İstanbul Üniversitesi İşletme Fakültesi Dergisi
Istanbul University Journal of the School of Business Administration
Cilt/Vol:39, Sayı/No:1, 2010, 1-20
ISSN: 1303-1732 – www.ifdergisi.org © 2010
Symmetric duality for multiobjective variational problems containing
support functions
Iqbal Husain 1
Department of Mathematics,
Jaypee Institute of Engineering and Technology
Guna, MP, India
Rumana G. Mattoo 2
Department of Statistics,
University of Kashmir
Srinagar, Kashmir, India
Abstract
Wolfe and Mond-Weir type symmetric dual models for multiobjective variational problems
with support functions are formulated. For these pairs of problems, weak, strong and
converse d uality t heorems a re v alidated u nder convexity-concavity a nd p seudoconvexity, p seudo-concavity a ssumptions o n certain c ombination o f f unctionals. S elf
duality theorems for both pairs are established. The problems with natural boundary
values are f ormulated. It i s also pointed out that our duality results can b e regarded as
dynamic g eneralizations o f n onlinear p rogramming p roblems h aving n ondifferentiable
terms as support functions.
Keywords: Multiobjective, variational problems, support functions, symmetric duality, self duality,
convexity-concavity, pseudoconvexity-pseudoconcavity.
Destek fonksiyonları içeren varyasyonel problemler için simetrik dualite
Özet
Destek fonksiyonlu çok amaçlı varyasyonel problemler için Wolfe ve Mond-Weir t ipi
simetrik dual modeller formüle edilmiştir. Bu çeşit problemler için; zayıf, güçlü ve karşıt
dualite teoremleri fonksiyonellerin belirli kombinasyonları üzerine konvekslik-konkavlık ve
pseudo-konvekslik, pseudo-konkavlık varsayımları altında geçerli kılınmıştır. İki çift için
de öz dualite teoremleri kurulmuştur. Doğal sınır değerli problemler formüle edilmiştir.
Ayrıca dualite sonuçlarımızın, destek fonksiyoları gibi diferansiyellenemeyen ifadelere
sahip olan nonlineer programlama problemlerinin dinamik genelleştirmeleri olarak kabul
edilebileceğine dikkat çekilmektedir.
Anahtar Sözcükler: Çok amaçlı, varyasyonel problemler, destek fonksiyonlar, simetrik dualite, öz
dualite, konvekslik-konkavlık, pseudokonvekslik- pseudokonkavlık.
1. Introduction
Following D orn [ 1], s ymmetric d uality r esults i n m athematical p rogramming h ave been
derived by a number of authors, notably, Dantzig et al. [2], Mond [3], Bazaraa and
Goode [ 4]. In t hese researches, t he a uthors have s tudied s ymmetric d uality u nder t he
hypothesis of convexity-concavity of the kernel function involved. Mond and Cottle [5]
presented self duality for the problems of [2] by assuming skew symmetric of the kernel
function. Later Mond-Weir [6] formulated a different pair of symmetric dual nonlinear
program with a view to generalize convexity-concavity of the kernel function to
pseudoconvexity-pseudoconcavity.
1
[email protected] (I. Husain)
2
[email protected] (Rumana G.Mattoo)
1
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
Symmetric duality for variational problems was first introduced by Mond and Hanson [7]
under
the
convexity-concavity
conditions o
f
a
scalar
functions
like
ψ ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) )
with
x ( t ) ∈ R n and y ( t ) ∈ R m . Bector, Chandra and Husain [8]
presented a different pair of symmetric dual variational problems in order to relax the
requirement of convexity-concavity to that of pseudoconvexity-pseudoconcavity. In [9]
Chandra and Husain gave a fractional analogue.
Bector a nd Husain [ 10] p robably w ere th e f irst t o s tudy d uality f or multiobjective
variational p roblems un der convexity a ssumptions. S ubsequently, G ulati, H usain a nd
Ahmed [11] presented two distinct pairs of symmetric dual multiobjective variational
problems and established various duality results under appropriate invexity
requirements. In this reference, self duality theorem is also given under skew symmetric
of the integrand of the objective functional.
The purpose of this research is to present Wolfe and Mond-Weir type symmetric dual
pairs o f m ultiobjective v ariational p roblems containing s upport f unctions i n o rder to
extend the results of Gulati, Husain and Ahmed [11] to nondifferentiable cases and hence
study sy mmetric a nd self d uality f or t hese pairs of n ondifferentiable m ultiobjective
variational p roblems. T he p roblems, t reated i n t his re search a re q uite h ard t o s olve. S o
to e xpect a ny i mmediate a pplication o f th ese p roblems w ould b e f ar f rom r eality.
Unfortunately, t here h as n ot a lways b een sufficient flow b etween t he researchers i n t he
multiple criteria decision making and the researchers applying it to their problems. Of
course, one can find optimal control applications in galore which reflect the utility of our
models. Further, the problems with natural boundary values are formulated and it is also
pointed o ut th at o ur r esults c an b e c onsidered a s d ynamic g eneralizations o f
corresponding (static) symmetric duality results of multiobjective nonlinear programming
problems i nvolving s upport f unctions. The re duced n ondiferentiable m ultiobjective
nonlinear p roblems a re n ot explicitly m entioned i n t he l iterature. The d uality re sults f or
them are immediate.
2. Notations And Preliminaries
The following notation will be used for vectors in R .
n
x < y,
⇔
xi < yi ,
i=
1, 2, , n.
x < y,
⇔
xi < yi ,
1, 2, , n.
i=
x ≤ y,
⇔
xi ≤ y=
i 1, 2, , n, but x ≠ y
i,
x ≤ y, is the negation of x ≤ y
Let
I = [ a, b ] be the real interval, and φ i ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) ) , i = 1, 2, , p be a scalar
x : I → R n and y : I → R n with derivatives
x and y . In order t o consider each φ i ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) ) denote the first partial
function and twice differentiable function where
derivatives of
φ i with re
spect t o
t , x ( t ) , x ( t ) , y ( t ) , y ( t ) respectively, b y φti , φxi , φxi , φ yi , φ yi ,
that is,
φti =
2
∂φ i
∂t
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
The t
wice p
φxi
 ∂φ i ∂φ i
 ∂φ i ∂φ i
∂φ i 
∂φ i 
i
=
,
,
,
,
,
,
,
φ




