠ i .
(10)
Using the second-order pseudoinvexity of
∫{ f
i
(t ,.,.) + y (t )T g (t ,.,.) + z (t )T h(t ,.,.)}dt from the relations (9) and (10), we get
I
∫ {η
T
(
(t , x, u ) fui (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u )
)
I
 dη 
+

 dt 
T
(f
i
) +
u (t , u , u
)
1
y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) − η T H i β (t )}dt < 0
2
for all i ∈ {1, 2,..., p}
(11)
multiplying each inequality of (11) by λ i > 0, i =
1, 2,..., p and adding , we have
(
0 > ∫ {η T (t , x, u ) λ T fui (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u )
)
I
 dη 
+

 dt 
T
∫η
=
T
(λ
T
)
fui (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) + η T H β (t )}dt
(t , x, u ){ ( λ T fui (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 ) ) + H β (t )}dt
+η T ( λ T fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) )
t =b
t=a
26
On Multiobjective Variational Problems
∫η
T
(t , x, u ){ ( λ T fui (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 ) ) + H β (t )}dt < 0
( u, y, z , λ , β )
This contradicts the feasibility of
for (WVED). Hence the result
follows.
Corollary 4.1 Assume that the weak duality (Theorem 4.1) holds between (VEP)
and (WVED). If x is feasible for (VEP) and ( x , y , z , λ , β ) is feasible for
(WVED) with y (t )T g ( t , x , x=
t ) 0, t ∈ I . Then x is efficient for (VEP)
) 0, β (=
and ( x , y , z , λ , β ) is efficient for (WVED).
Proof: Suppose x is not efficient for (VEP). Then there exist some feasible x̂
for (VEP) such that
∫f
i
(t , xˆ , xˆ )dt <
I
∫f
i
(t , x , x )dt , for some i ∈ {1, 2,..., p}
I
∫f
I
j
(t , xˆ , xˆ )dt <
∫f
j
(t , x , x )dt , for all j ∈ {1, 2,..., p}
I
Since y (t )T g ( t , x , x=
t ) 0, t ∈ I
) 0, z (t )T h ( t , x , x=) 0, t ∈ I and β (=
1


(t , x , x )dt < ∫  f i (t , x , x ) + y (t )T g (t , x , x ) + z (t )T h(t , x , x ) − β (t )T H i β (t ) dt
2

I
I 
for some i ∈ {1, 2,..., p} .
∫f
i
1


(t , x , x )dt < ∫  f j (t , x , x ) + y (t )T g (t , x , x ) + z (t )T h(t , x , x ) − β (t )T H j β (t ) dt
2

I
I 
for all j ∈ {1, 2,..., p} .
∫f
j
This contradicts weak duality. Hence x is efficient for (VEP).
Now suppose that ( x , λ , y , z , β ) is not efficient for (WVED) such that
I. Husain and Vikas K. Jain

∫  f
i
I
27
1

(t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x ) − β (t )T H i β (t ) dt
2

1


< ∫  f i (t , x , x ) + y (t )T g (t , x , x ) + z (t )T h(t , x , x ) − β (t )T H i β (t ) dt
2

I 
for some i ∈ {1, 2,..., p}

∫  f
I
j
1

(t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x ) − β (t )T H j β (t ) dt
2

1


< ∫  f j (t , x , x ) + y (t )T g (t , x , x ) + z (t )T h(t , x , x ) − β (t )T H j β (t ) dt
2

I 
for all j ∈ {1, 2,..., p} , j ≠ i .
 ) 0, z (t )T h ( t , x , x=
 ) 0, β (=
Since y (t )T g ( t , x , x=
t ) 0, t ∈ I , we have

∫  f
i
I
1

(t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x ) − β (t )T H i β (t ) dt <
2

∫f
i
(t , x , x )dt
I
for some i ∈ {1, 2,..., p} .

