2_534_geeta.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 5, 2010, 786-805

ISSN 1307-5543 – www.ejpam.com

Optimality and Duality for Second-order Multiobjective

Variational Problems

T. R. Gulati and Geeta Mehndiratta∗

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247 667, India

Abstract. In this work, we first modify a converse duality theorem for second-order dual of a scalar

variational problem. We then consider its multiobjective analogue and obtain necessary optimality

conditions and duality relations. At the end, the static case of our problems has also been discussed.

2000 Mathematics Subject Classifications: 90C26, 90C29, 90C30, 90C46

Key Words and Phrases: Multiobjective programming; Variational problem; Second-order duality;

Efficient solutions; Necessary optimality conditions

1. Introduction

Consider the variational problem:

(CP) Minimize

b∫

a

f (t, x , ẋ)d t

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

g(t, x , ẋ)≦ 0, t ∈ I ,

where I = [a, b] is a real interval, f : I × Rn × Rn → R, g : I × Rn × Rn → Rm, x(t) is an

n-dimensional piecewise smooth function of t and ẋ(t) is the derivative of x(t) with respect

to t in I . For notational simplicity, we write x(t) and ẋ(t) as x and ẋ , respectively.

Mond and Hanson [11] studied duality for the above problem under convexity. The

work in [11] was generalized to invex functions in [10, 12] and to multiobjective varia-

tional problems by Bector and Husain [2], Bhatia and Mehra [3], Kim and Kim [9], Ahmad

and Gulati [1] among others. In the work mentioned above, the boundary conditions are

∗Corresponding author.

Email addresses: trgulati�gmail.
om (T. Gulati), geeta.mehndiratta�rediffmail.
om (G.

Mehndiratta)

http://www.ejpam.com 786 c© 2010 EJPAM All rights reserved.



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 787

x(a) = α, x(b) = β .

Recently, Husain et al. [8] formulated the following second-order dual (CD) for (CP):

(CD) Maximize

b∫

a

{ f (t,u, u̇)−
1

2
β(t)T Fβ(t)}d t

Subject to u(a) = 0= u(b), (1)

fu + y(t)T gu− D( fu̇ + y(t)T gu̇) + (F +H)β(t) = 0, t ∈ I , (2)

b∫

a

{y(t)T g(t,u, u̇)−
1

2
β(t)T Hβ(t)}d t ≧ 0 (3)

y(t) ≧ 0, t ∈ I , (4)

where the symbols are as defined in [8].

They established the following converse duality theorem :

Theorem 1. [Converse duality] Suppose that f and g are thrice continuously differentiable. Let

( x̄(t), ȳ(t), β̄(t)) be an optimal solution of (CD) at which

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

(A2) ȳ(t)T gx − D ȳ(t)T g ẋ 6= 0,

(A3) (i)

b∫
a

β̄(t)T ( ȳ(t)T gx − D ȳ(t)T g ẋ)d t ≧ 0 and

b∫
a

β̄(t)T Hβ̄(t)d t > 0, or

(ii)

b∫
a

β̄(t)T ( ȳ(t)T gx − D ȳ(t)T g ẋ)d t ≦ 0 and

b∫
a

β̄(t)T Hβ̄(t)d t < 0.

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

f (t, ., .)d t is second-order pseudoinvex and

b∫
a

y(t)T g(t, ., .)d t is second-order quasiinvex with respect to the same η, then x̄(t) is an optimal

solution of (CP).

This result has been established by first proving that β̄(t) = 0, t ∈ I . It is obvious that the

assumption

b∫
a

β̄(t)T Hβ̄(t)d t > 0 (or < 0) and the conclusion β̄(t) = 0, t ∈ I , are inconsis-

tent. Moreover, assumption (A1) and equation (3.11) in [8], namely

(λ(t) + αβ̄(t))T F + (λ(t) + γβ̄(t))T H = 0, t ∈ I , do not imply λ(t) + αβ̄(t) = 0, t ∈ I and

λ(t) + γβ̄(t) = 0, t ∈ I . One needs to assume that the rows of F and H are linearly indepen-

dent.



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 788

This paper is organized as follows. In Section 2, we prove a converse duality theorem

modifying Theorem 1. In Section 3, we consider the multiobjective analogue of problem (CP).

This section also contains notations and preliminaries. The necessary optimality conditions for

an efficient solution are obtained in Section 4. The duality results are established in Section

5. The static case of our problems has been given in the last section.

2. Converse Duality

We denote the first partial derivatives of f with respect to t, x and ẋ , respectively, by ft , fx

and f ẋ such that fx = (
∂ f

∂ x1 ,
∂ f

∂ x2 , . . . ,
∂ f

∂ xn )
T and f ẋ = (

∂ f

∂ ẋ1 ,
∂ f

∂ ẋ2 , . . . ,
∂ f

∂ ẋn )
T . The matrices fx x

and gx are of order n×n and n×m, respectively. Similarly fx ẋ , f ẋ ẋ and the partial derivatives

of g j are also defined. All derivative of x , and all partial and total derivatives of f and g used

in this section are assumed to be continuous. Let the set M = {1,2, . . . , m}.

Remark 1. It may be noted that if we write

fx(t, x , ẋ)−
d

d t
f ẋ (t, x , ẋ) = L(t, x , ẋ , ẍ),

then the function F should be

F = Lx (t, x , ẋ , ẍ)−
d

d t
L ẋ(t, x , ẋ , ẍ) +

d2

d t2
L ẍ(t, x , ẋ , ẍ)

=
∂

∂ x
( fx(t, x , ẋ)−

d

d t
f ẋ(t, x , ẋ))−

d

d t
(
∂

∂ ẋ
( fx(t, x , ẋ)−

d

d t
f ẋ (t, x , ẋ)))

+
d2

d t2
(
∂

∂ ẍ
( fx (t, x , ẋ)−

d

d t
f ẋ (t, x , ẋ)))

= fx x − D f ẋ x − D fx ẋ + D2 f ẋ ẋ − D3 f ẋ ẍ

= fx x − 2D fx ẋ + D2 f ẋ ẋ − D3 f ẋ ẍ .

Thus F is a function of t, x(t), ẋ(t), ẍ(t),
...
x (t),

....
x (t) and is given by

F(t, x , ẋ , ẍ ,
...
x ,

....
x ) = fx x (t, x , ẋ)− 2D fx ẋ(t, x , ẋ)

+ D2 f ẋ ẋ(t, x , ẋ)− D3 f ẋ ẍ(t, x , ẋ), t ∈ I .

Therefore, it seems to us, that the function F should be as given above, while in Chen [6] and

Husain et al. [8], F has been taken as fx x−2D fx ẋ+D2 f ẋ ẋ and fx x−D fx ẋ+D2 f ẋ ẋ , respectively.

