Calcolo (2014) 51:393–421
DOI 10.1007/s10092-013-0092-6

Duality for multiobjective variational control problems
with

(, ρ)-invexity

Tadeusz Antczak

Received: 12 November 2012 / Accepted: 29 May 2013 / Published online: 29 November 2013
© The Author(s) 2013. This article is published with open access at Springerlink.com

Abstract In this paper, Mond-Weir and Wolfe type duals for multiobjective vari-
ational control problems are formulated. Several duality theorems are established
relating efficient solutions of the primal and dual multiobjective variational control
problems under

(, ρ)-invexity. The results generalize a number of duality results

previously established for multiobjective variational control problems under other
generalized convexity assumptions.

Keywords

Multiobjective variational problems

· Efficient solution ·

(, ρ)-invexity · Duality

Mathematics Subject Classification (2000)

65K10

· 90C29 · 90C46 · 26B25

1 Introduction

Duality constitute an essential part of study of mathematical programming in the sense
that these lay down the foundation of algorithms for a solution of an optimization
problem. In recent years, duality theory for multiobjective variational problems has
been of much interest, and several contributions have been made to its development.
This is also a consequence of its applications in physics, economics, management
sciences, engineering problems, etc.

Various generalized convexity notions have been used recently to prove duality

results for a larger class of variational control problems than those ones with convex
functions. Mond et al. [

] extended the concept of invexity introduced by Hanson

[

] to the continuous case and used it to generalize earlier duality results for a class

T. Antczak (

)

Faculty of Mathematics and Computer Science, University of Łód´z,
Banacha 22, 90-238 Łód´z, Poland
e-mail: antczak@math.uni.lodz.pl

123

394

T. Antczak

of variational problems. Mond and Smart [

] extended the duality theorems for a

class of static nondifferentiable problems with Wolfe type and Mond-Weir type duals,
and further extended these for the continuous analogues. In 1992, Bector and Husain
[

] first applied duality method of ordinary multiobjective optimization problem to

multiobjective variational problem, and obtained duality results for properly efficient
solution under convexity assumptions on involved functions. Nahak and Nanda for-
mulated Wolfe and Mond-Weir type duals for multiobjective variational problems
and proved several duality theorems under invexity assumptions [

] and also under

pseudo-invexity assumptions [

]. Bhatia and Kumar [

] have studied multiobjective

control problems under

ρ-pseudoinvexity, ρ-strictly pseudoinvexity, ρ-quasiinvexity,

strictly

ρ-quasiinvexity assumptions. They derived duality theorems for multiobjec-

tive control problems under generalized invexity notions mentioned above. Chen [

]

discussed and formulated Wolfe and Mond-Weir type duals for a class of nondif-
ferentiable multiobjective variational problems. Under invexity assumptions on the
objective and the constraint functions, he proved weak and strong duality theorems
to related properly efficient solutions for primal and dual multiobjective control prob-
lems. Nahak [

] formulated Wolfe and Mond-Weir type duals for multiobjective

control problems and, under pseudo-invexity/quasi-invexity, proved weak and strong
duality theorems to relate efficient solutions of the primal and dual problems. Mishra
and Mukherjee [

] generalized

(F, ρ)-convexity introduced by Preda [

], and estab-

lished duality results for a class of multiobjective variational control problems with
(F, ρ)-convex functions. Nahak and Nanda [

] used the concept of efficiency to

formulate Wolfe and Mond-Weir type duals for multiobjective variational control
problems and established weak and strong duality theorems under generalized

(F, ρ)-

convexity assumptions. Also Reddy and Mukherjee [

] have studied duality theorems

under

(F, ρ)-convexity assumptions and related efficient solutions of the primal and

dual problems for multiobjective fractional control problems. In [

], Mishra and

Mukherjee obtained Mond-Weir-type duality results for multiobjective control prob-
lems under V -invexity assumptions and their generalizations. Bhatia and Mehra [

]

extended the concepts of B-type I and generalized B-type I functions to the contin-
uous case and they used these concepts to establish sufficient optimality conditions
and duality results for multiobjective variational programming problems. Under the
assumption of invexity and its generalization, Xiuhong [

] proved duality theorems

through a parametric approach to related properly efficient solutions of the primal
and dual multiobjective control problems. Kim and Kim [

] introduced new classes

of generalized V -type I invex functions for multiobjective variational problems and
obtained duality results for Mond–Weir type duals under the generalized V -type I
invexity assumptions and their generalizations. In [

], Zhian and Qingkai discussed

duality for multiobjective control problems with the same objective functionals and
constraint conditions as in [

], but with the invexity defined in [

]. Ahmad and Sharma

[

] extended the generalized

(F, α, ρ, θ)-V -convex functions to multiobjective vari-

ational control problems and proved Wolfe type and Mond Weir type duality results
for multiobjective variational control programming problems with these functions.

In [

], for scalar differentiable optimization problems, Caristi et al. [

] introduced

(, ρ)-invexity notion. They proved optimality and duality results for smooth scalar
optimization problems with

(, ρ)-invex functions.

