On Mixed Type Duality for Nondifferentiable Multiobjective Variational Problems

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Vol. 3, No. 1, 2010, 81-97

ISSN 1307-5543 – www.ejpam.com

On Mixed Type Duality for Nondifferentiable Multiobjective

Variational Problems

I. Husain

1∗

_{and Rumana G. Mattoo}

2

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 mixed type dual to a nondifferentiable variational problem involving higher order deriva-tive is formulated and duality results are proved under generalized invexity conditions. Special cases are generated from our results.

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

Key Words and Phrases: Mixed type dual; Nondifferentiable Variational problem; Duality; Invexity; Generalized Invexity.

1. Introduction

In[9], Husain et al considered the following nondifferentiable multiobjective variational problem containing square root terms:

(VP) Minimize

Z

I

(f1(t,x, ˙x, ¨x) + (x(t)TB1(t)x(t))12)d t,

. . . ,

Z

I

(fp(t,x, ˙x, ¨x) + (x(t)TBp(t)x(t))12)d t

subject to

x(a) =0=x(b) ˙

x(a) =0=˙x(b)

g(t,x, ˙x, ˙x)≦0,t ∈I, where

∗_{Corresponding author.}

Email address:iqbal.husainjiet.a.in(I. Husain)

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I. Husain, R. Mattoo Eur. J. Pure Appl. Math,3(2010), 81-97 82

1 fi : I×Rn×Rn×Rn →R, (i = 1, 2, . . . ,p), g : I×Rn×Rn×Rn → Rm, j = 1, . . . ,m are assumed to be continuously differentiable functions, and for each t ∈ I, i ∈ P =

{1, 2, . . . ,p},Bi(t)is ann×npositive semidefinite (symmetric) matrix, withBi(·)continuous onI.

The following Wolfe type dual to the problem (VP) is presented in[9]:

(MWD) Maximize

Z

I

(f1(t,u, ˙u, ¨u) + (u(t)TB1(t)z1(t)) +y(t)Tg(t,u, ˙u, ¨u))d t

, . . . ,

Z

I

(fp(t,u, ˙u, ¨u) + (u(t)TBp(t)zp(t)) +y(t)Tg(t,u, ˙u, ¨u))d t

Subject to

u(a) =0=u(b) ˙

u(a) =0=u˙(b)

p

X

i=1

λif_ui(t,u, ˙u, ¨u)d t+Bi(t)zi(t) +y(t)T gu(t,u, ˙u, ¨u)

−DλTf˙u(t,u, ˙u, ¨u) + y(t)T gu˙(t,u, ˙u, ¨u)

+D2λTf_¨_u(t,u, ˙u, ¨u) +y(t)T g_u_¨(t,u, ˙u, ¨u)=0 , t∈I, y(t)≧0, t∈I,

λ >0, λTe=1.

¯

zi(t)TBi(t)¯zi(t)≦1, t∈I,i∈P, The problem (MWD) is a dual to (VP) assuming that

p

X

i=1

λi

Z

I

(fi(t, ., ., .) + (·)TBi(t)zi(t) +y(t)Tg(t, ., ., .))d t

is pseudoinvex with respect toη. The authors in[9]further weakened the invexity required in Wolfe type duality by constructing the following Mond- Weir type vector dual.

The following is the Mond-Weir vector type dual to (VP):

(M-WVD) Maximize

Z

I

(f1(t,u, ˙u, ¨u) + (u(t)TB1(t)z1(t)))d t

, . . . ,

Z

I

(fp(t,u, ˙u, ¨u) + (u(t)TBp(t)zp(t)))d t

Subject to

u(a) =0=u(b) ˙

u(a) =0=u˙(b)

p

X

i=1

λif_ui(t,u, ˙u, ¨u)d t+Bi(t)zi(t) +y(t)Tgu(t,u, ˙u, ¨u)

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−DλTfu˙(t,u, ˙u, ¨u) +y(t)Tgu˙(t,u, ˙u, ¨u)

+D2λTf_u_¨(t,u, ˙u, ¨u) +y(t)Tg_u_¨(t,u, ˙u, ¨u)=0 , t∈I, ¯

zi(t)TBi(t)z¯i(t)≦1, t∈I,i∈P

Z

I

y(t)Tg(t,u, ˙u, ¨u)d t≧0, y(t)≧0, t∈I,

λ >0.

