10_367_husain.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 4, 2009, (578-603)

ISSN 1307-5543 – www.ejpam.com

Mixed Type Symmetric and Self-Duality for Multiob-

jective Variational Problems

I. Husain1∗ and Rumana G. Mattoo2

1 Department of Mathematics, Jaypee Institute of Engineering and Technology, Guna, MP,

India. (A constituent centre of Jaypee University of Information Technology, Waknaghat,

Solan, HP, India)

2 Department of Statistics, University of Kashmir, Srinagar, Kashmir, India.

Abstract. In this paper, a new formulation of multiobjective symmetric dual pair, called

mixed type multiobjective symmetric dual pair, for multiobjective variational problems is pre-

sented. This mixed formulation unifies two existing Wolfe and Mond-Weir type symmetric

dual pairs of multiobjective variational problems. For this pair of mixed type multiobjec-

tive variational problems, various duality theorems are established under invexity-incavity

and pseudoinvexity-pseudoincavity of kernel functions appearing in the problems. Under ad-

ditional hypotheses, a self duality theorem is validated. It is also pointed that our duality

theorems can be viewed as dynamic generalization of the corresponding (static) symmetric

and self duality of multiobjective nonlinear programming already existing in the literature.

2000 Mathematics Subject Classifications: Primary 90C30, Secondary 90C11, 90C20, 90C26.

Key Words and Phrases: Efficiency; Mixed type multiobjective symmetric dual variational

problem; Mixed type symmetric duality; Mixed type self duality; Natural boundary values;

Multiobjective nonlinear programming.

∗Corresponding author.

http://www.ejpam.com 578 c© 2009 EJPAM All rights reserved.



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 579

1. Introduction

Following Dorn [7], symmetric duality results in mathematical programming have

been derived by a number of authors, notably, Dantzig et al [8], Mond [12], Bazaraa

and Goode [1]. In these researches, the authors have studied symmetric duality under

the hypothesis of convexity-concavity of the kernel function involved. Mond and

Cottle [13] presented self duality for the problems of [8] by assuming skew symmetric

of the kernel function. Later Mond-Weir [14] formulated a different pair of symmetric

dual nonlinear program with a view to generalize convexity-concavity of the kernel

function to pseudoconvexity-pseudoconcavity.

Symmetric duality for variational problems was first introduced by Mond and Han-

son [15] under the convexity-concavity conditions of a scalar functions like ψ(t , x(t),

ẋ(t), y(t), ẏ(t)) with x(t) ∈ Rn and y(t) ∈ Rm. Bector, Chandra and Husain [3] pre-

sented a different pair of symmetric dual variational problems in order to relax the

requirement of convexity-concavity to that of pseudoconvexity-pseudoconcavity while

in [6] Chandra and Husain gave a fractional analogue.

Bector and Husain [4] probably were the first to study duality for multiobjective

variational problems under appropriate convexity assumptions. Subsequently, Gulati,

Husain and Ahmed [9] presented two distinct pairs of symmetric dual multiobjec-

tive 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. Husain and Jabeen [10] for-

mulated a pair of mixed type symmetric dual variational problem in order to unify the

Wolfe and Mond-Weir symmetric dual pairs of variational problems studied in [9].

The purpose of this research is to unify formulations of Wolfe and Mond-Weir type

symmetric dual pairs of multiobjective variational problems incorporated by Gulati,

Husain and Ahmed [9] and also present multiobjective version of the formulation



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 580

of a pair of mixed type symmetric dual of Husain and Jabeen [10] and hence study

symmetric and self duality for a pair of mixed multiobjective variational problem. This

research is motivated by the work of Xu [18]. The problems, treated in this research

are quite hard to solve. So to expect any immediate application of these problems

would be far from reality. Unfortunately, there has not always been sufficient flow

between the researchers in the 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 model. Special cases are deduced and it

is also pointed out that our results can be considered as dynamic generalizations of

corresponding (static) symmetric duality results of multiobjective nonlinear nonlinear

treated by Bector et al. [3].

2. Notations and Preliminaries

The following notation will be used for vectors in Rn.

x < y ⇔ x i < yi, i = 1, 2, . . . , n.

x ≦ y ⇔ x i ≦ yi, i = 1, 2, . . . , n.

x ≤ y ⇔ x i ≤ yi, i = 1, 2, . . . , n, but x 6= y

x 6≤ y, is the negation of x ≤ y

Let I = [a, b] be the real interval, and φ i(t , x(t), ẋ(t), y(t), ẏ(t)) be a scalar

function and twice differentiable function for i = 1, 2, . . . , p where x : I → Rn and

y : I → Rn with derivatives ẋ and ẏ. In order to consider φ i(t , x(t), ẋ(t), y(t), ẏ(t))

denote the first partial derivatives of φ i with respect to t , x(t), ẋ(t), y(t), ẏ(t)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 581

respectively, by φ i
t
, φ i

x
, φ i

ẋ
, φ i

y
, φ i

ẏ
, that is,

φ i
t
=
∂ φ i

∂ t

φ i
x
=

�

∂ φ i

∂ x1

,
∂ φ i

∂ x2

, . . . ,
∂ φ i

∂ xn

�

, φ i
ẋ
=

�

∂ φ i

∂ ẋ1

,
∂ φ i

∂ ẋ2

, . . . ,
∂ φ i

∂ ẋn

�

φ i
y
=

�

∂ φ i

∂ y1

,
∂ φ i

∂ y2

, . . . ,
∂ φ i

∂ yn

�

, φ i
ẏ
=

�

∂ φ i
i

∂ ẏ1

,
∂ φ i

∂ ẏ2

, . . . ,
∂ φ i

∂ ẏn

�

.

The twice partial derivatives of φ i with respect to t , x(t), ẋ(t), y(t) and ẏ(t), respec-

tively are the matrices

φ i
x x
=

�

∂ 2φ i

∂ xk xs

�

n×n

, φ i
x ẋ
=

�

∂ 2φ i

∂ xk ẋs

�

n×n

, φ i
x y
=

�

∂ 2φ i

∂ xk ys

�

n×n

φ i
x ẏ
=

�

∂ 2φ i

∂ xk ẏs

�

n×n

, φ i
ẋ y
=

�

∂ 2φ i

∂ ẋk ys

�

n×n

, φ i
x ẏ
=

�

∂ 2φ i

∂ xk ẏs

�

n×n

φ i
y y
=

�

∂ 2φ i

∂ yk ys

�

n×n

, φ i
y ẏ
=

�

∂ 2φ i

∂ yk ẏs

�

n×n

, φ i
ẏ ẏ
=

�

∂ 2φ i

∂ ẏk ẏs

�

n×n

for i = 1, 2, . . . , p.