 

x
∂xn 
∂xn 
 ∂x1 ∂x2
 ∂x1 ∂x2
φ yi
 ∂φ i ∂φ i
 ∂φ i i ∂φ i
∂φ i 
∂φ i 
i
=
φ


,
,
,
,
,
,
,


 
.
y
∂yn 
∂y n 
 ∂y1 ∂y2
 ∂y1 ∂y 2
artial
derivatives
of φ ,
i = 1, 2, , p
i
with r
t, x (t ) ,
espect to
x ( t ) , y ( t ) and y ( t ) , respectively are the matrices
 ∂ 2φ i 
 ∂ 2φ i 
 ∂ 2φ i 
i
i
φxxi  =
φ
φ
=
,
=
,





xx
xy
 ∂xk xs  n×n
 ∂xk xs  n×n
 ∂xk ys  n×n
 ∂ 2φ i 
 ∂ 2φ i 
 ∂ 2φ i 
i
i
φxyi   =
φ
φ
=
,
=
,






xy
xy
 ∂xk y s  n×n
 ∂xk ys  n×n
 ∂xk y s  n×n
 ∂ 2φ i 
 ∂ 2φ i 
 ∂ 2φ i 
i
i
φ yyi  =
φ
φ
=
,
=
,






yy
yy
 ∂yk ys  n×n
 ∂yk y s  n×n
 ∂y k y s  n×n
Noting that
d i
x
φ y =φ yti + φ yyi y + φ yyi  y + φ yxi x + φ yxi  
dt
and hence
∂ d
d
φ y = φ yy ,
∂y dt
dt
∂ d
d
=
φ y
φ yy  + φ yy ,
∂y dt
dt
d d i
φ y = φ yyi

dy dt
d
∂ d
φ y = φ yx ,
dt
∂x dt
d
∂ d
=
φ y
φ yx  + φ yx ,
dt
∂x dt
∂ d
φ y = φ yx 
x dt
∂
In order to establish our main results, the following concepts are needed.
Definition 1. (Support f unction): Let
function of K is defined by
{
K be a c ompact s et i n R n , th en the s upport
=
s ( x ( t ) K ) max x ( t ) v ( t ) : v ( t ) ∈ K , t ∈ I
T
}
A support function, being convex everywhere finite, has a subdifferential in the sense of
convex analysis i.e., there exist z ( t ) ∈ R
n
, t ∈ I , such that
s ( y (t ) C ) − s ( x(t ) C ) ≥ ( y (t ) − x(t ) ) z ( t )
T
From [12], subdifferential of
s ( x ( t ) K ) is given by
∂s ( x ( t ) K ) =
For any set
{z (t ) ∈ K , t ∈ I such that x (t )
T
}
z (t ) = s ( x (t ) K ) .
Γ ⊂ R n , the normal cone to Γ at a point x ( t ) ∈ Γ is defined by
3
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
N Γ ( x=
(t ) )
{ y(t ) ∈ R
n
}
y (t ) ( z ( t ) − x ( t ) ) ≤ 0, ∀z ( t ) ∈ Γ
It can be verified that for a compact convex set K,
y (t ) ∈ N K ( x(t ) ) if and only if
=
s ( y (t ) K ) x ( t ) y (t ) , t ∈ I
T
Definition 2. (Skew Symmetric function): The function h : I × R
said to be skew symmetric if for all x and y in the domain of h if
n
× R n × R n × R n → R is
h ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) ) =
−h ( t , y ( t ) , y ( t ) , x ( t ) , x ( t ) ) , t ∈ I
where
x and y ( piecewise smooth are on I ) are of the same dimension.
Now consider the following multiobjective variational problem (VPo):
(VP 0 )
Minimize
∫ F ( t , x, x ) dt
I
Subject to
=
x ( a ) α=
,
x (b) β
g ( t , x, x ) < 0 , t ∈ I ,
where
F : I × R n × R n → R p and g : I × R n × R n → R m .
Definition 3. (Efficient Solution): A feasible solution
x is efficient for (VPo) if there
exists no other feasible x for (VP) such that for some i ∈ P =
{1, 2,..., p} ,
∫ F ( t , x, x ) dt < ∫ F ( t , x , x ) dt
i
i
I
for all
i ∈ P.
I
and
∫ F ( t , x, x ) dt < ∫ F ( t , x , x ) dt
j
j
I
for all
j∈P, j ≠ i.
I
3. Wolfe Type Symmetric Duality
We present the following pair of Wolfe type symmetric dual multiobjective variational
problems containing support functions:
∫(H , H
(SWP): Minimize:
1
2
,..., H p ) dt
I
Subject to:
x ( a )= 0= x ( b )
(1)
y ( a )= 0= y ( b )
(2)
∑ λ ( f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y ) ) < 0 , t ∈ I
p
i
i =1
4
i
y
i
i
y
(3)
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
zi (t ) ∈ C i , i =
1,..., p ,
t∈I
(4)
x(t )> 0 , t ∈ I
{λ ∈ R
+
λ ∈ Λ=
(5)
λ > 0, λ T e= 1, e=
p
(1,1, ,1)
T
∈ Rp
}
(6)
where,
i
1. H
=
f i ( t , x, x , y, y ) + s ( x(t ) C i )
∑ λ ( f ( t , x, x, y, y ) + z ( t ) − Df ( t , x, x, y, y ) ) − y ( t ) z ( t )
p
− y (t )
T
i
i
y
i =1
2.
i
T
i
y
f i : I × R n × R n → R , (i =
1, 2, , p) , is continuously differentiable function.
(SWD): Maximize:
∫ (G , G
1
2
)
,..., G p dt
I
Subject to:
u ( a )= 0= u ( b )
7)
v ( a )= 0= v ( b )
(8)
∑ λ ( f ( t , u, u, v, v ) + ω ( t ) − Df ( t , u, u, v, v ) ) > 0 , t ∈ I
(9)
ωi (t ) ∈ K i , i =
1,..., p
(10)
p
i
i
u
i =1
i
i
u
,
t∈I
v (t ) > 0 , t ∈ I
(11)
λ ∈ Λ+
(12)
where,
(
=
G i f i ( t , u , u , v, v ) + s v ( t ) K i
− u (t )
T
)
∑ λ ( f ( t , u, u, v, v ) + ω ( t ) − Df ( t , u, u, v, v ) ) − x ( t ) ω ( t )
p
i
i =1
i
u
i
T
i
u
We present various duality results under convexity-concavity assumption.
Theorem 1 (Weak D uality): L et
( u(t ), v(t ), ω (t ),..., ω (t ), λ )
∫ f ( t ,.,., y, y ) dt is convex
1
p
i
( x(t ), y(t ), z (t ),..., z
1
p
(t ), λ ) be f easible f or t he ( SWP) a nd
be feasible for the dual (SWD). Assume that for each i,
( x, x ) for
in
fixed
I
( y, y ) and ∫ f i ( t , x, x,.,.) dt is
concave in
I
( y, y ) for fixed ( x, x ) . Then,
∫ Hdt ≤ ∫ Gdt
I
I
5
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
where,
H = ( H 1 , H 2 ,..., H p )
and
G = ( G1 , G 2 ,..., G p ) .
∫ f ( t ,.,., y, y ) dt
Proof: Using the convexity of
i
in
( x, x ) for fixed ( y, y ) , we have
I
∫ f ( t , x, x, v, v ) d t− ∫ f ( t , u, u, v, v ) d t
i
i
I
I
T
≥ ∫ ( x ( t ) − u ( t ) ) fui ( t , u , u , v, v ) − ( x ( t ) − u ( t ) ) f ui ( t , u , u , v, v )  d t