We formulate the following dual problem for (CP) :

(ÓCD) Maximize

b∫

a

( f (t,u, u̇)−
1

2
β(t)T Fβ(t))d t

Subject to u(a) = 0= u(b), (5)



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 789

fx (t,u, u̇) + gx(t,u, u̇)y(t)− D( f ẋ (t,u, u̇) +

g ẋ(t,u, u̇)y(t)) + (F +H)β(t) = 0, t ∈ I , (6)

y(t)T g(t,u, u̇)−
1

2
β(t)T Hβ(t) ≧ 0, t ∈ I , (7)

y(t) ≧ 0, t ∈ I , (8)

where

H(t,u, u̇, ü,
...
u ,

....
u , y(t), ẏ(t), ÿ(t),

...
y (t)) = (gx(t,u, u̇)y(t))x − 2D(gx(t,u, u̇)y(t)) ẋ

+ D2(g ẋ(t,u, u̇)y(t)) ẋ − D3(g ẋ(t,u, u̇)y(t)) ẍ , t ∈ I ,

and F(t,u, u̇, ü,
...
u ,

....
u ), t ∈ I , is as given in Remark 1.

Like [8], we shall use Fritz John necessary optimality conditions [4] for the dual problem

to establish the converse duality theorem. Since the constraints in the problem considered in

[4] do not involve integrals, we have not taken integral in the dual constraint (7). Moreover,

some terms in (dC D) are in different form than in (CD). It has been done so to make all the

terms in an expression to be of the same dimension. However, the weak duality theorem given

in [8] holds for problems (CP) and (dC D).

Theorem 2. [Converse duality] Let (ū(t), ȳ(t), β̄(t)) be an optimal solution of (dC D). If for each

t ∈ I ,

(B1) the vectors {Fi , Hi , i = 1,2, . . . , n} are linearly independent, where Fi and Hi are the ith

rows of F(t, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū ) and H(t, ū, ˙̄u, ¨̄u,

...
ū ,

....
ū , ȳ(t), ˙̄y(t), ¨̄y(t),

...
ȳ (t)), respectively,

(B2) gx(t, ū, ˙̄u) ȳ(t)− D(g ẋ(t, ū, ˙̄u) ȳ(t)) 6= 0, and

(B3) either

(i) the n×n matrix H(t, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t), ˙̄y(t), ¨̄y(t),

...
ȳ (t))+(gx(t, ū, ˙̄u) ȳ(t))x is pos-

itive definite and β̄(t)T (gx(t, ū, ˙̄u) ȳ(t)) ≧ 0, or

(ii) the n×n matrix H(t, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t), ˙̄y(t), ¨̄y(t),

...
ȳ (t))+(gx (t, ū, ˙̄u) ȳ(t))x is neg-

ative definite and β̄(t)T (gx(t, ū, ˙̄u) ȳ(t)) ≦ 0,

then ū(t) is feasible for (CP) and the two objective functionals have same value. Also, if the weak

duality theorem [8] holds for all feasible solution of (CP) and (dC D), then ū(t) is an optimal

solution of (CP).

Proof. Since (ū(t), ȳ(t), β̄(t)) is an optimal solution of (dC D), there exist α ∈ R and piece-

wise smooth functions λ : I → Rn, γ : I → R and µ : I → Rm such that the following Fritz John

conditions [4] are satisfied at (ū(t), ȳ(t), β̄(t)) (for brevity, fx ≡ fx(t, ū, ˙̄u), g j ≡ g j(t, ū, ˙̄u),

g
j
x ≡ g

j
x(t, ū, ˙̄u) etc.):

−α( fx − D f ẋ −
1

2
(β̄(t)T F β̄(t))x +

1

2
D(β̄(t)T F β̄(t)) ẋ −

1

2
D2(β̄(t)T F β̄(t)) ẍ



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 790

+
1

2
D3(β̄(t)T F β̄(t))...x −

1

2
D4(β̄(t)T F β̄(t))....x ) + ( fx x − D fx ẋ + (gx ȳ(t))x − D(gx ȳ(t)) ẋ

−(D( f ẋ x + (g ẋ ȳ(t))x)− D(D( f ẋ ẋ + (g ẋ ȳ(t)) ẋ )) + D2(D( f ẋ ẍ + (g ẋ ȳ(t)) ẍ )))

+((F +H)β̄(t))x − D((F +H)β̄(t)) ẋ + D2((F +H)β̄(t)) ẍ − D3((F +H)β̄(t))...x

+D4((F +H)β̄(t))....x )λ(t)− γ(t)(gx ȳ(t)− D(g ẋ ȳ(t))−
1

2
(β̄(t)T Hβ̄(t))x

+
1

2
D(β̄(t)T Hβ̄(t)) ẋ −

1

2
D2(β̄(t)T Hβ̄(t)) ẍ +

1

2
D3(β̄(t)T Hβ̄(t))...x

−
1

2
D4(β̄(t)T Hβ̄(t))....x ) = 0, t ∈ I , (9)

λ(t)T (g j
x + g j

x x β̄(t))− γ(t)(g
j −

1

2
β̄(t)T g j

x x β̄(t))−µ
j(t) = 0, t ∈ I , j ∈ M , (10)

(λ(t) +αβ̄(t))T F + (λ(t) + γ(t)β̄(t))T H = 0, t ∈ I , (11)

γ(t)( ȳ(t)T g −
1

2
β̄(t)T Hβ̄(t)) = 0, t ∈ I , (12)

µ(t)T ȳ(t) = 0, t ∈ I , (13)

(α,γ(t),µ(t)) ≧ 0, t ∈ I , (14)

(α,λ(t),γ(t),µ(t)) 6= 0, t ∈ I . (15)

By Hypothesis (B1), equation (11) yields

λ(t) +αβ̄(t) = 0, t ∈ I and (16)

λ(t) + γ(t)β̄(t) = 0, t ∈ I . (17)

Using (6), (16) and (17) in (9), we have

(α− γ(t))(gx ȳ(t)− D(g ẋ ȳ(t)) +Hβ̄(t)) +
1

2
α((β̄(t)T F β̄(t))x − D(β̄(t)T F β̄(t)) ẋ

+D2(β̄(t)T F β̄(t)) ẍ − D3(β̄(t)T F β̄(t))...x + D4(β̄(t)T F β̄(t))....x ) + (((F +H)β̄(t))x −

D((F +H)β̄(t)) ẋ + D2((F +H)β̄(t)) ẍ − D3((F +H)β̄(t))...x + D4((F +H)β̄(t))....x )λ(t) +

1

2
γ(t)((β̄(t)T Hβ̄(t))x − D(β̄(t)T Hβ̄(t)) ẋ + D2(β̄(t)T Hβ̄(t)) ẍ − D3(β̄(t)T Hβ̄(t))...x +

D4(β̄(t)T Hβ̄(t))....x ) = 0, t ∈ I . (18)

Let γ(t) = 0 for some t. Suppose t0 ∈ I and γ(t0) = 0. Then by (17), we get λ(t0) = 0 and

so αβ̄(t0) = 0, by (16). Thus (18) yields α(gx ȳ(t0)− D(g ẋ ȳ(t0))) = 0, which by Hypothesis

(B2) implies α = 0. Since λ(t0) = 0 and γ(t0) = 0, from (10), we obtain µ j(t0) = 0, j ∈ M .