∫  f
I
j
1

(t , x, x ) + y (t )T g (t , x, x ) + z (t )T h(t , x, x ) − β (t )T H j β (t ) dt <
2

∫f
j
(t , x , x )dt
I
for all j ∈ {1, 2,..., p} .
This contradicts weak duality. Hence
( x , λ , y , z , β (t ) ) is efficient for (WVED).
Theorem 4.2 (Strong Duality) Let x be an efficient solution of (VEP) and x
satisfy the constraint qualification of Theorem 3.1 for Pk ( x ) , for at least one
k ∈ {1, 2,..., p} . Then there exist λ ∈ R k and piecewise smooth y : I → R m and
z : I → R l such
that
( x , λ , y , z , β = 0)
is
feasible
for
(WVED)
and
y (t )T g ( t , x , =
x ) 0 , t ∈ I . If the weak duality also holds between (VEP) and
(WVED) then
( x , λ , y , z , β = 0)
is efficient for (WVED).
Proof: As x satisfies the constraint qualification of Theorem 3.2 for Pk ( x ) , for
at least one k ∈ {1, 2,..., p} , it follows from Theorem 3.2 that there exist λ ′ ∈ R k −1
28
On Multiobjective Variational Problems
and piecewise smooth y′ : I → R m and z ′ : I → R l such that for all t ∈ I
f xi (t , x , x ) − Df xi (t , x , x ) + ∑ λ ′i ( f xi (t , x , x ) − Df xi (t , x , x ) )
p
i =1
i≠k
+ y′(t )T g x (t , x , x ) − Dy′(t )T g x (t , x , x ) + z ′(t )T hx (t , x , x ) − Dz ′(t )T hx (t , x , x ) =
0, t ∈ I
y′(t )T g x (t , x , =
x ) 0, t ∈ I
y′(t )T > 0, t ∈ I
λ ′i >=
0, i 1, 2,..., p, i ≠ k
Setting




p
p
i
k
i


λ=
λ ′ 1 + ∑ λ ′ , λ =+
1 1 ∑ λ′
 i 1=

 i1 
=
 i≠k

 i≠k





p
p
y (t ) =
y′(t ) 1 + ∑ λ ′i  , z (t ) =
z ′(t ) 1 + ∑ λ ′i 
 i 1=

 i1 
=
 i≠k

 i≠k

i


p
i

′
and dividing by 1 + ∑ λ , we finally get
 i =1 
 i≠k

λ T f x (t , x , 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
p
Also we get
∑λ
i
= 1 and z (t )T h(t , x , x=
) 0, t ∈ I . It implies that
i =1
( x , λ , y , z , β = 0 ) is feasible for (WVED).
The weak duality holds along y (t )T g (t , x , x ) = 0 and β (=
t ) 0, t ∈ I . This by
corollary 4.1 implies that
( x , λ , y , z , β = 0 ) is efficient for (WVED).
Theorem 4.3 (Strict- converse Duality) Let x be efficient and normal solution
of (VEP) and
( u, y, z , λ , β )
be efficient for (WVED) such that
I. Husain and Vikas K. Jain
p
∫∑λ
i
29
f i (t , x , x )dt
I i =1
p
1
= ∫ {∑ λ f (t , u , u ) + y (t ) g (t , u , u ) + z (t ) h(t , u , u ) − β (t )T H β (t )}dt
2
I i =1
i
If
∫ (λ
T
(12)
T
f x + y (t )T g + z (t )T h ) dt is second-order strictly pseudoinvex with respect
T
I
to η , then
=
x (t ) u (t ) , t ∈ I
Proof: By second-order strict pseudoinvexity of
∫ (λ
T
f (t ,.,.) + y (t )T g (t ,.,.) + z (t )T h(t ,.,.) ) dt , (12) implies
I
0 > ∫ {η T ( λ T fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) )
I

fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) ) + η T H β  dt

= ∫ η T { ( λ T fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) )
 dη 
+

 dt 
T
I
(λ
T
− D ( λ T fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) ) + H β (t )}dt
t =b
+ η T ( λ T fu (t , u , u ) + y (t )T gu (t , u , u ) + z (t )T hu (t , u , u ) )
t=a
Using η = 0, at=
t a=
, t b , we have
∫η { ( λ
T
T
fui (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 ) ) + H β (t )}dt < 0
=
x (t ) u (t ) , t ∈ I .
This contradicts the equality constraint of (WVED).Hence
Theorem 4.4 (Converse duality) Let
solution of (WVED) for which
(C1): H is non-singular
( x (t ), y (t ), z (t ), λ , β (t ) ) be
an efficient
30
On Multiobjective Variational Problems
( C2 ) : (σ (t )T H iσ (t ) ) x − D (σ (t )T H iσ (t ) ) x
+ D 2 (σ (t )T H iσ (t ) )  − D 3 (σ (t )T H iσ (t ) ) + D 4 (σ (t )T H iσ (t ) ) 
x
x
x 
+ 2 σ (t )T ( H σ (t ) ) x − σ (t )T D ( H σ (t ) ) x + σ (t )T D 2 ( H σ (t ) ) x
−σ (t )T D 3 ( H σ (t ) )x + σ (t )T D 4 ( H σ (t ) )x  =
0 ⇒ σ (t ) =
0,
where σ (t ) is a vector function.
Then x (t ) is feasible for (VEP) and the two objective functionals have the same
value. Also, if the weak duality theorem holds for all feasible solutions of (VEP)
and (WVED), then x (t ) is efficient for (VEP).
( x , y, z, λ , β (t ) ) is
Proof: Since
an efficient solution of (WVED), there exists
α , ξ ∈ R p , δ ∈ R and piecewise smooth
θ : I → R n , η : I → R m such that
following Fritz John conditions (Theorem (3.3)) are satisfied at ( x , y, z , λ , β (t ) ) :

∑ α ( f
p
i
i
i =1
+ yT g x + z T hx ) − D ( f i + yT g x + z T hx ) −
1
β (t )T H i β (t ) ) x
(
2
1
1
+ D ( β (t )T H i β (t ) )  − D 2 ( β (t )T H i β (t ) )  +
x
x
2
2
1
1
+ D 3 ( β (t )T H i β (t ) ) − D 4 ( β (t )T H i β (t ) ) ]
x
x
2
2
+θ (t )T  H + ( H β ) x − D ( H β ) x + D 2 ( H β ) x − D 3 ( H β )x + D 4 ( H β )x  =
0 (13)
(α e )  g
T
j
1

− β (t )T g xxj β (t ) 
2

+θ (t )T  g xj − Dg xj + ( g xxi − 2 Dg xxi  + D 2 g xxi  − D 3 g xxi ) β (t )  + η i (t ) =
0 (14)
(α e ) h
T
j
1

− β (t )T hxxj β (t ) 
2

0
+ θ (t )T  hxj − Dhxj + ( hxxi − 2 Dhxxi  + D 2 hxxi  − D 3 hxxi ) β (t )  =
(θ (t ) − (α e ) β )
T
T
(15)
H=
0
(16)
θ (t )T ( f xi − Df xi − Ai β (t ) ) + ξ i + δ i =
0
(17)
I. Husain and Vikas K. Jain
31
η (t )T y=
(t ) 0, t ∈ I
ξTλ = 0
(18)
(19)
 p

δ  ∑ λ i − 1 =
0
 i =1

(20)
(α ,η , ξ , δ ) > 0
(21)
(α , θ ,η , ξ , δ ) ≠ 0
(22)
Since λ > 0 , (19) implies ξ = 0 .
Since H is non singular, (16) implies
=
θ (t )
(α e ) β (t ) , t ∈ I
T
(23)
Using the equality constraint of the dual and (23) , we
1
1
1


H β (t )T − ( β (t )T H i β (t ) ) + D ( β (t )T H i β (t ) )  − D 2 ( β (t )T H i β (t ) )  

x
x
x
2
2
2
−∑ α i 

1
1
 + D 3 ( β (t )T H i β (t ) ) − D 4 ( β (t )T H i β (t ) )



x
x
 2

2
 H β (t )T + β (t )T ( H β ) x − β (t )T D ( H β ) x + β (t )T D 2 ( H β ) x 
T
=
+ (α e ) 
0
T
T
3
4
 − β (t ) D ( H β )x + β (t ) D ( H β )x