I
T
=
∫I ( x ( t ) − u ( t ) )
T
{ f ( t , u, u, v, v ) − f ( t , u, u, v, v )} d t
i
u
i
u
+ ( x ( t ) − u ( t ) ) f ui ( t , u , u , v, v )
T
t =a
t =b
.
This on using (1) and (7) gives,
∫ f ( t , x, x, v, v ) d t− ∫ f ( t , u, u, v, v ) d t
i
i
I
I
T
> ∫ ( x ( t ) − u ( t ) ) { f ui ( t , u , u , v, v ) − f ui ( t , u , u , v, v )} d t.


I
Also by concavity of
(13)
∫ f ( t , x, x,.,.) dt , we have
i
I
− ∫ f i ( t , x, x , v, v ) dt − ∫ f i ( t , x, x , y, y ) dt
I
I
T
> − ∫ ( v ( t ) − y ( t ) ) f yi ( t , x, x , y, y ) − ( v ( t ) − y ( t ) ) f yi ( t , x, x , y, y )  d t


I
T
T
=
− ∫ ( v ( t ) − y ( t ) ) { f yi ( t , x, x , y, y ) − f yi ( t , x, x , y, y )} d t


I
+ ( v ( t ) − y ( t ) ) f yi ( t , x, x , y, y )
T
t =a
t =b
which by using (2) and (8) we have,
− ∫ f i ( t , x, x , v, v ) dt − ∫ f i ( t , x, x , y, y ) dt
I
I
> − ∫ ( v ( t ) − y ( t ) )

I
T
{ f ( t , x, x, y, y ) − f ( t , x, x, y, y )} d t
i
y
Adding (13) and (14) we have,
6
i
y
(14)
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
∫ f ( t , x, x, y, y ) d −t∫ f ( t , u, u, v, v ) d ≥t ∫ ( x ( t ) − u ( t ) ) { f ( t , u, u, v, v ) − Df ( t , u, u, v, v )}
i
T
i
I
I
− (v (t ) − y (t ))
T
i
u
i
u
I
{ f ( t , x, x, y, y ) − Df ( t , x, x, y, y )} d .t
i
y
i
y
( x ( t ) )T { f i ( t , u , u , v, v ) − Df i ( t , u , u , v, v )} − ( u ( t ) )T { f i ( t , u , u , v, v ) − Df i ( t , u , u , v, v )}
u
u
u
u
∫I 
=
− (v (t ))
T
{ f ( t , x, x, y, y ) − Df ( t , x, x, y, y )} + ( y ( t ) ) { f ( t , x, x, y, y ) − Df ( t , x, x, y, y )} d .t
i
y
T
i
y
Multiplying this by λ and summing over
i
p
i
y
i
y
i , i = 1, 2,.., p , we get,
p
∑ λ i ∫ f i ( t , x, x, y, y ) dt − ∑ λ i ∫ f i ( t , u, u, v, v ) dt
i 1=
i 1
I
T

> ∫ ( x ( t ) )
I 
− (v (t ))
T
I
∑ λ i { fui ( t , u, u, v, v ) − Dfui ( t , u, u, v, v )} − ( u ( t ) )
p
T
∑ λ { f ( t , u, u, v, v ) − Df ( t , u, u, v, v )}
p
i
i
u
i 1 =i 1
∑ λ i { f yi ( t , x, x, y, y ) − Df yi ( t , x, x, y, y )} + ( y ( t ) )
p
T
i =1
i
u

∑ λ { f ( t , x, x, y, y ) − Df ( t , x, x, y, y )} dt
p
i
i
y
i =1
i
y


∫ ( x ( t ) ) ∑ λ { f ( t , u, u, v, v ) + ω ( t ) − Df ( t , u, u, v, v )}
p
T
i
i
u
i =1
I
i
u
i
− (u (t ))
∑ λ i { fui ( t , u, u, v, v ) + ωi ( t ) − Dfui ( t , u, u, v, v )} − ∑ λ i x ( t ) ωi ( t ) + ∑ λ iu ( t ) ωi ( t )
− (v (t ))
∑ λ { f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y )}
T
p
p
=i 1
T
T
=i 1 =i 1
p
i
i =1
+ ( y (t ))
p
i
y
i
y
i

∑ λ i { f yi ( t , x, x, y, y ) − zi ( t ) − Df yi ( t , x, x, y, y )} − ∑ λ i v ( t ) zi ( t ) + ∑ λ i y ( t ) zi ( t ) d . t
p
p
=i 1
p
=i 1 =i 1

Using (3), (5), (9) and (11), we have,
p
∑ λ { f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y )}
p
∑ λ i ∫  f i ( t , x, x, y, y ) − ( y ( t ) )
T
i 1=
i 1
I
p
i
y
i
i
y
i
p
+ ∑ λi ( x ( t ) ωi ( t ) ) − ∑ λi ( y ( t ) zi ( t ) )
i 1 =i 1
>
p
∑ λi ∫  f i ( t , u, u, v, v ) − ( u ( t ) )
T
∑ λ { f ( t , u, u, v, v ) + ω ( t ) − Df ( t , u, u, v, v )}
p
i 1=
i 1
I
i
i
u
i
u
i
+u ( t ) ωi ( t ) − y ( t ) zi ( t )  dt.
In v iew o f
(
)
(
)
s x (t ) C i > x (t ) ωi (t ) , i =
1,..., p and s v ( t ) K i > ( v ( t ) ) z i ( t ) , i =
1,..., p ,
T
T
this yields,
7
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
∑ λ ∫  f ( t , x, x, y, y ) + s ( x ( t ) C )
p
i
i
i =1
i
I
− ( y (t ))
T
>
∑ λ { f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y )} − ( y ( t ) ) z ( t ) d t
p
i
i =1
i
y
T
i
y
i
i
∑ λ ∫  f ( t , u, u, v, v ) − s ( v ( t ) K ( t ) )
p
i
i =1
i
i
I
− (u (t ))
T
∑ λ { f ( t , u, u, v, v ) + ω ( t ) − Df ( t , u, u, v, v )} +u ( t ) ω ( t ) d t .
p
i
i =1
i
u
i
u
i
i
That is,
p
p
∑ λi ∫ H i dt > ∑ λi ∫ G i dt .
=i 1 =
i 1
I
I
This yields,
∫ Hdt ≤ ∫ Gdt .
I
I
Theorem 2: (Strong Duality): Let
(SWP) and
T
T
p
(t ), λ ) be an efficient solution of
f yy ( t , x, x , y, y ) − Dλ T f yy ( t , x, x , y, y ) ) − D (φ ( t ) )