Mond-Weir established usual duality theorems under the hypotheses that

p

X

i=1

λi

Z

I

(fi(t, ., ., .) + (·)TBi(t)zi(t))d t

is pseudoinvex andR_Iy(t)Tg(t, ., ., .)d tis quasi-invex with respect to the sameη.

In this paper, we propose, in the spirit of Husain and Jabeen[7]and Xu[18], a mixed type dual to (VP) and established various duality results under generalized invexity requirements. Special cases are generalized and it is also pointed out that our duality results can be viewed as dynamic generalizations of those of static case, already existing in the literature.

2. Pre-requisites

Consider the real interval I = [a,b], and the continuously differentiable function φ : I×Rn×Rn×Rn→R. In order to considerφ(t,x, ˙x, ¨x), wherex :I→Rnis twice differentiable with its first and second order derivatives ˙xand ¨xrespectively, we denote the partial derivative

φbyφ_t.

φ_x =

_{∂ φ}

∂x1, . . . ,

∂ φ

∂xn

T

, φ_x_˙=

_{∂ φ}

∂˙x1, . . . ,

∂ φ

∂˙xn

T

, φ_¨_x =

_{∂ φ}

∂¨x1, . . . ,

∂ φ

∂x¨n

T

.

The partial derivative of other function will be written similarly. Let K designates the space of piecewise functions x : I → Rn possessing derivatives ˙x and ¨x with the norm kxk =

kxk_∞kxk_∞+kD2xk_∞, where the differentiation operatorDis given by

u=D x⇔x(t) =α+

Z t

a

u(s)ds,

Whereαis given boundary value; thusD≡ _{d t}d except at discontinuities.

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Before stating our mixed type multiobjective variational problem, we mention the follow-ing conventions for vectors x and y in n-dimensional Euclidian spaceRn to be used through-out the analysis of this research together with some definitions of invexity and generalized invexity for easy reference.

x < y, ⇔ xi< yi, i=1, 2, . . . ,n.

x ≦ y, ⇔ x_i≤ y_i, i=1, 2, . . . ,n. x ≤ y, ⇔ x_i≦ y_i, i=1, 2, . . . ,n. but x6= y

x y, is the negation of x ≤ y For x,y∈R, x ≤ y andx < y have the usual meaning.

Definition 1 (Invexity). If there exists vector function η(t,x,u) ∈Rn withη= 0and x(t) =

u(t), t ∈I and Dη=0for ˙x(t) =u˙(t), t∈I such that for a scalar functionφ(t,x, ˙x, ¨x), the functionalΦ(x, ˙x, ¨x) =R

Iφ(t,x, ˙x, ¨x)d t satisfies

Φ(x, ˙u, ¨u)−Φ(x, ˙x, ¨x)

≧

Z

I

{ηTφ_x(t,x, ˙x, ¨x) + (Dη)Tφ˙x(t,x, ˙x, ¨x) + (D2η)Tφ¨x(t,x, ˙x, ¨x)}d t,

Φis said to be invex in x, ˙x andx on I with respect to¨ η.

Definition 2(Pseudoinvexity). Φis said to be pseudoinvex in x, ˙x and x with respect to¨ ηif

Z

I

{ηTφ_x(t,x, ˙x, ¨x) + (Dη)Tφ_˙_x(t,x, ˙x, ¨x) + (D2η)Tφ_¨_x(t,x, ˙x, ¨x)}d t≧0

implies

Φ(x, ˙u, ¨u)≧Φ(x, ˙x, ¨x).

Definition 3 (Quasi-invex). The functionalΦ is said to quasi-invex in x, ˙x and ¨x with respect toηif

Φ(x, ˙u, ¨u)≦Φ(x, ˙x, ¨x)

implies

Z

I

{ηTφ_x(t,x, ˙x, ¨x) + (Dη)Tφ_˙_x(t,x, ˙x, ¨x) + (D2η)Tφ_¨_x(t,x, ˙x, ¨x)}d t≦0.