123

Duality for multiobjective variational control problems

395

In this paper, the notion of scalar

(, ρ)-invexity is extended to the continuous

case and it is defined for multiobjective variational control problems. By utilizing this
concept of generalized convexity, we prove Mond Weir type and Wolfe type duality
results for multiobjective variational control programming problem involving

(, ρ)-

invex functions with respect to, not necessarily, the same

ρ (with the exception of

those the equality constraints for which the associated piecewise smooth functions
satisfying the constraints of duals are negative). Thus, we generalize Mond Weir type
and Wolfe type duality results for multiobjective variational control programming
problems established in the literature under other generalized convexity notions.

2 Preliminaries, notations and statement of the problem

The following convention for equalities and inequalities will be used in the paper.

For any x

= (x

, x

, . . . , x

)

, y

= (y

, y

, . . . , y

)

, we define:

(i) x

= y if and only if x

= y

for all i

= 1, 2, . . . , n;

(ii) x

> y if and only if x

> y

for all i

= 1, 2, . . . , n;

(iii) x

y if and only if x

for all i

= 1, 2, . . . , n;

(iv) x

≥ y if and only if x y and x = y.

Let I

= [a, b] be a real interval and let A = {1, 2, . . . , p}, J = {1, 2, . . . , q} and

= {1, . . . , s}.
In this paper, we assume that x

(t) is an n-dimensional piecewise smooth function

of t, and

(t) is the derivative of x(t) with respect to t in [a, b].

Denote by X the space of piecewise smooth state functions x

: I → R

with

norm

x = x

∞

+ Dx

∞

, where the differentiation operator D is given by

= Dx ⇐⇒ x(t) = x(a) +

(s)ds, where x(a) is a given boundary value.

Therefore

≡ D except at discontinuities.

In the paper, consider the following multiobjective variational control problem

(

MVPP

Minimize

(t, x(t),

(t))dt

(MVPP)

⎛
⎝

(t, x(t),

(t)), . . . ,

(t, x(t),

(t))

⎞
⎠

subject to g

(t, x(t),

(t)) 0, t ∈ I,

(t, x(t),

(t)) = 0, t ∈ I,

(a) = α, x(b) = β,

where f

= ( f

, . . . , f

) : I × R

× R

→ R

, is a p-dimensional function and each

its component is a continuously differentiable real scalar function, g

= (g

, . . . , g

) :

123

396

T. Antczak

× R

→ R

and h

= (h

, . . . , h

) : I × R

× R

→ R

are assumed to be

continuously differentiable q-dimensional and s-dimensional functions, respectively.

For notational simplicity, we write x

(t) and

(t) as x and

x, respectively. We denote

the partial derivatives of f

with respect to t, x and

x, respectively, by f

, f

such that f

∂ f

∂x

, . . . ,

∂ f

∂x

and f

∂ f

∂

, . . . ,

∂ f

∂

. Similarly the partial

derivatives of the vector function g and the vector function h can be written, using
matrices with q rows and s rows instead of one, respectively.

Let S denote the set of all feasible points of (

MVPP

), i.e.:

= {x ∈ Xverifying the constraints of (MVPP)} .

For notational convenience, we use

ϕ(t, x,

) for ϕ(t, x(t),

(t)), x for x(t) and

for

(t).

Definition 1 A feasible solution x in the considered multiobjective variational control
problem (

MVPP

) is said to be efficient of (

MVPP

) if there exists no another x

∈ S

such that

(t, x,

)dt ≤

(t, x,

)dt

that is, there exists no another x

∈ S such that

(t, x,

)dt

(t, x,

)dt, ∀i ∈ A,

(t, x,

)dt <

(t, x,

)dt for some r ∈ A.

Now, we give a definition of convexity of a functional

: I × R

× R

× R → R.

Definition 2 Let

: I × R

× R

× R → R. A functional

(t, x,

, x,

; ·) is convex on R

if, for any x

, x,

∈ R

, the inequality

(t, x,

, x,

; (λ(ξ

, ρ

) + (1 − λ)(ξ

, ρ

)))

λ(t, x,

, x, x,

; (ξ

, ρ

)) + (1 − λ)(t, x,

, x,

; (ξ

, ρ

))

holds for any

, ξ

∈ R

, ρ

∈ R and for all λ ∈ [0, 1].

123

Duality for multiobjective variational control problems

397

Let

: X → R be defined by (x) =

ϕ(t, x,

)dt, where ϕ : I × X × X → R

and, for notational convenience, we use

ϕ(t, x,

) for ϕ(t, x(t),

(t)). The following

definitions introduce the concepts of

(, ρ)-invexity and of (, ρ)-incavity for the

functional

Definition 3 If there exist a real number

ρ and a functional : I ×R

×R

× R → R, where (t, x,

, x,

;·) is convex on R

(t, x,

, x,

; (o, a)) 0

for every x

∈ X and any a ∈ R

such that the relation

(x) − (x)

, x,

;

(t, x,

) −

(t, x,

), ρ

(>) (1)

holds for all x

∈ X, (x = x), then the functional is said to be (strictly) (, ρ)-invex

at x

∈ X on X.

If (

) is satisfied for every x

∈ X, then is said to be (strictly) (, ρ)-invex on X.

