9_husian.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 2, 2009, (278-295)

ISSN 1307-5543 – www.ejpam.com

Second-Order Duality for Variational Problems

I. Husain1∗, A. Ahmed and Mashoob Masoodi2

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. A Mond-Weir type second-order dual to a variational problem is constructed and

the notion of second-order invexity and second order generalized invexity are introduced

in variational problems. Under these second-order pseudoinvexity and second-order quasi-

invexity assumptions, weak, strong and converse duality results are established. It is pointed

out that our duality results can be viewed as dynamic generalizations of corresponding (static)

duality results in nonlinear programming.

AMS subject classifications: Primary 90C30, Secondary 90C11, 90C20, 90C26

Key words: Mond-Weir type second-order dual, Variational problem, Second-order invexity,

Natural boundary values, Nonlinear programming.

∗Corresponding author.

Email address: ihusain11�yahoo.
om (I. Husain)

http://www.ejpam.com 278 c© 2009 EJPAM All rights reserved.



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 279

1. Introduction

The calculus of variation has been one of the prominent branches of analysis, for

more than two centuries. It is a tool of great power that can be used to wide variety

of problems, in pure mathematics. It can also be used to express basic principles of

mathematical physics in forms of utmost simplicity and elegance. Hanson [6] pointed

out that some of the duality results in the mathematical programming have the ana-

logues in calculus of variations. Exploring this relationship between mathematical

programming and classical calculus of variation, Mond and Hanson [8] formulated

a constrained variational problem as mathematical programming problem in abstract

space and using Valentine [10] optimality conditions for the same, presented its Wolfe

dual variational problem for validating various duality results under usual convexity.

Later Bector, Chandra and Husain [2] studied Mond-Weir type duality for the prob-

lem of Mond and Hanson [8] for relaxing its convexity requirements. In [3] Chandra,

Craven and Husain studied optimality and duality for a class of nondifferentiable vari-

ational problems in which the integrand of the objective functional contains a term

of a square root of the quadratic form, while in [5], Husain and Jabeen studied op-

timality criteria and duality for variational problems in which integrand of objective

and constraint functions contains terms of support functions.

Second-order duality in mathematical programming has been extensively studied

in recent years. Mangasarian [7] was the first to identify a second-order dual for-

mulation for non-linear programs under the assumptions that are complicated and

somewhat difficult to verify. Mond [9] introduced the concept of second-order con-

vex functions (named as bonvex functions by Bector and Chandra [1]) and studied

second-order duality for nonlinear programs.

Recently Chen [4] formulated Wolfe type second-order dual problem to the ortho-

dox variational problem and studied usual duality results under invexity assumptions



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 280

on the functions that occur in the formulation of the problem along with some strange

assumptions. Mond [9] has pointed out that the second-order dual for a nonlinear

programming gives a tighter bound and has computational advantage over a first or-

der dual. Motivated with this of Mond [9] in this exposition, we construct Mond-Weir

type second-order dual to the variational problem and derive usual duality results

under second-order pseudo-invexity and second order quasi-invexity assumptions.

The relationship of our results to second-order duality results in nonlinear pro-

gramming reported in [1] is indicated. In essence it is shown that our duality results

can be viewed as dynamic generalizations of corresponding (static) duality theorems

of nonlinear programming already in the literature.

2. Definitions and Related Pre-requisites

Let I = [a, b] be a real interval, f : I × Rn × Rn → R and g : I × Rn × Rn → Rm

be twice continuously differentiable functions. In order to consider f (t , x (t) , ẋ (t)),

where x : I → Rn is differentiable with derivative ẋ , denoted by fx and f ẋ the partial

derivative of f with respect to x and ẋ , respectively, that is,

fx =





























∂ f

∂ x1

∂ f

∂ x2

...

∂ f

∂ x n





























, f ẋ =





























∂ f

∂ ẋ1

∂ f

∂ ẋ2

...

∂ f

∂ ẋ n































I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 281

denote by fx x the Hessian matrix of f with respect to x , that is,

fx x =





























∂ 2 f

∂ x1∂ x1

∂ 2 f

∂ x1∂ x2
· · ·

∂ 2 f

∂ x1∂ x n

∂ 2 f

∂ x2∂ x1

∂ 2 f

∂ x2∂ x2
· · ·

∂ 2 f

∂ x2∂ x n

...
...

. . .
...