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Definition 4(Efficiency). A point¯x ∈X is said to be efficient solution of (VP) if for all feasible ¯

x ∈X

Z

I

(fi(t,x(t), ˙x(t), ¨x(t))d t+ (x(t)TBi(t)x(t))12)d t

Z

I

(fi(t, ¯x(t), ˙¯x(t), ¨¯x(t))d t+ (¯x(t)TBi(t)¯x(t))12)d t

for all i∈P.

In order to prove the strong duality theorem we will invoke the following lemma due to Changkong and Haimes[5].

Lemma 1. A point ¯x ∈X be an efficient solution of (VP) if and only if ¯x ∈X is an optimal solution of the following problem(P_k(¯x))for all k

(P_k(¯x)) Minimize

Z

I

(fk(t,x, ˙x, ¨x)d t+ (x(t)TBk(t)x(t))12

d t

Subject to

x(a) =0=x(b), ˙

x(a) =0=x˙(b),

g(t,x, ˙x, ˙x)≦0, t∈I,

Z

I

(fi(t,x(t), ˙x(t), ¨x(t))d t+ (x(t)TBi(t)x(t))12)d t

≤ Z

I

(fi(t, ¯x(t), ˙x¯(t), ¨x¯(t))d t+ (¯x(t)TBi(t)x¯(t))12)d t, i6=k

3. Mixed Type Duality

We formulate the following type dual (Mix D) to (VP):

(Mix D) Maximize

Z

I

f1(t,u, ˙u, ¨u) +u(t)TB1(t)z1(t) +X

j∈J◦

yj(t)gj(t,u, ˙u, ¨u)

d t

, . . . ,

Z

I

fp(t,u, ˙u, ¨u) + (u(t)TBp(t)zp(t)) +X

j∈J◦

d t

Subject to

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

˙

u(a) =0=u˙(b), (2)

p

X

i=1

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−D(λTf_u_˙+y(t)Tg_˙_u) +D2(λTf_¨_u+y(t)Tg_u_¨) =0, t∈I (3)

X

j∈Jα Z

I

yj(t)gj(t,u, ˙u, ¨u)d t≧0, α=1, 2, . . . ,r (4) ¯

zi(t)TBi(t)z¯i(t)≦1, t∈I,i∈P (5)

y(t)≧0, t∈I, (6)

λ∈Λ+ (7)

where

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

(ii) J_α⊆M ={1, 2, . . . ,m},α=0, 1, 2, . . . ,r with

r

S

α=0

J_α=M andJ_αTJ_β =φ, ifα6=β.

IfJ◦=M, then (Mix D) becomes Wolfe type dual considered in[9], ifJ◦=φandJα=M for

someα∈ {1, 2, . . . ,r}, then (Mix D) becomes Mond-Weir type dual considered in[9].

Theorem 1 (Weak Duality). Let ¯x be feasible for (VP) and(u,y,z1, . . . ,zp,λ) be feasible for (Mix D). If for feasible(x,u,z1, . . . ,zp,λ),

p

X

i=1

λi

Z

I

fi(t, ., ., .) +X

j∈J◦

yj(t)gj(t, ., ., .) + (·)TBi(t)zi(t)

d t

is pseudoinvex and

X

j∈Jα Z

I

yj(t)gj(t, ., ., .)d t, α=1, 2, . . . ,r

is quasi-invex with respect to sameη, then the following cannot hold:

Z

I

(fi(t,x, ˙x, ¨x)d t+ (x(t)TBi(t)x(t))12)d t

≦

Z

I

fi(t,u, ˙u, ¨u)d t+ (u(t)TBi(t)zi(t)) +X

j∈J◦

d t (8)

For all i∈P, and

Z

I

(fk(t,x, ˙x, ¨x)d t+ (x(t)TBk(t)x(t))12)d t

≦

Z

I

fk(t,u, ˙u, ¨u)d t+ (u(t)TBk(t)zk(t)) +X

j∈J◦

d t (9)

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Proof. Suppose contrary to the result, that (8) and (9) hold. In view of y(t) ≧0, t ∈I, ¯

zi(t)TBi(t)¯zi(t)≦1,t∈I,i∈Pand g(t,x, ˙x, ˙x)≦0,t∈I, these inequalities yield,

Z

I

fi(t,x, ˙x, ¨x)d t+ (x(t)TBi(t)zi(t)) +X

j∈J◦

d t ≦ Z I

j∈J◦

d t

for alli∈P, and

Z

I

fk(t,x, ˙x, ¨x)d t+ (x(t)TBk(t)zk(t)) +X

j∈J◦

d t ≦ Z I

fk(t,u, ˙u, ¨u)d t+ (u(t)TBk(t)zk(t)) +X

j∈J◦

d t

for somek.