Consequently, we obtain
( β (t )T H i β (t ) ) − D ( β (t )T H i β (t ) ) + D 2 ( β (t )T H i β (t ) ) 

x
x
x


T
i
T
i
3
4
 − D ( β (t ) H β (t ) ) + D ( β (t ) H β (t ) )



x
x


T
T
T
2
 β (t ) ( H β ) x − β (t ) D ( H β ) x + β (t ) D ( H β ) x 
+ 2
0
=
T
T
3
4
 − β (t ) D ( H β )x + β (t ) D ( H β )x

β (t ) 0 , t ∈ I ,
This, because of the hypothesis (C2) implies =
β (t ) 0 , t ∈ I in (23), we have
Using =
=
θ (t ) 0 , t ∈ I .
β (t ) 0 , t ∈ I . From (14), we have
Using θ (t ) = 0 and =
(α e ) g
T
j
+ η i (t ) =
0
(24)
32
On Multiobjective Variational Problems
Let α = 0 , then (14) and (17) respectively give δ i = 0 and η i = 0 .
Consequently (α ,θ ,η ,ξ ,δ ) = 0 leading to a contradiction to (22). Hence α > 0 .
From (24), we have
η i (t )
< 0, t ∈ I
(α T e )
gj =
−
(25)
g ( t , x , x ) < 0, t ∈ I
or
the relations (24) and (15) respectively imply that
y (t )T g ( t , x , x=
) 0, t ∈ I
and
h(t , x, x ) = 0
0 , t ∈ I imply that x is feasible for (VEP).
Also g (t , x , x ) < 0 and h(t , x , x ) =
Now, we have
∫( f
i
(t , x , x ) + y (t )T g (t , x , x ) + z (t )T h(t , x , x ) ) dt =
I
∫f
i
(t , x , x ) dt , i ∈ K ,
I
implying the equality of objective values. By Corollary 4.1 the efficiency of x
for (VEP) follows.
5 Multiobjective variational problems with natural boundary
values
The following is a pair of Wolfe type second- order Multiobjective
variational problems with natural values:


(VEP)N: Minimize  ∫ f 1 (t , x, x )dt ,..., ∫ f p (t , x, x )dt 
I
I

subject to
(WVED)N: Maximize
g ( t , x, x ) < 0,
t∈I
h ( t=
, x, x ) 0,
t∈I
I. Husain and Vikas K. Jain
33
  1
1


T
T
T
1
 ∫  f (t , x, x ) + y (t ) g (t , x, x ) + z (t ) h(t , x, x ) − 2 β (t ) H β (t )  dt ,..., 

I

  p

1

T
T
T
p
 ∫  f (t , x, x ) + y (t ) g (t , x, x ) + z (t ) h(t , x, x ) − β (t ) H β (t )  dt 
2

I

subject to
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 ) + H β (t ) =
y (t ) > 0 , t ∈ I
λ > 0, λ T e= 1, where e= (1,1,...,1) ∈ R k
at t a=
λ T fu (t , u, u ) + y (t )T gu (t , u, u ) + z (t )T hu (t , u, u ) =
0,=
and t b
The theorems established in the proceeding sections can easily be proved for the
above pair of problems.
6 Wolfe type second–order Multiobjective nonlinear
programming problems
If the problems (VEP) and WVED) are independent of t, i.e. if f, g and h do
not depend explicitly on t, then these problems essentially reduce to the static
cases of nonlinear programming studied in [7], namely
(VEP)0: Minimize
(f
1
( x), f 2 ( x),..., f p ( x) )
subject to g ( x) < 0 , h( x) =
0
(WVED)0: Maximize
1 T 2 1
 1

T
T
T
T
 f (u ) + y g (u ) + z h(u ) − 2 β ∇ ( f (u ) + y g (u ) + z h(u ) ) β ,..., 


 f p (u ) + yT g (u ) + z T h(u ) − 1 β T ∇ 2 f p (u ) + yT g (u ) + z T h(u ) β 
(
) 

2

34
On Multiobjective Variational Problems
subject to
0
λ T fu (u ) + yT gu (u ) + z T hu (u ) + ∇ 2 ( λ T fu (u ) + yT gu (u ) + z T hu (u ) ) β =
y>0
λ > 0, λ T e =
1
Acknowledgements. The authors acknowledge the valuable comments of the
anonymous referees which have substantially improved the presentation of this
research work.
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