∂ 2 f

∂ x n∂ x1

∂ 2 f

∂ x n∂ x2
· · ·

∂ 2 f

∂ x n∂ x n





























n×n

It is obvious that fx x is a symmetric n× n matrix. Denote by gx the m× n matrix

with respect to x , that is,

gx =





























∂ g1

∂ x1

∂ g1

∂ x2
· · ·

∂ g1

∂ x n

∂ g2

∂ x1

∂ g2

∂ x2
· · ·

∂ g2

∂ x n

...
...

. . .
...

∂ gm

∂ x1

∂ gm

∂ x2
· · ·

∂ gm

∂ x n





























m×n

Similarly f ẋ , f ẋ x , fx ẋ and g ẋ can be defined.

Denote by X , the space of piecewise smooth functions x : I → Rn, with the norm

‖x‖ = ‖x‖∞ + ‖Dx‖∞ + ‖D
2 x‖∞, where the differentiation operator D is given by

u= D x⇔ x (t) = α+

∫ t

a

u (s) ds,

where α is given boundary value; thus
d

d t
= D except at discontinuities.

We introduce the following definitions which are needed for the duality results to

hold.

Definition 2.1 (Second-order Invexity). If there exists a vector function η= η(t , x , x̄) ∈

Rn where η : I × Rn × Rn → Rn and with η = 0 at t = a and t = b, such that for the



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 282

functional
∫

I
φ(t , x , ẋ)d t where φ : I × Rn× Rn→ R satisfies

∫

I

φ(t , x , ẋ)d t −

∫

I

�

φ(t , x̄, ˙̄x)−
1

2
β (t)T Gβ (t)

�

d t

≥

∫

I

n

ηTφx + (Dη)
Tφ ẋ +η

T Gβ(t)
o

d t ,

then
∫

I
φ(t , x , ẋ)d t is second-order invex with respect to η where G = φx x − Dφx ẋ +

D2φ ẋ ẋ and β ∈ C(I , Rn), the space of continuous n-dimensional vector function. The

function β is analogous to the auxiliary vector p in [1].

Definition 2.2 (Second-order Pseudoinvex). If the functional
∫

I
φ(t , x , ẋ)d t satisfies

∫

I

¦

ηTφx + (Dη)
Tφ ẋ +η

T Gβ(t)
©

d t ≥ 0

⇒

∫

I

φ(t , x , ẋ)d t ≥

∫

I

�

φ(t , x̄, ˙̄x)−
1

2
β (t)

T
Gβ (t)

�

d t ,

then
∫

I
φ (t , x , ẋ) d t is said to be second-order pseudoinvex with respect to η.

Definition 2.3 (Second-order Quasi-invex). If the functional
∫

I
φ(t , x , ẋ)d t satisfies

∫

I

φ(t , x , ẋ)d t ≤

∫

I

�

φ(t , x̄, ˙̄x)−
1

2
β (t)

T
Gβ (t)

�

d t

⇒

∫

I

n

ηTφx + (Dη)
Tφ ẋ +η

T Gβ(t)
o

d t ≤ 0,

then
∫

I
φ(t , x , ẋ) is said to be second-order quasi-invex with respect to η.

If φ does not depend on t , then the above definitions reduce to those given in [1]

for static cases.

Consider the following constrained variational problem:

(VP) : Minimize

∫

I

f (t , x , ẋ)d t



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 283

Subject to

x(a) = 0 = x(b),

g(t , x , ẋ) ≤ 0, t ∈ I ,

h(t , x , ẋ) = 0, t ∈ I ,

where f : I × Rn × Rn → R, g : I × Rn × Rn → Rm and h : I × Rn × Rn → Rk are

continuously differentiable.

Proposition 2.1 ([3] (Fritz-John Conditions)). If (CP) attains a local (or) global min-

imum at x = x̄ ∈ X then there exist Lagrange multiplier τ ∈ R, z : I → Rk and piecewise

smooth y : I → Rm such that

τ fx(t , x̄ , ˙̄x) + y(t)T gx(t , x̄, ˙̄x) + z(t)Thx(t , x̄, ˙̄x)

−D[ f ẋ(t , x̄ , ˙̄x) + y(t)T g ẋ(t , x̄, ˙̄x) + z(t)Th ẋ(t , x̄, ˙̄x)] = 0, , t ∈ I ,

y(t)T g(t , x̄ , ˙̄x) = 0, t ∈ I

(τ, y(t)) ≥ 0, t ∈ I

(τ, y(t), z(t)) 6= 0, t ∈ I

The Fritz John necessary conditions for (CP), become the Karush-Kuhn-Tucker

conditions [7] if τ = 1. If τ= 1, the solution x̄ is said to be normal.