Definition 4 If there exist a real number

ρ and a functional : I ×R

×R

× R → R, where (t, x,

, x,

, ·) is convex on R

(t, x,

, x,

; (o, a)) 0

for every x

∈ X and any a ∈ R

such that the relation

(x) − (x)

, x,

;

(t, x,

) −

(t, x,

), ρ

(<) (2)

holds for all x

∈ X, (x = x), then the functional is said to be (strictly) (, ρ)-

incave at x

∈ X on X. If (

) is satisfied for every x

∈ X, then is said to be (strictly)

(, ρ)-incave on X.

Now, we give an example of a multiobjective variational control problem in which

the functions involved are

(, ρ)-invex.

Example 5 Let n

= 2 and consider the following multiobjective variational control

problem:

Minimize

(t, x(t),

(t))dt

(MVPP1)

⎛
⎝

(t) + x

(t))dt,

(t) + x

(t)x

(t) + x

(t))dt

⎞
⎠

subject to g

(t, x(t),

(t)) = x

(t)x

(t) − 1 0, t ∈ I.

123

398

T. Antczak

Now, we show that the functions constituting the considered multiobjective varia-

tional control problem (

MVPP1

) are

(, ρ)-invex. Indeed, we set

, x,

; (ϕ

(t, x,

) −

(t, x,

), ρ

)

= −x

(t, x,

) − x

(t, x,

) + 2(2

− 1)(|x

| + |x

|),

= 0, ρ

= −1,

where

ϕ is equal to f

, f

or g, respectively,

is equal to

, respectively.

Then, by Definition

, it can be proved that all functions constituting the considered

multiobjective variational control problem (

MVPP1

) are

(, ρ)-invex with respect to

and

defined above.

Remark 6 Note that, in general, the functions constituting optimization problem are
(, ρ)-invex with respect to more than one and ρ. Indeed, it is not difficult to
show that, in the case of the considered multiobjective variational control problem
(

MVPP1

), the functions constituting it are

(, ρ)-invex also with respect to

and

defined as follows:

, x,

;

(t, x,

) −

(t, x,

), ρ

= −x

(t, x,

) − x

(t, x,

) + 2(2

− 1) |x

+ x

| ,

= 1, ρ

= −1.

Remark 7 Note that not all functions constituting the considered multiobjective vari-
ational control problem (

MVPP1

) are invex (see [

]). Indeed, the constraint g is not

invex with respect to any function

η : I × R

× R

→ R

, since a stationary

point is not its global minimizer (see [

]). Therefore, it is not possible to prove the dual-

ity results established in the paper under invexity assumptions imposed on the functions
constituting the considered multiobjective variational control problem (

MVPP1

). Fur-

ther, since most of the generalized convex functions possess the fundamental property
invexity mentioned above, therefore, they can not be used to prove duality result for
such nonconvex multiobjective variational control problems as problem (

MVPP1

However, we are in a position to prove various duality results for such nonconvex
multiobjective variational control problems under

(, ρ)-invexity. In other words, the

duality results established in this paper under

(, ρ)-invexity are true also for such

nonconvex multiobjective variational control problems as problem (

MVPP1

) consid-

ered in Example

for which most of the generalized convexity notions may avoid.

3 Mond-Weir type vector duality

In this section, we prove duality results between the considered multiobjective vari-
ational control problem (

MVPP

) and its Mond-Weir type vector variational control

dual problem (

VMWD

) defined as follows

123

Duality for multiobjective variational control problems

399

Maximize

(t, y(t),

(t))dt

(VMWD)

subject to

(t, y(t),

(t)) + ξ(t)

(t, y(t),

(t)) + ζ(t)

(t, y(t),

(t))

(t, y(t),

(t))+ξ(t)

(t, y(t),

(t))+ζ(t)

(t, y(t),

(t))

, t ∈ I,

ξ(t)

(t, y(t),

(t))dt 0,

ζ(t)

(t, y(t),

(t))dt = 0,

(a) = α, y(b) = β,

λ ≥ 0, λ

= 1, ξ(t) 0,

where e

= (1, . . . , 1) ∈ R

is a p-dimensional vector. It may be noted here that the

above dual constraints are written using the Karush-Kuhn-Tucker necessary optimal-
ity conditions for the considered multiobjective programming problem (

MVPP

) (see

Theorem

). For notational convenience, we use

ξ for ξ(t) and ζ for ζ(t).

Let

M W

be the set of all feasible solutions

(y, λ, ξ, ζ ) in Mond-Weir type dual

problem (

VMWD

). We denote by Y the set Y

= {y ∈ X : (y, λ, ξ, ζ ) ∈

M W

Theorem 8 (Weak duality): Let x and

(y, λ, ξ, ζ ) be any feasible solutions in multi-

objective variational control problems (

MVPP

) and (

VMWD

), respectively. Further,

assume that the following hypotheses are fulfilled:

(a)

(t, ·, ·)dt, i = 1, . . . , k, is strictly (, ρ

)-invex at y on S ∪ Y ,

(b)

(t, ·, ·)dt, j = 1, . . . , q, is (, ρ

)-invex at y on S ∪ Y ,

(c)

(t, ·, ·)dt, k ∈ K

(t) = {k ∈ K : ζ

(t) > 0}, is (, ρ

)-invex at y on S∪Y ,

(d)

(−h

(t, ·, ·))dt, k ∈ K

−

(t) = {k ∈ K : ζ

(t) < 0}, is (, ρ

)-invex at y on

∪ Y ,

(e)

∈K

(t)

−

∈K

−

(t)

Then, the following cannot hold

(t, x,

)dt ≤

(t, y,

)dt,

that is, the following cannot hold

(t, x,

)dt

(t, y,

)dt, ∀i ∈ A,

(3)

123

400

T. Antczak

and

(t, x,

)dt <

(t, y,

)dt for some r ∈ A.