Noting that

d

d t
φ i

ẏ
= φ i

ẏ t
+φ i

ẏ y
ẏ +φ i

ẏ ẏ
ÿ +φ i

ẏ x
ẋ +φ i

ẏ ẋ
ẍ

and hence

∂

∂ y

d

d t
φ i

ẏ
=

d

d t
φ i

y ẏ
,
∂

∂ ẏ

d

d t
φ i

ẏ
=

d

d t
φ i

ẏ ẏ
+φ i

ẏ y
,
∂

∂ ÿ

d

d t
φ i

ẏ
= φ i

ẏ y

∂

∂ x

d

d t
φ i

ẏ
=

d

d t
φ i

ẏ x
,
∂

∂ ẋ

d

d t
φ i

ẏ
=

d

d t
φ i

ẏ ẋ
+φ i

ẏ x
,
∂

∂ ẍ

d

d t
φ i

ẏ
= φ i

ẏ ẋ

In order to establish our main results, the following are needed.

Definition 1 (Partially Invex). If there exists a vector function η(t , x(t), y(t), u(t),

v(t)) ∈ Rn
+

with η = 0 at x(t) = u(t) or y(t) = v(t), such that for the scalar function

h(t , x(t), ẋ(t), y(t), ẏ(t)) the functional

H(x , ẋ , y, ẏ) =

∫

I

h(t , x(t), ẋ(t), y(t), ẏ(t))d t



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 582

satisfies

H(x , ẋ , y, ẏ)−H(u, u̇, v, v̇) ≥

∫

I

[ηT hx(t , x(t), ẋ(t), y(t), ẏ(t))

+ (Dη)T h ẋ(t , x(t), ẋ(t), y(t), ẏ(t))]d t

then H(x , ẋ , y, ẏ) is said to be partially invex in x and ẋ on I with respect to η, for fixed

y. if H satisfies

H(x , ẋ , y, ẏ)−H(x , ẋ , v, v̇) ≥

∫

I

[ηT hy(t , x(t), ẋ(t), v(t), v̇(t))

+(Dη)T h ẏ(t , x(t), ẋ(t), v(t), v̇(t))]d t ,

then H(x , ẋ , y, ẏ) is said to be partially invex in y and ẏ on I with respect to η, for fixed

x. If −H is partially invex in x and ẋ (or in y and ẏ) on I with respect to η, for fixed

y(or for fixed x), then H is said to be partially incave in x and ẋ (or in y and ẏ) on I

with respect to η, for fixed y (or for fixed x).

Definition 2 (Partially Pseudoinvex). The functional H is said to be partially pseudoin-

vex in x and ẋ with respect to η, for fixed y if H satisfies

∫

I

[ηT hx(t , x , ẋ, y, ẏ) + (Dη)T h ẋ(t , x , ẋ, y, ẏ)]d t ≧ 0

implies

H(x , ẋ , u, u̇)≧ H(x , ẋ , y, ẏ)

and H is said to be partially pseudoinvex in y and ẏ with respect to η, for fixed x If H

satisfies

∫

I

[ηT hy(t , x , ẋ, y, ẏ) + (Dη)T h ẏ(t , x , ẋ, y, ẏ)]d t ≧ 0

implies

H(x , ẋ , v, v̇)≧ H(x , ẋ , y, ẏ).



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 583

Definition 3 (Partially Quasi-invex). The functional H is said to be partially quasi-invex

in x and ẋ with respect to η, for fixed y if H satisfies

H(x , ẋ , u, u̇)≦ H(x , ẋ , y, ẏ)

implies

∫

I

[ηT hx(t , x , ẋ, y, ẏ) + (Dη)T h ẋ(t , x , ẋ, y, ẏ)]d t ≦ 0

and H is said to be partially quasi-invex in y and ẏ with respect to η, for fixed x if H

satisfies

H(x , ẋ , v, v̇)≦ H(x , ẋ , y, ẏ)

implies

∫

I

[ηT hy(t , x , ẋ, y, ẏ) + (Dη)T h ẏ(t , x , ẋ, y, ẏ)]d t ≦ 0.

If h is independent of t , then the above definitions become the usual definitions

of invexity and generalized invexity, discussed by several authors, notably Ben-Israel

and Mond [5], Martin [11], and Rueda and Hanson [16].

Definition 4 (Skew Symmetry). The function h : I × Rn× Rn× Rn × Rn→ R is said to

be skew symmetric if for all x and y in the domain of h if

h(t , x(t), ẋ(t), y(t), ẏ(t)) =−h(t , y(t), ẏ(t), x(t), ẋ(t)), t ∈ I

where x and y are piecewise smooth on I.

Now consider the following multiobjective variational problem considered in [4]:

(VP0) Minimize

�
∫

I

φ1(t , x , ẋ)d t ,

∫

I

φ2(t , x , ẋ)d t , . . . ,

∫

I

φp(t , x , ẋ)d t ,

�

Subject to



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 584

x(a) = α, x(b) = β

h(t , x , ẋ)≦ 0, t ∈ I ,

where φ i : I × Rn × Rn × Rn → R (i = 1, 2, . . . , p) and h : I × Rn × Rn × Rn → Rm. Let

the set of feasible solution of (V P0) be represented by K .

Definition 5 (Efficiency). A point x̄ ∈ K is an efficient (Pareto optimal) solution of

(VP0) if for all x ∈ K,
∫

I

φ i(t , x , ẋ)d t 6≤

∫

I

φ i(t , x̄ , ˙̄x)d t, (i = 1, 2, . . . , p)

3. Statement of the Problems

For N = {1, 2, . . . , n} and M = {1, 2, . . . , m}, let J1 ⊂ N , K1 ⊂ M , J2 = N \ J1 and

K2 = M \ K1. Let |J1| denote the number of elements in the subset J1. The other

symbol |J2|, |K1| and |K2| are similarly defined. Let x1 : I → R|J1| and x2 : I → R|J2|,

then any x : I → Rn can be written as x = (x1, x2). Similarly for y1 : I → R|K1|

and y2 : I → R|K2| can be written as y = (y1, y2) where x : I → Rn, y : I → Rm. Let

f : I×R|J1|×R|K1|→ Rp and g : I×R|J2|×R|K2|→ Rp be twice continuously differentiable

functions.

We state the following pair of mixed type multiobjective symmetric dual varia-

tional problems involving vector functions f and g.