3. Second-Order Duality

Consider the following variational problem (CP) by ignoring the equality con-

straint of (VP):

(CP) : Minimize

∫

I

f (t , x , ẋ)d t



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 284

Subject to

x (a) = 0= x (b) , (3.1)

g (t , x , ẋ)≤ 0 , t ∈ I , (3.2)

Chen [4] presented the following Wolfe type second-order dual problem for (CP)

analogous to that for nonlinear programming by Mangasarian [7] and established

various duality results under somewhat strange invexity-like conditions.

Maximize :

∫ b

a

n

f (t , u(t), u̇(t)) +α(t)T g(t , u(t), u̇(t))

1

2
β(t)T[ fuu(t , u(t), u̇(t)) + (gu(t , u(t), u̇(t))Tα(t))u

−2D( fuu̇(t , u(t), u̇(t)) + (gu(t , u(t), u̇(t))Tα(t))u̇)

+D2( fu̇u̇(t , u(t), u̇(t)) + (gu̇(t , u(t), u̇(t))Tα(t))u̇)]β(t)
o

d t

Subject to

u(a) = 0= u(b), u̇(a) = 0 = u̇(b)

fu(t , u(t), u̇(t)) + gu(t , u(t), u̇(t))Tα(t)

−D[ fu̇(t , u(t), u̇(t)) + gu̇(t , u(t), u̇(t))Tα(t)]

+[ fuu(t , u(t), u̇(t)) + (gu(t , u(t), u̇(t))Tα(t))u

−2D( fuu̇(t , u(t), u̇(t)) + (gu(t , u(t), u̇(t))Tα(t))u̇)

+D2( fu̇u̇(t , u(t), u̇(t)) + (gu̇(t , u(t), u̇(t))Tα(t))u̇)]β(t) = 0,

t ∈ I ,



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 285

α(t) ∈ Rm
+

, β(t) ∈ Rn, t ∈ I

where Rm
+

designates the non-negative orthant of the Euclidean space Rn.

Let

H = fuu(t , u(t), u̇(t)) + (gu(t , u(t), u̇(t))Tα(t))u

−2D( fuu̇(t , u(t), u̇(t)) + (gu(t , u(t), u̇(t))Tα(t))u̇)

+D2( fu̇u̇(t , u(t), u̇(t)) + (gu̇(t , u(t), u̇(t))Tα(t))u̇).

Then the above dual problem can be expressed in a much simpler form which is

given below.

(VD) Maximize :

∫ b

a

{ f (t , u(t), u̇(t)) +α(t)T g(t , u(t), u̇(t))

−
1

2
β(t)T H(t , u(t), u̇(t),α(t),β(t))}d t

Subject to

u(a) = 0= u(b), u̇(a) = 0 = u̇(b)

fu(t , u(t), u̇(t)) + gu(t , u(t), u̇(t))Tα(t)

−D[ fu̇(t , u(t), u̇(t)) + gu̇(t , u(t), u̇(t))Tα(t)]

+H(t , u(t), u̇(t))α(t)β(t) = 0, t ∈ I

α(t) ∈ Rm
+

, β(t) ∈ Rn, t ∈ I

It is remarked here that if f and g are independent of t , then (VD) becomes

second-order dual problem studied by Mangasarian [7].

Now we present the following Mond-Weir type second-order dual (CD) in the

spirit of [1] to relax second-order invexity requirements and establish various duality



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 286

results between the problems (CP) and (CD) under generalized second-order invexity

hypothesis.

(CD): Maximize

∫

I

{ f (t , u, u̇)−
1

2
β(t)T Fβ(t)}d t

Subject to

u(a) = 0= u(b) (3.3)

fu + y(t)T gu − D( fu̇ + y(t)T gu̇) + (F +H)β(t) = 0, t ∈ I (3.4)

∫

I

�

y(t)T g(t , u, u̇)−
1

2
β(t)T Hβ(t)

�

d t ≥ 0 , (3.5)

y(t)≥ 0, t ∈ I (3.6)

where F = fuu− D fuu̇+ D2 fu̇u̇ and H = (y(t)T gu)u− D(y(t)T gu)u̇+ D2(y(t)T gu̇)u̇ and

define D =
d

d t
as defined earlier.