(4)

Proof We proceed by contradiction. Suppose, contrary to the result, that the inequal-
ities (

) and (

) are satisfied. Since the hypotheses (a)–(d) are fulfilled, therefore, by

Definitions

and

, the following inequalities

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

, i ∈ A,

(5)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

, j ∈ J,

(6)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

, k ∈ K

(t),

(7)

−

(t, x,

)dt +

(t, y,

)dt

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

, k ∈ K

−

(t)

(8)

holds. Combining (

)–(

) and taking into account that

λ ≥ 0, we get

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

0, i ∈ A (9)

123

Duality for multiobjective variational control problems

401

and

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

< 0

(10)

for at least one r

∈ A.

Since

(y, λ, ξ, ζ) ∈

M W

, adding both sides of the inequalities (

)–(

), we obtain

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

(11)

∈K

(t)∪K

−

(t)

(t, x,

)dt −

∈K

(t)∪K

−

(t)

(t, y,

)dt

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

(12)

Using the feasibility of x and of

(y, λ, ξ, ζ) in problems (

MVPP

) and (

VMWD

respectively, the inequalities (

) and (

) yield, respectively,

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

(13)

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

)−

−h

(t, y,

)

, ρ

(14)

123

402

T. Antczak

Combining (

), (

) and (

), we get

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

)−

−h

(t, y,

)

, ρ

<0.

(15)

We denote

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, i ∈ A, (16)

(t) =

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, j ∈ J, (17)

(t) =

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, k ∈ K

(t),

(18)

(t) =

−ζ

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, k ∈ K

−

(t).

(19)

By (

)–(

), it follows that 0

1, i ∈ A, but λ

> 0 for at least one i ∈ A,

(t) 1, j ∈ J, 0

(t) 1, k ∈ K , and, moreover,

(t) +

∈K

(t)

(t) +

∈K

−

(t)

(t) = 1.

(20)

Combining (

)–(

), we get

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

123

Duality for multiobjective variational control problems

403

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

), ρ

∈K

−

(t)

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

<0.

(21)

Then, by Definition

, it follows that the functional

(t, x,

, y,

, ·) is convex on

. Thus, by (

) and (

), Definition

implies

⎛
⎝t, x,

, y,

;

⎛
⎝

⎡
⎣

(t, y,

) +

(t, y,

)

∈K

(t)

(t, y,

) +

∈K

−

(t)

(−

(t, y,

)

⎤
⎦

−

⎡
⎣

(t, y,

) +

(t, y,

) +

∈K

(t)

(t, y,

)

∈K

−

(t)

(−

(t, y,

)

⎤
⎦ ,

λρ

∈K

(t)∪K

−

(t)

⎞
⎠

⎞
⎠ dt <0.

Hence, the first constraint of (

VMWD

) yields

⎛
⎝t, x,

, y,

;

⎛
⎝0,

∈K

(t)∪K

−

(t)

⎞
⎠

⎞
⎠ dt < 0.

(22)

From the hypothesis e), it follows that

(t)ρ

(23)

123

404

T. Antczak

By Definition

, it follows that

(t, x,

, y,

; (0, a)) 0 for any a ∈ R

. Thus, (

)

implies that the following inequality

, x,

, y,

;

⎛
⎝0,

∈K

(t)∪K

−

(t)

⎞
⎠

⎞
⎠ dt 0

holds, contradicting (

In order to formulate strong duality theorem, we give the Karush-Kuhn-Tucker

necessary optimality conditions for the considered multiobjective variational control
problem (

MVPP

). This theorem is the continuous version of Theorem 2.2 [

] (see

also [

]).

Theorem 9 Let x be a normal efficient solution in problem (

MVPP

) at which the

Kuhn–Tucker constraint qualification is satisfied. Then, there exist

λ ∈ R

and the

piecewise smooth functions

ξ(·) : I → R

and

ζ (·) : I → R

such that

(t, x,

) + ξ

(t, x,

) + ζ

(t, x,

)

(t, x,

) + ξ

(t, x,

) + ζ

(t, x,

)

, t ∈ I,

(24)

(t, x,

)dt = 0,

(25)

λ ≥ 0, λ

= 1, ξ(t) 0.

(26)

Theorem 10 (Strong duality): Let x be an efficient solution in the considered multi-
objective variational control problem (

MVPP

). Further, assume that the Kuhn–Tucker

constraint qualification is satisfied for (

MVPP

). Then, there exist

λ ∈ R

and the piece-

wise smooth functions

ξ(·) : I → R

and

ζ (·) : I → R

such that

(x, λ, ξ(t), ζ (t)) is

a feasible solution for problem (

VMWD

). If also the weak duality theorem (Theorem

)

holds between (

MVPP

) and (

VMWD

), then

, λ, ξ(t), ζ (t)

is an efficient solution in

Mond-Weir type dual problem (

VMWD

) and the objective functions values are equal.