(Mix SP) Minimize F(x1, x2, y1, y2) =

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)

−y1(t)T(λT f y1(t , x1, ẋ1, y1, ẏ1)

−DλT f ẏ1(t , x1, ẋ1, y1, ẏ1))e}d t

Subject to

x1(a) = 0= x1(b), y1(a) = 0 = y1(b), (1)

x2(a) = 0= x2(b), y2(a) = 0 = y2(b), (2)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 585

λT f y1(t , x1, ẋ1, y1, ẏ1)− DλT f ẏ1(t , x1, ẋ1, y1, ẏ1)≦ 0, t ∈ I , (3)

λT g y2(t , x2, ẋ2, y2, ẏ2)− DλT g ẏ2(t , x2, ẋ2, y2, ẏ2) ≦ 0, t ∈ I , (4)
∫

I

y2(t)T(λT g y2(t , x2, ẋ2, y2, ẏ2)

−DλT g ẏ2(t , x2, ẋ2, y2, ẏ2))≧ 0, (5)

λ ∈ Λ+. (6)

(Mix SD) Maximize G(u1, u2, v1, v2) =

∫

I

{ f (t , u1, u̇1, v1, v̇1) + g(t , u2, u̇2, v2, v̇2)

−u1(t)T(λT f y1(t , u1, u̇1, v1, v̇1)

−DλT f ẏ1(t , u1, u̇1, v1, v̇1))e}d t

Subject to

u1(a) = 0 = u1(b), v1(a) = 0= v1(b), (7)

u2(a) = 0 = u2(b), v2(a) = 0= v2(b), (8)

λT fu1(t , u1, u̇1, v1, v̇1)− DλT fu̇1(t , u1, u̇1, v1, v̇1)≧ 0, t ∈ I , (9)

λT gu2(t , u2, u̇2, v2, v̇2)− DλT gu̇2(t , u2, u̇2, v2, v̇2)≧ 0, t ∈ I , (10)
∫

I

u2(t)T(λT gu2(t , u2, u̇2, v2, v̇2)

−DλT gu̇2(t , u2, u̇2, v2, v̇2))≧ 0, (11)

λ ∈ Λ+. (12)

where Λ+ = {λ ∈ Rp|λ > 0,λT e = 1, e = (1, 1, . . . , 1)T ∈ Rp}.

4. Mixed Type Multiobjective Symmetric Duality

In this section, we present various duality results and the appropriate invexity and

generalized invexity assumptions.

Theorem 1 (Weak Duality). Let (x1, x2, y1, y2,λ) be feasible for (Mix SP) and

(u1, u2, v1, v2,λ) be feasible for (Mix SD).



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 586

Let

H1

∫

I
f (t , ., ., y1(t), ẏ1(t))d t be partially invex in x1, ẋ1 on I for fixed y1, ẏ1 with re-

spect to η1(t , x1, u1) ∈ R|J1|.

∫

I
f (t , x1(t), ẋ1(t), ., .)d t be partially incave in y1, ẏ1 on I for fixed x1, ẋ1 with

respect to η2(t , y1, v1) ∈ R|K1|.

H2

∫

I
λT g(t , ., ., y2(t), ẏ2(t))d t be partially pseudoinvex in x2, ẋ2 on I for fixed y2, ẏ2

with respect η3(t , x2, u2) ∈ R|J2| and
∫

I
λT g(t , x2, ẋ2, . . .)d t be partially pseudoin-

cave in y2, ẏ2 on I for fixed x2, ẋ2 with respect to η4(t , y2, v2) ∈ R|K2|.

H3

η1(t , x1, u1) + u1(t)≧ 0, t ∈ I , (13)

η2(t , v1, y1) + y1(t)≧ 0, t ∈ I , (14)

η3(t , x2, u2) + u2(t)≧ 0, t ∈ I , (15)

η4(t , v2, y2) + y2(t)≧ 0, t ∈ I , (16)

then

F(x1, x2, y1, y2) 6≤ G(u1, u2, v1, v2).

Proof. Because of the partial invexity-incavity of the function f , we have for each

i = {1, 2, . . . , p}.
∫

I

f i(t , x1, ẋ1, v1, v̇1)d t −

∫

I

f i(t , u1, u̇1, v1, v̇1)d t

≧

∫

I

{ηT
1

f i

x1(t , u1, u̇1, v1, v̇1) + (Dη1)
T f i

ẋ1(t , u1, u̇1, v1, v̇1)}d t (17)

∫

I

f i(t , x1, ẋ1, v1, v̇1)d t −

∫

I

f i(t , x1, ẋ1, y1, ẏ1)d t

≦

∫

I

{ηT
2

f i

y1(t , x1, ẋ1, y1, ẏ1) + (Dη2)
T f i

ẏ1(t , x1, ẋ1, y1, ẏ1)}d t (18)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 587

Multiplying (17) by λi > 0 and summing over i.
∫

I

λT f (t , x1, ẋ1, v1, v̇1)d t −

∫

I

λT f (t , u1, u̇1, v1, v̇1)d t

≧

∫

I

{ηT
1
(λT fx1(t , u1, u̇1, v1, v̇1) + (Dη1)

TλT f ẋ1(t , u1, u̇1, v1, v̇1))}d t

Integrating by parts, the above inequality becomes
∫

I

λT f (t , x1, ẋ1, v1, v̇1)−

∫

I

λT f (t , u1, u̇1, v1, v̇1)d t

≧

∫

I

ηT
1
λT fx1(t , u1, u̇1, v1, v̇1)d t +ηT

1
λT f ẋ1(t , u1, u̇1, v1, v̇1)|

t=b

t=a

−

∫

I

ηT
1

DλT f ẋ1(t , u1, u̇1, v1, v̇1)d t

Using the boundary conditions which at t = a, t = b gives η1 = 0, we have
∫

I

λT f (t , x1, ẋ1, v1, v̇1)−

∫

I

λT f (t , u1, u̇1, v1, v̇1)d t

≧

∫

I

ηT
1
[λT fx1(t , u1, u̇1, v1, v̇1)d t − D(λT f ẋ1(t , u1, u̇1, v1, v̇1))]d t (19)

Multiplying (18) by λi, i ∈ {1, 2, . . . , p} and summing over i, we get,
∫

I

λT f (t , x1, ẋ1, v1, v̇1)−

∫

I

λT f (t , x1, ẋ1, y1, ẏ1)d t

≦

∫

I

{ηT
2
(λT f y1(t , x1, ẋ1, y1, ẏ1)) + (Dη2)

TλT f ẏ1(t , x1, ẋ1, y1, ẏ1)}d t

On integrating by parts the R.H.S of the above inequality and using the boundary

conditions which at t = a, t = b gives η2 = 0, we have
∫

I

λT f (t , x1, ẋ1, v1, v̇1)−

∫

I

λT f (t , x1, ẋ1, y1, ẏ1)d t

≦

∫

I

ηT
2
[(λT f y1(t , x1, ẋ1, y1, ẏ1))− D(λT f ẏ1(t , x1, ẋ1, y1, ẏ1))]d t (20)

