Mixed type duality for multiobjective fractional variational control problems

VARIATIONAL CONTROL PROBLEMS

Received 20 July 2003 and in revised form 2 August 2003

The concept of mixed-type duality has been extended to the class of multiobjective frac-tional variafrac-tional control problems. A number of duality relations are proved to relate the efficient solutions of the primal and its mixed-type dual problems. The results are obtained forρ-convex (generalizedρ-convex) functions. The results generalize a num-ber of duality results previously obtained for finite-dimensional nonlinear programming problems under various convexity assumptions.

1. Introduction

Duality for multiobjective fractional variational problems has been of great interest in recent years [6,7,19]. Under different assumptions of convexity (convexity, generalized convexity, generalizedρ-convexity), Weir and Mond [16], Weir [15], and Egudo [4] have used efficiency to establish some duality results, where Wolfe [17] and Mond-Weir [11] duals are considered. Recently, Xu [18] introduced a mixed-type duality model, which contains Wolfe and Mond-Weir duality models as special cases and established various duality results by relating “efficient” solutions of his mixed-type dual pair of problems.

Preda [13] introduced the concept of generalized (F,ρ)-convexity, an extension ofF -convexity defined by Hanson and Mond [5], and generalizedρ-convexity defined by Vial [14], and he used the concept to obtain some duality results. In [8], Mishra and Mukher-jee discussed duality for multiobjective variational problems containing generalized (F, ρ)-convex functions. Some duality results for a class of differentiable multiobjective vari-ational problems were studied in [3]. Mond and Hanson [10] have obtained duality the-orems for control problems. Mond and Hanson [9] considered a dual formulation for a class of variational problems. Nahak and Nanda [12] used the concept of efficiency to for-mulate Wolfe and Mond-Weir type duals for multiobjective variational control problems and established weak and strong duality theorems under generalized (F,ρ)-convexity as-sumptions.

In this paper, we introduce a continuous analog of the (static) mixed-type dual of Xu [18] in a class of multiobjective fractional variational control problems and establish a large number of duality results by relating efficient solutions between this mixed-type Copyright©2005 Hindawi Publishing Corporation

dual pair. The results are obtained for differentiable ρ-convex (generalized ρ-convex) functions in their continuous version.

2. Definitions and preliminaries

We use the following notations for vector inequalities. Forx,y∈Rn_,

x≤y⇐⇒xi≤yi fori=1, 2,. . .,n,

x < y⇐⇒xi< yi fori=1, 2,. . .,n. (2.1)

Let I =[a,b] be a real interval and let K = {1, 2,. . .,k} and M= {1, 2,. . .,m}. Let φ:I×Rn_×Rn_×Rm_×Rm_{→Rbe continuously differentiable function. In order to}

con-siderφ(t,x(t), ˙x(t),u(t), ˙u(t)), wherex(t) :I→Rn_{,u(t) :I→R}m _{are differentiable with}

derivatives ˙x(t) and ˙u(t), respectively. For notational simplicity, we writex(t), ˙x(t),u(t), ˙

u(t) asx, ˙x,u, ˙u, respectively, as and when necessary. We denote the partial derivatives of φbyφxandφx˙, where

φx=

_∂φ

∂x1,

∂φ ∂x1,. . .,

∂φ ∂xn

,

φx˙=

_∂φ

∂x˙1,

∂φ ∂x˙2,. . .,

∂φ ∂x˙n

.

(2.2)

The partial derivatives of other functions used will be written similarly. Let PS(I,Rn₎

denote the space of piecewise smooth functions x with norm x = x∞+Dx∞, where the differentiation operatorDis given by

y=Dx⇐⇒x(t)=α+ b

a y(s)ds (2.3)

in whichαis a given boundary value. Therefore,D=d/dtexcept at discontinuities. Consider the following multiobjective variational control programming problem.

Minimize _b

a f(t,x, ˙x,u, ˙u)dt=

_b

a f

1_{(t,x, ˙x,u, ˙u)dt,. . .,}

_b

a f

k_{(t,x, ˙x,u, ˙u)dt,}

subject tox(a)=α, x(b)=β, b

ah(t,x, ˙x,u, ˙u)dt≤0, t∈[a,b], x∈PS

I,Rn_{, y∈PSI,R}m_,

f : [a,b]×Rn_×Rn_×Rm_×Rm_{−→ ×R}k_,

h: [a,b]×Rn_×Rn_×Rm_×Rm_{−→ ×R}m

(2.4)

Definition 2.1. A point (x0_,u0_{) inX is said to be an efficient solution of (2.4) if for all}

(x,u) inX,

b

a f

i_t,x0_{, ˙x}0_,u0_{, ˙u}0_dt≥

b

a f

i_{(t,x, ˙x,u, ˙u)dt}

=⇒ b

a f

i_t,x0_{, ˙x}0_,u0_{, ˙u}0_dt=

b

a f

i_{(t,x, ˙x,u, ˙u)dt}

∀i∈ {1, 2,. . .,k}.