T
+ D 2  (φ ( t ) )

T
( − f ( t , x, x, y, y ) )} (φ ( t ) )=

yy
( − Dλ
T
f yy  ( t , x, x , y, y ) ) 

0, t ∈ I ⇒ φ ( t )= 0, t ∈ I
f yi ( t , x, x , y, y ) − z i ( t ) − Df yi ( t , x, x , y, y ) ,=
i 1,. . p. , t, ∈ I are linearly independent.
(C 2 ):
Then
1
λ = λ be fixed in (SWD). Furthermore, assume that
(C 1 ):
{(φ (t )) ( λ
( x(t ), y(t ), z (t ),..., z
( x(t ), y(t ), ω (t ),..., ω
1
p
(t ), λ ) is feasible for (SWD) and the objective functional values
are equal. If, in addition, the hypotheses of Theorem 1
hold, then there exists
( u(t ), v(t ), λ , ω (t ),. . ω. , (t ) ) = ( x(t ), y(t ), λ , ω (t ),. . ω. , (t ) ) is a n
ω1 (t ), ω2 (t ),..., ω p (t ) such t hat
1
p
1
p
efficient solution of the dual (SWD).
Proof: Since
( x(t ), y(t ), z (t ),..., z
1
p
(t ), λ ) is e fficient f or th e p roblem (S WP), i t is w eak
minimum [ 13]. H ence t here exists τ
∈ R p , η ∈ R m , γ ∈ R , θ ( t ) : I → R n and α ( t ) ∈ R n such
that the following Fritz-John optimality conditions [14], are satisfied
∑τ ( f ( t , x, x, y, y ) + ω ( t ) − Df ( t , x, x, y, y ) )
p
i
i
x
i =1
(
+ θ ( t ) − (τ T e ) y ( t )
i
) ∑ (λ
p
T
8
T
i =1

+ D  θ ( t ) − (τ T e ) y ( t )

(
i
x
f yxi ( t , x, x , y, y ) − Dλ T f yxi ( t , x, x , y, y ) )
) ∑λ ( f
T
p
i =1
i
i
yx

( t , x, x, y, y ) − Df yxi ( t , x, x, y, y ) − f yxi ( t , x, x, y, y ) )

I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010

+ D 2  θ ( t ) − (τ T e ) y ( t )

(
) ∑λ ( f

0, t ∈ I ,
( t , x, x, y, y ) ) =
p
T
i
i =1
i

yx
(15)

∑ (τ − (τ e ) λ ) ( f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y ) )
p
i
T
i
i
y
i =1
(
+ θ ( t ) − (τ T e ) y ( t )
i
) ∑λ ( f
p
T
i
i
yy
i =1

− D  θ ( t ) − (τ T e ) y ( t )

(
( t , x, x, y, y ) − Df yyi ( t , x, x, y, y ) )
) ∑ λ ( − Df
( t , x, x, y, y ) )
i

yy
0, t I ,
( t , x, x, y, y ) ) =∈
i
i =1

+ D 2  θ ( t ) − (τ T e ) y ( t )

(
(θ ( t ) − (τ e ) y ( t ) ) ( f
T
) ∑ λ (− f
i
y
p
T
i
i =1

i

yy
p
T
i
y


( t , x, x, y, y ) − z i ( t ) − Df yi ( t , x, x, y, y ) ) − η=
(
(16)

0 , t ∈ I,
)
=
xT ( t ) z i ( t ) s x ( t ) C i , t ∈ I
(17)
θ ( t ) ∑ ( f xi ( t , x, x , y, y ) − z i ( t ) − Df xi ( t , x, x , y, y ) ) = 0 , t ∈ I ,
(18)
p
(19)
i =1
(
)
τ i y ( t ) − θ ( t ) − (τ T e ) λ i ∈ N C ( z i ( t ) ) , t ∈ I ,
i
xT ( t ) α=
(t ) 0 , t ∈ I
ηT λ = 0
Since
λ > 0,
(22) implies η
(θ ( t ) − (τ e ) y ( t ) ) ( f
T
i
y
 p i
T
i
∑ τ − (τ e ) λ
 i =1
(
(
)( f
+ θ ( t ) − (τ T e ) y ( t )
i
y
t∈I .
(23)
(24)
= 0 . Consequently, (17) reduces to
(25)
(θ ( t ) − (τ e ) y ( t ) ) , we get,
T
( t , x, x, y, y ) − z i ( t ) − Df yi ( t , x, x, y, y ) )
) ∑λ ( f
p
T
i
i =1

− D  θ ( t ) − (τ T e ) y ( t )

(
,
0 , t∈I ,
( t , x, x, y, y ) − zi ( t ) − Df yi ( t , x, x, y, y ) ) =
Post multiplying (16) by
(21)
(22)
(τ ,θ ( t ) , α ( t ) ,η ) > 0 , t ∈ I ,
(τ ,θ ( t ) , α ( t ) ,η ) ≠ 0
(20)
i
yy
( t , x, x, y, y ) − Df yyi ( t , x, x, y, y ) )
) ∑ λ ( − Df
T
p
i
i =1
i

yy

0
( t , x, x, y, y ) )  =

9
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
(
+ D 2 θ ( t ) − (τ T e ) y ( t )
) ∑ λ (− f
p
T
i
i =1
Premultiplying (25) by
i

yy

0, t ∈I ,
( t , x, x, y, y ) ) (θ ( t ) − (τ T e ) y ( t ) ) =
(26)