Multiplying (20) by (-1) and adding to (19), we have
∫

I

λT f (t , x1, ẋ1, y1, ẏ1)d t −

∫

I

λT f (t , u1, u̇1, v1, v̇1)d t



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 588

≧

∫

I

ηT
1
[(λT fx1(t , u1, u̇1, v1, v̇1))− D(λT f ẋ1(t , u1, u̇1, v1, v̇1))]d t

−

∫

I

ηT
2
[(λT f y1(t , u1, u̇1, v1, v̇1))− D(λT f ẏ1(t , u1, u̇1, v1, v̇1))]d t (21)

Now from the inequality (9) along with (13), it follows

∫

I

ηT
1
(λT fu1(t , u1, u̇1, v1, v̇1)− DλT fu̇1(t , u1, u̇1, v1, v̇1))d t

≧−

∫

I

u1(t)T[λT fu1(t , u1, u̇1, v1, v̇1)− DλT fu̇1(t , u1, u̇1, v1, v̇1)]d t (22)

Also from the inequality (3) together with (14) implies

−

∫

I

ηT
2
(λT f y1(t , x1, ẋ1, y1, ẏ1)− DλT f ẏ1(t , x1, ẋ1, y1, ẏ1))d t

≧

∫

I

y1(t)T[λT f y1(t , x1, ẋ1, y1, ẏ1)− DλT f ẏ1(t , x1, ẋ1, y1, ẏ1)]d t (23)

Using (22) and (23), in (21), we have

∫

I

λT f (t , x1, ẋ1, y1, ẏ1)d t − y1(t)T
∫

I

λT f y1(t , x1, ẋ1, y1, ẏ1)d t

≧−

∫

I

u1(t)T[(λT fu1(t , u1, u̇1, v1, v̇1))− D(λT fu̇1(t , u1, u̇1, v1, v̇1))]d t

+

∫

I

y1(t)T[(λT f y1(t , x1, ẋ1, y1, ẏ1))

−D(λT f ẏ1(t , x1, ẋ1, y1, ẏ1))]d t , (24)

which implies

∫

I

{λT f (t , x1, ẋ1, y1, ẏ1)− y1(t)T (λT f y1(t , x1, ẋ1, y1, ẏ1)

−DλT f ẏ1(t , x1, ẋ1, y1, ẏ1))}d t

≧

∫

I

{λT fu1(t , u1, u̇1, v1, v̇1)− u1(t)T(λT fx1(t , u1, u̇1, v1, v̇1)

−DλT f ẋ1(t , u1, u̇1, v1, v̇1))}d t (25)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 589

Now from the inequality (10) along with (15), we have
∫

I

(ηT
3
(t , x2, ẋ2, u2, u̇2) + u2(t))(λT gu2(t , u2, u̇2, v2, v̇2)

−DλT gu̇2(t , u2, u̇2, v2, v̇2))≧ 0.

This implies
∫

I

ηT
3
(λT gu2(t , u2, u̇2, v2, v̇2)− D(λT gu̇2(t , u2, u̇2, v2, v̇2)))d t

≧ −

∫

I

u2(t)T[λT gu2(t , u2, u̇2, v2, v̇2)− D(λT gu̇2(t , u2, u̇2, v2, v̇2))]d t

Integrating by parts and using the boundary conditions which at t = a, t = b gives

η3 = 0, we have
∫

I

{ηT
3
(λT gu2(t , u2, u̇2, v2, v̇2) + (Dη3)

T (λT gu̇2(t , u2, u̇2, v2, v̇2)))}d t ≧ 0

Because of the partial pseudo-invexity of
∫

I
λT gu2 d t , this gives

∫

I

λT g(t , x2, ẋ2, y2, ẏ2)d t ≧

∫

I

λT g(t , u2, u̇2, v2, v̇2)d t (26)

Also from (4) together with (16), we have
∫

I

(ηT
4
(t , v2, v̇2, y2, ẏ2) + y2(t))(λT g y2(t , x2, ẋ2, y2, ẏ2)

−DλT g ẏ2(t , x2, ẋ2, y2, ẏ2))d t ≧ 0

This implies,
∫

I

ηT
4
(λT g y2(t , x2, ẋ2, y2, ẏ2))− D(λT g ẏ2(t , x2, ẋ2, y2, ẏ2))d t

≦ −

∫

I

y2(t)T[λT g y2(t , x2, ẋ2, y2, ẏ2)− D(λT g ẏ2(t , x2, ẋ2, y2, ẏ2))]d t

This in view of (5) yields,
∫

I

ηT
4
{λT g y2(t , x2, ẋ2, y2, ẏ2)− D(λT g ẏ2(t , x2, ẋ2, y2, ẏ2))}d t ≦ 0



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 590

On integrating by parts and using the boundary conditions which at t = a, t = b gives

η4 = 0, we have,

∫

I

ηT
4
{λT g y2(t , x2, ẋ2, y2, ẏ2) + (Dη4)

T (λT g ẏ2(t , x2, ẋ2, y2, ẏ2))}d t ≦ 0

Because of partial pseudo-incavity of
∫

I
λT g y2 d t , we have

∫

I

(λT g(t , x2, ẋ2, v2, v̇2))d t ≦

∫

I

(λT g(t , x2, ẋ2, y2, ẏ2))d t (27)

From (26) and (27), we get,

∫

I

(λT g(t , x2, ẋ2, y2, ẏ2))d t ≧

∫

I

(λT g(t , u2, u̇2, v2, v̇2))d t (28)

Combining (25) and (28), we get

∫

I

{λT f (t , x1, ẋ1, y1, ẏ1)− y1(t)T(λT f y1(t , x1, ẋ1, y1, ẏ1))

−DλT f ẏ1(t , x1, ẋ1, y1, ẏ1) +λT g(t , x2, ẋ2, y2, ẏ2)}d t

≧

∫

I

{λT f (t , u1, u̇1, v1, v̇1)− u1(t)T(λT fx1(t , u1, u̇1, v1, v̇1))

−DλT f ẋ1(t , u1, u̇1, v1, v̇1) +λT g(t , x2, ẋ2, y2, ẏ2)}d t

This implies,

λT

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)

−y1(t)T(λT f y1(t , x1, ẋ1, y1, ẏ1)−DλT f ẏ1(t , x1, ẋ1, y1, ẏ1))e}d t

≧ λT

∫

I

{ f (t , u1, u̇1, v1, v̇1) + g(t , u2, u̇2, v2, v̇2)

−u1(t)T(λT fx1(t , u1, u̇1, v1, v̇1)−DλT f ẋ1(t , u1, u̇1, v1, v̇1))e}d t

This implies,

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 591

−y1(t)T(λT f y1(t , x1, ẋ1, y1, ẏ1)−DλT f ẏ1(t , x1, ẋ1, y1, ẏ1))e}d t

6≤

∫

I

{ f (t , u1, u̇1, v1, v̇1) + g(t , u2, u̇2, v2, v̇2)

−u1(t)T(λT fx1(t , u1, u̇1, v1, v̇1)−DλT f ẋ1(t , u1, u̇1, v1, v̇1))e}d t

This was to be proved.