Therefore (α,λ(t0),γ(t0),µ(t0)) = 0, contradicting (15). Hence γ(t) > 0, t ∈ I .

Now, multiplying (10) by ȳ j(t), t ∈ I , summing over j, and then using (12), (13), (17)

and γ(t) > 0, t ∈ I , we get

2β̄(t)T (gx ȳ(t)) + β̄(t)T (H + (gx ȳ(t))x)β̄(t) = 0, t ∈ I ,



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 791

which contradicts Hypothesis (B3) unless

β̄(t) = 0, t ∈ I . (19)

Thus from (16), λ(t) = 0, t ∈ I . Therefore for j ∈ M , equation (10) gives

g j = −
µ j(t)

γ(t)
≦ 0, t ∈ I .

Thus ū(t) is feasible for (CP). Also, in view of (19), the two objectives are equal.

Now, assume that ū(t) is not an optimal solution of (CP). Then, there exists û(t) ∈ X such

that
b∫

a

f (t, û, ˙̂u)d t <

b∫

a

f (t, ū, ˙̄u)d t.

As β̄(t) = 0, t ∈ I , we have

b∫

a

f (t, û, ˙̂u)d t <

b∫

a

( f (t, ū, ˙̄u)−
1

2
β̄(t)T F β̄(t))d t,

a contradiction to the weak duality theorem [8]. Hence ū(t) is an optimal solution for (CP).

3. Multiobjective Variational Problem

We consider the following multiobjective variational problem (P):

(P) Minimize (

b∫

a

f 1(t, x , ẋ)d t,

b∫

a

f 2(t, x , ẋ)d t, . . . ,

b∫

a

f k(t, x , ẋ)d t)

Subject to x(a) = α, x(b) = β , (20)

g(t, x , ẋ)≦ 0, t ∈ I , (21)

where f i : I × S × S → R(i ∈ K), g = (g1, g2, . . . , gm) : I × S × S → Rm and S is an open set

in Rn. We denote the first partial derivatives of f i, with respect to t, x and ẋ respectively by

f i
t , f i

x and f i
ẋ

such that f i
x = (

∂ f i

∂ x1 ,
∂ f i

∂ x2 , . . . ,
∂ f i

∂ xn )
T and f i

ẋ
= (

∂ f i

∂ ẋ1 ,
∂ f i

∂ ẋ2 , . . . ,
∂ f i

∂ ẋn )
T . The Hessian

matrix f i
x x , is an n × n symmetric matrix. Similarly f i

x ẋ
, f i

ẋ ẋ
and the partial derivatives of

g j are also defined. All the partial and total derivatives of f i and g used here onwards are

assumed to be continuous. We shall use X for the set of all feasible solutions of (P). For the

sake of convenience, we shall not write the limits a and b in the integrals, i.e.,
∫

f id t shall



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 792

mean

b∫
a

f id t. Let K = {1,2, . . . , k} and for r ∈ K , the set Kr = K − r. For a and b in Rn, we

shall use the following three inequalities :

a ≧ b⇔ ai ≧ bi (i = 1,2, . . . , n)

a ≥ b⇔ (a ≧ b, a 6= b)

a > b⇔ ai > bi (i = 1,2, . . . , n).

Definition 1. A point x̄(t) ∈ X is said to be an efficient solution of (P) if there exists no x(t) ∈ X

such that
∫

f r(t, x , ẋ)d t <

∫
f r(t, x̄ , ˙̄x)d t for some r ∈ K

and

∫
f i(t, x , ẋ)d t ≦

∫
f i(t, x̄ , ˙̄x)d t for i ∈ Kr .

Let p : I → Rn and Ai(t, x , ẋ , ẍ ,
...
x ,

....
x ) = f i

x x(t, x , ẋ) − 2D f i
x ẋ
(t, x , ẋ) + D2 f i

ẋ ẋ
(t, x , ẋ) −

D3 f i
ẋ ẍ
(t, x , ẋ), t ∈ I , i ∈ K .

Definition 2. A functional G : I ×S×S×Rn→ R is said to be sublinear, if for all x(t),u(t) ∈ S,

G(t, x ,u;ξ1 + ξ2)≦ G(t, x ,u;ξ1) + G(t, x ,u;ξ2) for all ξ1,ξ2 ∈ Rn (22)

and

G(t, x ,u; aξ) = aG(t, x ,u;ξ) for all a ∈ R, a ≧ 0 and ξ ∈ Rn. (23)

From (23), it follows that G(t, x ,u; 0) = 0.

Let ρ ∈ R and d : I × S × S→ Rn be a pseudometric on Rn.

Definition 3. The functional
∫

f i(t, x , ẋ)d t is said to be second-order (G,ρ)-convex at u(t) ∈ S,

if there exists a sublinear functional G : I ×S×S×Rn→ R such that for all x(t) ∈ S, p(t) ∈ Rn,

∫
f i(t, x , ẋ)d t −

∫
f i(t,u, u̇)d t +

1

2

∫
p(t)T Ai p(t)d t ≧

∫
G(t, x ,u; f i

x (t,u, u̇)− D f i
ẋ (t,u, u̇) + Ai p(t))d t +ρ

∫
d2(t, x ,u)d t.

If in the above definition, inequality is satisfied as strict inequality, then we say that the functional∫
f i(t, x , ẋ)d t is second-order strictly (G,ρ)-convex at u(t) ∈ S.

Definition 4. The functional
∫

f i(t, x , ẋ)d t is said to be second-order (G,ρ)-pseudoconvex at

u(t) ∈ S, if there exists a sublinear functional G : I × S × S × Rn → R such that for all x(t) ∈
S, p(t) ∈ Rn,

∫
G(t, x ,u; f i

x (t,u, u̇) − D f i
ẋ (t,u, u̇) + Ai p(t))d t +ρ

∫
d2(t, x ,u)d t ≧ 0



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 793

⇒

∫
f i(t, x , ẋ)d t ≧

∫
f i(t,u, u̇)d t −

1

2

∫
p(t)T Ai p(t)d t.

