Optimality and Duality for Second-order Multiobjective Variational Problems

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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 Z

a

f(t,x, ˙x)d t

Subject to x(a) =0=x(b), g(t,x, ˙x)≦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 ˙x(t)is the derivative of x(t)with respect tot inI. For notational simplicity, we writex(t)and ˙x(t)as x and ˙x, 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:trgulatigmail.om(T. Gulati),geeta.mehndirattarediffmail.om(G.

Mehndiratta)

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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 Z

a

{f(t,u, ˙u)−1

2β(t)

T_F_β₍_t₎_}_{d t}

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

f_u+y(t)Tg_u−D(f_˙_u+y(t)Tg_u_˙) + (F+H)β(t) =0, t∈I, (2) b

Z

a

{y(t)Tg(t,u, ˙u)−1 2β(t)

T_H_β₍_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 (¯x(t), ¯y(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)Tg_x−D¯y(t)Tg_˙_x 6=0,

(A3) (i)

b R

a ¯

β(t)T(¯y(t)Tg_x−D¯y(t)Tg_˙_x)d t≧0and

b R

a ¯

β(t)THβ¯(t)d t>0, or

(ii)

b R

a ¯

β(t)T₍_¯_y₍_t₎T_g

x−D¯y(t)Tg˙x)d t≦0and b R

a ¯

β(t)T_H_β_¯₍_t₎_{d t}_<_0.

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

a

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

b R

a

y(t)T_g₍_t_{, ., .)}_{d t is second-order quasiinvex with respect to the same}_η_{, then}_x_¯₍_t₎_{is an optimal}

solution of (CP).

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

b R

a ¯

β(t)T_H_β_¯₍_t₎_{d t} _>_{0 (or}_<_{0) and the conclusion ¯}_β₍_t_{) =}_0, _t _∈_I_{, are} inconsis-tent. Moreover, assumption (A1) and equation (3.11) in[8], namely

(λ(t) +αβ¯(t))TF+ (λ(t) +γβ¯(t))TH= 0, t ∈I, do not implyλ(t) +αβ¯(t) =0, t ∈I and

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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 tot,x and ˙x, respectively, byf_t,f_x and f_x_˙ such that f_x = (_∂∂f

x1,

∂f

∂x2, . . . ,

∂f

∂xn)

T _and _f

˙

x = (

∂f

∂˙x1,

∂f

∂x˙2, . . . ,

∂f

∂˙xn)

T_{. The matrices} _f x x andgx are of ordern×nandn×m, respectively. Similarly fx˙x,f˙x˙x and the partial derivatives ofgjare also defined. All derivative of x, and all partial and total derivatives of f andgused in this section are assumed to be continuous. Let the setM ={1, 2, . . . ,m}.

Remark 1. It may be noted that if we write

fx(t,x, ˙x)−

d

d tf˙x(t,x, ˙x) =L(t,x, ˙x, ¨x), then the function F should be

F = L_x(t,x, ˙x, ¨x)− d

d tLx˙(t,x, ˙x, ¨x) + d2

d t2L¨x(t,x, ˙x, ¨x)

= ∂

∂x(fx(t,x, ˙x)− d

d tf˙x(t,x, ˙x))− d d t(

∂

∂˙x(fx(t,x, ˙x)− d

d tf˙x(t,x, ˙x))) + d

2

d t2( ∂

∂¨x(fx(t,x, ˙x)− d

d tf˙x(t,x, ˙x))) = f_{x x}−D f_˙_{x x}−D f_x_x_˙+D2f_˙_x_˙_x−D3f_˙_x_¨_x = f_{x x}−2D f_x_˙_x+D2f_˙_x_x_˙−D3f_x_˙_¨_x.

Thus F is a function of t,x(t), ˙x(t), ¨x(t),...x(t),....x(t)and is given by F(t,x, ˙x, ¨x,...x,....x ) = fx x(t,x, ˙x)−2D fx˙x(t,x, ˙x)

+ D2f_˙_x_˙_x(t,x, ˙x)−D3f_˙_x_x_¨(t,x, ˙x), 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 f_{x x}−2D f_x_˙_x+D2f_˙_x_˙_x and f_{x x}−D f_x_x_˙+D2f_x_˙_˙_x, respectively.

We formulate the following dual problem for (CP) :

(ÓCD) Maximize b Z

a

(f(t,u, ˙u)−1 2β(t)

T_F_β₍_t₎₎_{d t}

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f_x(t,u, ˙u) +g_x(t,u, ˙u)y(t)−D(f_˙_x(t,u, ˙u) +

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

y(t)Tg(t,u, ˙u)−1

2β(t)

T_H_β₍_t₎_≧_0, _t_∈_I_, ₍₇₎

y(t)≧0, t∈I, (8)

where

H(t,u, ˙u, ü,...u,....u,y(t), ˙y(t), ÿ(t),...y(t)) = (g_x(t,u, ˙u)y(t))_x−2D(g_x(t,u, ˙u)y(t))_˙_x +D2(gx˙(t,u, ˙u)y(t))˙x−D3(gx˙(t,u, ˙u)y(t))¨x, t∈I, andF(t,u, ˙u, 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(¯u(t), ¯y(t), ¯β(t))be an optimal solution of(dC D). If for each t∈I,