If f and g are independent of t then F = fuu and H = (yT gu)u and consequently

(CD) will reduce to the second-order dual problem introduced in [1].

Theorem 3.1 (Weak duality). Let x(t) ∈ X be a feasible solution of (CP) and

(u(t), y(t),β(t)) be feasible solution of (CD).If
∫

I
f (t , ., .)d t be second-order pseudoin-

vex and
∫

I
y (t)

T
g(t , ., .) d t be second-order quasi-invex with respect to the same η :

I × Rn× Rn→ Rn satisfying η= 0 at t = a and t = b, then
∫

I

f (t , x , ẋ)d t ≥

∫

I

�

f (t , u, u̇)−
1

2
β(t)T Fβ (t)

�

d t

Proof. The relations, g(t , x , ẋ) ≤ 0, y(t)≥ 0, t ∈ I and (3.5) imply
∫

I

y(t)T g(t , x , ẋ)d t ≤

∫

I

�

y(t)T g(t , u, u̇)−
1

2
β(t)T Hβ(t)

�

d t ,

This, because of second-order quasi-invexity of
∫

I
y(t)T g(t , ., .)d t , implies that

∫

I

{ηT (y(t)T gu) + (Dη)
T (y(t)T gu̇) +η

T Hβ(t)}d t ≤ 0



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 287

i.e.,
∫

I

ηT (y(t)T gu)d t +

∫

I

(Dη)T (y(t)T gu̇)d t +

∫

I

ηT Hβ(t)d t ≤ 0

This, by integration by parts, this inequality yields,
∫

I

ηT (y(t)T gu)d t +ηy(t)T gu̇|
b
a
−

∫

I

ηT D(y(t)T gu̇)d t +

∫

I

ηT Hβ(t)d t ≤ 0

Using η = 0 at t = a and t = b in the above inequality, we obtain,
∫

I

η[(y(t)T gu)− D(y(t)T gu̇) +η
T Hβ(t)]d t ≤ 0,

Using (3.4), this gives
∫

I

[ηT ( fu − D fu̇) +η
T Fβ(t)]d t ≥ 0.

Integrating by parts, gives
∫

I

(ηT fu + (Dη)
T fu̇ +η

T Fβ(t)d t)≥ 0.

This, in view of second-order pseudoinvexity of
∫

I
f (t , ., .)d t implies

∫

I

f (t , x , ẋ)d t ≥

∫

I

�

f (t , u, u̇)−
1

2
β(t)T Fβ(t)

�

d t .

This implies,

infimum(CP) ≥ supremum(CD).

Theorem 3.2 (Strong Duality). If x̄(t) ∈ X is an optimal solution of (CP) and meets

the normality conditions, then there exists a piece wise smooth ȳ : R → Rm such that

( x̄(t), ȳ(t),β(t) = 0) is a feasible for (CD) and the two objective values are equal.

Furthermore, if the hypothesis of Theorem 1 holds, then ( x̄(t), ȳ(t),β(t)) is an optimal

solution for (CD).



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 288

Proof. From Proposition 1, there exists a piece wise smooth function ȳ : R→ Rm

satisfying the following conditions:

( fx(t , x̄ , ˙̄x) + ȳ(t)T gx(t , x̄ , ˙̄x))− D( f ẋ(t , x̄ , ˙̄x) + ȳ(t)T g ẋ(t , x̄ , ˙̄x)) = 0, t ∈ I

i.e,

( fx(t , x̄ , ˙̄x) + ȳ(t)T gx(t , x̄ , ˙̄x))

−D( f ẋ(t , x̄ , ˙̄x) + ȳ(t)T g ẋ(t , x̄ , ˙̄x)) + (F + H)β(t) = 0, (3.7)

where

β(t) = 0, t ∈ I

ȳ(t)T g(t , x̄ , ˙̄x) = 0

i.e.,
∫

I

{ ȳ(t)T g(t , x̄ , ˙̄x)−
1

2
β(t)T Hβ(t)}d t = 0, where β(t) = 0, t ∈ I (3.8)

ȳ(t)≥ 0, t ∈ I (3.9)

From (3.7), (3.8) and (3.9), it implies that ( x̄(t), ȳ(t),β(t) = 0) is feasible for

(CD) and the objective value of (CP) and (CD) are equal. The optimality of

( x̄(t), ȳ(t),β(t)) follows by an application of Theorem 1.