Definition 5. The functional
∫

f i(t, x , ẋ)d t is said to be second-order (G,ρ)-quasiconvex at

u(t) ∈ S, if there exists a sublinear functional G : I × S × S × Rn→ R such that for all x(t) ∈ S,

p(t) ∈ Rn,
∫

f i(t, x , ẋ)d t ≦

∫
f i(t,u, u̇)d t −

1

2

∫
p(t)T Ai p(t)d t

⇒

∫
G(t, x ,u; f i

x (t,u, u̇)− D f i
ẋ (t,u, u̇) + Aip(t))d t +ρ

∫
d2(t, x ,u)d t ≦ 0.

4. Necessary optimality conditions

We now prove three results. The first gives Fritz John type necessary conditions for (P)

and the remaining two are Kuhn-Tucker type necessary conditions for (P). To establish these

necessary conditions, we shall use the corresponding result for single objective variational

problems obtained by Chandra et al. [4] in their Theorem 1. As stated in the remarks after

Theorem 1 in [4], to use their Kuhn-Tucker conditions, we shall assume Slater’s or Robinson

condition.

The following result relates an efficient solution of (P) with an optimal solution of k-scalar

objective variational problems.

Lemma 1 (Chankong and Haimes [5]). A point x̄(t) ∈ X is an efficient solution of (P) if and

only if x̄(t) is an optimal solution of (Pr) for each r ∈ K.

(Pr) Minimize
∫

f r(t, x , ẋ)d t

Subject to x(a) = α, x(b) = β ,

g(t, x , ẋ)≦ 0, t ∈ I ,∫
f i(t, x , ẋ)d t ≦
∫

f i(t, x̄ , ˙̄x)d t, i ∈ Kr .

The k-problems (Pr) involve integral in the constraints, while to obtain necessary optimal-

ity conditions for (P) via (Pr) we need necessary optimality conditions for scalar variational

problem [4]. Since the variational problem in [4] does not involve the integral in the con-

straints, we first derive the result relating an efficient solution of (P) with the optimal solution

of the following k-single objective problems:

(P̂r) Minimize
∫

f r(t, x , ẋ)d t

Subject to x(a) = α, x(b) = β ,

g(t, x , ẋ)≦ 0, t ∈ I ,

f i(t, x , ẋ)≦ f i(t, x̄ , ˙̄x), t ∈ I , i ∈ Kr .

Lemma 2. Let x̄(t) ∈ X be an efficient solution of (P). Then x̄(t) is an optimal solution of (P̂r)

for each r ∈ K.



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 794

Proof. Let x̄(t) be an efficient solution of (P) and suppose, to the contrary, that x̄(t) is not

an optimal solution of (P̂r), for some r ∈ K . Then there exists an x̂(t) ∈ X such that

g(t, x̂ , ˙̂x)≦ 0, t ∈ I , (24)

f i(t, x̂ , ˙̂x)≦ f i(t, x̄ , ˙̄x), t ∈ I , i ∈ Kr , (25)

and

∫
f r(t, x̂ , ˙̂x)d t <

∫
f r(t, x̄ , ˙̄x)d t. (26)

Inequality (25) implies
∫

f i(t, x̂ , ˙̂x)d t ≦

∫
f i(t, x̄ , ˙̄x)d t, i ∈ Kr . (27)

Inequalities (26) and (27) contradict the fact that x̄(t) is an efficient solution of (P). Hence

x̄(t) is an optimal solution of (P̂r) for each r ∈ K .

Theorem 3 (Fritz John type necessary conditions). Let x̄(t) be an efficient solution of (P). Then

there exist λ̄i ∈ R, i ∈ K and a piecewise smooth function ȳ : I → Rm such that

k∑

i=1

λ̄i( f i
x (t, x̄ , ˙̄x)− D f i

ẋ (t, x̄ , ˙̄x))+ gx(t, x̄ , ˙̄x) ȳ(t)− D(g ẋ(t, x̄ , ˙̄x) ȳ(t)) = 0, t ∈ I , (28)

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I , (29)

(λ̄, ȳ(t)) ≥ 0, t ∈ I . (30)

Proof. Since x̄(t) is an efficient solution of (P), by Lemma 2, x̄(t) is an optimal solution

of (P̂r) for each r ∈ K and hence in particular of (P̂1). Therefore, by [4], there exist λ̄i, i ∈ K

and a piecewise smooth function ȳ(t) ∈ Rm such that

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

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I ,

(λ̄1, λ̄2, . . . , λ̄k, ȳ(t)) ≥ 0, t ∈ I ,

which give (28) to (30).

Theorem 4 (Kuhn-Tucker type necessary conditions). Let x̄(t) be an efficient solution of (P)

and let for some r ∈ K, the constraints of (P̂r) satisfy Slater’s or Robinson condition at x̄(t).

Then there exist λ̄ ∈ Rk and a piecewise smooth function ȳ : I → Rm such that

k∑

i=1

λ̄i( f i
x (t, x̄ , ˙̄x)− D f i

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

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I ,

λ̄≥ 0

ȳ(t)≧ 0, t ∈ I .



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 795

Proof. Since x̄(t) is an efficient solution of (P), by Lemma 2, x̄(t) is an optimal solution

of (P̂r) for each r. As for some r, the constraints of (P̂r) satisfy Slater’s or Robinson condition

at x̄(t), by the Kuhn-Tucker necessary conditions in [4], there exist 0 < λ̄r ∈R, 0 ≦ λ̄i ∈ R,

i ∈ Kr and a piecewise smooth function ȳ(t) ∈ Rm such that

λ̄r( f r
x (t, x̄ , ˙̄x)− D f r

ẋ (t, x̄ , ˙̄x))+
∑

i∈Kr

λ̄i( f i
x (t, x̄ , ˙̄x)− D f i

ẋ (t, x̄ , ˙̄x))+

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

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I ,

λ̄r > 0,0≦ λ̄i ∈ R, i ∈ Kr ,

ȳ(t)≧ 0, t ∈ I .

Or equivalently

k∑

i=1

λ̄i( f i
x (t, x̄ , ˙̄x)− D f i

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

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I ,

λ̄≥ 0, λ̄ ∈ Rk,

ȳ(t)≧ 0, t ∈ I .

In Theorem 4, we assumed Slater’s or Robinson condition for some (P̂r), which gave us

λ̄ ≥ 0. In the following Theorem, we assume Slater’s or Robinson condition for every (P̂r)

and obtain λ̄ > 0.