Theorem 2 (Strong Duality). Let ( x̄1, x̄2, ȳ1, ȳ2, λ̄) be an efficient solution of (Mix SP).

Let λ= λ̄ be fixed in (Mix SD) and

(C1)

∫

I

[{(φ1(t))T (λT f y1 y1 − DλT f y1 ẏ1)− Dφ1(t)T(−DλT f ẏ1 ẏ1)

+D2φ1(t)T(−λT f ẏ1 ẏ1)}φ1(t)]d t > 0,

and
∫

I

[{(φ2(t))T (λT g y2 y2 − DλT g y2 ẏ2)− Dφ2(t)T(−DλT g ẏ2 ẏ2)

+D2φ2(t)T(−λT g ẏ2 ẏ2)}φ2(t)]d t > 0,

(C2)

∫

I

[{(φ1(t))T (λT f y1 y1 − DλT f y1 ẏ1)− Dφ1(t)T(−DλT f ẏ1 ẏ1)

+D2φ1(t)T(−λT f ẏ1 ẏ1)}φ1(t)]d t = 0, t ∈ I ⇒ φ1(t) = 0, t ∈ I ,

and
∫

I

[{(φ2(t))T (λT g y2 y2 − DλT g y2 ẏ2)− Dφ2(t)T(−DλT g ẏ2 ẏ2)

+D2φ2(t)T(−λT g ẏ2 ẏ2)}φ2(t)]d t = 0, t ∈ I ⇒ φ1(t) = 0, t ∈ I .

and

(C3) g i

y2 − Dg i

ẏ2 = 0, i = 1, 2, . . . , p are linearly independent.

Let
∫

I
f d t and
∫

I
λT gd t satisfy the invexity and generalized invexity as stated in Theo-

rem 1, then ( x̄1, x̄2, ȳ , ȳ2, λ̄) and (ū1, ū2, v̄, v̄2, λ̄) are efficient solution of (Mix SP) and

(Mix SD) respectively.



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 592

Proof. Since ( x̄1, x̄2, ȳ1, ȳ2, λ̄) is efficient, it is weak minimum. Hence there exists

τ ∈ Rp,η ∈ Rp, γ ∈ R and piecewise smooth functions θ 1(t) : I → R|K1|,θ 2(t) : I →

R|k2| and µ : I → Rm such that the following Fritz-John optimality conditions, in view

of the analysis on [13, 9, 14], are satisfied

H = τ( f + g) + (θ 1(t)− (τT e)y1(t)T)(λT f y1 − DλT f ẏ1)

+(θ 2(t)− γy2(t)T)(λT g y2 − DλT g ẏ2) +ηTλ

Satisfying

Hx1 − DH ẋ1 + D2H ẍ1 = 0, t ∈ I (29)

Hx2 − DH ẋ2 + D2H ẍ2 = 0, t ∈ I (30)

H y1 − DH ẏ1 + D2H ÿ1 = 0, t ∈ I (31)

H y2 − DH ẏ2 + D2H ÿ2 = 0, t ∈ I (32)

(θ 1(t)− (τT e)y1(t))T ( f y1 − D f ẏ1) + (θ 2(t)− γy2(t))T(g y2 − Dg ẏ2)−η= 0, t ∈ I

(33)

θ 1(t)(λT f y1 − DλT f ẏ1) = 0, t ∈ I (34)

θ 2(t)(λT g y2 − DλT g ẏ2) = 0, t ∈ I (35)

γ

∫

I

y2(t)T(λT g y2 − DλT g ẏ2) = 0 (36)

ηT λ̄ = 0 (37)

(τ,θ 1(t),θ 2(t),η,γ)≧ 0, t ∈ I (38)

(τ,θ 1(t),θ 2(t),η,γ) 6= 0, t ∈ I (39)

hold throughout I (except at the corners of ( x̄1(t), x̄2(t), ȳ1(t), ȳ2(t)) where (29)-

(32) are valid for unique right and left hand limits). Here θ 1 and θ 2 are continuous

except possibly at corner of ( x̄1(t), x̄2(t), ȳ1(t), ȳ2(t)).



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 593

The relations (29)-(32) are all deducible from the classical Euler-Lagrange and

Clebsch necessary optimality conditions. Particularly, the equations (29)-(32) are the

famous Euler-Lagrange differential equation when second order derivatives appear in

H. Using the analogies of the observation of Dφ ÿ from the notational section, the

equations (29)-(32) become,

τ( fx1 − D f ẋ1)− (θ 1(t)− (τT e) ȳ1(t))T (λT f y1 x1 − DλT f ẏ1 x1)

−D(θ 1(t)− (τT e) ȳ1(t))T (λT f y1 ẋ1 − DλT f ẏ1 ẋ1 −λT f ẏ1 x1)

+D2((θ 1(t)− (τT e) ȳ1(t))T (−λT f ẏ1 ẋ1)) = 0 (40)

τ(gx2 − Dg ẋ2) + (θ 2(t)− γ ȳ2(t))T (λT g y2 x2 − DλT g ẏ2 x2)

−D(θ 2(t)− γ ȳ2(t))T (λT g y2 ẋ2 − DλT g ẏ2 ẋ2 −λT g ẏ2 x2)

+D2((θ 2(t)− γ ȳ2(t))T (−λT g ẏ2 ẋ2)) = 0 (41)

(τ− (τT e)λ)T ( f y1 − D f ẏ1) + (θ 1(t)− (τT e) ȳ1(t))T

×(λT f y1 y1 − DλT f ẏ1 y1)

−D(θ 1(t)− (τT e) ȳ1(t))T(−DλT f ẏ1 ẏ1)

+D2((θ 1(t)− (τT e) ȳ1(t))T(−λT f ẏ1 ẏ1)) = 0 (42)

(τ− γλ)T (g y2 − Dg ẏ2) + (θ 2(t)− γ ȳ2(t))T(λT g y2 y2 − DλT g ẏ2 y2)