(τ − (τ e ) λ ) and summing over i , we have
i
T
i
0
∑ (τ − (τ e ) λ ) (θ ( t ) − (τ e ) y ( t ) ) ( f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y ) ) =
p
i
T
i
T
i
y
i =1
i
i
y
(27)
t∈I
Using (27) in (26) we have,
T p

T
−
θ
τ
t
e
y
t
(
)
(
)
( λ T f yy ( t , x, x, y, y ) − Dλ T f yy ( t , x, x, y, y ) )
( )

∑
i =1

T p


T
+ D  θ ( t ) − (τ e ) y ( t ) ∑ ( − Dλ T f yy  ( t , x, x , y, y ) ) 
i =1


p
T

+ D 2 θ ( t ) − (τ T e ) y ( t ) ∑ ( −λ T f yy  ( t , x, x , y, y ) )  θ ( t ) − (τ T e ) y ( t ) =∈
0, t I
i =1

(
)
(
)
(
)
(
)
This in view of hypothesis (C 1 ) we have,
(
)
φ (t ) =
θ ( t ) − (τ T e ) y ( t ) =
0, t ∈ I .
(28)
Hence from (15) we have,
0 , t∈I .
∑τ ( f ( t , x, x, y, y ) + ω ( t ) − Df ( t , x, x, y, y ) ) − α ( t ) =
p
i
i
x
i =1
Let
i
(29)
i
x
τ = 0 . From (29) we have α =
(t ) 0 , θ =
(t ) 0 , t ∈ I .
(τ ,θ ( t ) , α ( t ) ,η ) = 0
Therefore,
,
t ∈ I , but this contradicts (24). Hence τ > 0 .
From (16) we have,
0 , t∈I .
∑ (τ − (τ e ) λ ) ( f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y ) ) =
p
i
T
i
i =1
i
y
i
From hypothesis (C 2 ),
i
y
( f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y ) ) is
i
y
i
i
y
linearly independent,
hence,
τ i = (τ T e ) λ i
i = 1, 2..., p .
From (29), we have
0 , t∈I
∑ (τ e ) λ ( f ( t , x, x, y, y ) + ω ( t ) − Df ( t , x, x, y, y ) ) − α ( t ) =
p
T
i =1
yielding,
10
i
i
x
i
i
x
(30)
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
0 , t∈I
∑ λ ( f ( t , x, x, y, y ) + ω ( t ) − Df ( t , x, x, y, y ) ) − α ( t ) =
p
i
i
x
i =1
i
i
x
(31)
This, in view of (23) implies
xT ( t ) ∑ λ i ( f xi ( t , x, x , y, y ) + ω i ( t ) − Df xi ( t , x, x , y, y ) ) > 0 , t ∈ I
p
(32)
i =1
Again (31) together with (21) gives
xT ( t ) ∑ λ i ( f xi ( t , x, x , y, y ) + ω i ( t ) − Df xi ( t , x, x , y, y ) ) =
0 , t∈I
p
(33)
i =1
(28) along with (19) yields,
yT ( t ) ∑ λ i ( f yi ( t , x, x , y, y ) − z i ( t ) − Df yi ( t , x, x , y, y ) ) =0 , t ∈ I
p
(34)
i =1
Now from (20), with
τ i > 0 , we have,
(
y ( t ) ∈ NCi z i ( t )
)
, i=
1,... p , t ∈ I
(35)
This implies,
(
yT ( t ) z i ( t ) = s y ( t ) K i
)
,
t∈I
(36)
Also from (28) we have,
θ (t )
≥ 0 ,t ∈ I
(τ T e )
y ( t )=
ω i ∈ K i yield
The relations (33), (37) and
(37)
that
( x (t ), y (t ), ω (t ),..., ω
1
p
(t ), λ ) is feasible for
(SWD).
Consider,
(
=
H i f i ( t , x , x , y , y ) + s x ( t ) C i
− y (t )
T
)
∑ λ ( f ( t , x , x , y , y ) − z ( t ) − Df ( t , x , x , y , y ) ) − y ( t ) z ( t )
p
i
i
y
i =1
i
y
(
)
= f i ( t , x , x , y , y ) + x T ( t ) ω i ( t ) − s y ( t ) K i
− x (t )
T
T
i
∑ λ ( f ( t , x , x , y , y ) + ω ( t ) − Df ( t , x , x , y , y ) )
p
i
i
x
i =1
i
i
x
or
i
=
H i G=
, i 1, 2,... p ,
implying
i
=
∫ H dt
I
G dt , i
∫=
i
t ∈ I,
1, 2,... p
I
or
11
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
∫ Hdt = ∫ Gdt
I
I
This by Theorem 1 establishes the efficiency of
( x (t ), y (t ), ω (t ),..., ω
1
p
(t ), λ ) for the dual
problem (SWD).
Now we state the converse duality theorem whose proof follows b y s ymmetry of the
formulations of the pair of problems.
( x(t ), y(t ), ω (t ),..., ω
THEOREM 3: (Converse d uality): Let
for (SWP) and
(A 1 ):
{(ψ (t )) (λ
T
T
f xx ( t , x, x , y, y ) − Dλ T f xx ( t , x, x , y, y ) ) − D (ψ ( t ) )

(t ), λ ) be an efficient solution
T
( −λ
T
Then
p
λ = λ be fixed in (SWD). Furthermore, assume that
+ D 2 (ψ ( t ) )

(A 2 ):
1
T
( − Dλ
T
f xx  ( t , x, x , y, y ) ) 