Theorem 5 (Kuhn-Tucker type necessary conditions). Let x̄(t) be an efficient solution of (P)

and let for each r ∈ K, the constraints of (P̂r) satisfy Slater’s or Robinson condition at x̄(t). Then

there exist λ̄ ∈ Rk and a piecewise smooth function ȳ : I → Rm such that

k∑

i=1

λ̄i( f i
x (t, x̄ , ˙̄x)− D f i

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

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I ,

λ̄ > 0,

k∑

i=1

λ̄i = 1,

ȳ(t)≧ 0, t ∈ I .

Proof. Since x̄(t) is an efficient solution of (P), by Lemma 2, x̄(t) is an optimal solution of

(P̂r) for each r ∈ K . As for each r, the constraints of (P̂r) satisfy Slater’s or Robinson condition

at x̄(t), by the Kuhn-Tucker necessary conditions in [4], for each r ∈ K , there exist v̄ i
r ∈ R

(i ∈ Kr) and piecewise smooth functions µ̄
j
r(t) ∈ R ( j ∈ M) such that



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 796

f r
x (t, x̄ , ˙̄x)− D f r

ẋ (t, x̄ , ˙̄x)+
∑

i∈Kr

v̄ i
r( f

i
x (t, x̄ , ˙̄x)− D f i

ẋ (t, x̄ , ˙̄x))+

m∑

j=1

(g j
x(t, x̄ , ˙̄x)µ̄ j

r(t)− D(g
j

ẋ
(t, x̄ , ˙̄x)µ̄ j

r(t))) = 0, t ∈ I ,

m∑

j=1

µ̄ j
r(t)g

j(t, x̄ , ˙̄x) = 0, t ∈ I ,

v̄ i
r ≧ 0, i ∈ Kr ,

ȳ(t)≧ 0, t ∈ I .

Summing over r ∈ K , we get

k∑

i=1

(v̄ i
1+ v̄ i

2 + ...+ v̄ i
k
)( f i

x (t, x̄ , ˙̄x)− D f i
ẋ (t, x̄ , ˙̄x))+

m∑

j=1

(g j
x(t, x̄ , ˙̄x)(µ̄

j

1(t) + µ̄
j

2(t) + . . .+

µ̄
j

k
(t))− D(g

j

ẋ
(t, x̄ , ˙̄x)(µ̄

j

1(t) + µ̄
j

2(t) + . . .+ µ̄
j

k
(t)))) = 0, t ∈ I ,

m∑

j=1

(µ̄
j

1
(t) + µ̄

j

2
(t) + . . .+ µ̄

j

k
(t))g j(t, x̄ , ˙̄x) = 0, t ∈ I ,

where v̄ i
i = 1 for each i ∈ K .

Equivalently,

k∑

i=1

v̄ i( f i
x (t, x̄ , ˙̄x)−D f i

ẋ(t, x̄ , ˙̄x))+

m∑

j=1

(g j
x(t, x̄ , ˙̄x)µ̄ j(t)−D(g

j

ẋ
(t, x̄ , ˙̄x)µ̄ j(t))) = 0, t ∈ I , (31)

m∑

j=1

µ̄ j(t)g j(t, x̄ , ˙̄x) = 0, t ∈ I , (32)

where v̄ i = 1+
∑

r∈Ki

v̄ i
r > 0,i ∈ K , and µ̄ j(t) =

k∑
r=1

µ̄
j
r(t) ≧ 0, t ∈ I , j ∈ M .

Dividing (31) and (32) by
k∑

i=1

v̄ i and setting

λ̄i =
v̄ i

k∑
i=1

v̄ i

, i ∈ K , ȳ j(t) =
µ̄ j(t)

k∑
i=1

v̄ i

, j ∈ M ,

we get

k∑

i=1

λ̄i( f i
x (t, x̄ , ˙̄x)− D f i

ẋ (t, x̄ , ˙̄x)) +

m∑

j=1

(g j
x(t, x̄ , ˙̄x) ȳ j(t)− D(g

j

ẋ
(t, x̄ , ˙̄x) ȳ j(t))) = 0, t ∈ I ,



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 797

m∑

j=1

ȳ j(t)g j(t, x̄ , ˙̄x) = 0, t ∈ I ,

or

k∑

i=1

λ̄i( f i
x (t, x̄ , ˙̄x)− D f i

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

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I ,

λ̄= (λ̄1, λ̄2, . . . , λ̄k)> 0,

k∑

i=1

λ̄i = 1,

ȳ(t) = ( ȳ1(t), ȳ2(t), . . . , ȳm(t)) ≧ 0, t ∈ I .

5. Second-order Mond-Weir Type Duality

We present the following multiobjective variational dual problem for (P):

(MWD) Maximize (

∫
( f 1(t,u, u̇)−

1

2
p(t)T A1p(t))d t, . . . ,

∫
( f k(t,u, u̇)−

1

2
p(t)T Akp(t))d t)

Subject to u(a) = α,u(b) = β , (33)

k∑

i=1

λi( f i
x (t,u, u̇)− D f i

ẋ (t,u, u̇) + Aip(t)) +

gx(t,u, u̇)y(t)− D(g ẋ(t,u, u̇)y(t)) + Bp(t) = 0, t ∈ I , (34)

y(t)T g(t,u, u̇)−
1

2
p(t)T Bp(t) ≧ 0, t ∈ I , (35)

λ ≥ 0, (36)

y(t) ≧ 0, t ∈ I , (37)

where y : I → Rm, p : I → Rn, λ = (λ1,λ2, . . . ,λk) ∈ Rk, Ai(t,u, u̇, ü,
...
u ,

....
u ), t ∈ I , i ∈ K (as

defined earlier) and

B(t,u, u̇, ü,
...
u ,

....
u , y(t), ẏ(t), ÿ(t),

...
y (t)) = (gx y(t))x−2D(gx y(t)) ẋ+D2(g ẋ y(t)) ẋ−D3(g ẋ y(t)) ẍ ,

t ∈ I , are n× n symmetric matrices. Let Y be the set of all feasible solutions of the above

problem.

Theorem 6. (Weak duality) Let x(t) ∈ X and (u(t),λ, y(t), p(t)) ∈ Y such that



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 798

(i)
∫ k∑

i=1

λi f i(t, ., .)d t is second-order (G,ρ1)-pseudoconvex at u(t),

(ii)
∫

y(t)T g(t, ., .)d t is second-order (G,ρ2)-quasiconvex at u(t),

(iii) λ > 0 and ρ1+ρ2 ≧ 0.