(B1) the vectors{F_i,H_i,i= 1, 2, . . . ,n}are linearly independent, where F_i and H_i are the ith rows of F(t, ¯u, ˙u¯, ü¯,...u¯,....u¯)and H(t, ¯u, ˙u¯, ü¯,...u¯,....u¯, ¯y(t), ˙y¯(t), ÿ¯(t),...¯y(t)), respectively, (B2) g_x(t, ¯u, ˙¯u)¯y(t)−D(g_˙_x(t, ¯u, ˙u¯)¯y(t))6=0,and

(B3) either

(i) the n×n matrix H(t, ¯u, ˙u¯, ¨u¯,...u¯,....u¯, ¯y(t), ˙¯y(t), ¨y¯(t),...¯y(t))+(g_x(t, ¯u, ˙u¯)¯y(t))_x is pos-itive definite andβ¯(t)T(g_x(t, ¯u, ˙u¯)¯y(t))≧0, or

(ii) the n×n matrix H(t, ¯u, ˙u¯, ¨u¯,...¯u,....u¯, ¯y(t), ˙¯y(t), ¨¯y(t),...¯y(t))+(g_x(t, ¯u, ˙u¯)¯y(t))_x is neg-ative definite andβ¯(t)T₍_g

x(t, ¯u, ˙u¯)¯y(t))≦0,

thenu¯(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 u¯(t) is an optimal solution of (CP).

Proof. Since(¯u(t), ¯y(t), ¯β(t))is an optimal solution of(dC D), there existα∈Rand piece-wise smooth functionsλ:I→Rn, γ:I→Randµ:I→Rmsuch that the following Fritz John conditions[4]are satisfied at(¯u(t), ¯y(t), ¯β(t))(for brevity, f_x ≡ f_x(t, ¯u, ˙u¯), gj ≡ gj(t, ¯u, ˙u¯), gxj ≡gxj(t, ¯u, ˙u¯)etc.):

−α(f_x −D f_x_˙−1 2(

¯

β(t)TFβ¯(t))_x+ 1 2D(

¯

β(t)TFβ¯(t))_˙_x−1 2D

2₍_β_¯₍_t₎T_F_β_¯₍_t₎₎

¨

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+1 2D

3₍_β_¯₍_t₎T_F_β_¯₍_t₎₎_... x −

1 2D

4₍_β_¯₍_t₎T_F_β_¯₍_t₎₎_....

x ) + (fx x−D fxx˙+ (gx¯y(t))x−D(gx¯y(t))˙x

−(D(f_˙_{x x}+ (g_x_˙¯y(t))_x)−D(D(f_˙_x_˙_x+ (g_x_˙¯y(t))_˙_x)) +D2(D(f_˙_x_¨_x+ (g_˙_x¯y(t))_¨_x))) +((F+H)β¯(t))_x −D((F+H)β¯(t))_˙_x+D2((F+H)β¯(t))_¨_x−D3((F+H)β¯(t))...x +D4((F+H)β¯(t))....x )λ(t)−γ(t)(gx¯y(t)−D(g˙x¯y(t))−

1 2(β¯(t)

T_H_β_¯₍_t₎₎ x

+1 2D(β¯(t)

T_H_β_¯₍_t₎₎

˙

x− 1 2D

2₍_β_¯₍_t₎T_H_β_¯₍_t₎₎

¨

x+ 1 2D

3₍_β_¯₍_t₎T_H_β_¯₍_t₎₎_... x

−1

2D

4₍_β_¯₍_t₎T_H_β_¯₍_t₎₎_....

x ) =0, t∈I, (9)

λ(t)T(g_xj +g_{x x}j β¯(t))−γ(t)(gj−1 2

¯

β(t)Tg_{x x}j β¯(t))−µj(t) =0, t∈I, j∈M, (10) (λ(t) +αβ¯(t))TF+ (λ(t) +γ(t)β¯(t))TH=0, t∈I, (11)

γ(t)(¯y(t)Tg−1 2β¯(t)

T_H_β_¯₍_t_{)) =}_0, _t_∈_I_, ₍₁₂₎

µ(t)T¯y(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) +αβ¯(t) =0, t∈I and (16)

λ(t) +γ(t)β¯(t) =0, t∈I. (17)

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

(α−γ(t))(gx¯y(t)−D(g˙x¯y(t)) +Hβ¯(t)) + 1 2α((

¯

β(t)TFβ¯(t))_x −D(β¯(t)TFβ¯(t))_˙_x +D2(β¯(t)TFβ¯(t))_¨_x−D3(β¯(t)TFβ¯(t))..._x +D4(β¯(t)TFβ¯(t))...._x ) + (((F+H)β¯(t))_x−

D((F+H)β¯(t))_˙_x+D2((F+H)β¯(t))_¨_x−D3((F+H)β¯(t))...x +D4((F+H)β¯(t))....x )λ(t) + 1

2γ(t)(( ¯

β(t)THβ¯(t))_x−D(β¯(t)THβ¯(t))_˙_x+D2(β¯(t)THβ¯(t))_¨_x−D3(β¯(t)THβ¯(t))...x +

D4(β¯(t)THβ¯(t))...._x ) =0, t∈I. (18)

Letγ(t) =0 for some t. Suppose t₀∈I andγ(t₀) =0. Then by (17), we getλ(t₀) =0 and soαβ¯(t₀) =0, by (16). Thus (18) yieldsα(g_x¯y(t₀)−D(g_˙_x¯y(t₀))) =0, which by Hypothesis (B2) impliesα=0. Sinceλ(t0) =0 andγ(t0) =0, from (10), we obtainµj(t0) =0, j ∈M.