}
f xx  ( t , x, x , y, y ) )  (ψ ( t ) )= 0, t ∈ I ⇒ ψ ( t )= 0, t ∈ I

f xi ( t , x, x , y, y ) + ω i ( t ) − Df xi ( t , x, x , y, y ) , i =
1,. . p.are
, linearly independent.
( x(t ), y(t ), z (t ),..., z
1
p
(t ), λ ) is feasible for (SWD) and the objective functional values
are equal. I f, i n a ddition, t he h ypothesis of T heorem 1 h old, t hen t here e xist
( u(t ), v(t ), λ , z (t ),. . z. ,(t ) ) = ( x(t ), y(t ), λ , z (t ),. . z. ,(t ) ) is a
z1 (t ), z2 (t ),..., z p (t ) such t hat
1
p
1
p
n
efficient solution of dual (SWD).
4. Mond-Weir Type Dualıty
In t his s ection, w e p resent t he f ollowing p air o f M ond-Weir d ual p roblems, ( SM-WP) a nd
(SM-WD):
∫ ( Φ , Φ ,..., Φ ) dt
(SM-WP): Maximize:
1
2
p
I
Subject to:
x ( a )= 0= x ( b )
y ( a )= 0= y ( b )
∑ λ ( f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y ) ) < 0 , t ∈ I
p
i
i =1
i
y
i
i
x
T
i
i
i
i
∫ y ( t ) ∑ λ ( f y ( t , x, x, y, y ) − z ( t ) − Df x ( t , x, x, y, y ) ) d t> 0
p
I
i =1
zi (t ) ∈ C i , i =
1,..., p , t ∈ I
x (t ) > 0 , t ∈ I
λ >0
12
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
(SM-WD): Minimize:
∫ (ψ
1
,ψ 2 ,...,ψ
p
)dt
I
Subject to:
u (a ) = 0 = u (b )
v(a ) = 0 = v(b )
∑ λ ( f ( t , u, u, v, v ) + ω ( t ) − Df ( t , u, u, v, v ) ) > 0 , t ∈ I
p
i
i =1
i
u
i
i
u
∫ y ( t ) ∑ λ ( f ( t , u, u, v, v ) + ω ( t ) − Df ( t , u, u, v, v ) ) d t< 0
p
T
i
i =1
I
i
u
i
i
u
ω i (t ) ∈ K i , i = 1,..., p
v (t ) > 0 , t ∈ I
λ >0
where,
i
1. Φ
=
f i ( t , x, x , y, y ) + s ( x(t ) C i ) − y (=
t ) z ( t ) , i 1,..., p
T
(
)
ψ=i f i ( t , u, u , v, v ) − s v ( t ) K i + u ( t ) ω ( t ) ,=
i 1,..., p
2.
T
The d uality t heorems f or t hese p roblems will be m erely stated b elow f or completeness a s
their proofs follow on the lines of [15]:
Theorem 4 (Weak Duality): Let
( x(t ), y(t ), z (t ),..., z
1
p
( u(t ), v(t ), ω (t ),..., ω (t ), λ ) be feasible for the d ual
∑ λ ∫ ( f ( t ,.,., y, y ) dt + (.) z ) dt is pseudoconvex
1
p
(t ), λ ) be feasible for the (SM-WP) and
(SM-WD). Assume that, for each i,
p
i
i =1
i
i
in
( x, x ) for
fixed
( y, y ) and
I
∑ λ ∫ ( f ( t , x, x,.,.) − (.) z ) dt is pseudoconcave in ( y, y ) for fixed ( x, x ) . Then,
p
i
i =1
i
i
I
∫ φ ( t , x, x, y, y , λ ) dt ≤ ∫ψ ( t , x, x, y, y , λ ) dt
I
I
Theorem 5 (Strong Duality): Let
(SM-WP) and
( x(t ), y(t ), z (t ),..., z
1
p
(t ), λ ) be an efficient solution for
λ = λ be fixed in (SM-WD). Furthermore, assume that
(H 1 ):
∫ {(φ ( t ) ) ( λ
T
− D  (φ ( t ) )

( − Dλ
T
f yy ( t , x, x , y, y ) − Dλ T f yy ( t , x, x , y, y ) )
I
T
T
f yy  ( t , x, x , y, y ) ) 

13
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
( − f ( t , x, x, y, y ) )} (φ ( t ) ) d =
+ D 2  (φ ( t ) )

T

yy
f yi ( t , x, x , y, y ) − z i (t ) − Df yi ( t , x, x , y, y ) , i =
1,. . p.are
, linearly independent.
(H 2 ):
( x(t ), y(t ), ω (t ),..., ω
Then
t0, t ∈ I ⇒ φ ( t )= 0, t ∈ I
1
p
(t ), λ ) is feasible for (SM-WD) and the objective functional values
are e qual. I f, i n a ddition, t he h ypotheses of T heorem 5 h old, t hen t here e xist
ω1 (t ), ω2 (t ),..., ω p (t ) such t hat
( u(t ), v(t ), λ , ω (t ),. . ω. , (t ) ) = ( x(t ), y(t ), λ , ω (t ),. . ω. , (t ) ) is a n
1
p
1
p
efficient solution of dual (SM-WD).
Theorem 6 (Converse duality): Let
( x(t ), y(t ), ω (t ),..., ω
1
p
(t ), λ ) be an efficient solution
λ = λ be fixed in (SM-WD). Furthermore, assume that
for (SM-WP) and
(A 1 ):

T
T
T
 ∫ (ψ ( t ) ) ( λ f xx ( t , x, x , y, y ) − Dλ f xx ( t , x, x , y, y ) )
I
T
− D (ψ ( t ) ) ( − Dλ T f xx  ( t , x, x , y, y ) ) 


+ D 2 (ψ ( t ) )