Proof By assumption, x is an efficient solution in the considered multiobjective vari-
ational control problem (

MVPP

). Hence, by Theorem

, there exist

λ ∈ R

and the

piecewise smooth functions

ξ(·) : I → R

and

ζ (·) : I → R

such that the Karush-

Kuhn-Tucker optimality conditions (

)–(

) are satisfied. Thus,

(x, λ, ξ(t), ζ (t)) is

a feasible solution in Mond-Weir dual problem (

VMWD

) and the two objective func-

tionals have the same values. Efficiency of

(x, λ, ξ(t), ζ (t)) in problem (

VMWD

)

follows directly from weak duality theorem (Theorem

Proposition 11 Let

(x, λ, ξ(t), ζ (t)) be a feasible solution in Mond-Weir type dual

problem (

VMWD

) such that x

∈ S. Further, assume that the following hypotheses are

fulfilled:

123

Duality for multiobjective variational control problems

405

(a)

(t, ·, ·)dt, i = 1, . . . , k, is strictly (, ρ

)-invex at x on S,

(b)

(t, ·, ·)dt, j = 1, . . . , q, is (, ρ

)-invex at x on S,

(c)

(t, ·, ·)dt, k ∈ K

(t) =

∈ K : ζ

(t) > 0

, is

(, ρ

)-invex at x on S,

(d)

(−h

(t, ·, ·))dt, k ∈ K

−

(t) =

∈ K : ζ

(t) < 0

, is

(, ρ

) -invex at x on

(e)

∈K

(t)

−

∈K

−

(t)

Then x is efficient in the considered multiobjective variational control problem

(

MVPP

Proof Proof follows from the weak duality theorem (Theorem

The following result follows directly from the above proposition.

Theorem 12 (Converse duality): Let

(x, λ, ξ(t), ζ (t)) be an efficient solution in

Mond-Weir type vector variational control dual problem (

VMWD

) and x

∈ S. Further,

assume that the hypotheses (a)–(e) of Proposition

are fulfilled. Then x is efficient

in the considered multiobjective variational control problem (

MVPP

Theorem 13 (Strict converse duality): Let x and

(y, λ, ξ(t), ζ (t)) be any feasible

solutions in problems (

MVPP

) and (

VMWD

), respectively, such that

(t, x,

)dt

(t, y,

)dt.

(27)

Further, assume that the following hypotheses are fulfilled:

(a)

(t, ·, ·) dt, i = 1, . . . , k, is strictly (, ρ

)-invex at y on S ∪ Y ,

(b)

(t, ·, ·)dt, j = 1, . . . , q, is (, ρ

)-invex at y on S ∪ Y ,

(c)

(t, ·, ·)dt, k ∈ K

(t) =

∈ K : ζ

(t) > 0

, is

(, ρ

)-invex at y on

∪ Y ,

(d)

(−h

(t, ·, ·))dt, k ∈ K

−

(t) =

∈ K : ζ

(t) < 0

, is

(, ρ

) -invex at y on

∪ Y ,

(e)

∈K

(t)

−

∈K

−

(t)

Then x

= y and y is efficient in the considered multiobjective variational control

problem (

MVPP

Proof Suppose, contrary to the result, that x

= y. Since x and (y, λ, ξ(t), ζ (t)) are

any feasible solutions in problems (

MVPP

) and (

VMWD

), we have

123

406

T. Antczak

(t, x,

)dt

(t, y,

)dt,

(28)

∈K

(t, x,

)dt =

∈K

(t, y,

)dt,

(29)

By Definition

, the hypotheses (b)–(d) yield

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, j ∈ J,

(30)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, k ∈ K

(t),

(31)

−

(t, x,

)dt +

(t, y,

)dt

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

, k ∈ K

−

(t).

(32)

After multiplying both sides of (

) by

0, j ∈ J, adding both sides of the

obtained inequalities, by (

), we get

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

(33)

After multiplying both sides of (

) by

> 0, k ∈ K

(t), and (

) by

−ζ

, k ∈

−

(t), where ζ

< 0, k ∈ K

−

(t), adding both sides of the obtained inequalities, we

get

123

Duality for multiobjective variational control problems

407

∈K

(t)∪K

−

(t)

(t, x,

)dt −

∈K

(t)∪K

−

(t)

(t, y,

)dt

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

(34)

Thus, (

) and (

) give

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

)

−

−h

(t, y,

)

, ρ

(35)

We denote

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, i ∈ A,

(36)

(t) =

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, j ∈ J,

(37)

(t) =

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, k ∈ K

(t),

(38)

(t) =

−ζ

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, k ∈ K

−

(t).

(39)

By (

)–(

), it follows that 0

1, i ∈ A, but λ

> 0 for at least one i ∈ A,

(t) 1, j ∈ J, 0

(t) 1, k ∈ K

(t) ∪ K

−

(t), and, moreover,

123

408

T. Antczak

(t) +

∈K

(t)

(t) +

∈K

−

(t)

(t) = 1.

(40)

Then, hypothesis (e) implies

(t)ρ

∈K

(t)∪K

−

(t)

(t)ρ

(41)

By Definition

, it follows that

(t, x,

, y,

; (0, a)) 0 for any a ∈ R

. Thus,

(

) gives

⎛
⎝t, x,

, y,

;

⎛
⎝0,

(t)ρ

∈K

(t)∪K

−

(t)

(t)ρ

⎞
⎠

⎞
⎠ dt 0.