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

Now, multiplying (10) by ¯yj(t), t ∈I, summing over j, and then using (12), (13), (17) andγ(t)>0, t∈I, we get

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which contradicts Hypothesis (B3) unless ¯

β(t) =0, t∈I. (19)

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

gj=−µ

j₍_t₎

γ(t) ≦0, t∈I.

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

Now, assume that ¯u(t)is not an optimal solution of (CP). Then, there existsuˆ(t)∈X such that

b Z

a

f(t,uˆ, ˙uˆ)d t<

b Z

a

f(t, ¯u, ˙u¯)d t.

As ¯β(t) =0, t∈I, we have b Z

a

f(t,uˆ, ˙uˆ)d t<

b Z

a

(f(t, ¯u, ˙u¯)− 1

2β¯(t)

T_F_β_¯₍_t₎₎_{d t}_,

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

3. Multiobjective Variational Problem

We consider the following multiobjective variational problem (P):

(P) Minimize ( b Z

a

f1(t,x, ˙x)d t, b Z

a

f2(t,x, ˙x)d t, . . . , b Z

a

fk(t,x, ˙x)d t)

Subject to x(a) =α,x(b) =β, (20)

g(t,x, ˙x)≦0, t∈I, (21)

where fi :I×S×S→R(i∈K), g= (g1,g2, . . . ,gm):I×S×S →Rm andS is an open set inRn. We denote the first partial derivatives of fi, with respect to t,x and ˙x respectively by

f_ti,f_xi and f_x_˙i such that f_xi = (∂fi

∂x1,

∂fi

∂x2, . . . ,

∂fi

∂xn)T and f_˙_xi = (

∂fi

∂˙x1,

∂fi

∂˙x2, . . . ,

∂fi

∂˙xn)T. The Hessian

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mean b R

a

fid t. LetK ={1, 2, . . . ,k}and for r ∈K, the setK_r= K−r. For aand bin Rn, we shall use the following three inequalities :

a≧ b⇔ai≧ bi (i=1, 2, . . . ,n) a≥b⇔(a≧b, a6=b) a>b⇔ai> bi (i=1, 2, . . . ,n).

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

Z

fr(t,x, ˙x)d t<

Z

fr(t, ¯x, ˙¯x)d t for some r∈K

and Z

fi(t,x, ˙x)d t≦ Z

fi(t, ¯x, ˙x¯)d t for i∈K_r.

Let p : I → Rn and Ai(t,x, ˙x, ¨x,...x,....x ) = f_{x x}i (t,x, ˙x)−2D f_xi_˙_x(t,x, ˙x) + D2f_˙_xi_˙_x(t,x, ˙x)− D3f_˙_xi_¨_x(t,x, ˙x), 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 allξ₁,ξ₂∈Rn (22) and

G(t,x,u;aξ) =aG(t,x,u;ξ)for all a∈R, a≧0andξ∈Rn. (23) From (23), it follows that G(t,x,u; 0) =0.

Letρ∈Randd:I×S×S→Rn be a pseudometric onRn.

Definition 3. The functionalR fi(t,x, ˙x)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,

Z

fi(t,x, ˙x)d t− Z

fi(t,u, ˙u)d t+1 2

Z

p(t)TAip(t)d t≧ Z

G(t,x,u;f_xi(t,u, ˙u)−D f_˙_xi(t,u, ˙u) +Aip(t))d t+ρ

Z

d2(t,x,u)d t.

If in the above definition, inequality is satisfied as strict inequality, then we say that the functional R

fi(t,x, ˙x)d t is second-order strictly(G,ρ)-convex at u(t)∈S.

Definition 4. The functionalR fi(t,x, ˙x)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, Z

G(t,x,u;f_xi(t,u, ˙u) − D f_˙_xi(t,u, ˙u) +Aip(t))d t+ρ

Z

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⇒

Z

fi(t,x, ˙x)d t≧ Z

fi(t,u, ˙u)d t−1 2 Z

p(t)TAip(t)d t.

Definition 5. The functionalR fi(t,x, ˙x)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,

Z

fi(t,x, ˙x)d t ≦ Z

fi(t,u, ˙u)d t−1 2

Z

p(t)TAip(t)d t

⇒

Z

G(t,x,u;f_xi(t,u, ˙u)−D f_˙_xi(t,u, ˙u) +Aip(t))d t+ρ

Z

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 ofk-scalar objective variational problems.

Lemma 1(Chankong and Haimes[5]). A point ¯x(t)∈X is an efficient solution of (P) if and only if ¯x(t)is an optimal solution of (P_r) for each r∈K.

(Pr) Minimize

R

fr(t,x, ˙x)d t Subject to x(a) =α,x(b) =β,

g(t,x, ˙x)≦0, t∈I, R

fi(t,x, ˙x)d t≦R fi(t, ¯x, ˙¯x)d t, i∈K_r.

Thek-problems (P_r) 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 followingk-single objective problems:

(ˆP_r) Minimize R fr(t,x, ˙x)d t Subject to x(a) =α,x(b) =β,

g(t,x, ˙x)≦0, t ∈I,

fi(t,x, ˙x)≦ fi(t, ¯x, ˙x¯), t ∈I, i∈K_r.