Now usingλ >0 andλTe=1, these inequalities yield

p

X

i=1

λi

Z

I

fi(t,x, ˙x, ¨x)d t+ (x(t)TBi(t)zi(t)) +X

j∈J◦

yj(t)gj(t,x, ˙x, ¨x)

d t

<

p

X

i=1

λi

Z

I

j∈J◦

yj(t)gj(t,x, ˙x, ¨x)

d t

By pseudo invexity of

p

X

i=1

λi

Z

I

fi(t, ., ., .) +X

j∈J◦

yj(t)gj(t, ., ., .) + (·)TBi(t)zi(t)

d t

with respect toη, we have

0>

p

X

i=1

λi

Z

I

ηT

f_xi+Bi(t)zi(t) +X

j∈J◦

yj(t)g_xj

+(Dη)T

f_˙_xi+X

j∈J◦

yj(t)g_˙_xj

+ (D2η)T

f_¨_xi+X

j∈J◦

yj(t)g_¨_xj

d t

This, by integration by parts, gives,

=

p

X

i=1

λi

Z

I

ηT

(f_xi +Bi(t)zi(t) +X

j∈J◦

yj(t)g_xj)−D

f_˙_xi+X

j∈J◦

yj(t)g_˙_xj

d t

+ηT

f_x_˙i+X

j∈J◦

yj(t)g_˙_xj

t=b

t=a

+ (Dη)T

f_¨_xi+X

j∈J◦

yj(t)g_¨_xj

t=b

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+

Z

I

(Dη)TD

f_¨_xi+X

j∈J◦

yj(t)g_¨_xj

d t

Using the boundary conditions which at t=a,t =bgivesDη=0=η

=

p

X

i=1

λi Z I ηT

j∈J◦

yj(t)g_xj

−D

f_x_˙i+X

j∈J◦

yj(t)g_˙_xj

d t

+

Z

I

(Dη)TD

f_x_¨i+X

j∈J◦

yj(t)g_¨_xj

d t

=

p

X

i=1

λi Z I ηT

j∈J◦

yj(t)g_xj

−D

f_x_˙i+X

j∈J◦

yj(t)g_˙_xj

+D2

f_¨_xi+X

j∈J◦

yj(t)g_¨_xj

d t−ηT

f_¨_xi+X

j∈J◦

yj(t)g_¨_xj

t=b

t=a

0 >

p

X

i=1

λi Z I ηT

j∈J◦

yj(t)g_xj

−D

f_x_˙i+X

j∈J◦

yj(t)g_˙_xj

+D2

f_¨_xi+X

j∈J◦

yj(t)g_¨_xj

d t (10)

Now, from the feasibility of (VP) and (Mix-D), we have

X

j∈Jα Z

I

yj(t)gj(t,x, ˙x, ¨x)d t≦X

j∈Jα Z

I

yj(t)gj(t,u, ˙u, ¨u)d t

This, because of quasi-invexity of

X

j∈Jα Z

I

yj(t)gj(t, ., ., .)d t

withηimplies

X

j∈Jα Z

I

{ηT(yj(t)g_xj) + (Dη)Tyj(t)g_x_˙j + (D2η)Tyj(t)g_¨_xj}d t≦0

As earlier, integrating by parts and using the boundary conditions, we have

X

j∈Jα Z

I

ηT{(yj(t)g_xj) +DTyj(t)g_˙_xj +D2yj(t)g_¨_xj}d t≦0 (11)

Combining (10) and (11), we have

Z

I

ηT

p X

i=1

λi(f_xi+Bi(t)zi(t) +y(t)Tgx

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−D(λTf_˙_x+y(t)Tg_˙_x) +D2(λTf_¨_x+y(t)Tg_¨_x))d t<0 From the equality constraint of the dual, we have

Z

I

ηT(

p

X

i=1

λi(f_xi+Bi(t)zi(t) +y(t)Tg_x)

−D(λTf_˙_x+y(t)Tg_˙_x) +D2(λTf_¨_x+y(t)Tg_¨_x))d t=0 which is a contradiction. Hence the conclusion of the theorem is true.