−D(θ 2(t)− γ ȳ2(t))T(−DλT g ẏ2 ẏ2)

+D2(θ 2(t)− γ ȳ2(t))T (−λT g ẏ2 ẏ2) = 0 (43)

Since λ > 0, (37) implies η= 0. Consequently, (33) reduces to

(θ 1(t)− (τT e)y1(t))T ( f y1 − D f ẏ1) + (θ 2(t)− γy2(t))T (g y2 − Dg ẏ2) = 0, t ∈ I (44)

Postmultiplying (42) by (θ 1(t)− (τT e)y1(t)), (43) by (θ 2(t)− γy2(t)) and then

adding, we have

{(τ− (τT e)λ)T ( f y1 − D f ẏ1) + (θ 1(t)− (τT e) ȳ1(t))T (λT f y1 y1 − DλT f ẏ1 y1)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 594

−D[(θ 1(t)− (τT e) ȳ1(t))T (−DλT f ẏ1 ẏ1)]

+D2[(θ 1(t)− (τT e) ȳ1(t))T (−λT f ẏ1 ẏ1)]}(θ 1(t)− (τT e) ȳ1(t))

+{(τ− γλ)T (g y2 − Dg ẏ2) + (θ 2(t)− γ ȳ2(t))T (λT g y2 y2 − DλT g ẏ2 y2)

−D[(θ 2(t)− γ ȳ2(t))T (−DλT g ẏ2 ẏ2)]

+D2[(θ 2(t)− γ ȳ2(t))T (−λT g ẏ2 ẏ2)]}(θ 2(t)− γ ȳ2(t)) = 0 (45)

Now multiplying (44) by λ̄ and then using (35) and (36) we have

∫

I

(θ 1(t)− (τT e) ȳ1(t))T (λT f y1 − DλT f ẏ1)d t = 0

that is

∫

I

(θ 1(t)− (τT e) ȳ1(t))T (λT f y1 − DλT f ẏ1)(τT e)d t = 0 (46)

Multiplying (44) by τ, we have

∫

I

[(θ 1(t)− (τT e)y1(t))T (τ f y1 − Dτ f ẏ1)

+(θ 2(t)− γy2(t))T (τg y2 − Dτg ẏ2)]d t = 0 (47)

Subtracting (46) and (47) and using (35) and (36), we have

∫

I

[(θ 1(t)− (τT e)y1(t))T ( f y1 − D f ẏ1)(τ− (τT e)λ̄)

+(θ 2(t)− γy2(t))T (τg y2 − Dτg ẏ2)(τ− γλ̄)]d t = 0 (48)

From (45) and (48), we obtain

∫

I

[{(θ 1(t)− (τT e) ȳ1(t))T (λT f y1 y1 − DλT f y1 ẏ1)

−D[(θ 1(t)− (τT e) ȳ1(t))T (−DλT f ẏ1 ẏ1)]

+D2[(θ 1(t)− (τT e) ȳ1(t))T (−λT f ẏ1 ẏ1)]}.(θ 1(t)− (τT e) ȳ1(t))]d t

+

∫

I

[{(θ 2(t)− γ ȳ2(t))T (λT g y2 y2 − DλT g y2 ẏ2)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 595

−D[(θ 2(t)− γ ȳ2(t))T (−DλT g ẏ2 ẏ2)]

+D2[(θ 2(t)− γ ȳ2(t))T (−λT g ẏ2 ẏ2)]}(θ 2(t)− γ ȳ2(t))]d t = 0

In view of the hypothesis (C1), we have

∫

I

[{(θ 1(t)− (τT e) ȳ1(t))T (λT f y1 y1 − DλT f y1 ẏ1)

−D[(θ 1(t)− (τT e) ȳ1(t))T (−DλT f ẏ1 ẏ1)]

+D2[(θ 1(t)− (τT e) ȳ1(t))T (−λT f ẏ1 ẏ1)]}.(θ 1(t)− (τT e) ȳ1(t))]d t = 0

and

∫

I

[{(θ 2(t)− γ ȳ2(t))T (λT g y2 y2 − DλT g y2 ẏ2)

−D[(θ 2(t)− γ ȳ2(t))T (−DλT g ẏ2 ẏ2)]

+D2[(θ 2(t)− γ ȳ2(t))T (−λT g ẏ2 ẏ2)]}(θ 2(t)− γ ȳ2(t))]d t = 0

This in view of the hypothesis (C2) yields,

φ1(t) = θ 1(t)− (τT e) ȳ1(t) = 0, t ∈ I (49)

φ2(t) = θ 2(t)− γ ȳ2(t) = 0, t ∈ I (50)

From (50) and (43), we have

(τ− γλ)T (g y2 − Dg ẏ2) = 0

that is

p
∑

i=1

(τi − γλi)T (g y2 − Dg ẏ2) = 0

This in view of the (C3) yields

τi = γλi, i = 1, 2, . . . , p (51)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 596

Let if possible, γ = 0. Then from (51), we have τ = 0 and therefore, from (49) and

(50), we have

φ1(t) = 0,θ 2(t) = 0, t ∈ I

Hence (τ,θ 1(t),θ 2(t),η,γ) = 0, contradicting Fritz-John conditions (39). Hence

γ > 0 and consequently τ > 0.

From (40) and (4.29) along with (51), we obtain

(λ̄T fx1 − Dλ̄T f ẋ1) = 0, t ∈ I (52)

(λ̄T gx2 − Dλ̄T g ẋ2) = 0, t ∈ I (53)

which implies

∫

I

x2(t)T(λ̄T gx2 − Dλ̄T g ẋ2)d t = 0 (54)

From (52)-(54) together with (49), we have

y1(t)T (λ̄T f y1 − Dλ̄T f ẏ1) = 0, t ∈ I (55)

From the primal objective with (55)

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)

− y1(t)T(λT f y1−DλT f ẏ1)}d t

=

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)}d t (56)

From the dual objective in view of (52), we have

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)

− x1(t)T(λT fx1−DλT f ẋ1)}d t

=

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)}d t (57)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 597

From (55) and (57), the equality of objective values is evident. Consequently, in view

of the hypothesis of Theorem 1, the efficiency of ( x̄1, x̄2, ȳ1, ȳ2, λ̄) follows.

We now state converse duality whose proof follows by symmetry.

Theorem 3 (Converse Duality). Let ( x̄1, x̄2, ȳ1, ȳ2, λ̄) be an efficient solution of (Mix SP).