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Proof. Let ¯x(t)be an efficient solution of (P) and suppose, to the contrary, that ¯x(t)is not an optimal solution of (Pˆ_r), for some r∈K. Then there exists an xˆ(t)∈X such that

g(t,xˆ, ˙xˆ)≦0, t∈I, (24)

fi(t,xˆ, ˙ˆx)≦ fi(t, ¯x, ˙x¯), t ∈I, i∈K_r, (25)

and Z

fr(t,ˆx, ˙ˆx)d t<

Z

fr(t, ¯x, ˙¯x)d t. (26)

Inequality (25) implies Z

fi(t,ˆx, ˙ˆx)d t≦ Z

fi(t, ¯x, ˙¯x)d t, i∈Kr. (27)

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

x(t)is an optimal solution of (Pˆ_r) for eachr ∈K.

Theorem 3(Fritz John type necessary conditions). Let¯x(t)be an efficient solution of (P). Then there existλ¯i∈R, i∈K and a piecewise smooth function ¯y :I→Rmsuch that

k X

i=1

¯

λi(f_xi(t, ¯x, ˙x¯)−D f_x_˙i(t, ¯x, ˙¯x)) +gx(t, ¯x, ˙¯x)¯y(t)−D(g˙x(t, ¯x, ˙¯x)¯y(t)) =0, t ∈I, (28)

¯

y(t)Tg(t, ¯x, ˙x¯) =0, t∈I, (29) (λ¯, ¯y(t))≥0, t∈I. (30) Proof. Since ¯x(t)is an efficient solution of (P), by Lemma 2, ¯x(t)is an optimal solution of (Pˆ_r) for each r∈K and hence in particular of (Pˆ₁). Therefore, by[4], there exist ¯λi, i∈K and a piecewise smooth function ¯y(t)∈Rmsuch that

g_x(t, ¯x, ˙¯x)¯y(t)−D(g_˙_x(t, ¯x, ˙¯x)¯y(t)) =0, t ∈I, ¯

y(t)Tg(t, ¯x, ˙x¯) =0, t∈I, (λ¯1, ¯λ2, . . . , ¯λk, ¯y(t))≥0, t∈I,

which give (28) to (30).

Theorem 4(Kuhn-Tucker type necessary conditions). Let ¯x(t) be an efficient solution of (P) and let for some r ∈ K, the constraints of (Pˆ_r) satisfy Slater’s or Robinson condition at ¯x(t). Then there existλ¯∈Rkand a piecewise smooth function ¯y:I→Rm such that

k X

i=1

¯

λi(f_xi(t, ¯x, ˙¯x)−D f_x_˙i(t, ¯x, ˙x¯)) +g_x(t, ¯x, ˙¯x)¯y(t)−D(g_˙_x(t, ¯x, ˙x¯)¯y(t)) =0, t∈I,

¯

y(t)Tg(t, ¯x, ˙x¯) =0, t∈I, ¯

λ≥0 ¯

(10)

Proof. Since ¯x(t)is an efficient solution of (P), by Lemma 2, ¯x(t)is an optimal solution of (Pˆ_r) for eachr. As for somer, the constraints of (Pˆ_r) satisfy Slater’s or Robinson condition at ¯x(t), by the Kuhn-Tucker necessary conditions in [4], there exist 0<λ¯r ∈R, 0≦λ¯i ∈R, i∈K_r and a piecewise smooth function ¯y(t)∈Rmsuch that

¯

λr(f_xr(t, ¯x, ˙x¯)−D f_˙_xr(t, ¯x, ˙x¯)) +X i∈Kr

¯

λi(f_xi(t, ¯x, ˙¯x)−D f_x_˙i(t, ¯x, ˙¯x))+

g_x(t, ¯x, ˙¯x)¯y(t)−D(g˙x(t, ¯x, ˙¯x)¯y(t)) =0, t∈I,

¯

y(t)Tg(t, ¯x, ˙x¯) =0, t∈I, ¯

λr>0, 0≦λ¯i∈R, i∈K_r, ¯

y(t)≧0, t∈I. Or equivalently

k X

i=1

¯

y(t)Tg(t, ¯x, ˙x¯) =0, t∈I, ¯

λ≥0, ¯λ∈Rk, ¯

y(t)≧0, t∈I.

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

λ ≥0. In the following Theorem, we assume Slater’s or Robinson condition for every (Pˆr) and obtain ¯λ >0.

Theorem 5(Kuhn-Tucker type necessary conditions). Let ¯x(t) be an efficient solution of (P) and let for each r∈K, the constraints of (Pˆr) satisfy Slater’s or Robinson condition atx¯(t). Then

there existλ¯∈Rkand a piecewise smooth function ¯y:I→Rmsuch that

k X

i=1

¯

λi(f_xi(t, ¯x, ˙¯x)−D f_x_˙i(t, ¯x, ˙x¯)) +gx(t, ¯x, ˙¯x)¯y(t)−D(g˙x(t, ¯x, ˙x¯)¯y(t)) =0, t∈I,

¯

y(t)Tg(t, ¯x, ˙x¯) =0, t∈I,

¯

λ >0, k X

i=1

¯

λi=1,

¯

y(t)≧0, t∈I.