(2.5)

Definition 2.2. A point (x0_,u0_{) inXis said to be a weak minimum for problem (2.4) if}

there exists no other (x,u) inXfor which b

a f

i_t,x0_{, ˙x}0_,u0_{, ˙u}0_{dt >}

b

a f

i_{(t,x, ˙x,u, ˙u)dt. (2.6)}

From the above two definitions, it follows that if (x,u) inXis an efficient solution for (2.4), then it is also a weak minimum for (2.4).

Definition 2.3. The functionalF:I×Rn_×Rn_×Rm_×Rm_×Rn_×Rn_×Rm_×Rm_→Ris

sublinear if for anyx,x0_∈Rn_{, ˙x, ˙x}0_∈Rn_,u,u0_∈Rm_{, ˙u, ˙u}0_∈Rm_,

Ft,x, ˙x,u, ˙u,x0_{, ˙x}0_,u0_{, ˙u}0_;α 1+α2

≤Ft,x, ˙x,u, ˙u,x0_{, ˙x}0_,u0_{, ˙u}0_;α 1

+Ft,x, ˙x,u, ˙u,x0_{, ˙x}0_,u0_{, ˙u}0_;α 2

, Ft,x, ˙x,u, ˙u,x0_{, ˙x}0_,u0_{, ˙u}0_{;αa=αFt,x, ˙x,u, ˙u,x}0_{, ˙x}0_,u0_{, ˙u}0_;a

(2.7)

for anyα1,α2∈Rn,α∈R,α≥0 anda∈Rn.

Definition 2.4. LetF[x,u]=abf(t,x, ˙x,u, ˙u)dtbe Fr´echet differentiable. Letρ be a real

number. Then the functionalFat a point (x0_,u0_{) inXis said to be}

(a)ρ-convex if there exists a real numberρsuch that b

a f

i_{(t,x, ˙x,u, ˙u)dt−}

b

a f

i_t,x0_{, ˙x}0_,u0_{, ˙u}0_dt

≥ b

a

x−x0_f

x0t,x0, ˙x0,u0, ˙u0+Dx−x0f_x˙0t,x0, ˙x0,u0, ˙u0dt+ρ x−x0 2; (2.8) the functionf is said to be strictlyρ-convex if strict inequality holds;

(b)ρ-pseudoconvex if there exists a real numberρsuch that b

a

x−x0_f

x0t,x0, ˙x0,u0, ˙u0+Dx−x0f_x˙0t,x0, ˙x0,u0, ˙u0dt≥ −ρ x−x0 2 =⇒

b

a f

i_{(t,x, ˙x,u, ˙u)dt≥}

b

a f

i_t,x0_{, ˙x}0_,u0_{, ˙u}0_dt;

the functionf is said to be strictlyρ-pseudoconvex if strict inequality holds in the right-hand inequality of the above implication;

(c)ρ-quasiconvex if there exists a real numberρsuch that b

a f

i_{(t,x, ˙x,u, ˙u)dt≤}

b

a f

i_t,x0_{, ˙x}0_,u0_{, ˙u}0_dt

=⇒ b

a

x−x0_f

x0t,x0, ˙x0,u0, ˙u0+Dx−x0f_x˙0t,x0, ˙x0,u0, ˙u0dt≤ −ρ x−x0 2. (2.10)

Now we use the term “generalizedρ-convexity” to indicateρ-pseudoconvexity,ρ -quas-iconvexity, and so forth. Letl=(l1_,l2_{,. . .,l}n_{) be then-dimensional vector function and let}

each of its components beρ-convex (generalizedρ-convex) at the same point (x0_,u0_).

Also let q=(q1,q2,. . .,qn) be a vector constant such that qi≥0 for all i=1, 2,. . .,n.

Then

(a)ΣNli(t,·,·,·,·) isΣNρi-convex at (x0,u0);

(b) eachqifi(t,·,·,·,·) isqiρi-convex at (x0,u0); and hence

(c)l(t,·,·,·,·) isΣNρi-convex at (x0,u0).

These properties will be used frequently throughout the paper. We state the continuous version of [16, Theorem 3.2] in the form of the following proposition, which will be needed in the proof of the strong duality theorem.

Proposition2.5. Let( ¯x, ¯u)be a weak minimum for (2.4) at which the Kuhn-Tucker con-straint qualification is satisfied. Then there existsλinRk_{and a piecewise smoothβ(·) :I→}

Rk_{such that}

λT_f

¯

x

t, ¯x, ¯˙x, ¯u, ¯˙u+βT_h

¯

x

t, ¯x, ¯˙x, ¯u, ¯˙u=DλT_f

¯˙

x

t, ¯x, ¯˙x, ¯u, ¯˙x+βT_h

¯˙

x

t, ¯x, ¯˙x, ¯u, ¯˙u, b

a β

T_{ht, ¯x, ¯˙x, ¯u, ¯˙udt=0, β(t)≥0,λ}T_e=1,λ≥0, (2.11)

whereeis the vector ofRk_{, the components of which are all ones.}

We divide the index setM of the constraint function of the problem (2.4) into two distinct subsets, namelyJandQsuch thatJ∪Q=M, and let

βT

jhj(t,x, ˙x,u, ˙u)=

j

βjhj(t,x, ˙x,u, ˙u),

βT

QhQ(t,x, ˙x,u, ˙u)=

Q

βQhQ(t,x, ˙x,u, ˙u).