Theorem 2 (Strong Duality). Let ¯x ∈ X be an efficient solution of (VP) and for at least one i∈P,x satisfies the regularity condition¯ [3]for the problem(P_k(¯x)). Then there exist multipliers ¯

λ∈Rp, piecewise smooth ¯y ∈Rmand zi(t)∈Rn, i={1, 2, . . . ,p}such that(¯x, ¯y, ¯z1, . . . , ¯zp,λ)

is feasible for (Mix D) and the objectives of (VP) and (Mix D) are equal.

Further, if the hypotheses of Theorem 1 are met, then(¯x, ¯y, ¯z1, . . . , ¯zp,λ)is an efficient solu-tion of (Mix D).

Proof. ¯x ∈X is an optimal solution of(P_k(¯x)). This implies that there exist ¯ξ∈Rp with ¯

ξ1, . . . , ¯ξp,zi(t)∈Rn, i={1, 2, . . . ,p} and piecewise smooth ¯υ∈Rmsuch that, the following optimality conditions[3]hold:

¯

ξk(f_xk+Bk(t)zk(t)−D f_˙_xk+D2f_¨_xk)

+

p

X

i=1

i6=k

¯

ξi(f_xi+Bi(t)zi(t)−D f_˙_xi+D2f_¨_xi)

+υ¯(t)Tg_x−D(υ¯(t)Tg_˙_x) +D2(υ¯(t)Tg_¨_x) =0 (12) ¯

υ(t)Tg(t, ¯x, ˙¯x, ¨¯x)d t=0 (13)

(¯x(t)TBi(t)¯x(t))12 = (x¯(t)TBi(t)z¯i(t)), i=1, . . . ,p (14) (¯z(t)TBi(t)z¯i(t))≦1, t∈I, i=1, 2, . . . ,p (15)

¯

ξ >0, ¯υ(t)≧0, t ∈I (16)

Dividing (12), (13) and (16) by

p

P

i=1

¯

ξi, and setting ¯λi= pξ¯

P

i=1

¯ ξi

,i=1, . . . ,pand

¯

y(t) = υ¯p(t)

P

i=1

¯ ξi

, we have

p

X

i=1

¯

λi(f_xi +Bi(t)z¯i(t)−D f_x_˙i+D2f_¨_xi)

+¯y(t)Tg_x−D(¯y(t)Tg_˙_x) +D2(¯y(t)Tg_¨_x) =0 (17) ¯

y(t)Tg(t, ¯x, ˙¯x, ¨x¯)d t=0 (18) ¯

(10)

The equation (18) implies

X

j∈J◦

yj(t)gj(t,x, ˙x, ¨x) =0 and X

j∈Jα

yj(t)gj(t,x, ˙x, ¨x) =0, α=1, . . . ,r (20)

This implies,

X

j∈Jα Z

I

yj(t)gj(t,x, ˙x, ¨x) =0, α=1, . . . ,r (21)

Consequently (15), (17), (19) and (21) implies that(¯x, ¯y, ¯z1, . . . , ¯zp, ¯λ)is feasible for (Mix D).

Z

I

fi(t, ¯x, ˙¯x, ¨¯x) + (¯x(t)TBi(t)¯x(t))12+

X

j∈J◦

yj(t)gj(t, ¯x, ˙x¯, ¨¯x)

d t

=

Z

I

(fi(t, ¯x, ˙x¯, ¨¯x) + (¯x(t)TBi(t)¯zi(t)))d t, i=1, 2, . . . ,p

This in view of Theorem 1, the efficiency of(¯x, ¯y, ¯z1, . . . , ¯zp, ¯λ)follows.