Proof. Since ¯x(t)is an efficient solution of (P), by Lemma 2, ¯x(t)is an optimal solution of (Pˆ_r) for eachr∈K. As for eachr, the constraints of (Pˆ_r) satisfy Slater’s or Robinson condition at ¯x(t), by the Kuhn-Tucker necessary conditions in[4], for each r ∈ K, there exist ¯v_ri ∈R (i∈Kr) and piecewise smooth functions ¯µ

j

(11)

f_xr(t, ¯x, ˙¯x)−D f_˙_xr(t, ¯x, ˙x¯) +X i∈Kr

¯

v_ri(f_xi(t, ¯x, ˙x¯)−D f_˙_xi(t, ¯x, ˙x¯))+

m X

j=1

(g_xj(t, ¯x, ˙¯x)¯µ_rj(t)−D(g_x_˙j(t, ¯x, ˙¯x)µ¯_rj(t))) =0, t∈I,

m X

j=1

¯

µ_rj(t)gj(t, ¯x, ˙¯x) =0, t∈I,

¯

v_ri ≧0, i∈K_r, ¯

y(t)≧0, t∈I. Summing overr∈K, we get

k X

i=1

(¯v₁i+¯v₂i +...+¯v_ki)(f_xi(t, ¯x, ˙¯x)−D f_x_˙i(t, ¯x, ˙x¯)) + m X

j=1

(g_xj(t, ¯x, ˙¯x)(¯µ₁j(t) +µ¯₂j(t) +. . .+

¯

µ_kj(t))−D(g_x_˙j(t, ¯x, ˙¯x)(¯µ₁j(t) +µ¯₂j(t) +. . .+µ¯_kj(t)))) =0, t∈I, m

X

j=1

(µ¯₁j(t) +µ¯₂j(t) +. . .+µ¯_kj(t))gj(t, ¯x, ˙x¯) =0, t∈I,

where ¯v_ii=1 for eachi∈K.

Equivalently, k

X

i=1

¯

vi(f_xi(t, ¯x, ˙¯x)−D f_˙_xi(t, ¯x, ˙¯x)) + m X

j=1

(g_xj(t, ¯x, ˙¯x)µ¯j(t)−D(g_x_˙j(t, ¯x, ˙x¯)µ¯j(t))) =0, t∈I, (31)

m X

j=1

¯

µj(t)gj(t, ¯x, ˙¯x) =0, t∈I, (32)

where ¯vi=1+ P r∈Ki

¯

v_ri >0,i∈K, and ¯µj(t) = k P

r=1

¯

µrj(t)≧0,t ∈I,j∈M.

Dividing (31) and (32) by k P

i=1

¯

vi and setting

¯

λi = ¯v i

k P

i=1

¯ vi

, i∈K, ¯yj(t) =µ¯ j₍_t₎

k P

i=1

¯ vi

, j∈M,

we get k X

i=1

¯

λi(f_xi(t, ¯x, ˙¯x)−D f_x_˙i(t, ¯x, ˙¯x)) + m X

j=1

(12)

m X

j=1

¯

yj(t)gj(t, ¯x, ˙x¯) =0, t∈I,

or k X

i=1

¯

y(t)Tg(t, ¯x, ˙x¯) =0, t∈I,

¯

λ= (λ¯1, ¯λ2, . . . , ¯λk)>0, k X

i=1

¯

λi =1,

¯

y(t) = (¯y1(t), ¯y2(t), . . . , ¯ym(t))≧0, t ∈I.

5. Second-order Mond-Weir Type Duality

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

(MWD) Maximize ( Z

(f1(t,u, ˙u)−1 2p(t)

T_A1_p₍_t₎₎_{d t}_{, . . . ,}

Z

(fk(t,u, ˙u)−1

2p(t)

T_Ak_p₍_t₎₎_{d t}₎

Subject to u(a) =α,u(b) =β, (33)

k X

i=1

λi(f_xi(t,u, ˙u)−D f_x_˙i(t,u, ˙u) +Aip(t)) +

g_x(t,u, ˙u)y(t)−D(gx˙(t,u, ˙u)y(t)) +Bp(t) =0, t∈I, (34)

y(t)Tg(t,u, ˙u)−1

2p(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, ˙u, ¨u,...u,....u), t ∈I, i∈K (as defined earlier) and

B(t,u, ˙u, ¨u,...u,....u,y(t), ˙y(t), ¨y(t),...y(t)) = (gxy(t))x−2D(gxy(t))˙x+D2(g˙xy(t))˙x−D3(g˙xy(t))x¨,

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

(13)

(i) R

k P

i=1

λifi(t, ., .)d t is second-order(G,ρ1)-pseudoconvex at u(t),

(ii) R y(t)Tg(t, ., .)d t is second-order(G,ρ₂)-quasiconvex at u(t), (iii) λ >0andρ1+ρ2≧0.

Then

Z

fr(t,x, ˙x)d t<

Z

(fr(t,u, ˙u)−1 2p(t)

T_Ar_p₍_t₎₎_{d t for some r}_∈_K ₍₃₈₎

and Z

fi(t,x, ˙x)d t≦ Z

(fi(t,u, ˙u)−1 2p(t)

T_Ai_p₍_t₎₎_{d t}_, _i_∈_K

r (39)

cannot hold.