(2.12)

(FP)

Minimize b

a f(t,x, ˙x,u, ˙u)dt

b

ag(t,x, ˙x,u, ˙u)dt

= b

a f1(t,x, ˙x,u, ˙u)dt

b

ag1(t,x, ˙x,u, ˙u)dt

,. . ., b

a fk(t,x, ˙x,u, ˙u)dt

b

agk(t,x, ˙x,u, ˙u)dt

, (2.13)

subject tox(a)=α, x(b)=β, b

ah(t,x, ˙x,u, ˙u)dt≤0, t∈[a,b], (2.14)

x∈PSI,Rn_{, y∈PS(I,R}m_{), (2.15)}

f : [a,b]×Rn_×Rn_×Rm_×Rm_−→Rk_{, g: [a,b]×R}n_×Rn_×Rm_×Rm_−→Rk_,

(2.16) h: [a,b]×Rn×Rn×Rm×Rm−→Rm (2.17) are assumed to be continuously differentiable vector functions. Also we assume that

_b

a f(t,·,·,·,·)dt≥0,

_b

ag(t,·,·,·,·)dt >0, i=1, 2,. . .,k. (2.18)

Following Bector et al. [1], the problem (FP)vstated below is associated with the given

problem (FP) forv∈Rk

+, whereRk+is the positive orthant ofRk.

(FP)v

Minimize _b

a

f(t,x, ˙x,u, ˙u)−vTg(t,x, ˙x,u, ˙u)dt, subject to (2.14).

(2.19)

The following lemma connecting (FP) and (FP)vhas been proved in [2].

Lemma2.6. Let(x0_,u0_{)be an efficient solution to (FP). Then there existsv}0_∈Rk

+such that

(x0_,u0_{)is efficient to the problem (FP)}

v, whereRk+is the positive orthant ofRk.

3. Duality

Now we introduce the continuous analog of the static mixed-type dual [8] for the primal problem (2.4).

(FD)

Maximize b

a

ft,x0_{, ˙x}0_,u0_{, ˙u}0_−vT_gt,x0_{, ˙x}0_,u0_{, ˙u}0_+βT j(t)hj

t,x0_{, ˙x}0_,u0_{, ˙u}0_edt,

subject tox(a)=α, x(b)=β,

(3.1)

λTfx0t,x0, ˙x0,u0, ˙u0−vTg_x0t,x0, ˙x0,u0, ˙u0+βTh_x0t,x0, ˙x0,u0, ˙u0 =DλT_f

˙

x0t,x0, ˙x0,u0, ˙u0−vTg_x˙0t,x0, ˙x0,u0, ˙u0+βTh_x˙0t,x0, ˙x0,u0, ˙u0, (3.2) b

a

_b

aβ T Q(t)hQ

t,x0, ˙x0,u0, ˙u0dt≥0, (3.4) β(t)≥0, λT_{e=1, λ≥0, (3.5)}

whereeis the vector ofRk_{, the components of which are all ones.}

Here we present a number of duality results between (FP)vand (FD) by imposing

var-iousρ-convexity (generalizedρ-convexity) conditions upon the objective and constraint functions. We begin with a situation in which all of the functions areρ-convex. Subse-quently, we formulate more general duality criteria in which the generalizedρ-convexity requirements are placed on certain combinations of the objective and constraint func-tions.

LetYdenote the set of all feasible solutions of (FD).

Theorem3.1. Assume that for all feasible(x,u)for (FP)vand for all feasible(x0,u0,λ,v,β)

for (FD),

(a)for eachi∈K,fi_{(t,·,·,·,·),−g}i_{(t,·,·,·,·)isρ}

i-convex, and for eachj∈M,hj(t,·,·,

·,·)isγj-convex.

Further if either

(b)for eachi∈K,λi>0withΣKλiρi+ΣMβjγj≥0, or

(c)ΣKλiρi+ΣMβjγj>0,

then

b

a

fi(t,x, ˙x,u, ˙u)−vigi(t,x, ˙x,u, ˙u)dt ≤

b

a

fi_t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi_t,x0_{, ˙x}0_,u0_{, ˙u}0

+βT_jhjt,x0_{, ˙x}0_,u0_{, ˙u}0_{dt, ∀i∈ {1, 2,. . .,k},}

b

a

fi(t,x, ˙x,u, ˙u)−vigi(t,x, ˙x,u, ˙u)dt <

b

a

fit,x0, ˙x0,u0, ˙u0−vigit,x0, ˙x0,u0, ˙u0 +βT

jhj

t,x0_{, ˙x}0_,u0_{, ˙u}0_{dt, for somej∈ {1, 2,. . .,k}}

(3.6)

cannot hold.

Proof. Ifx=x0_{, then a weak duality theorem holds trivially, so we assume thatx=x}0_.