Theorem 3.3 (Converse duality). Suppose that f and g are thrice continuously differ-

entiable. Let ( x̄(t), ȳ(t),β(t)) be an optimal solution of (CD) at which

(A1): the Hessian matrices F and H are not the multiple of each other.

(A2): y(t)T gx − D y(t)T g ẋ 6= 0,

(A3): (i)
∫

I
β(t)T(y(t)T gx − D y(t)T g ẋ)d t ≥ 0 and

∫

I
β(t)T Hβ(t)d t > 0

or



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 289

(ii)
∫

I
β(t)T(y(t)T gx − D y(t)T g ẋ)d t ≤ 0 and

∫

I
β(t)T Hβ(t)d t < 0

If, for all feasible (x(t), y(t),β(t)),
∫

I
f (t , ., .)d t be second order pseudoinvex and

∫

I
y(t)T g(t , ., .)d t be second-order quasi-invex with respect to the same η, then x̄(t) is

an optimal solution of (P).

Proof. Since ( x̄(t), ȳ(t),β(t)) is an optimal solution for (CD), by proposition 1,

there exist Lagrange multiplier α ∈ R, and piece wise smooth λ : I → Rn, γ ∈ R and

µ : I → Rm such that Fritz-John conditions hold at
�

x̄ (t) , ȳ (t) ,β (t)
�

:

−α
��

fx −
1

2
(β(t)T Fβ(t))x

�

− D

�

f ẋ −
1

2
(β(t)T Fβ(t)) ẋ

��

+λ(t)T
n

fx x + (y(t)
T gx)x − D( f ẋ x + (y(t)

T g ẋ)x) + ((F +H)β(t))x

−D( fx ẋ + (y(t)
T gx) ẋ)− ( f ẋ ẋ + (y(t)

T g ẋ) ẋ) + ((F +H)β(t)) ẋ

o

γ

�

y(t)T gx −
1

2
(β(t)T Fβ(t))x

−D(y(t)T g ẋ −
1

2
(β(t)T Fβ(t)) ẋ)

�

= 0, t ∈ I , (3.10)

(λ(t) +αβ(t))F + (λ(t) + γβ(t))H = 0, t ∈ I (3.11)

λ(t)T
�

g j x − Dg j ẋ + (g j x x − Dg j x ẋ + D2 g ẋ ẋ)β(t)
�

+γ

�

g j +
1

2
β(t)T(g j x x − Dg j x ẋ + D2 g ẋ ẋ)β(t)

�

+µ j(t) = 0, t ∈ I (3.12)

( fx + y(t)T g ẋ)− D( f ẋ + y(t)T g ẋ) + (F +H)β(t) = 0, t ∈ I (3.13)

γ

∫

I

�

y(t)T g −
1

2
β(t)Hβ(t)

�

d t = 0, t ∈ I (3.14)

µT (t) ȳ(t) = 0, t ∈ I (3.15)

(α,γ,µ(t)) ≥ 0, t ∈ I (3.16)



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 290

(α,γ,λ(t),µ(t)) 6= 0, t ∈ I (3.17)

In view of hypothesis (A1), the equation (3.11) yields,

λ(t) +αβ(t) = 0, t ∈ I

λ(t) + γβ(t) = 0, t ∈ I







(3.18)

Multiplying (3.12) by y j(t) and summing over j, we have

λ(t)T
h

y(t)T gx − D(y(t)T g ẋ) + ((y(t)
T gx)x − D(y(t)T gx) ẋ + D2(y(t)T g ẋ) ẋ)β(t)

i

−γ
�

y(t)T gx −
1

2
β(t)T((y(t)T gx)x − D(y(t)T gx) ẋ + D2(y(t)T g ẋ) ẋ)β(t)

�

+µT (t)y(t) = 0,

Using (3.15) and then integrating, we have
∫

I

λ(t)T{y(t)T gx − D(y(t)T g ẋ) +Hβ(t)}d t

−γ

∫

I

{y(t)T gx −
1

2
β(t)T Hβ(t)}d t = 0

This, because of (3.14), yields,
∫

I

λ(t)T{y(t)T gx − D(y(t)T g ẋ) +Hβ(t)}d t = 0 (3.19)

If (α,γ) = 0 i.e. α = 0 = γ, then (3.18) implies λ(t) = 0, t ∈ I and µ(t) = 0 from

(3.12).