As in[15], by employing chain rule in calculus, it can be easily seen that the expression

p

X

i=1

λi(f_xi(t,u, ˙u, ¨u)d t+Bi(t)zi(t) +y(t)Tg_x(t,u, ˙u, ¨u))

−D(λTf˙x+y(t)Tg˙x) +D2(λTf¨x+ y(t)Tgx¨) =0, ,t∈I

may be regarded as a functionθ of variables t,x, ˙x, ¨x,...x,y, ˙y, ¨y andλ, where...x =D3x and ¨

y=D2y. That is, we can write

θ(t,x, ˙x, ¨x,...x,y, ˙y, ¨y,λ) =

p

X

i=1

λi(f_xi(t,u, ˙u, ¨u)d t+Bi(t)zi(t) +y(t)Tg_x(t,u, ˙u, ¨u))

−D(λTf˙x+ y(t)Tg˙x) +D2(λTf¨x+y(t)Tg¨x) =0

In order to prove converse duality between (VP) and (Mix D), the spaceX is now replaced by a smaller spaceX₂ of piecewise smooth thrice differentiable function x :I →Rn with the normkxk_∞+kD xk_∞+kD2xk_∞+kD3xk_∞. The problem (Mix D) may now be briefly written as,

Minimize

−

Z

I

(f1(t,u, ˙u, ¨u) +u(t)TB1(t)z1(t) +X

j∈J◦

d t

, . . . ,

Z

I

−(fp(t,u, ˙u, ¨u) + (u(t)TBp(t)zp(t)) +X

j∈J◦

d t

(11)

u(a) =0=u(b) ˙

u(a) =0=u˙(b)

θ(t,x, ˙x, ¨x,...x,y, ˙y, ¨y,λ) =0, t∈I

X

j∈Jα Z

I

yj(t)gj(t,u, ˙u, ¨u)d t≧0, α=1, 2, . . . ,r ¯

zi(t)TBi(t)¯zi(t)≦1, t∈I,i∈P y(t)≧0, t∈I

λ >0, λTe=1

where

θ(t,x, ˙x, ¨x,...x,y, ˙y, ¨y,λ) =

p

X

i=1

λi(f_xi(t,u, ˙u, ¨u) +Bi(t)zi(t) +y(t)Tgx(t,u, ˙u, ¨u))

−D(λTf_˙_x+y(t)Tg_˙_x) +D2(λTf_¨_x+ y(t)Tg_¨_x) =0

Considerθ(t,x(·), ˙x(·), ¨x(·),...x(·),y(·), ˙y(·), ¨y(·),λ) =0 as defining a mapping ψ: X×Y× Rp → B where Y is a space of piecewise twice differentiable function and B is the Banach Space. In order to apply Theorem 1 [10]to the problem (Mix D), the infinite dimensional inequality must be restricted. In the following theorem, we useψ′ to represent the Frèchèt derivativeψ_x(x,y,λ),ψ_y(x,y,λ),ψ_λ(x,y,λ).

Theorem 3 (Converse Duality). Let (¯x, ¯y, ¯z1, . . . , ¯zp, ¯λ) be an efficient solution for (Mix D). Assume that

A1 The Frèchèt derivativeψ′has a (weak*) closed range,

A₂ f and g are twice continuously differentiable,

A3

f_xi+Bi(t)zi(t) + P

j∈J◦

yj(t)gi_x

−D

f_˙_xi+ P

j∈J◦

yj(t)gi_˙_x

+D2

f_x_˙i+ P

j∈J◦

yj(t)gi_¨_x

,

i∈ {1, 2, . . . ,p}are linearly independent, and

A₄ (β(t)Tθ_x−Dβ(t)Tθ_x_˙+D2β(t)Tθ_¨_x−D3β(t)Tθ..._x)β(t) =0,⇒β(t) =0, t∈I Further, if the hypotheses of Theorem 1 are met thenx is an efficient solution of (VP).¯

Proof. Since(¯x, ¯y, ¯z1, . . . , ¯zp, ¯λ) withψ′having a (weak∗) closed range, is an efficient so-lution of (Mix D), then by Theorem 1[9], there exist multipliersτ∈Rp, piecewise smooth

β(t) : I → Rn, zi(t) ∈Rn, i = 1, . . . ,p, γ ∈R for each of rconstraints, η ∈Rp, δ ∈ R and piecewise smoothµ(t):I →Rmsatisfying the following Fritz-John optimality conditions[9].