From (3.2), we have _b

a

x−x0λTfx0

t,x0, ˙x0,u0, ˙u0−vTgx0

t,x0, ˙x0,u0, ˙u0+βThx0

t,x0, ˙x0,u0, ˙u0dt =

b

a

x−x0DλTfx˙0

t,x0, ˙x0,u0, ˙u0−vTgx˙0

t,x0, ˙x0,u0, ˙u0 +βT_h

˙

x0t,x0, ˙x0,u0, ˙u0dt.

Contrary to the result, we assume that (3.6) holds. Then in view of feasibility of (x,u) for (FP)v, we have

b

a

fi_{(t,x, ˙x,u, ˙u)−v}i_gi_{(t,x, ˙x,u, ˙u)+β}T jhj

t,x, ˙x,u, ˙udt ≤

_b

a

fit,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi_t,x0_{, ˙x}0_,u0_{, ˙u}0

+βT jhj

t,x0_{, ˙x}0_,u0_{, ˙u}0_{dt, ∀i∈K,}

b

a

fi_{(t,x, ˙x,u, ˙u)−v}i_gi_{(t,x, ˙x,u, ˙u)+β}T jhj

t,x, ˙x,u, ˙udt <

b

a

fi_t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi_t,x0_{, ˙x}0_,u0_{, ˙u}0

+βT jhj

t,x0_{, ˙x}0_,u0_{, ˙u}0_{dt, for somei∈K.}

(3.8)

From the strict positivity of each componentλiofλand the fact thatλTe=1, it follows

that b

a

λT_fi_{(t,x, ˙x,u, ˙u)−v}i_gi_{(t,x, ˙x,u, ˙u)+β}T

jhj(t,x, ˙x,u, ˙u)

dt

< b

a

λT_fi_t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi_t,x0_{, ˙x}0_,u0_{, ˙u}0_+βT jhj

t,x0_{, ˙x}0_,u0_{, ˙u}0_dt.

(3.9)

Using the definitions ofρi-convexity of (fi(t,·,·,·,·)−vigi(t,·,·,·,·)),i∈K, andγj

-convexity ofhj_{(t,·,·,·,·), j∈M, we have}

b

a

fi_{(t,x, ˙x,u, ˙u)−v}i_gi_{(t,x, ˙x,u, ˙u)−f}i_t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi_t,x0_{, ˙x}0_,u0_{, ˙u}0_dt

≥ b

a

x−x0_fi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0

+Dx−x0fx˙i0

t,x0, ˙x0,u0, ˙u0−vigxi˙0

t,x0, ˙x0,u0, ˙u0dt +ρi x−x0

2

, ∀i∈K,

(3.10) b

a

hj_{(t,x, ˙x,u, ˙u)−h}j_t,x0_{, ˙x}0_,u0_{, ˙u}0_dt

≥ _b

a

x−x0_hj

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_+Dx−x0_hj ˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_dt

+γ x−x0 2, ∀j∈M.

Now multiplying (3.10) byλi, (3.11) byβjand adding the resulting inequalities, we get

b

a

λTf(t,x, ˙x,u, ˙u)−vTg(t,x, ˙x,u, ˙u)+βThj(t,x, ˙x,u, ˙u)

−λT_ft,x0_{, ˙x}0_,u0_{, ˙u}0_−vT_gt,x0_{, ˙x}0_,u0_{, ˙u}0_+βT_hj_t,x0_{, ˙x}0_,u0_{, ˙u}0_dt

≥ b

a

x−x0_λT_f

x0t,x0, ˙x0,u0, ˙u0−vTg_x0t,x0, ˙x0,u0, ˙u0+βTh_x0t,x0, ˙x0,u0, ˙u0 +Dx−x0_λT_f

˙

x0t,x0, ˙x0,u0, ˙u0−vTg_x˙0t,x0, ˙x0,u0, ˙u0 +βT_h

˙

x0t,x0, ˙x0,u0, ˙u0dt+

K

αiρi+

M

βjγj

_x−x0 2_.

(3.12) By integration by parts, the right-hand side reduces to the following via (b):

b

a

x−x0_λT_f

x0t,x0, ˙x0,u0, ˙u0−vTg_x0t,x0, ˙x0,u0, ˙u0+βTh_x0t,x0, ˙x0,u0, ˙u0dt +λT_f

˙

x0t,x0, ˙x0,u0, ˙u0−vig_x˙0t,x0, ˙x0,u0, ˙u0+βTh_x˙0t,x0, ˙x0,u0, ˙u0x−x0t=β

t=α

− _b

a

x−x0_DλT_f

˙

x0t,x0, ˙x0,u0, ˙u0−vTg_x˙0t,x0, ˙x0,u0, ˙u0 +βT_h

˙

x0t,x0, ˙x0,u0, ˙u0dt.

(3.13) On using the boundary conditions (3.7), it yields

b

a

λT_{f(t,x, ˙x,u, ˙u−v}T_{gt,x, ˙x,u, ˙u+β}T_{ht,x, ˙x,u, ˙u}

−λTft,x0_{, ˙x}0_,u0_{, ˙u}0_−vT_gt,x0_{, ˙x}0_,u0_{, ˙u}0_−βT_ht,x0_{, ˙x}0_,u0_{, ˙u}0_dt≥0.