(42)

Thus, the first constraint of Mond-Weir type vector variational control dual problem

(

VMWD

) yields

⎛
⎝t, x,

, y,

;

⎛
⎝

(t, y,

) +

(t, y,

)

∈K

(t)

(t, y,

) +

∈K

−

(t)

(−

(t, y,

)

−

⎡
⎣

(t, y,

) +

(t, y,

)

∈K

(t)

ζ h

(t, y,

) +

∈K

−

(t)

(−

(t, y,

)

⎤
⎦ ,

∈K

(t)∪K

−

(t)

⎞
⎠

⎞
⎠ dt 0.

(43)

By Definition

, it follows that the functional

(t, x,

, y,

; ·) is convex on R

By (

)–(

), it follows that

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

123

Duality for multiobjective variational control problems

409

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−

)

, x,

, y,

;

−h

(t, y,

)−

−h

(t, y,

)

, ρ

(44)

Again by (

)–(

), it follows that

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

)−

−h

(t, y,

)

, ρ

(45)

Combining (

), (

) and (

), we get

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

0. (46)

By hypothesis (a) and Definitions

, it follows that

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, i ∈ A.

Multiplying both sides of the above inequalities by

, i

= 1, . . . , p, where λ ≥ 0,

= 1, we obtain

123

410

T. Antczak

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

)−

(t, y,

)

, ρ

, i ∈ A, (47)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

(48)

for at least one r

∈ A.

Adding both sides of the inequalities (

) and (

), we get

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

. (49)

By (

) and (

), it follows that the following inequality

(t, x,

)dt >

(t, y,

)dt

holds, contradicting (

). Hence, x

= y and efficiency of y in the multiobjective

variational control problem (

MVPP

) follows by the weak duality theorem (Theorem

). Thus, the theorem is proved.

4 Wolfe duality

In this section, we prove duality results between the considered multiobjective vari-
ational programming problem (

MVPP

) and its Wolfe type vector variational control

dual problem (

VWD

) defined as follows

Maximize

(t, y(t),

(t)) + ξ(t)

(t, y(t),

(t))e

(VWD)

+ζ(t)

(t, y(t),

(t))e

123

Duality for multiobjective variational control problems

411

subject to

(t, y(t),

(t)) + ξ(t)

(t, y(t),

(t))

+ζ(t)

(t, y(t),

(t))

(t, y(t),

(t)) + ξ(t)

(t, y,

)

+ζ(t)

(t, y,

)

, t ∈ I,

(a) = α, x (b) = β,

λ ≥ 0, λ

= 1, ξ (t) 0,

where e

= (1, . . . , 1) ∈ R

is a p-dimensional vector. It may be noted here that the

above dual constraints are written using the Karush–Kuhn–Tucker necessary optimal-
ity conditions for the problem (

MVPP

Let

be the set of all feasible solutions

(y, λ, ξ, ζ ) in Wolfe type dual problem

(

VWD

). We denote by Y the set Y

= {y ∈ X : (y, λ, ξ, ζ ) ∈

Theorem 14 (Weak duality): Let x and

(y, λ, ξ, ζ) be any feasible solutions in prob-

lems (

MVPP

) and (

VWD

). Further, assume that the following hypotheses are fulfilled:

(a)

(t, ·, ·)dt, i = 1, . . . , k, is strictly (, ρ

)-invex at y on S ∪ Y ,

(b)

(t, ·, ·)dt, j = 1, . . . , q, is (, ρ

)-invex at y on S ∪ Y ,

(c)

(t, ·, ·)dt, k ∈ K

(t) = {k ∈ K : ζ

(t) > 0}, is (, ρ

)-invex at y on S∪Y ,

(d)

(−h

(t, ·, ·))dt, k ∈ K

−

(t) = {k ∈ K : ζ

(t) < 0}, is (, ρ

)-invex at y on

∪ Y ,

(e)

∈K

(t)

−

∈K

−

(t)

Then, the following cannot hold

(t, x,

)dt ≤

(t, y,

) + ξ(t)

(t, y,

)e + ζ(t)

(t, y,

that is, the following cannot hold

(t, x,

)dt

{ f (t, y,

) + ξ(t)

(t, y,

)e + ζ(t)

(t, y,

)e}dt,

∀i ∈ A,

(50)

and

(t, x,

)dt <

( f

(t, y,

) + ξ(t)

(t, y,

) + ζ(t)

(t, y,

))dt

for at least one r

∈ A.

(51)

123

412

T. Antczak

Proof Suppose, contrary to the result of the theorem, that the inequalities (

) and

(

) are satisfied. Since

λ ≥ 0, λ

= 1, the inequalities (

) and (

) yield

(t, x,

)dt

(t, y,

)dt +

(t, y,

)dt

(t, y,

)dt 0.

(52)

By assumption, the hypotheses (a)–(d) are fulfilled. Thus, by Definition

, we have

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, i ∈ A, (53)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, j ∈ J, (54)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, k ∈ K

(t),

(55)

−

(t, x,

)dt +

(t, y,

)dt

−

, x,

, y,

;

−h

(t, y,

)−

−h

(t, y,

)

, ρ

, k ∈ K

−

(t).