Proof. Suppose, to the contrary, that (38) and (39) hold. Sinceλ >0, the above inequali-ties give

Z _Xk

i=1

λifi(t,x, ˙x)d t<

Z _Xk

i=1

λi(fi(t,u, ˙u)− 1

2p(t)

T_Ai_p₍_t₎₎_{d t}_. ₍₄₀₎

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

y(t)Tg(t,x, ˙x)d t≦ Z

(y(t)Tg(t,u, ˙u)− 1 2p(t)

T_Bp₍_t₎₎_{d t}

AsR y(t)T_g₍_t_{, ., .)}_{d t} _{is second-order}₍_G_,_ρ

2)-quasiconvex atu(t), we get

Z

(G(t,x,u;g_x(t,u, ˙u)y(t)−D(g_˙_x(t,u, ˙u)y(t)) +Bp(t)) +ρ₂d2(t,x,u))d t≦0. (41)

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

(G(t,x,u; k X

i=1

λi(f_xi(t,u, ˙u) − D f_˙_xi(t,u, ˙u) +Aip(t)) +g_x(t,u, ˙u)y(t)

− D(g_˙_x(t,u, ˙u)y(t)) +Bp(t)))d t=0, which on using inequality (41), Hypothesis (iii) and the sublinearity ofG, yield

Z

G(t,x,u; k X

i=1

λi(f_xi(t,u, ˙u)−D f_x_˙i(t,u, ˙u) +Aip(t)))d t+ Z

ρ₁d2(t,x,u)d t≧0.

By Hypothesis (i), it implies Z _Xk

i=1

λifi(t,x, ˙x)d t≧ Z _Xk

i=1

λi(fi(t,u, ˙u)− 1

2p(t)

T_Ai_p₍_t₎₎_{d t}_,

(14)

Remark 2. If we drop the assumptionλ >0from Theorem 6, then we need to replace Hypothesis (i) by :R

k P

i=1

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

Theorem 7(Strong duality). Letx¯(t)be normal and is an efficient solution of (P). Then, there exist λ¯ ∈ Rk, a piecewise smooth function ¯y : I → Rm such that (x¯(t), ¯λ, ¯y(t), ¯p(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(¯x(t), ¯λ, ¯y(t), ¯p(t) =0) is an efficient solution of the problem (MWD).

Proof. Since ¯x(t)is normal and an efficient solution of (P), therefore by Theorem 4, there exist ¯λ∈Rkand a piecewise smooth function ¯y(t)∈Rmsatisfying

k X

i=1

¯

λi(f_xi(t, ¯x, ˙¯x)−D f_x_˙i(t, ¯x, ˙x¯)) +gx(t, ¯x, ˙¯x)¯y(t)−D(g˙x(t, ¯x, ˙x¯)¯y(t)) =0, t∈I,

¯

y(t)Tg(t, ¯x, ˙x¯) =0, t∈I, ¯

λ≥0, t∈I, ¯

y(t)≧0, t∈I.

Hence(x¯(t), ¯λ, ¯y(t), ¯p(t) =0)satisfies the constraints of (MWD) and thus the two objective functionals have the same value.

Now, we claim that(¯x(t), ¯λ, ¯y(t), ¯p(t) =0)is an efficient solution of (MWD). If not, then there exists(ˆu(t),λˆ,ˆy(t),ˆp(t))∈Y, such that

( Z

(f1(t,uˆ, ˙ˆu)−1

2ˆp(t)

T_A1₍_t_,_u_ˆ_{, ˙}_u_ˆ_{, ¨}_u_ˆ_,..._ˆ_u_,...._u_ˆ_)ˆ_p₍_t₎₎_{d t}_{, . . . ,}

Z

(fk(t,uˆ, ˙uˆ)−1 2ˆp(t)

T_Ak₍_t_,_u_ˆ_{, ˙}_u_ˆ_{, ¨}_u_ˆ_,..._u_ˆ_,...._u_ˆ_)ˆ_p₍_t₎₎_{d t}₎

≥( Z

(f1(t, ¯x, ˙¯x)−1 2¯p(t)

T_A1₍_t_{, ¯}_x_{, ˙}_¯_x_{, ¨}_¯_x_,..._¯_x_,...._¯_x_)¯_p₍_t₎₎_{d t}_{, . . . ,}

Z

(fk(t, ¯x, ˙¯x)−1 2¯p(t)

T_Ak₍_t_{, ¯}_x_{, ˙}_¯_x_{, ¨}_¯_x_,..._¯_x_,...._¯_x_)¯_p₍_t₎₎_{d t}_).

As ¯p(t) =0, we have

( Z

(f1(t,uˆ, ˙ˆu)−1

2ˆp(t)

T_A1₍_t_,_u_ˆ_{, ˙}_u_ˆ_{, ¨}_u_ˆ_,..._ˆ_u_,...._u_ˆ_)ˆ_p₍_t₎₎_{d t}_{, . . . ,}

Z

(fk(t,uˆ, ˙uˆ)−1

2ˆp(t)

T_Ak₍_t_,_u_ˆ_{, ˙}_u_ˆ_{, ¨}_u_ˆ_,..._ˆ_u_,...._u_ˆ_)ˆ_p₍_t₎₎_{d t}₎_≥₍ Z

f1(t, ¯x, ˙¯x)d t, . . . , Z

(15)

which contradicts the weak duality theorem. Hence (¯x(t), ¯λ, ¯y(t), ¯p(t) =0) is an efficient solution of (MWD).