(3.14)

SinceM=J∪Q, so

βT_{h(t,·,·,·,·)=β}T

jhj(t,·,·,·,·) +βTQhj(t,·,·,·,·), (3.15)

and hence the above inequality implies, along with (3.9), that b

a

βQT(t)hQ(t,x, ˙x,u, ˙u)−βTQ(t)hQ

t,x0, ˙x0,u0, ˙u0dt >0. (3.16)

Now sincex0_,u0_{,λ,v,β∈Y, from (3.4),}

b

aβ T

Q(t)hQ(t,x, ˙x,u, ˙u)dt >0, (3.17)

which is a contradiction to the fact that (x,u) is feasible for (FP)v, and therefore (3.6)

(c) In this case, the multipliersλiof the objective function (fi(t,·,·,·,·)−vigi(t,·,·,

·,·)) need not be strictly positive, and it gives≤in place of<of (3.9). If we assume the condition in (c), we get>in place of≥of (3.14). So, we get (3.16) and we conclude the theorem as in the case of (b). And it completes the proof. The above theorem has a number of important special cases, which can be identified by the properties ofρ-convex functions. Here we state some of them as corollaries. Corollary3.2. Assume that for all feasible(x,u)for (FP)v and for all feasible(x0,u0,λ,

v,β)for (FD),

(a)for eachi∈K,fi_{(t,·,·,·,·),−g}i_{(t,·,·,·,·)isρ}

i-convex, and for eachj∈M,βTjhj(t,·,

·,·,·)isγj-convex.

Further if either

(b)for eachi∈K,λi>0withΣKλiρi+ΣMγj≥0, or

(c)ΣKλiρi+ΣMγj>0,

then (3.6) cannot hold. Proof. Sincehj_{(t,·,·,·,·) isγ}

j-convex, wheneverβTjhj(t,·,·,·,·) isβjγj-convex andβj≥

0, the proof is similar toTheorem 3.1.

Corollary3.3. For all feasible(x,u)for (FP)vand for all feasible(x0,u0,λ,v,β)for (FD),

assume thatCorollary 3.2holds, except that instead ofβT

jhj(t,·,·,·,·)beingγj-convex, the

function (t,x0_{, ˙x}0_,u0_{, ˙u}0_)→Σ

MβTjhj(t,x0, ˙x0,u0, ˙u0)isγ-convex, and instead ofCorollary

3.2(b) and (c), the following conditions hold, respectively, (b)for eachi∈K,λi>0withΣKλiρi+γ≥0,

(c)ΣKλiρi+γ >0.

Then (3.6) cannot hold.

We note that, inTheorem 3.1, each constraint functionhj_{(t,·,·,·,·) is assumed to be}

γj-convex whereas, inCorollary 3.3, all constraint functions are aggregated into oneγ

-convex function. So, it is possible to consider a situation intermediate between these two extreme cases (keeping in view the partition of the constraint function in the objective function of the dual problem (FD)) in which some of the constraint functions can be combined intoγ-convex function while the rest are individuallyγ-convex. Situations of this type are presented in the next two corollaries.

Corollary3.4. Assume that for all feasible(x,u)for (FP)v and for all feasible(x0,u0,λ,

v,β)for (FD),

(a)for eachi∈K, fi_{(t,·,·,·,·),−g}i_{(t,·,·,·,·)isρ}

i-convex andβTjhj(t,·,·,·,·)isγj

-convex, whereasβT

Qhj(t,·,·,·,·)isγQ-convex.

Further if either

(b)for eachi∈K,λi>0withΣKλiρi+γj+ΣQγj≥0, or

(c)ΣKλiρi+γj+ΣQγj>0,

Corollary3.5. Assume that for all feasible(x,u)for (FP)v and for all feasible(x0,u0,λ,

v,β)for (FD),

(a)for eachi∈K, fi_{(t,·,·,·,·),−g}i_{(t,·,·,·,·)isρ}

i-convex andβTjhj(t,·,·,·,·)isγj

-convex whereasβT_Qhj_{(t,·,·,·,·)isγ}

Q-convex.

Further if either

(b)for eachi∈K,λi>0withΣKλiρi+γj+γQ≥0, or

(c)ΣKλiρi+γj+γQ>0,

then (3.6) cannot hold.

Corollary3.6. Assume that for all feasible(x,u)for (FP)v and for all feasible(x0,u0,λ,

v,β)for (FD),

(a)λT_{[f(t,·,·,·,·)−v}T_{g(t,·,·,·,·)] +β}T_{h(t,·,·,·,·)isγ-convex.}

Further if either

(b)for eachi∈K,λi>0withρ≥0, or

(c)ρ >0,

then (3.6) cannot hold.

In the rest of this section, we use the generalizedρ-convexity. So, we restrict ourselves in most of the cases to situations in which only scalarizations of the objective and con-straint functions are considered. The related corollaries can also be seen as in the case of

Theorem 3.1. Therefore, we do not state these corollaries explicitly.