(56)

Using the last constraint of (

VWD

) together with (

)–(

), we get

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, i ∈ A,

(57)

123

Duality for multiobjective variational control problems

413

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

for at least one r

∈ A,

(58)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

)−

(t, y,

)

, ρ

, j ∈ J,

(59)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

)−

(t, y,

)

, ρ

, k ∈ K

(t), (60)

(t, x,

)dt −

(t, y,

)dt

(−ζ

)

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

∈ K

−

(t).

(61)

Adding both sides of the inequalities (

)–(

) and then, using the feasibility of x

in problem (

MVPP

) and the feasibility of

(y, λ, ξ, ζ ) in problem (

VWD

), we obtain,

respectively,

(t, y,

)dt

, x,

, y,

; g

(t, y,

) −

(t, y,

)

, ρ

, j ∈ J,

(62)

123

414

T. Antczak

∈K

(t, y,

)dt

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

)−

−h

(t, y,

)

, ρ

. (63)

Combining (

), (

) and, (

), we get

(t, x,

)dt >

(t, y,

)dt

(t, y,

)dt +

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

)−

−h

(t, y,

)

, ρ

. (64)

By (

) and (

), it follows that

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

123

Duality for multiobjective variational control problems

415

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

(−ζ

)

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

<0.

(65)

We denote

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, i ∈ A,

(66)

(t) =

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, j ∈ J,

(67)

(t) =

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, k ∈ K

(t),

(68)

(t) =

−ζ

(t)

(t) +

∈K

(t)

(t) −

∈K

−

(t)

, k ∈ K

−

(t).

(69)

By (

)–(

), it follows that

(t) +

∈K

(t)

(t) +

∈K

−

(t)

(t) = 1.

(70)

Combining (

)–(

), we get

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

<0.

(71)

By Definition

, it follows that the functional

, x,

, y,

; ·

is convex on R

Thus, by (

) and (

), Definition

implies

123

416

T. Antczak

⎛
⎝t, x,

, y,

;

⎛
⎝

(t, y,

) +

(t, y,

)

∈K

(t)

(t, y,

) +

∈K

−

(t)

(−

(t, y,

)

−

⎡
⎣

λf

(t, y,

) +

(t, y,

) +

∈K

(t)

(t, y,

)

∈K

−

(t)

(−

(t, y,

)

⎤
⎦,

λρ

∈K

(t)∪K

−

(t)

⎞
⎠

⎞
⎠ dt <0.

Hence, the first constraint of (

VWD

) yields

⎛
⎝t, x,

, y,

;

⎛
⎝0,

(t)ρ

∈K

(t)∪K

−

(t)

(t)ρ

⎞
⎠

⎞
⎠ dt <0.

(72)

From the hypothesis (e), it follows that

(t)ρ

(73)

By Definition

, it follows that

(t, x,

, y,

, (0, a))

0 for any a

∈ R

. Thus, (

)

implies that the following inequality

⎛
⎝t, x,

, y,

;

⎛
⎝0,

(t) ρ

∈K

(t)∪K

−

(t)

(t)ρ

⎞
⎠

⎞
⎠ dt

holds, contradicting (

Theorem 15 (Strong duality): Let x be an efficient solution in the considered multi-
objective variational control problem (

MVPP

). Further, assume that the Kuhn–Tucker

constraint qualification is satisfied for (

MVPP

). Then there exist

λ ∈ R

and the piece-

wise smooth functions

ξ(·) : I → R

and

ζ (·) : I → R

such that

, λ, ξ(t), ζ (t)

is a feasible solution for problem (

VWD

). If also the weak duality theorem (Theorem

) holds between (

MVPP

) and (

VWD

), then

, λ, ξ(t), ζ (t)

is an efficient solution

in Wolfe type dual problem (

VWD

) and the objective function values are equal.

Proof By assumption, x is an efficient solution in the considered multiobjective vari-
ational control problem (

MVPP

). Hence, by Theorem

, there exist

λ ∈ R

and the

piecewise smooth functions

ξ(·) : I → R

and

ζ (·) : I → R

such that the Karush–

Kuhn–Tucker optimality conditions (

)–(

) are satisfied. Thus,

, λ, ξ(t), ζ (t)

123

Duality for multiobjective variational control problems

417

a feasible solution in Wolfe dual problem (

VWD

) and the two objective functionals

have the same values.

Now, we show that

, λ, ξ(t), ζ (t)

is an efficient solution in Wolfe type dual

problem (

VWD

). We proceed by contradiction. Suppose that

, λ, ξ(t), ζ (t)

is not

efficient in problem (

VWD

). Then, there exists

,λ,ξ(t),

ζ (t)

∈

such that

{ f

(t,

) +ξ(t)

(t,

) +

ζ (t)

(t,

)}dt

{ f

(t, x,

) + ξ(t)

(t, x,

) + ζ(t)

(t, x,

)}dt, i ∈ A,

and

{ f

(t,

) +ξ(t)

(t,

) +

ζ (t)

(t,

)}dt > 0

{ f

(t, x,

) + ξ(t)

(t, x,

) + ζ (t)

(t, x,

)}dt for some r ∈ A.