Then
∫

f r(t, x , ẋ)d t <

∫
( f r(t,u, u̇)−

1

2
p(t)T Ar p(t))d t for some r ∈ K (38)

and

∫
f i(t, x , ẋ)d t ≦

∫
( f i(t,u, u̇)−

1

2
p(t)T Aip(t))d t, i ∈ Kr (39)

cannot hold.

Proof. Suppose, to the contrary, that (38) and (39) hold. Since λ > 0, the above inequali-

ties give ∫ k∑

i=1

λi f i(t, x , ẋ)d t <

∫ k∑

i=1

λi( f i(t,u, u̇)−
1

2
p(t)T Ai p(t))d t. (40)

Now, by the constraints (21), (35) and (37), we have
∫

y(t)T g(t, x , ẋ)d t ≦

∫
(y(t)T g(t,u, u̇)−

1

2
p(t)T Bp(t))d t

As
∫

y(t)T g(t, ., .)d t is second-order (G,ρ2)-quasiconvex at u(t), we get

∫
(G(t, x ,u; gx (t,u, u̇)y(t)− D(g ẋ(t,u, u̇)y(t)) + Bp(t)) +ρ2d2(t, x ,u))d t ≦ 0. (41)

From the constraint (34) and the fact that G(t, x ,u; 0) = 0, we have

∫
(G(t, x ,u;

k∑

i=1

λi( f i
x (t,u, u̇) − D f i

ẋ (t,u, u̇) + Ai p(t)) + gx(t,u, u̇)y(t)

− D(g ẋ(t,u, u̇)y(t)) + Bp(t)))d t = 0,

which on using inequality (41), Hypothesis (iii) and the sublinearity of G, yield

∫
G(t, x ,u;

k∑

i=1

λi( f i
x (t,u, u̇)− D f i

ẋ (t,u, u̇) + Ai p(t)))d t +

∫
ρ1d2(t, x ,u)d t ≧ 0.

By Hypothesis (i), it implies

∫ k∑

i=1

λi f i(t, x , ẋ)d t ≧

∫ k∑

i=1

λi( f i(t,u, u̇)−
1

2
p(t)T Ai p(t))d t,

a contradiction to (40). Hence the result.



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 799

Remark 2. If we drop the assumption λ > 0 from Theorem 6, then we need to replace Hypothesis

(i) by :
∫ k∑

i=1

λi f i(t, ., .)d t is second-order strictly (G,ρ1)-pseudoconvex at u(t).

Theorem 7 (Strong duality). Let x̄(t) be normal and is an efficient solution of (P). Then, there

exist λ̄ ∈ Rk, a piecewise smooth function ȳ : I → Rm such that ( x̄(t), λ̄, ȳ(t), p̄(t) = 0) is

feasible for (MWD) and the two objective functionals are equal. Furthermore, if the weak duality

holds for all feasible solutions of (P) and (MWD), then ( x̄(t), λ̄, ȳ(t), p̄(t) = 0) is an efficient

solution of the problem (MWD).

Proof. Since x̄(t) is normal and an efficient solution of (P), therefore by Theorem 4, there

exist λ̄ ∈ Rk and a piecewise smooth function ȳ(t) ∈ Rm satisfying

k∑

i=1

λ̄i( f i
x (t, x̄ , ˙̄x)− D f i

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

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I ,

λ̄≥ 0, t ∈ I ,

ȳ(t)≧ 0, t ∈ I .

Hence ( x̄(t), λ̄, ȳ(t), p̄(t) = 0) satisfies the constraints of (MWD) and thus the two objective

functionals have the same value.

Now, we claim that ( x̄(t), λ̄, ȳ(t), p̄(t) = 0) is an efficient solution of (MWD). If not, then

there exists (û(t), λ̂, ŷ(t), p̂(t)) ∈ Y , such that

(

∫
( f 1(t, û, ˙̂u)−

1

2
p̂(t)T A1(t, û, ˙̂u, ¨̂u,

...
û ,

....
û )p̂(t))d t, . . . ,

∫
( f k(t, û, ˙̂u)−

1

2
p̂(t)T Ak(t, û, ˙̂u, ¨̂u,

...
û ,

....
û )p̂(t))d t)

≥ (

∫
( f 1(t, x̄ , ˙̄x)−

1

2
p̄(t)T A1(t, x̄ , ˙̄x , ¨̄x ,

...
x̄ ,

....
x̄ )p̄(t))d t, . . . ,

∫
( f k(t, x̄ , ˙̄x)−

1

2
p̄(t)T Ak(t, x̄ , ˙̄x , ¨̄x ,

...
x̄ ,

....
x̄ )p̄(t))d t).

As p̄(t) = 0, we have

(

∫
( f 1(t, û, ˙̂u)−

1

2
p̂(t)T A1(t, û, ˙̂u, ¨̂u,

...
û ,

....
û )p̂(t))d t, . . . ,

∫
( f k(t, û, ˙̂u)−

1

2
p̂(t)T Ak(t, û, ˙̂u, ¨̂u,

...
û ,

....
û )p̂(t))d t) ≥ (

∫
f 1(t, x̄ , ˙̄x)d t, . . . ,

∫
f k(t, x̄ , ˙̄x)d t),



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 800

which contradicts the weak duality theorem. Hence ( x̄(t), λ̄, ȳ(t), p̄(t) = 0) is an efficient

solution of (MWD).

Theorem 8 (Converse duality). Let (ū(t), λ̄, ȳ(t), p̄(t)) be an efficient solution of (MWD) for

which

(C1) the vectors {Ai
m, Bm, t ∈ I , i ∈ K , m = 1,2, . . . , n} are linearly independent, where Ai

m is the

mth row of Ai(t, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū ) and Bm is the mth row of

B(t, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t), ˙̄y(t), ¨̄y(t),

...
ȳ (t)),

(C2) f i
x (t, ū, ˙̄u)− D f i

ẋ
(t, ū, ˙̄u), t ∈ I , i ∈ K , are linearly independent, and

(C3) for t ∈ I , either

(a) the n×n matrix B(t, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t), ˙̄y(t), ¨̄y(t),

...
ȳ (t))+(gx(t, ū, ˙̄u) ȳ(t))x is pos-

itive definite and p̄(t)T (gx(t, ū, ˙̄u) ȳ(t)) ≧ 0, or

(b) the n×n matrix B(t, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t), ˙̄y(t), ¨̄y(t),

...
ȳ (t))+(gx(t, ū, ˙̄u) ȳ(t))x is neg-

ative definite and p̄(t)T (gx(t, ū, ˙̄u) ȳ(t))≦ 0.

Then ū(t) is feasible for (P) and the two objective functionals have same value. Also, if the

weak duality theorem holds for all feasible solutions of (P) and (MWD), then ū(t) is an efficient

solution of (P).