Theorem3.7. Assume that for all feasible(x,u)for (FP)vand for all feasible(x0,u0,λ,v,β)

for (FD),

(a)βT_QhQ_{(t,·,·,·,·)isρ-quasiconvex,}

(b)for i∈K, λi>0 and(fi(t,·,·,·,·)−vigi(t,·,·,·,·)) +βTjhj(t,·,·,·,·) is bothγi

-quasiconvex andγi-pseudoconvex withΣKλiγi+ρ≥0.

Then (3.6) cannot hold.

Proof. Ifx=x0_{, then a weak duality theorem holds trivially, so we assume thatx=x}0_.

Since (x,u)∈Xand (x0_,u0_{,λ,v,β)∈Y, we have}

b

a β T

QhQ(t,x, ˙x,u, ˙u)dt≤0≤

b

aβ T Q(t)hQ

t,x0, ˙x0,u0, ˙u0dt. (3.18)

ρ-quasiconvexity in (a), in view of the above theorem, implies that _b

a

x−x0_βT QhQx0

t,x0_{, ˙x}0_,u0_{, ˙u}0_+Dx−x0_βT QhQx˙0

t,x0_{, ˙x}0_,u0_{, ˙u}0_dt

≤ −ρ x−x0 2.

The substitution of the duality constraint (3.2) in the first term of the above implication gives us, along with (3.15),

b

a

x−x0DλTfx˙0

t,x0, ˙x0,u0, ˙u0−vTgx˙0

t,x0, ˙x0,u0, ˙u0+βTjh j

˙

x0

t,x0, ˙x0,u0, ˙u0 +βT_QhQx˙0

t,x0, ˙x0,u0, ˙u0−λTfx0t,x0, ˙x0,u0, ˙u0−vTg_x0t,x0, ˙x0,u0, ˙u0 +βTjh

j

˙

x0

t,x0, ˙x0,u0, ˙u0dt+ b

a

Dx−x0βTQhQx˙0

t,x0, ˙x0,u0, ˙u0dt ≤ −ρ x−x0 2_.

(3.20) By integration by parts and making use of boundary conditions, we get

b

a

x−x0_λT_f

x0t,x0, ˙x0,u0, ˙u0−vTg_x0t,x0, ˙x0,u0, ˙u0+βT_jhj

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0

+Dx−x0λTfx˙0

t,x0, ˙x0,u0, ˙u0−vTgx˙0

t,x0, ˙x0,u0, ˙u0 +βT_jhxj˙0

t,x0, ˙x0,u0, ˙u0dt ≥ρ x−x0 2.

(3.21) So making use of the conditionΣKλiγi+ρ≥0 and asλTe=1, we get

K λi b a

x−x0_fi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_+βT jh

j x0

t,x0_{, ˙x}0_,u0_{, ˙u}0

+Dx−x0_λ

i fi ˙ x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0

+βT jh

j

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_{dt≥ −}

K

λiγi

_x−x0 2_.

(3.22) Sinceλi>0,i∈K, it follows from the above that

b

a

x−x0_fi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_+βT jh

j x0

t,x0_{, ˙x}0_,u0_{, ˙u}0

+Dx−x0fx˙i0

t,x0, ˙x0,u0, ˙u0−vigxi˙0

t,x0, ˙x0,u0, ˙u0+βTjh j

˙

x0

t,x0, ˙x0,u0, ˙u0dt ≥ −γi x−x0

2

, ∀i∈K,

(3.23) b

a

x−x0_fi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_+βT jh

j x0

t,x0_{, ˙x}0_,u0_{, ˙u}0

+Dx−x0_fi

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_+βT jh

j

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_dt

>−γi x−x0 2, for somei∈K.

Suppose (3.23) holds, then theγi-pseudoconvexity assumption in (b) gives along with

the feasibility of (x,u) for (FP)v, for alli∈K,

b

a

fi(t,x, ˙x,u, ˙u)−vigi(t,x, ˙x,u, ˙u)dt ≥

b

a

fit,x0, ˙x0,u0, ˙u0−vigit,x0, ˙x0,u0, ˙u0+βTjhj

t,x0, ˙x0,u0, ˙u0dt. (3.25)

Now, suppose (3.24) holds, then the equivalent form of theγi-quasiconvexity assumption

in (b) gives, along with the feasibility of (x,u) for (FP)v, for somei∈K,

b

a

fi_{(t,x, ˙x,u, ˙u)−v}i_gi_{(t,x, ˙x,u, ˙u)dt}

> b

a

fi_t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi_t,x0_{, ˙x}0_,u0_{, ˙u}0_+βT jhj

t,x0_{, ˙x}0_,u0_{, ˙u}0_dt.

(3.26)

So (3.25) and (3.26) show that (3.6) cannot hold, and this completes the proof. The followingTheorem 3.8is stated without proof.