T
(A 2 ):
Then
}
( − f xx  ) (ψ ( t ) ) d =
t0, t ∈ I ⇒ ψ ( t )= 0, t ∈ I
f xi ( t , x, x , y, y ) + z i ( t ) − Df xi ( t , x, x , y, y ) , i =
1,. . p., ,t ∈ I are linearly independent.
( x(t ), y(t ), z (t ),..., z
1
p
(t ), λ ) is feasible for (SM-WD) and the objective functional values
are e qual. I f, i n a ddition, t he h ypotheses of T heorem 5 h old, t hen t here e xist
( u(t ), v(t ), λ , z (t ),. . z. ,(t ) ) = ( x(t ), y(t ), λ , z (t ),. . z. ,(t ) ) is a
z1 (t ), z2 (t ),..., z p (t ) such t hat
1
p
1
p
efficient solution of dual (SM-WD).
5. Self Duality
A p roblem i s s aid t o b e s elf-dual if it is formally id entical with it s dual, in g eneral, the
problems (S P) a nd (S WD) a re n ot f ormally identical if the k ernel f unction d oes n ot
possess any special characteristics. Hence, skew symmetry of each
f i is assumed in
order to validate the following self-duality theorems for the two pairs of problems treated
in the preceding sections.
Theorem 7. (Self Duality): Let
ωi (t ) = zi (t )
f i , i = 1, 2, , p , be skew symmetric and C i = K i and
. Then the problem (SP) is self dual. If the problems (SWP) and (SWD) are
( x ( t ) , y ( t ) , z ( t ) ,..., z ( t ) , λ ) is a joint optimal solution of (SWP) and
(SWD), then so is ( y ( t ) , x ( t ) , z ( t ) ,..., z ( t ) , λ ) , i.e.
Minimum (SWP) = ∫ ( f ( t , x, x , y, y ) , f ( t , x, x , y, y ) , , f ( t , x, x , y, y ) ) d t= 0 .
dual problems and
1
p
1
1
p
2
I
Proof: By skew symmetric of
14
f i , we have
p
n
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
f xi ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) ) = − f yi ( t , y ( t ) , y ( t ) , x ( t ) , x ( t ) )
f yi ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) ) = − f xi ( t , y ( t ) , y ( t ) , x ( t ) , x ( t ) )
f xi ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) ) = − f yi ( t , y ( t ) , y ( t ) , x ( t ) , x ( t ) )
f yi ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) ) = − f xi ( t , y ( t ) , y ( t ) , x ( t ) , x ( t ) )
Recasting th e d ual p roblem (S WD) a s a m inimization p roblem a nd u sing th e a bove
relations, we have,
− ∫ ( G1 , G 2 ,..., G p ) dt
(SWD 1 ): Minimize
I
Subject to:
x ( a )= 0= x ( b ) ,
y ( a )= 0= y ( b )
p
−∑ λ i  − f xi ( t , y, y , x, x ) + ω i ( t ) − Df xi ( t , y, y , x, x )  < 0 , t ∈ I ,
i =1
p
∑λ
=
i =1
i
i
i
i
 f y ( t , x, x , y, y ) − ω ( t ) − Df y ( t , x, x , y, y )  < 0 , t ∈ I
v (t ) > 0 , t ∈ I
ω i ( t ) ∈ K i ,=
i 1,..., p, t ∈ I
λ ∈ Λ+
(
)
−G i =
− f i ( t , x, x , y, y ) − s y ( t ) K i − x ( t ) ωi ( t )
p
− x ( t ) ∑ λ i  f xi ( t , x, x , y, y ) − ωi ( t ) − Df xi ( t , x, x , y, y ) 
i =1
(
)
= f i ( t , y, y , x, x ) − s y ( t ) K i − x ( t ) ωi ( t )
p
− x ( t ) ∑ λ i  f yi ( t , x, x , y, y ) − ωi ( t ) − Df yi ( t , x, x , y, y ) 
i =1
= H ( t , y, y , x, x , ω1 ,..., ω p , λ )
i
Hence we have,
(SWP-1):
∫(H , H
1
2
,..., H p ) dt
I
Subject to:
x ( a )= 0= x ( b ) ,
p
∑λ
i =1
i
y ( a )= 0= y ( b )
 f yi ( t , x, x , x, y ) − zi ( t ) − Df yi ( t , x, x , y, y )  < 0 , t ∈ I ,
15
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
y (t ) > 0 , t ∈ I
zi (t ) ∈ C i ,
t∈I
λ > 0, λ T e =
1 where eT = (1,.....,1) ,
( x ( t ) , y ( t ) , z ( t ) ,..., z ( t ) , λ ) is a n
efficient solution of d ual p roblem im plies t hat ( y ( t ) , x ( t ) , z ( t ) ,..., z ( t ) , λ ) is a n e fficient
solution of the primal. Similarly ( x ( t ) , y ( t ) , z ( t ) ,..., z ( t ) , λ ) is an efficient solution of (SP)
implies ( y ( t ) , x ( t ) , z ( t ) ,..., z ( t ) , λ ) is an efficient solution of the dual problem (SWD). In
which is j ust t he p rimal p roblem ( SWP). T herefore
1
1
1
p
p
p
p
1
view of (18), (33), (34) and (36), we get,
Minimum (SWP) =
∫ { f ( t , x ( t ) , x ( t ) , y ( t ) , y ( t ) ) ,.
1
}
f p .( t , x (.t ) , x (,t ) , y ( t ) , y ( t ) ) d t
I
Corresponding, to the solution
Minimum (SWP) =
( y ( t ) , x ( t ) , z ( t ) ,..., z ( t ) , λ ) , we have,
1
p
∫ { f ( t , y ( t ) , y ( t ) , x ( t ) , x ( t ) ) ,.
1
}
f p (.t , y ( t,) , y ( t ) , x ( t ) , x ( t ) ) d t
I
By the skew-symmetry of each
fi
∫ { f ( t , x , x , y , y ) ,..., f ( t , x , x , y , y )} d t= ∫ { f ( t , y , y , x , x ) ,..., f ( t , y , y , x , x )} d
1
1
p
I
I
p
{
t
}
0
=
− ∫ f 1 ( t , x , x , y , y ) ,. . f .p ( t,, x , x , y , y ) d t=
I
The following self duality theorem for the pair of Mond-Weir type self d uality theorem will
be merely stated for completeness and its proof runs parallel to that of Theorem 4.
Theorem 8. (Self Duality) Let
ωi (t ) = zi (t )
f i , i = 1, 2, , p , be skew s ymmetric and C i = K i and
. Then the problem (SM-WP) is self dual. If the problems (SM-WP) and
( x ( t ) , y ( t ) , z ( t ) ,..., z ( t ) , λ ) is a joint optimal solution
of (SM-WP) and (SM-WD), then so is ( y ( t ) , x ( t ) , z ( t ) ,..., z ( t ) , λ ) , i.e.
(SM-WD) are dual problems and
1
p
1
Minimum (SM-WP) =
p
∫ f ( t , x, x, y, y ) dt = 0 .
I
6. Natural Boundary Values
The pairs of Wolfe type and Mond-Weir type symmetric multiobjective variational
problem can be formulated with natural boundary values rather than fixed end points.
The problems w ith na tural boundary c onditions are needed t o establish well d efined
relationship between the pairs of continuous programming problems and nonlinear