−

p

X

i=1

τi

j∈J◦

yj(t)g_xj

−D

f_˙_xi+X

j∈J◦

yj(t)g_˙_xj

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+D2

f_¨_xi+X

j∈J◦

yj(t)g_¨_xj

−γ

r

X

α=1 X

j∈Jα

(yj(t)g_xj −D yj(t)g_x_˙j +D2yj(t)g_x_¨j)

+β(t)Tθ_x−Dβ(t)Tθ_˙_x+D2β(t)Tθ_¨_x−D3β(t)Tθ...x =0, t∈I (22)

−(τTe)gj+β(t)Tθ_yj−Dβ(t)Tθ_y_˙j +D2β(t)Tθ_¨_yj−µj(t) =0, j∈J_◦ (23) −γgj+β(t)Tθ_yj−Dβ(t)Tθ_˙_yj+D2β(t)Tθ_¨_yj−µj(t) =0, j∈J_α,α=1, . . . ,r (24) −τix(t)TBi(t) +λiβ(t)TBi(t) +2δiBi(t)zi(t) =0, i=1, . . . ,p (25)

j∈J◦

yj(t)g_xj −D

f_˙_xi+X

j∈J◦

yj(t)g_˙_xj

+D2

f_¨_xi+X

j∈J◦

yj(t)g_x_¨j

−ηi=0 (26)

µT(t)¯y(t) =0, t∈I (27)

ηTλ=0 (28)

γX

j∈Jα Z

I

y(t)Tg(t, ¯x, ˙x¯, ¨x¯)d t=0, α=1, 2, . . . ,p (29)

δi(zi(t)TBi(t)zi(t)−1) =0, t∈I (30)

(τ,γ,µ(t),δ,η)≧0 (31)

(τ,β(t),γ,µ(t),δ,η)6=0 (32) Sinceλ >0, (28) impliesη=0. Consequently (26) implies

j∈J◦

yj(t)g_xj −D(f_˙_xi+X

j∈J◦

yj(t)g_˙_xj) +D2(f_x_¨i+X

j∈J◦

yj(t)g_¨_xj)

=0 (33)

Using the duality constraint of (Mix D) equation (22) reduces to

−

p

X

i=1

(τi−γλi)

j∈J◦

yj(t)g_xj)−D(f_x_˙i+X

j∈J◦

yj(t)g_˙_xj

+D2

f_¨_xi+X

j∈J◦

yj(t)g_¨_xj

+β(t)Tθ_x −Dβ(t)Tθ_˙_x

+D2β(t)Tθ¨x−D3β(t)Tθ...x =0, t∈I (34)

Post multiplying (34) byβ(t)and then using (33), we obtain,

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This because of the hypothesis (A4) yields,

β(t) =0, t ∈I (35)

Using (35) in (34) in (22), we have

−

p

X

i=1

(τi−γλi){(f_xi+Bi(t)zi(t) +X

j∈J◦

yj(t)g_xj)−D(f_x_˙i+X

j∈J◦

yj(t)g_˙_xj)

+D2(f_¨_xi+X

j∈J◦

yj(t)g_¨_xj)}=0

This, due to the hypothesis (A₃) gives,

τi−γλi=0, i=1, 2, . . . ,p (36)

Supposeγ=0, then from (36) we haveτ=0. Consequently from (23) and (24) implies

µ(t) =0, t ∈I.

Also from (25) we have δiBi(t)zi(t) = 0, which together with (30) implies δi = 0, i.e.

δ=0. Thus,(τ,β(t),µ(t),η,γ,δ,) =0, which is a contradiction to (32). Henceγ >0. From (34) it implies thatτ >0.