Theorem3.8. Assume that for all feasible(x,u)for (FP)vand for all feasible(x0,u0,λ,v,β)

for (FD), (a)βT

QhQ(t,·,·,·,·)isρ-quasiconvex,

(b)for eachi∈K, λi>0 and(fi(t,·,·,·,·)−vigi(t,·,·,·,·)) +βTjhj(t,·,·,·,·) isγi

-quasiconvex and there exists someq∈Ksuch that it is strictlyγq-pseudoconvex (with

the corresponding componentλqofλpositive) withΣKλiγi+ρ≥0.

Then (3.6) cannot hold.

Theorem3.9. Assume that for all feasible(x,u)for (FP)vand for all feasible(x0,u0,λ,v,β)

for (FD), (a)βT

QhQ(t,·,·,·,·)isρ-quasiconvex,

(b)for eachi∈K,λi>0andλT(f(t,·,·,·,·)−vTg(t,·,·,·,·)) +βTjhj(t,·,·,·,·)isγ

-pseudoconvex withρ+γ≥0. Then (3.6) cannot hold.

Proof. As in the case ofTheorem 3.7, we assume thatx=x0_{and get (3.21). Now using}

the conditionρ+γ≥0 and by ourρ-pseudoconvexity assumption in (b), we get b

a

λT_{f(t,x, ˙x,u, ˙u)−v}T_{g(t,x, ˙x,u, ˙u)dt+β}T jhj

t,x, ˙x,u, ˙udt ≥

b

a

λT_ft,x0_{, ˙x}0_,u0_{, ˙u}0_−vT_gt,x0_{, ˙x}0_,u0_{, ˙u}0_+βT jhj

t,x0_{, ˙x}0_,u0_{, ˙u}0_dt.

The feasibility of (x,u) for (FP)vand the fact thatλTe=1 imply

λT _b

a

f(t,x, ˙x,u, ˙u)−vTg(t,x, ˙x,u, ˙u)dt ≥λT

b

a

ft,x0, ˙x0,u0, ˙u0−vTgt,x0, ˙x0,u0, ˙u0+βTjhj

t,x0, ˙x0,u0, ˙u0dt. (3.28) This concludes the theorem, sinceλi≥0 for eachi∈K.

Theorem3.10. Assume that for all feasible(x,u)for (FP)vand for all feasible(x0,u0,λ,v,β)

for (FD),

(a)for eachi∈K,λi>0and (fi(t,·,·,·,·)−vigi(t,·,·,·,·)) +βTh(t,·,·,·,·)is both

ρ-pseudoconvex andρ-quasiconvex withΣKλiρi≥0.

Then (3.6) cannot hold.

Proof. We assume thatx=x0_{. From the duality constraint (3.2), we get (3.7). Now by}

integration by parts, b

a

x−x0_λT_f

x0t,x0, ˙x0,u0, ˙u0−vTg_x0t,x0, ˙x0,u0, ˙u0+βTh_x0t,x0, ˙x0,u0, ˙u0dt =x−x0_λT_f

˙

x0t,x0, ˙x0,u0, ˙u0−vTg_x˙0t,x0, ˙x0,u0, ˙u0+βTh_x˙0t,x0, ˙x0,u0, ˙u0t=β

t=α

− _b

a

Dx−x0λTfx˙0

t,x0, ˙x0,u0, ˙u0−vTgx˙0

t,x0, ˙x0,u0, ˙u0 +βT_h

˙

x0t,x0, ˙x0,u0, ˙u0dt.

(3.29) Sinceλi>0,i∈K, and by the fact thatλTe=1, we get

k

λi

b

a

x−x0_fi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_+βT_h

x0t,x0, ˙x0,u0, ˙u0 +Dx−x0_fi

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0

+βT_h

˙

x0t,x0, ˙x0,u0, ˙u0dt=0.

(3.30) Given thatΣKλiρi≥0 andx−x02is always positive,

k

λi

b

a

x−x0_fi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_+βT_h

x0t,x0, ˙x0,u0, ˙u0 +Dx−x0_fi

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi

˙

x0

t,x0_{, ˙x}0_,u0_{, ˙u}0

+βT_h

˙

x0t,x0, ˙x0,u0, ˙u0dt≥ −Σ_Kλ_iρ_i x−x0 2.

Again using the nonnegativity of eachλi,i∈K, andρ-pseudoconvexity and the

equiv-alent form ofρ-quasiconvexity inTheorem 3.10(a), it follows from the above inequal-ity that

b

a

fi_{(t,x, ˙x,u, ˙u)−v}i_gi_{(t,x, ˙x,u, ˙u)+β}T_{h(t,x, ˙x,u, ˙u)dt}

≥ b

a

fi_t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi_t,x0_{, ˙x}0_,u0_{, ˙u}0

+βTht,x0, ˙x0,u0, ˙u0dt, ∀i∈K, b

a

fi_{(t,x, ˙x,u, ˙u)−v}i_gi_{(t,x, ˙x,u, ˙u)+β}T_{h(t,x, ˙x,u, ˙u)dt}

> b

a

fi_t,x0_{, ˙x}0_,u0_{, ˙u}0_−vi_gi_t,x0_{, ˙x}0_,u0_{, ˙u}0

+βTht,x0, ˙x0,u0, ˙u0dt, for somei∈K.