Let λ= λ̄ be fixed in (Mix SD) and

(A1)

∫

I

[{ψ1(t)T (λT fx1 x1 − DλT fx1 ẋ1)− Dψ1(t)T (−DλT f ẋ1 ẋ1)

+D2ψ1(t)T (−λT f ẋ1 ẋ1)}ψ1(t)]d t > 0,

and

∫

I

[{ψ2(t)T(λT gx2 x2 − DλT gx2 ẋ2)− Dψ2(t)T(−DλT g ẋ2 ẋ2)

+D2ψ2(t)T(−λT g ẋ2 ẋ2)}ψ2(t)]d t > 0,

(A2)

∫

I

[{ψ1(t)T(λT fx1 x1 − DλT fx1 ẋ1)− Dψ1(t)T(−DλT f ẋ1 ẋ1)

+D2ψ1(t)T(−λT f ẋ1 ẋ1)}ψ1(t)]d t = 0, t ∈ I ⇒ψ1(t) = 0, t ∈ I ,

and

∫

I

[{ψ2(t)T(λT gx2 x2 − DλT gx2 ẋ2)− Dψ2(t)T(−DλT g ẋ2 ẋ2)

+D2ψ2(t)T(−λT g ẋ2 ẋ2)}ψ2(t)]d t = 0, t ∈ I ⇒ψ2(t) = 0, t ∈ I

and

(A3) g i

x2 − Dg i

ẋ2 = 0, i = 1, 2, . . . , p are linearly independent.

Let
∫

I
f d t and
∫

I
λT gd t satisfy the invexity and generalized invexity as stated in

Theorem 1, then ( x̄1, x̄2, ȳ, ȳ2, λ̄) and (ū1, ū2, v̄, v̄2, λ̄) are efficient solution of (Mix

SP) and (Mix SD) respectively.



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 598

5. Self Duality

A problem is said to be self-dual if it is formally identical with its dual, in general,

the problems (Mix SP) and (Mix SD) are not formally in the absence of an additional

restrictions of the function f and g. Hence skew symmetric of f and g is assumed in

order to validate the following self-duality theorem.

Theorem 4 (Self Duality). Let f i and g i, i = 1, 2, . . . , p, be skew symmetric. Then the

problem (Mix SP) is self dual. If the problems (Mix SP) and (Mix SD) are dual problems

and ( x̄1(t), x̄2(t), ȳ(t), ȳ2(t), λ̄) is a joint optimal solution of (Mix SP) and (Mix SD),

then so is ( ȳ(t), ȳ2(t), x̄1(t), x̄2(t), λ̄), and the common functional value is zero, i.e.

Minimum(Mix SP) =

∫

I

{ f (x1, ẋ1, y1, ẏ1) + g(x2, ẋ2, y2, ẏ2)}d t = 0

Proof. By skew symmetric of f i and g i, we have

f i

x1(t , x1(t), ẋ1(t), y1(t), ẏ1(t)) = − f i

y1(t , y1(t), ẏ1(t), x1(t), ẋ1(t))

g i

x2(t , x2(t), ẋ2(t), y(t), ẏ2(t)) = −g i

ẏ2(t , y2(t), ẏ2(t), x2(t), ẋ2(t))

f i

y1(t , x1(t), ẋ1(t), y1(t), ẏ1(t)) =− f i

x1(t , y1(t), ẏ1(t), x1(t), ẋ1(t))

g i

y2(t , x2(t), ẋ2(t), y(t), ẏ2(t)) =−g i

ẋ2(t , y2(t), ẏ2(t), x2(t), ẋ2(t))

f i

x1(t , x1(t), ẋ1(t), y1(t), ẏ1(t)) = − f i

ẏ1(t , y1(t), ẏ1(t), x1(t), ẋ1(t))

g i

ẋ2(t , x2(t), ẋ2(t), y(t), ẏ2(t)) = −g i

ẏ2(t , y2(t), ẏ2(t), x2(t), ẋ2(t))

f i

y1(t , x1(t), ẋ1(t), y1(t), ẏ1(t)) =− f i

ẋ1(t , y1(t), ẏ1(t), x1(t), ẋ1(t))

g i

ẏ2(t , x2(t), ẋ2(t), y(t), ẏ2(t)) =−g i

ẋ2(t , y2(t), ẏ2(t), x2(t), ẋ2(t))

Recasting the dual problem (Mix SD) as a minimization problem and using the above

relations, we have

(Mix SD1) Minimize−

∫

I

{ f (t , y1, ẏ1, x1, ẋ1) + g(t , y2, ẏ2, x2, ẋ2)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 599

x1(t)T(λT fx1(t , y1, ẏ1, x1, ẋ1)

−DλT f ẋ1(t , y1, ẏ1, x1, ẋ1))e}d t

Subject to

x1(a) = 0 = x1(b), y1(a) = 0 = y1(b)

x2(a) = 0 = x2(b), y2(a) = 0 = y2(b)

λT fx1(t , y1, ẏ1, x1, ẋ1)− DλT f ẋ1(t , y1, ẏ1, x1, ẋ1)≦ 0, t ∈ I

λT gx2(t , y2, ẏ2, x2, ẋ2)− DλT g ẋ2(t , y2, ẏ2, x2, ẋ2)≦ 0, t ∈ I
∫

I

x2(t)T(λT gx2(t , y2, ẏ2, x2, ẋ2)

−DλT g ẋ1(t , y2, ẏ2, x2, ẋ2))d t ≧ 0

λ ∈ Λ+

This shows that the problem (Mix SD1) is just the primal problem (Mix SP). There-

fore, ( x̄1(t), x̄2(t), ȳ1(t), ȳ2(t), λ̄) is an optimal solution of (Mix SD) implies that

( ȳ1(t), ȳ2(t), x̄1(t), x̄2(t), λ̄) is an optimal solution for (Mix SP), and by symmetric

duality also for (Mix SD).

Now from (55)

Minimum (Mix SP)=

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)}d t

Correspondingly with the solution ( ȳ(t), ȳ2(t), x̄1(t), x̄2(t), λ̄), we have

Minimum (Mix SP)=

∫

I

{ f (t , y1, ẏ1, x1, ẋ1) + g(t , y2, ẏ2, x2, ẋ2)}d t

By the skew symmetric of f i and g i, we have

Minimum (Mix SP) =

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)}d t

=

∫

I

{ f (t , y1, ẏ1, x1, ẋ1) + g(t , y2, ẏ2, x2, ẋ2)}d t

= −

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)}d t



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 600

this yields,

Minimum (Mix SP) =

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2)}d t = 0

This accomplishes the proof of the theorem.

6. Natural Boundary Conditions

The pair of mixed symmetric multiobjective variational problem with natural bound-

ary values rather than fixed points may be formulated as,

Primal problem (Mix SP0)

Minimize

∫

I

{ f (t , x1, ẋ1, y1, ẏ1) + g(t , x2, ẋ2, y2, ẏ2).