Theorem 8(Converse duality). Let(¯u(t), ¯λ, ¯y(t), ¯p(t))be an efficient solution of (MWD) for which

(C1) the vectors{Ai_m,B_m,t∈I,i∈K,m=1, 2, . . . ,n}are linearly independent, where Ai_mis the mth row of Ai(t, ¯u, ˙¯u, ¨u¯,...u¯,....u¯)and Bm is the mth row of

B(t, ¯u, ˙¯u, ¨u¯,...u¯,....u¯, ¯y(t), ˙¯y(t), ¨¯y(t),...¯y(t)),

(C2) f_xi(t, ¯u, ˙u¯)−D f_x_˙i(t, ¯u, ˙u¯), t∈I, i∈K,are linearly independent, and (C3) for t∈I, either

(a) the n×n matrix B(t, ¯u, ˙u¯, ¨u¯,...u¯,....u¯, ¯y(t), ˙¯y(t), ¨y¯(t),...¯y(t)) + (g_x(t, ¯u, ˙u¯)¯y(t))_x is pos-itive definite and¯p(t)T(gx(t, ¯u, ˙u¯)¯y(t))≧0, or

(b) the n×n matrix B(t, ¯u, ˙¯u, ¨u¯,...u¯,....u¯, ¯y(t), ˙¯y(t), ¨¯y(t),...¯y(t)) + (gx(t, ¯u, ˙u¯)¯y(t))x is

neg-ative definite and¯p(t)T₍_g

x(t, ¯u, ˙u¯)¯y(t))≦0.

Then ¯u(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¯u(t)is an efficient solution of (P).

Proof. Since(¯u(t), ¯λ, ¯y(t), ¯p(t))is an efficient solution of (MWD), there existα,η∈Rkand piecewise smooth functionsβ :I →Rn, γ: I →R, µ:I →Rm, such that the following Fritz John conditions (Theorem 3) are satisfied at(¯u(t), ¯λ, ¯y(t), ¯p(t))(for brevity, f_xi≡ f_xi(t, ¯u, ˙u¯), gj≡ gj(t, ¯u, ˙¯u), gxj ≡g

j

x(t, ¯u, ˙u¯),Ai≡Ai(t, ¯u, ˙u¯, ¨u¯, ...

¯ u,....¯u), B≡B(t, ¯u, ˙¯u, ¨u¯,...u¯,....u¯, ¯y(t), ˙¯y(t), ¨y¯(t),...¯y(t))etc.):

k X

i=1

αi(f_xi−D f_˙_xi−1 2(¯p(t)

T_Ai_¯_p₍_t₎₎ x+

1 2D(¯p(t)

T_Ai_¯_p₍_t₎₎

˙

x− 1 2D

2_(¯_p₍_t₎T_Ai_¯_p₍_t₎₎

¨

x +

1 2D

3_(¯_p₍_t₎T_Ai_¯_p₍_t₎₎_... x −

1 2D

4_(¯_p₍_t₎T_Ai_¯_p₍_t₎₎_.... x )−(

k X

i=1

¯

λi(Ai+ (Ai¯p(t))_x−D(Ai¯p(t))_˙_x+

D2(Ai¯p(t))_¨_x−D3(Ai¯p(t))...x +D4(Ai¯p(t))....x ) +B+ (B¯p(t))x−D(B¯p(t))˙x+

D2(B¯p(t))_¨_x−D3(B¯p(t))..._x +D4(B¯p(t))...._x )β(t) +γ(t)(g_xy(t)−D(g˙x¯y(t))− 1

2(¯p(t)

T_B_¯_p₍_t₎₎ x+

1 2D(¯p(t)

T_B_¯_p₍_t₎₎

˙

x − 1 2D

2_(¯_p₍_t₎T_B_¯_p₍_t₎₎

¨

x+ 1 2D

3_(¯_p₍_t₎T_B_¯_p₍_t₎₎_... x − 1

2D

4_(¯_p₍_t₎T_B_¯_p₍_t₎₎_....

x ) =0, t∈I, (42)

β(t)T(f_xi−D f_˙_xi+Aip¯(t))−ηi=0, t∈I, i∈K, (43)

β(t)T(g_xj+g_{x x}j ¯p(t))−γ(t)(gj−1 2¯p(t)

T_gj

x x¯p(t))−µ

(16)

k X

i=1

αiAi¯p(t) + ( k X

i=1

¯

λiAi+B)β(t) +γ(t)B¯p(t) =0, t∈I, (45)

γ(t)(¯y(t)Tg−1 2¯p(t)

T_B_¯_p₍_t_{)) =}_0, _t_∈_I_, ₍₄₆₎

µ(t)T¯y(t) =0, t∈I, (47)

ηTλ¯=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

X

i=1

Ai(αi¯p(t) +λ¯iβ(t)) +B(β(t) +γ(t)¯p(t)) =0, t∈I,

which by Hypothesis (C1), yield

αi¯p(t) +λ¯iβ(t) =0, t ∈I, i∈K, (51) and

β(t) +γ(t)¯p(t) =0, t∈I. (52)

Now, supposeγ(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¯p(t₀) = 0. Therefore for t = t₀, equation (42) reduces to

k X

i=1

αi(f_xi−D f_˙_xi) =0.

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

αi =0, i∈K.

Also, equations (43) and (44) giveηi =0, i∈K andµj(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 ¯yj(t), summing over jand then using (46), (47), (52) andγ(t)>0, we have

2¯p(t)T(g_xy¯(t)) +¯p(t)T(B+ (g_x¯y(t))_x)¯p(t) =0, t ∈I, which contradicts Hypothesis (C3) unless

¯

(17)

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

gj= −µj(t)

γ(t) ≦0, t∈I, j∈M. Thus ¯u(t)is feasible for (P).