(3.32)

Now using the feasibility of (x,u) for (FP)vand (x0,u0,λ,v,β) for (FD) provides us with

the desired conclusion that (3.6) cannot hold.

Next we state the last weak duality theorem. The proof is on similar lines as earlier. Theorem3.11. Assume that for all feasible(x,u)for (FP)vand for all feasible(x0,u0,λ,v,β)

for (FD),

(a)for eachi∈K,λi>0 andλT(f(t,·,·,·,·)−vTg(t,·,·,·,·)) +βTh(t,·,·,·,·)is ρ

-convex withρ≥0, or

(b)λT_{(f(t,·,·,·,·)−v}T_{g(t,·,·,·,·)) +β}T_{h(t,·,·,·,·)is strictlyρ-convex withρ≥0.}

Then (3.6) cannot hold.

Assumption (b) above thatλT_{(f(t,·,·,·,·)−v}T_{g(t,·,·,·,·)) +β}T_{h(t,·,·,·,·) is strictly}

ρ-convex can be replaced by much weaker conditions. This leads to the following corol-lary.

Corollary3.12. For all feasible(x,u)for (FP)vand for all feasible(x0,u0,λ,v,β)for (FD),

assume thatTheorem 3.11holds, except that instead of (b), for eachi∈K,(fi_{(t,·,·,·,·)−}

vi_gi_{(t,·,·,·,·)) +β}T_{h(t,·,·,·,·)isρ}

i-convex, and for at least oneq∈K,(fq(t,·,·,·,·)−

vq_gq_{(t,·,·,·,·)) +β}T_{h(t,·,·,·,·)is strictlyρ}

q-convex (with the corresponding componentλq

ofλpositive) withΣKλiρi≥0. Then (3.6) cannot hold.

Before proving strong duality theorem, we give the following lemma.

Lemma3.13. Assume that weak duality (any of the Theorems3.1–3.11or any of the Corol-laries 3.2–3.12) holds between (FP)v and (FD). If ( ¯x, ¯u, ¯λ, ¯v, ¯β) is feasible for (FD) with

b

aβ¯Th(t, ¯x, ¯˙x, ¯u, ¯˙u)dt=0and( ¯x, ¯u)is feasible for (FP)v, then( ¯x, ¯u)is efficient for (FP)vand

Proof. On the contrary, we assume that ( ¯x, ¯u) is not efficient for (FP)v, then there exists a

feasible (x,u) for (FP)vsuch that

b

a

fi_{(t,x, ˙x,u, ˙u)−v}i_gi_{(t,x, ˙x,u, ˙u)dt}

≤ b

a

fi_{t, ¯x, ¯˙x, ¯u, ¯˙u−v}i_gi_{t, ¯x, ¯˙x, ¯u, ¯˙udt, ∀i∈K,}

_b

a

fi(t,x, ˙x,u, ˙u)−vigi(t,x, ˙x,u, ˙u)dt <

_b

a

fit, ¯x, ¯˙x, ¯u, ¯˙u−vigit, ¯x, ¯˙x, ¯u, ¯˙udt, for somei∈K.

(3.33)

Since _abβ¯T_{h(t, ¯x, ¯˙x, ¯u, ¯˙u)dt=0 (⇒}b

aβ¯Tjhj(t, ¯x, ¯˙x, ¯u, ¯˙u)dt=0), we can write (fi

t, ¯x, ¯˙x, ¯u, ¯˙

u)−v¯i_fi_{(t, ¯x, ¯˙x, ¯u, ¯˙u)) + ¯β}T

jhj(t, ¯x, ¯˙x, ¯u, ¯˙u) in place of (fi(t, ¯x, ¯˙x, ¯u, ¯˙u)−v¯ifi(t, ¯x, ¯˙x, ¯u, ¯˙u)) in

the right-side terms of (3.33).

Now using the feasibility of (x,u) for (FP)v and ( ¯x, ¯u, ¯λ, ¯v, ¯β) for (FD), we get a

con-tradiction to the weak duality. So, ( ¯x, ¯u) is efficient for (FP)v. Similarly, we can show that

( ¯x, ¯u, ¯λ, ¯v, ¯β) is efficient for (FD).

Now using this lemma in conjunction with the necessary optimality conditions ( Prop-osition 2.5) ofSection 2, we establish the following strong duality theorem.

Theorem3.14. Let( ¯x, ¯u)be an efficient solution for (FP)vand assume that( ¯x, ¯u)satisfies

the Kuhn-Tucker constraint qualification for (FP)v. Then there existsλ¯∈Rkand a piecewise

smooth functionβ¯:I→Rm _{such that( ¯x, ¯u, ¯λ, ¯v, ¯β)is feasible for (FD), along with the}

con-dition_abβ¯T_{h(t, ¯x, ¯˙x, ¯u, ¯˙u)dt=0. Further, if weak duality (Theorems3.1–3.11or Corollaries}

3.2–3.12) also holds between (FP)vand (FD), then( ¯x, ¯u, ¯λ, ¯v, ¯β)is efficient for (FD).

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Raman Patel: Department of Statistics, South Gujarat University, Surat 395007, India