−y1(t)T(λT f y1(x1, ẋ1, y1, ẏ1)

−DλT f ẏ1(x1, ẋ1, y1, ẏ1))e}d t

Subject to

λT f y1(t , x1, ẋ1, y1, ẏ1)− DλT f ẏ1(t , x1, ẋ1, y1, ẏ1) ≦ 0

λT g y2(t , x2, ẋ2, y2, ẏ2)− DλT g ẏ2(t , x2, ẋ2, y2, ẏ2)≦ 0
∫

I

y2(t)T(λT g y2(t , x2, ẋ2, y2, ẏ2)

−DλT g ẏ2(t , x2, ẋ2, y2, ẏ2))≧ 0

λT f y1(t , u1, u̇1, v1, v̇1)|t=a = 0, λT f ẏ1(t , u1, u̇1, v1, v̇1)|t=b = 0

λT g y2(t , x2, ẋ2, y2, ẏ2)|t=a = 0, λT g ẏ2(t , x2, ẋ2, y2, ẏ2)|t=b = 0

λ ∈ Λ+

Dual problem (Mix SD0)

Maximize

∫

I

{ f (u1, u̇1, v1, v̇1) + g(u2, u̇2, v2, v̇2)

−u1(t)T(λT fu1(u1, u̇1, v1, v̇1)



I. Husain and R. Mattoo / Eur. J. Pure Appl. Math, 2 (2009), (578-603) 601

−DλT fu̇1(u1, u̇1, v1, v̇1))e}d t

Subject to

λT fu1(t , u1, u̇1, v1, v̇1)− DλT fu̇1(t , u1, u̇1, v1, v̇1)≧ 0

λT gu2(t , u2, u̇2, v2, v̇2)− DλT gu̇2(t , u2, u̇2, v2, v̇2)≧ 0
∫

I

u2(t)T(λT gu2(t , u2, u̇2, v2, v̇2)

−DλT gu̇2(t , u2, u̇2, v2, v̇2))d t ≦ 0,

λT f ẋ1(t , u1, u̇1, v1, v̇1)|t=a = 0, λT f ẋ1(t , u1, u̇1, v1, v̇1)|t=b = 0

λT g ẋ2(t , x2, ẋ2, y2, ẏ2)|t=a = 0, λT g ẋ2(t , x2, ẋ2, y2, ẏ2)|t=b = 0

λ ∈ Λ+

For these problems, Theorem 1-3 will remain true except that some slight modifica-

tions in the arguments for these theorems are to be indicated.

7. Mathematical Programming

If the time dependency of (Mix SP) and (Mix SD) is removed and b − a = 1, we

obtain following pair of static mixed type multiobjective dual problems studied by

Bector, Chandra and Abha [2].

Primal (Mix SP1) Minimize f (x1, y1) + g(x2, y2)− (y1)T (λT f y1(x1, y1)

Subject to

λT f y1(x1, y1)≦ 0,

λT g y2(x2, y2)≦ 0,

y2(t)T(λT g y2(x2, y2))≧ 0,

λ ∈ Λ+

Dual (Mix SD1) Maximize f (u1, v1) + g(u2, v2)− u1(t)T(λT fu1(u1, v1))



REFERENCES 602

Subject to

λT fu1(u1, v1) ≧ 0,

λT gu2(u2, v2)≧ 0,

(u2)T (λT gu2(u2, v2).≦ 0,

λ ∈ Λ+.

ACKNOWLEDGEMENTS The authors are grateful to the anonymous referee for

his/her valuable comments that have substantially improved the presentation of this

research.

References

[1] M.S. Bazaraa and J.J. Goode, On Symmetric Duality in Nonlinear Programming, Oper-

ations Research 21(1) (1973), 1–9.

[2] C.R. Bector, Chandra and Abha, On Mixed Type Symmetric Duality in Multiobjective

Programming, Opsearch 36(4) (1999), 399–407.

[3] C.R. Bector, S. Chandra and I. Husain, Generalized Concavity and Duality in Continuous

Programming, Utilitas, Mathematica 25(1984), 171–190.

[4] C.R. Bector and I. Husain, Duality for Multiobjective Variational Problems, Journal of

Math. Anal and Appl. 166(1) (1992), 214–224.

[5] A. Ben-Israel and B. Mond, What is Invexity? J. Austral. Math. Soc. Ser. B 28(1986),

1–9.

[6] S. Chandra and I. Husain, Symmetric Dual Continuous Fractional Programming, J. Inf.

Opt. Sc. 10(1989), 241–255.

[7] W.S. Dorn, Asymmetric Dual Theorem for Quadratic Programs, Journal of Operations

Research Society of Japan 2(1960) 93–97.

[8] G.B. Dantzig, Eisenberg and R.W. Cottle, Symmetric Dual Nonlinear Programs, Pacific

Jounal of Mathematics 15(1965) 809–812.



REFERENCES 603

[9] Gulati, I. Husain and A. Ahmed, Multiobjective Symmetric Duality with Invexity, Bulletin

of the Australian Mathematical Society 56(1997) 25–36.

[10] I. Husain and Z. Jabeen, Mixed Type Symmetric and Self Duality for Variational Prob-

lems, Congressus Numerantum 171(2004), 77–103.

[11] D.H. Martin, The Essence of Invexity, Journal of Optimization Theory and Applicatins

47(1) (1985), 65–76.

[12] B. Mond, A Symmetric Dual Theorem for Nonlinear Programs, Quaterly Jounal of ap-

plied Mathematics 23(1965) 265–269.

[13] B. Mond and R.W. Cottle, Self Duality in Mathematical Programming, SIAM J. Appl.

Math. 14(1966), 420–423.

[14] B. Mond and T. Weir, Generalized Concavity and Duality, in : S.Sciable, W.T.Ziemba

(Eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York,

(1981).

[15] B. Mond and M.A. Hanson, Symmetric Duality for Variational problems, J. Math. Anal.

Appl. 18(1967) 161-172.

[16] N.G. Rueda and M.A. Hanson, Optimality Criteria in Mathematical Programming In-

volving Generalized Invexity. Journal of Mathematical Analysis and Applications 130(2),

375–385.

[17] F.A. Valentine, The Problem of Lagrange with Differential Inequalities as Added Side

Conditions, Contributions to calculus of variations, 1933-37, Univ. Of Chicago Press,

(1937), 407–448.

[18] Z. Xu, Mixed Type Duality in Multiobjective Programming Problems, Journal of Mathe-

matical Analysis and Applications 198(1996), 621–663