Proof. Since (ū(t), λ̄, ȳ(t), p̄(t)) is an efficient solution of (MWD), there exist α,η ∈ Rk and

piecewise smooth functions β : I → Rn, γ : I → R, µ : I → Rm, such that the following Fritz

John conditions (Theorem 3) are satisfied at (ū(t), λ̄, ȳ(t), p̄(t)) (for brevity, f i
x ≡ f i

x (t, ū, ˙̄u),

g j ≡ g j(t, ū, ˙̄u), g
j
x ≡ g

j
x(t, ū, ˙̄u), Ai ≡ Ai(t, ū, ˙̄u, ¨̄u,

...
ū ,

....
ū ),

B ≡ B(t, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t), ˙̄y(t), ¨̄y(t),

...
ȳ (t)) etc.):

k∑

i=1

αi( f i
x − D f i

ẋ −
1

2
(p̄(t)T Ai p̄(t))x +

1

2
D(p̄(t)T Ai p̄(t)) ẋ −

1

2
D2(p̄(t)T Ai p̄(t)) ẍ +

1

2
D3(p̄(t)T Ai p̄(t))...x −

1

2
D4(p̄(t)T Ai p̄(t))....x )− (

k∑

i=1

λ̄i(Ai + (Ai p̄(t))x − D(Ai p̄(t)) ẋ +

D2(Ai p̄(t)) ẍ − D3(Ai p̄(t))...x + D4(Ai p̄(t))....x ) + B + (Bp̄(t))x − D(Bp̄(t)) ẋ +

D2(Bp̄(t)) ẍ − D3(Bp̄(t))...x + D4(Bp̄(t))....x )β(t) + γ(t)(gx y(t)− D(g ẋ ȳ(t))−
1

2
(p̄(t)T Bp̄(t))x +

1

2
D(p̄(t)T Bp̄(t)) ẋ −

1

2
D2(p̄(t)T Bp̄(t)) ẍ +

1

2
D3(p̄(t)T Bp̄(t))...x −

1

2
D4(p̄(t)T Bp̄(t))....x ) = 0, t ∈ I , (42)

β(t)T ( f i
x − D f i

ẋ + Ai p̄(t))−ηi = 0, t ∈ I , i ∈ K , (43)

β(t)T (g j
x + g j

x x p̄(t))− γ(t)(g j −
1

2
p̄(t)T g j

x x p̄(t))−µ j(t) = 0, t ∈ I , j ∈ M , (44)



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 801

k∑

i=1

αiAi p̄(t) + (

k∑

i=1

λ̄iAi + B)β(t) + γ(t)Bp̄(t) = 0, t ∈ I , (45)

γ(t)( ȳ(t)T g −
1

2
p̄(t)T Bp̄(t)) = 0, t ∈ I , (46)

µ(t)T ȳ(t) = 0, t ∈ I , (47)

ηT λ̄ = 0, (48)

(α,β(t),γ(t),µ(t),η) 6= 0, t ∈ I , (49)

(α,γ(t),µ(t),η) ≥ 0, t ∈ I . (50)

On rearranging (45), we get

k∑

i=1

Ai(αi p̄(t) + λ̄iβ(t)) + B(β(t) + γ(t)p̄(t)) = 0, t ∈ I ,

which by Hypothesis (C1), yield

αi p̄(t) + λ̄iβ(t) = 0, t ∈ I , i ∈ K , (51)

and

β(t) + γ(t)p̄(t) = 0, t ∈ I . (52)

Now, suppose γ(t) = 0 for some t, i.e., γ(t0) = 0 for some t0 ∈ I . Then equation (52) gives

β(t0) = 0 and so equation (51) implies αi p̄(t0) = 0. Therefore for t = t0, equation (42)

reduces to

k∑

i=1

αi( f i
x − D f i

ẋ ) = 0.

On using Hypothesis (C2) in the above equation, we obtain

αi = 0, i ∈ K .

Also, equations (43) and (44) give ηi = 0, i ∈ K and µ j(t0) = 0, j ∈ M , respectively. Hence

(α,β(t0),γ(t0),µ(t0),η) = 0, which contradicts (49). Therefore

γ(t) > 0, t ∈ I . (53)

Multiplying (44) by ȳ j(t), summing over j and then using (46), (47), (52) and γ(t) > 0, we

have

2p̄(t)T (gx ȳ(t)) + p̄(t)T (B+ (gx ȳ(t))x)p̄(t) = 0, t ∈ I ,

which contradicts Hypothesis (C3) unless

p̄(t) = 0, t ∈ I . (54)



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 802

And thus relation (52) gives β(t) = 0, t ∈ I . For j ∈ M , equation (44) yield

g j =
−µ j(t)

γ(t)
≦ 0, t ∈ I , j ∈ M .

Thus ū(t) is feasible for (P).

As p̄(t) = 0, t ∈ I , the two objectives functionals are equal.

Now, suppose that ū(t) is not an efficient solution of (P). Then, there exists û(t) ∈ X such

that ∫
f r(t, û, ˙̂u)d t <

∫
f r(t, ū, ˙̄u)d t for some r ∈ K

and ∫
f i(t, û, ˙̂u)d t ≦

∫
f i(t, ū, ˙̄u)d t, i ∈ Kr .

Using (54), we get

∫
f r(t, û, ˙̂u)d t <

∫
( f r(t, ū, ˙̄u)−

1

2
p̄(t)T Ar(t, ū, ˙̄u, ¨̄u,

...
ū ,

....
ū )p̄(t))d t for some r ∈ K

and ∫
f i(t, û, ˙̂u)d t ≦

∫
( f i(t, ū, ˙̄u)−

1

2
p̄(t)T Ai(t, ū, ˙̄u, ¨̄u,

...
ū ,

....
ū )p̄(t))d t, i ∈ Kr ,

which contradicts the weak duality theorem. Hence ū(t) is an efficient solution for (P).

Remark 3. For the single objective problems in Section 2, the Hypothesis (C2) reduces to

(C2′) fx − D f ẋ 6= 0, t ∈ I .

Therefore following the above proof, Theorem 2 can also be proved if Hypothesis (B2) is replaced

by (C2′). This also makes the proof simpler as it does not require equation (18). However, we

proved Theorem 2 assuming (B2), which is same as Hypothesis (A2) in [8].