programming problems.
Following is th e p air o f W olfe t ype s ymmetric d ual p roblems w ith natural boundary
values.
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I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
Primal (SWPo): Maximize
∫(H , H
1
2
,..., H p ) dt
I
Subject to:
p
∑λ
i =1
i
 f yi ( t , x, x , y, y ) − z i ( t ) − Df yi ( t , x, x , y, y )  < 0 ,
( x (t )) > 0 , t ∈ I
zi (t ) ∈ C i ,
t∈I
λ > 0, λ T e =1 , eT =(1,.....,1)
=
f ( t , x, x , y, y )
0=
, f ( t , x, x , y, y )
0
i
i
y
y
=
t a=
t b
∫ (G , G
Dual (SWD 0 ) : Maximize
1
2
,..., G p ) dt
I
Subject to:
p
∑λ
i =1
i
 fui ( t , u , u , v, v ) + ωi ( t ) − Dfui ( t , u , u , v, v )  > 0 , t ∈ I ,
v (t ) > 0 , t ∈ I
ωi (t ) ∈ K i , t ∈ I
λ > 0, λ T e =1 , eT =(1,.....,1)
=
f ( t , x, x , y, y )
0=
, f ( t , x, x , y, y )
0
i
i
x
x
=
t a=
t b
The d uality re sults f or each of t he a bove p airs of d ual p roblems c an b e p roved e asily on
the lines of the proofs of the Theorems 1-8, with slight modifications in the arguments, as
in Mond and Hanson [7].
Following is the pair of Mond-Weir type symmetric dual problems with natural boundary
values.
Primal (SM-WP 0 ): Maximize
∫ ( Φ , Φ ,..., Φ ) dt
1
p
2
I
Subject to:
∑ λ ( f ( t , x, x, y, y ) − z ( t ) − Df ( t , x, x, y, y ) ) < 0 , t ∈ I
p
i
i =1
i
y
i
i
y
T
i
i
i
i
∫ y ( t ) ∑ λ ( f y ( t , x, x, y, y ) − z ( t ) − Df y ( t , x, x, y, y ) ) d t> 0
p
I
i =1
zi (t ) ∈ C i , i =
1,..., p ,
t∈I
x (t ) > 0 , t ∈ I
17
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
λ >0
=
λ f ( t , x, x, y, y )
0=
, λ f ( t , x, x , y, y )
0
T
T
y
y
=
t a=
t b
∫ (ψ
Dual (SM-WD 0 ): Minimize:
1
,ψ 2 ,...,ψ
p
)dt
I
Subject to:
∑ λ ( f ( t , u, u, v, v ) + ω ( t ) − Df ( t , u, u, v, v ) ) > 0 , t ∈ I
p
i
i
u
i =1
i
i
u
(
p
)
yT ( t ) ∑ λ i fui ( t , u , u , v, v ) + ω i ( t ) − Df ui ( t , u , u , v, v ) < 0 , t ∈ I
i =1
ω i (t ) ∈ K i , i = 1,..., p
v (t ) > 0 , t ∈ I
λ ∈ Λ+
=
λ f ( t , u, u, v, v )
0=
, λ f ( t , u , u , v, v )
0
T
T
u
u
=
t a=
t b
7. Nondifferentiable Multiobjective Nonlinear Programming
If the time dependency of Wolfe type symmetric pair of dual problems, (SWPo) and
(SWDo) i s re moved a nd b − a =
1 , w e o btain f ollowing p air o f W olfe ty pe s tatic
nondifferentiable m ultiobjective d ual pr oblems w ith s upport f unctions w hich a re n ot
explicitly reported in literature.
(Primal SWP-2): Minimize
Hˆ i = Hˆ 1 , Hˆ 2 ,..., Hˆ p
Subject to:
p
∑λ
i =1
i
 f yi ( x, y ) − z i  < 0 ,
zi ∈ K i , i =
1,..., p
λ ∈ Λ+
where,
=
Hˆ i f i ( x, y ) + s ( x C i ) − yT ∑ λ i ( f yi ( x, y ) − z i ) − yT z ,
p
i =1
(Dual SWD-2): Maximize:
Subject to:
18
(
Gˆ i = Gˆ 1 , Gˆ 2 ,..., Gˆ p
)
i = 1,..., p
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
p
∑λ
i =1
i
 fui ( u , v ) + ω i  > 0 ,
ωi ∈ Ci
i = 1,..., p
,
λ ∈ Λ+
Where
=
Gˆ i f i ( u , v ) + s ( v K i ) − u T ∑ λ i ( f ui ( u , v ) + ω i ) − xT ω
p
i =1
As in the case of pair of Wolfe type dual problems, the pair of Mond-Weir type dual
problems (SM-WPo) and (SM-WDo) reduce to the following static counterparts in
nonlinear programming.
(
ˆi = Φ
ˆ 1, Φ
ˆ 2 ,..., Φ
ˆp
Φ
Primal (SM-WP-2): Maximize
)
Subject to:
p
∑λ
 f yi ( x, y ) − z i  < 0 ,
i
i =1
∑ λ ( f ( x, y ) − z ) > 0 ,
p
y
T
i
i =1
i
y
i
zi ∈ C i , i =
1,..., p ,
x >0 ,
λ >0
where
ˆ i f i ( x, y ) + s ( x C i ) − y T z
=
Φ
Dual (SM-WD-2): Maximize:
ψˆ i = ψˆ 1 ,ψˆ 2 ,...,ψˆ p
Subject to:
p
∑λ
i =1
p
i
 fui ( u , v ) + ω i  > 0 ,
(
)
yT ∑ λ i fui ( u , v ) + ω i < 0
i =1
ωi ∈ K i
i = 1,..., p
v >0
λ ∈ Λ+
where,
19
I. Husain, R.G. Mattoo / İstanbul Üniversitesi İşletme Fakültesi Dergisi 39, 1, (2010) 1-20 © 2010
ψˆ i = f i ( x, y ) + s ( y K i ) − xT w
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[1]
W.S. Dorn, “ Asymmetric d ual theorem f or q uadratic p rograms. Journal o f
Operations Research Society of Japan. 2, 93-97 (1960).
[2]
G.B. Dantzig, E isenberg a nd R. W.Cottle, “ Symmetric d ual n onlinear p rograms”.
Pacific Journal of Mathematics. 15, 809-812 (1965).
[3]
B. Mond, “A symmetric dual theorem for nonlinear programs”. Quaterly Jounal of
Applied Mathematics. 23, 265-269 (1965).
[4]
M.S.
Bazaraa,
J.J.
Goode,
“On
symmetric
duality
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[5]
B. Mond, R.W. Cottle, “Self duality i n m athematical pr ogramming”, SI AM.
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[6]
B. Mond, T.Weir, “Generalized concavity and duality”, in: S.Sciable, W.T.Ziemba
(Eds.), Generalized Concavity in Optimization and Economics, Academic Press, New
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[7]
B. Mond, M.A. Hanson, “Symmetric Duality for Variational problems”. J.Math. Anal.
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C.R. Bector, S. Chandra, I. Husain, “Generalized concavity and duality in continuous
programming. Utilitas Mathematica. 25, 171-190 (1984).
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S. Chandra, I. Husain, “Symmetric dual continuous fractional programming”. J. Inf.
Opt. Sc. 10, 241-255 (1989).
in
nonlinear
[10] C.R. Bector, I. Husain, “ Duality f or mu ltiobjective v ariational p roblems.
Journal of Math. Anal and Appl. 166, No.1, 214-224 (1992).
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20

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