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

Now, suppose that ¯u(t)is not an efficient solution of (P). Then, there existsuˆ(t)∈X such

that _Z

fr(t,uˆ, ˙uˆ)d t<

Z

fr(t, ¯u, ˙¯u)d tfor somer∈K

and _Z

fi(t,uˆ, ˙uˆ)d t≦ Z

fi(t, ¯u, ˙u¯)d t, i∈Kr.

Using (54), we get Z

fr(t,uˆ, ˙uˆ)d t<

Z

(fr(t, ¯u, ˙u¯)−1 2¯p(t)

T_Ar₍_t_{, ¯}_u_{, ˙}_u_¯_{, ¨}_u_¯_,..._u_¯_,...._u_¯_)¯_p₍_t₎₎_{d t}_{for some}_r_∈_K

and _Z

fi(t,uˆ, ˙uˆ)d t≦ Z

(fi(t, ¯u, ˙u¯)−1 2¯p(t)

T_Ai₍_t_{, ¯}_u_{, ˙}_u_¯_{, ¨}_u_¯_,..._u_¯_,...._u_¯_)¯_p₍_t₎₎_{d t}_, _i_∈_K r, which contradicts the weak duality theorem. Hence ¯u(t)is an efficient solution for (P).

Remark 3. For the single objective problems in Section 2, the Hypothesis (C2) reduces to (C2′) f_x−D f_˙_x 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 ¯x(t)and(¯u(t), ¯λ, ¯y(t), ¯p(t))be efficient solutions of (P) and (MWD) respectively, such that

Z _Xk

i=1

¯

λifi(t, ¯x, ˙¯x)d t= Z _Xk

i=1

¯

λi(fi(t, ¯u, ˙u¯)−1 2¯p(t)

T_Ai₍_t_{, ¯}_u_{, ˙}_u_¯_{, ¨}_u_¯_,..._¯_u_,...._u_¯_)¯_p₍_t₎₎_{d t} ₍₅₅₎

If

(i) R

k P

i=1

¯

λifi(t, ., .)d t is second-order strictly(G,ρ1)-convex atu¯(t),

(18)

(iii) ρ₁+ρ₂≧0. Then ¯x(t) =u¯(t).

Proof. Suppose ¯x(t0)6=u¯(t0)for somet0∈I. By Hypothesis (i), we have

Z _Xk

i=1

¯

λifi(t₀, ¯x, ˙¯x)d t−

Z _Xk

i=1

¯

λifi(t₀, ¯u, ˙¯u)d t>

Z

(G(t₀, ¯x, ¯u; k X

i=1

¯

λi(f_xi(t₀, ¯u, ˙u¯)−D f_˙_xi(t₀, ¯u, ˙u¯)

+Ai(t0, ¯u, ˙u¯, ¨u¯,

... ¯

u,....u¯)¯p(t0)))−

1 2

k X

i=1

¯

λi¯p(t0)TAi(t0, ¯u, ˙u¯, ¨u¯,

... ¯

u,....¯u)¯p(t0) +ρ1d2(t0, ¯x, ¯u))d t.

(56) By Hypothesis (ii), we get

Z ¯

y(t0)Tg(t0, ¯x, ˙¯x)d t−

Z ¯

y(t0)Tg(t0, ¯u, ˙u¯)d t≧

Z

(G(t0, ¯x, ¯u;gx(t0, ¯u, ˙u¯)¯y(t0)−

D(g_˙_x(t₀, ¯u, ˙u¯)¯y(t₀)) +B(t₀, ¯u, ˙u¯, ¨u¯,...u¯,....u¯, ¯y(t₀), ˙¯y(t₀), ¨y¯(t₀),...¯y(t₀))¯p(t₀))−

1 2¯p(t0)

T_B₍_t

0, ¯u, ˙u¯, ¨u¯,

... ¯

u,....¯u, ¯y(t₀), ˙y¯(t₀), ¨¯y(t₀),...y¯(t₀))¯p(t₀) +ρ₂d2(t₀, ¯x, ¯u))d t. (57)

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

Z

G(t0, ¯x, ¯u;

k X

i=1

¯

λi(f_xi(t0, ¯u, ˙¯u)−D f˙xi(t0, ¯u, ˙¯u) +Ai(t0, ¯u, ˙u¯, ¨u¯,

... ¯

u,....u¯)¯p(t0)) +gx(t0, ¯u, ˙u¯)¯y(t0)−

D(g_˙_x(t₀, ¯u, ˙u¯)¯y(t₀)) +B(t₀, ¯u, ˙u¯, ¨u¯,...u¯,....u¯, ¯y(t₀), ˙¯y(t₀), ¨¯y(t₀),...¯y(t₀))¯p(t₀))d t<0,

which by (34) impliesRG(t0, ¯x, ¯u; 0)d t<0, a contradiction to the fact thatG(t, ¯x, ¯u; 0) =0,

t∈I. Hence ¯x(t) =u¯(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].

(19)

(ND) Minimize (f1(u)−1 2p

T_∇2_f1₍_u₎_p_,_f2₍_u₎₋1

2p

T_∇2_f2₍_u₎_p_{, . . . ,}_fk₍_u₎₋1 2p

T_∇2_fk₍_u₎_p₎

Subject to k X

i=1

λi(∇fi(u) +∇2fi(u)p) + m X

j=1

yj(∇gj(u) +∇2gj(u)p) =0,

yTg(u)−1

2p

T_∇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.

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