Theorem 9 (Strict converse duality). Let x̄(t) and (ū(t), λ̄, ȳ(t), p̄(t)) be efficient solutions of

(P) and (MWD) respectively, such that

∫ k∑

i=1

λ̄i f i(t, x̄ , ˙̄x)d t =

∫ k∑

i=1

λ̄i( f i(t, ū, ˙̄u)−
1

2
p̄(t)T Ai(t, ū, ˙̄u, ¨̄u,

...
ū ,

....
ū )p̄(t))d t (55)

If

(i)
∫ k∑

i=1

λ̄i f i(t, ., .)d t is second-order strictly (G,ρ1)-convex at ū(t),

(ii)
∫

ȳ(t)T g(t, ., .)d t is second-order (G,ρ2)-convex at ū(t), and



T. Gulati and G. Mehndiratta / Eur. J. Pure Appl. Math, 3 (2010), 786-805 803

(iii) ρ1 +ρ2 ≧ 0.

Then x̄(t) = ū(t).

Proof. Suppose x̄(t0) 6= ū(t0) for some t0 ∈ I . By Hypothesis (i), we have

∫ k∑

i=1

λ̄i f i(t0, x̄ , ˙̄x)d t −

∫ k∑

i=1

λ̄i f i(t0, ū, ˙̄u)d t >

∫
(G(t0, x̄ , ū;

k∑

i=1

λ̄i( f i
x (t0, ū, ˙̄u)− D f i

ẋ (t0, ū, ˙̄u)

+Ai(t0, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū )p̄(t0)))−

1

2

k∑

i=1

λ̄i p̄(t0)
T Ai(t0, ū, ˙̄u, ¨̄u,

...
ū ,

....
ū )p̄(t0) +ρ1d2(t0, x̄ , ū))d t.

(56)

By Hypothesis (ii), we get

∫
ȳ(t0)

T g(t0, x̄ , ˙̄x)d t −

∫
ȳ(t0)

T g(t0, ū, ˙̄u)d t ≧

∫
(G(t0, x̄ , ū; gx(t0, ū, ˙̄u) ȳ(t0)−

D(g ẋ(t0, ū, ˙̄u) ȳ(t0))+ B(t0, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t0), ˙̄y(t0), ¨̄y(t0),

...
ȳ (t0))p̄(t0))−

1

2
p̄(t0)

T B(t0, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t0), ˙̄y(t0), ¨̄y(t0),

...
ȳ (t0))p̄(t0) +ρ2d2(t0, x̄ , ū))d t. (57)

Adding (56), (57) and using Hypothesis (iii), (21), (35), (37), (55), we obtain

∫
G(t0, x̄ , ū;

k∑

i=1

λ̄i( f i
x (t0, ū, ˙̄u)− D f i

ẋ (t0, ū, ˙̄u) + Ai(t0, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū )p̄(t0)) + gx(t0, ū, ˙̄u) ȳ(t0)−

D(g ẋ(t0, ū, ˙̄u) ȳ(t0)) + B(t0, ū, ˙̄u, ¨̄u,
...
ū ,

....
ū , ȳ(t0), ˙̄y(t0), ¨̄y(t0),

...
ȳ (t0))p̄(t0))d t < 0,

which by (34) implies
∫

G(t0, x̄ , ū; 0)d t < 0, a contradiction to the fact that G(t, x̄ , ū; 0) = 0,

t ∈ I . Hence x̄(t) = ū(t), t ∈ I .

Remark 4. The above Theorem also holds true if we replace “efficient solutions” by “feasible

solutions”.

6. Related Problem

If the time dependency of problems (P) and (MWD) is removed, then these problems

reduce to the following second-order multiobjective nonlinear problems studied in Mond and

Zhang [13] and Gulati and Agarwal [7]. The assumptions in our converse duality theorem

are similar to the assumptions in [7].

(NP) Minimize ( f 1(x), f 2(x), . . . , f k(x))

Subject to g(x)≦ 0,



REFERENCES 804

(ND) Minimize ( f 1(u)−
1

2
pT∇2 f 1(u)p, f 2(u)−

1

2
pT∇2 f 2(u)p, . . . , f k(u)−

1

2
pT∇2 f k(u)p)

Subject to

k∑

i=1

λi(∇ f i(u) +∇2 f i(u)p) +

m∑

j=1

y j(∇g j(u) +∇2 g j(u)p) = 0,

yT g(u)−
1

2
pT∇2(yT g(u))p ≧ 0, λ≥ 0, y ≧ 0.

ACKNOWLEDGEMENTS The second author is thankful to the University Grants Commis-

sion, New Delhi (India) for providing financial support.

References

[1] I. Ahmad, T. R. Gulati, Mixed type duality for multiobjective variational problems with

generalized (F,ρ)-convexity, Journal of Mathematical Analysis and Applications 306

(2005) 669-683.

[2] C. R. Bector, I. Husain, Duality for multiobjective variational problems, Journal of Math-

ematical Analysis and Applications 166 (1992) 214-229.

[3] D. Bhatia, A. Mehra, Optimality conditions and duality for multiobjective variational

problems with generalized B-invexity, Journal of Mathematical Analysis and Applica-

tions 234 (1999) 341-360.

[4] S. Chandra, B. D. Craven, I. Husain, A class of nondifferentiable continuous program-

ming problems, Journal of Mathematical Analysis and Applications 107 (1985) 122-131.

[5] V. Chankong, Y. Y. Haimes, "Multiobjective Decision Making : Theory and Methodology,"

North-Holland, New York, 1983.

[6] X. H. Chen, Second-order duality for the variational problems, Journal of Mathematical

Analysis and Applications 286 (2003) 261-270.

[7] T. R. Gulati, D. Agarwal, On Huard type second-order converse duality in nonlinear

programming, Applied Mathematics Letters 20 (2007) 1057-1063.

[8] I. Husain, A. Ahmed, M. Masoodi, Second-order duality for variational problems, Euro-

pean Journal of Pure and Applied Mathematics 2 (2009) 278-295.

[9] D. S. Kim, A. L. Kim, Optimality and duality for nondifferentiable multiobjective vari-

ational problems, Journal of Mathematical Analysis and Applications 274 (2002) 255-

278.

[10] B. Mond, S. Chandra, I. Husain, Duality for variational problems with invexity, Journal

of Mathematical Analysis and Applications 134 (1988) 322-328.



REFERENCES 805

[11] B. Mond, M. A. Hanson, Duality for variational problems, Journal of Mathematical Anal-

ysis and Applications 18 (1967) 355-364.

[12] B. Mond, I. Smart, Duality and sufficiency in control problems with invexity, Journal of

Mathematical Analysis and Applications 136 (1988) 325-333.

[13] B. Mond, J. Zhang, Duality for multiobjective programming involving second order V-

invex functions, Proceedings of the Optimization Miniconference, B. M. Glover and V.

Jeyakumar, Editors, University of New South Wales, Sydney, Australia (1985) 89-100.