Thus, we have

(α,γ,λ(t),µ(t)) = 0.

This contradicts (3.17). Hence

(α,γ) 6= 0 i.e. α > 0 or γ > 0.



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 291

We claim β(t) = 0, t ∈ I . Suppose that β(t) 6= 0, t ∈ I .

From (3.18) we have

(α− γ)β(t) = 0

implying α = γ > 0. Using (3.18) in (3.19), we have
∫

I

αβ(t)T{y(t)T gx − D(y(t)T g ẋ) +Hβ(t)}d t = 0

implies
∫

I

β(t)T{y(t)T gx − D(y(t)T g ẋ)}d t +

∫

I

β(t)T Hβ(t)d t = 0 (3.20)

In view of the hypothesis (A3) i.e.,
∫

I

{β(t)T y(t)T gx − D(y(t)T g ẋ)}d t ≥ 0

and
∫

I

β(t)T Hβ(t)d t > 0.

We have
∫

I

β(t)T{y(t)T gx − D(y(t)T g ẋ) +Hβ(t)}d t > 0

This contradicts (3.20). Hence β(t) = 0, t ∈ I . Consequently (3.18) implies λ(t) = 0,

t ∈ I .

From (3.10), we have

−α( fx − D f ẋ) + γ(y(t)
T gx − D y(t)T g ẋ) = 0 (3.21)

Also from (3.4), we have

( fx − D f ẋ) = −(y(t)
T gx − D(y(t)T g ẋ))

Using this in (3.21), we have

(α− γ)(y(t)T gx − D y(t)T g ẋ) = 0



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 292

In view of the hypothesis (A2), this gives

α = γ > 0.

From (3.12) we have

γg j + µ j(t) = 0

Because γ > 0, this gives

g j = −
µ j(t)

γ
≤ 0

g(t , x̄ , ˙̄x) ≤ 0

x̄ is feasible to (CP). In view of β(t) = 0, t ∈ I gives the equality of two objective

values follows. The optimality of x̄ for (CP) follows from Theorem 1.

4. Natural Boundary Values

In this section, we formulate dual variational problem with natural boundary val-

ues rather than fixed end points.

(C P0) : Minimize

∫

I

f (t , x , ẋ)d t

Subject to

g(t , x , ẋ)≤ 0, t ∈ I

(C D0) : Maximize

∫

I

{F(t , x , ẋ)−
1

2
β(t)T Fβ(t)}d t

Subject to

fx + y(t)T gx − D( f ẋ + y(t)T g ẋ) + (F +H)β(t) = 0 t ∈ I



I. Husain, A. Ahmed, and M. Massodi / Eur. J. Pure Appl. Math, 2 (2009), (278-295) 293

y(t)≥ 0, t ∈ I

y(t)T g ẋ |t=a = 0,

y(t)T g ẋ |t=b = 0,

We shall not repeat the proofs of Theorem 3.1-3.3, as these follow on the lines of

the analysis given in [1].

5. Nonlinear Programming

If all functions in (C P0) and (C D0) are independent of t , then these problems will

reduce to following pair of dual problems, treated by Bector and Chandra [1].

(P1) : Minimize f (x)

Subject to

g(x) ≤ 0,

(D1) : Maximize f (x)−
1

2
pT∇2 f (x)p

Subject to

∇( f + yT g) +∇2( f + yT g)p = 0

yT g(x)−
1

2
pT∇2(yT g(x))p ≥ 0

y ≥ 0

where

fx(x) =∇ f (x), yT g(x) =∇(yT g), fx x(x) =∇
2 f (x)



REFERENCES 294

∇2(yT g(x)) = (yT gx)x and β = p

6. Conclusion

We have considered a pair of Mond-Weir type second order dual that relaxes the

invexity requirement in Chen [4] to validate duality theorems. The approach chosen

here is to render the problem analogous to the second-order dual problems intro-

duced in [1] as a mathematical programming problem in infinite dimensional space.

Our dual model presents simpler dual objective function and also allows the weaking

of the invexity assumption of [4]. There is a rich scope to study this problem in multi-

objective setting. One can also formulate a fractional analogue of our model to study

duality results.

ACKNOWLEDGEMENTS. The authors are grateful to the anonymous referee for

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

research.

References

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problems", J. Math. Anal. Appl.107: 122-131 (1985).

[4] X.Chen, "Second order duality for the variational problems", J. Math. Anal. Appl. 286

261-270 (2003).



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