Using the feasibility of x in problem (

MVPP

) together with (

), we get

{ f

(t,

) +ξ(t)

(t,

) +

ζ (t)

(t,

)}dt

(t, x,

)dt, i ∈ A,

and

{ f

(t,

) +ξ(t)

(t,

) +

ζ (t)

(t,

)}dt

(t, x,

)dt, for some r ∈ A.

The inequalities above contradict the weak duality theorem (Theorem

). Thus,

(x, λ, ξ(t), ζ (t)) is efficient in problem (

VWD

Proposition 16 Let

(x, λ, ξ(t), ζ (t)) be a feasible solution in Wolfe type dual problem

(

VWD

) such that x

∈ S. Further, assume that the following hypotheses are fulfilled:

(a)

(t, ·, ·)dt, i = 1, . . . , k, is strictly (, ρ

)-invex at x on S,

(b)

(t, ·, ·)dt, j = 1, . . . , q, is (, ρ

)-invex at x on S,

(c)

(t, ·, ·)dt, k ∈ K

(t) =

∈ K : ζ

(t) > 0

, is

(, ρ

)-invex at x on S,

123

418

T. Antczak

(d)

(−h

(t, ·, ·))dt, k ∈ K

−

(t) =

∈ K : ζ

(t) < 0

, is

(, ρ

) -invex at x on

(e)

∈K

(t)

−

∈K

−

(t)

Then x is efficient in the considered multiobjective variational control problem

(

MVPP

Theorem 17 (Converse duality): Let

(x, λ, ξ(t), ζ (t)) be an efficient solution in Wolfe

type dual problem (

VWD

) and x

∈ S. Further, assume that the hypotheses (a)–(e)

of Proposition

are fulfilled. Then x is efficient in the considered multiobjective

variational control problem (

MVPP

Proof Proof follows directly from Proposition

Theorem 18 (Strict converse duality): Let x and

(y, λ, ξ(t), ζ (t)) be any feasible

solutions in problems (

MVPP

) and (

VMWD

), respectively, such that

(t, x,

)dt

(t, y,

)dt +

(t, y,

)dt

∈K

(t, y,

)dt.

(74)

Further, assume that the following hypotheses are fulfilled:

(a)

(t, ·, ·)dt, i = 1, . . . , k, is strictly (, ρ

)-invex at y on S ∪ Y ,

(b)

(t, ·, ·)dt, j = 1, . . . , q, is (, ρ

)-invex at y on S ∪ Y ,

(c)

(t, ·, ·)dt, k ∈ K

(t) =

∈ K : ζ

(t) > 0

, is

(, ρ

)-invex at y on

∪ Y ,

(d)

(−h

(t, ·, ·))dt, k ∈ K

−

(t) =

∈ K : ζ

(t) < 0

, is

(, ρ

)-invex at y on

∪ Y ,

(e)

∈K

(t)

−

∈K

−

(t)

Then x

= y and y is efficient in the considered multiobjective variational control

problem (

MVPP

Proof Suppose, contrary to the result, that x

= y. By hypotheses (a)–(d), Definition

gives

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

(75)

123

Duality for multiobjective variational control problems

419

(t, x,

)dt −

(t, y,

)dt.

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, j ∈ J,

(76)

(t, x,

)dt −

(t, y,

)dt

, x,

, y,

(t, y,

) −

(t, y,

)

, ρ

, k ∈ K

(t),

(77)

−

(t, x,

)dt +

(t, y,

)dt

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

, k ∈ K

−

(t).

(78)

Hence, by x

∈ S and (y, λ, ξ(t), ζ (t)) ∈

, (

)–(

) yield

(t, x,

)dt −

⎡
⎣

(t, y,

)dt

(t, y,

)dt

∈K

(t)

(t, y,

)dt +

∈K

−

(t)

(t, y,

)dt

⎤
⎦

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

(79)

123

420

T. Antczak

Combining (

) and (

), we get

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

(t)

, x,

, y,

;

(t, y,

) −

(t, y,

)

, ρ

∈K

−

(t)

, x,

, y,

;

−h

(t, y,

) −

−h

(t, y,

)

, ρ

<0.

(80)

The last part of proof is similar to the proof of Theorem

5 Conclusion

Many works of variational problems have been focused on looking for solutions for
these problems from duality results, studying the properties of the classes of function-
als which are involved, and from the relationship between multiobjective variational
programming problems and their duals.

In this paper, the concept of

(, ρ)-invexity was extended to multiobjective varia-

tional control problems. Further, the concept of efficiency has been used to formulate
multiobjective variational control dual problems in the sense of Mond-Weir and in the
sense of Wolfe for the considered multiobjective variational programming problem.
Under

(, ρ)-invexity assumptions imposed on the functions involved in the con-

sidered multiobjective variational control problem, weak, strong, converse and strict
converse dual theorems in the sense of Mond-Weir and in the sense of Wolfe have
been proved between a new class of multiobjective variational control problems and
their duals. Since

(, ρ)-invexity notion unifies several concepts of generalized con-

vex functions, therefore, the results established in the paper extend duality results for
multiobjective variational control problems in a fairly large number of earlier works.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License

which permits any use, distribution, and reproduction in any medium, provided the original author(s) and
the source are credited.

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123

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Duality for multiobjective variational control problems with (Φ, ρ)-invexity

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