By the generalized Schwartz inequality[17],

(x¯(t)TBi(t)z¯i(t))¶(x¯(t)TBi(t)x¯(t))12(¯zi(t)Bi(t)z¯i(t)) 1

2, i=1, 2, . . . ,p (37) Now let δ_τii =αi. Thenαi≧0 and from (25), we have

Bi(t)¯x(t) =2αiBi(t)¯zi(t), i=1, 2, . . . ,p which, is the condition for the equality in (37). Therefore, we have

(¯x(t)TBi(t)z¯i(t)) = (¯x(t)TBi(t)x¯(t))12(zi(t)Bi(t)zi(t)) 1 2

From (30), eitherδi=0 or ¯zi(t)TBi(t)¯zi(t) =1 orδi=0,i=1, . . . ,pand hence Bi(t)¯x(t) =0. Therefore, in either case

(x(t)TBi(t)zi(t)) = (x(t)TBi(t)x(t))12 , i=1, 2, . . . ,p (38) From (23) and (24) we readily obtain,

g(t, ¯x, ˙x¯, ¨¯x)≦0, t∈I

which gives the feasibility of ¯x for (P). Using (27) in (23) and (24) we have ¯

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This obviously gives

X

j∈J◦ ¯

yj(t)gj=0, j=1, 2, . . . ,m (39)

Hence

Z

I

(fi(t, ¯x, ˙¯x, ¨¯x)d t+x¯(t)TBi(t)¯zi(t) +X

j∈J◦ ¯

yj(t)gj)d t

=

Z

I

(fi(t, ¯x, ˙x¯, ¨¯x)d t+ (¯x(t)TBi(t)¯x(t))12)d t, i=1, 2, . . . ,p (by using (38) and (39)) The efficiency of ¯x for (VP) follows by an application of Theorem 1.

4. Variational Problems with Natural Boundary Conditions

It is possible to extend the duality theorems established in the previous sections to the corresponding variational problems with natural boundary values rather that fixed points.

(VP)₀ Minimize

Z

I

f1(t,x, ˙x, ¨x)d t+ (x(t)TB1(t)x(t))12

d t

, . . . ,

Z

I

fp(t,x, ˙x, ¨x)d t+ (x(t)TBp(t)x(t))12

d t

Subject to

gj(t,x, ˙x, ˙x)≦0, t∈I, j=1, . . . ,m

(M I X D)₀ Maximize

Z

I

f1(t,u, ˙u, ¨u) +u(t)TB1(t)z1(t) +X

j∈J◦

d t

, . . . ,

Z

I

fp(t,u, ˙u, ¨u) + (u(t)TBp(t)zp(t)) +X

j∈J◦

d t

Subject to

p

X

i=1

λi(f_ui(t,u, ˙u, ¨u) +Bi(t)zi(t) +y(t)Tg_u(t,u, ˙u, ¨u))

−D(λTf˙u(t,u, ˙u, ¨u) +y(t)Tg˙u(t,u, ˙u, ¨u))

+D2(λTfu¨(t,u, ˙u, ¨u) +y(t)Tgu¨(t,u, ˙u, ¨u)) =0, t∈I,

f_˙_xi+X

j∈J◦

yj(t)g_˙_xj

t=a

=0

f_˙_xi+X

j∈J◦

yj(t)g_¨_xj

t=b

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f_¨_xi+X

j∈J◦

yj(t)g_˙_xj

t=a

=0

f_¨_xiX

j∈J◦

yj(t)g_¨_xj

t=b

=0

∀ i∈ {1, 2, . . . ,p} ¯

zi(t)TBi(t)¯zi(t)≦1, t∈I,i∈P

X

j∈Jα Z

I

yj(t)gj(t,u, ˙u, ¨u)d t≧0, α=1, 2, . . . ,r, y(t)≧0, t∈I.

5. Nonlinear Programming

If all the functions are independent of t then the problems (VP)0and (MIX D)0 become

the following problems.

(VP)₁ Minimize(f1(x) + (xTB1x)12, . . . ,fp(x) + (xTBpx) 1 2) Subject to

g(x)≦0 (MixD)₁ Maximize

f1(u) +uTB1z1+X

j∈J◦

yjgj(u), . . . ,fP(u) +uTBPzP+X

j∈J◦

yjgj(u)

Subject to

p

X

i=1

λi(f_ui(u)d t+Bizi+yTgu(u)) =0

¯

ziTBi¯zi≦1, i∈P

X

j∈Jα

yjgj(u)≧0, α=1, 2, . . . ,r y(t)≧0, t∈I

λ∈Λ+

whereΛ+={λ∈Rp|λ >0,λTe=1,e= (1, 1, . . . , 1)T ∈Rp}

The duality results for this pair of problems are not explicitly reported in the literature but can be derived easily on the lines of the analysis of this research.

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