The Modified Objective Function Method for Univex ... and Nanda [25] discussed duality theorems and...

Bulletin of the Iranian Mathematical Society (2019) 45:267–282https://doi.org/10.1007/s41980-018-0131-9

ORIG INAL PAPER

The Modified Objective Function Method for UnivexMultiobjective Variational Problems

Tadeusz Antczak1 · Anurag Jayswal2 · Shalini Jha2

Received: 5 May 2016 / Accepted: 21 February 2018 / Published online: 24 July 2018© The Author(s) 2018

AbstractIn this paper, we use the modified objective function method for a class of noncon-vex multiobjective variational problems involving univex functions. Under univexityhypotheses, we prove the equivalence between an (weakly) efficient solution of theconsidered multiobjective variational problem and an (weakly) efficient solution ofthe associatedmodifiedmultiobjective variational problem constructed in themodifiedobjective function method.

Keyword Multiobjective variational problem, Modified objective function method,Efficient solution, Univexity

Mathematics Subject Classification Primary 65K10; Secondary 90C29 · 90C30 ·90C46 · 90C26

1 Introduction

During the last three decades, multiobjective variational problems have been consid-ered in flight control design, control of space structures, industrial process control,impulsive control problems, control of production and inventory, mechanics, eco-

Communicated by Maziar Salahi.

B Tadeusz [email protected]

Anurag [email protected]

Shalini [email protected]

1 Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz,Poland

2 Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines),Dhanbad, Jharkhand 826 004, India

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268 Bulletin of the Iranian Mathematical Society (2019) 45:267–282

nomics, mechanical engineering problems, and other diverse fields. Multiobjectivevariational programming problems with equality and inequality restrictions were con-sidered by many authors (see, for example, [1,2,4–6,8–18,20–29], and others).

Bector and Husain [10] applied duality method of ordinary vector optimizationproblem to multiobjective variational problem and obtained duality results for a prop-erly efficient solution under convexity assumptions on the involved functions. Craven[13] obtained the Kuhn–Tucker type necessary conditions for the considered multiob-jective variational problem and he proved that they are also sufficient if the objectivefunctions are pseudo-convex and constraints are quasi-convex. Bhatia and Kumar [11]established duality results forWolfe aswell asMond–Weir-type duals underρ-invexityassumptions and their generalizations for nonconvex multiobjective control problems.Mukherejee and Mishra [22] defined the concept of V -invexity for multiobjectivevariational problems and they established optimality conditions and duality results formultiobjective control problems under (generalized) V -invexity assumptions. Thus,they extended the results of Bhatia and Kumar [11] to a wider class of nonconvexmultiobjective control problems. In [23], Mukherejee and Rao extended mixed-typeduality to the class of multiobjective variational problems and proved duality resultsunder generalized ρ-invexity. Bhatia and Mehra [12] extended the concepts of B-type I and generalized B-type I functions to the continuous case and they used theseconcepts to establish sufficient optimality conditions and duality results for multiob-jective variational programming problems. Nahak and Nanda [25] discussed dualitytheorems and related efficient solutions to the primal and dual problems for multiob-jective variational control problemswith (F, ρ)-convexity. Reddy andMukherjee [27]studied duality theorems and related efficient solutions of the primal and dual prob-lems for multiobjective fractional control problems under (F, ρ)-convexity. Ahmadand Gulati [2] defined mixed-type duality for multiobjective variational problems,obtaining new optimality results also under (F, ρ)-convexity. Using the relationshipbetween an efficient solution of the multiobjective control problem and an optimalsolution of the associated scalar control problem, Gramatovici [14] derived the nec-essary optimality conditions for the multiobjective variational problems with invexfunctions. Kim and Kim [18] introduced new classes of generalized V -type I invexfunctions for variational problems and they proved a number of sufficiency results andduality theorems using Lagrange multiplier conditions under various types of general-ized V -type I invexity requirements. Further, under the generalized V -type I invexityassumptions and their generalizations, they obtained duality results for Mond–Weir-type duals. Also Hachimi and Aghezzaf [15] obtained several mixed-type dualityresults for multiobjective variational programming problems, but under a new intro-duced concept of generalized type I functions. Khazafi and Rueda [16] extended theconcept of V -univexity type I tomultiobjective variational programming problems andthey derived various sufficient optimality conditions and mixed-type duality resultsunder generalized V -univexity type I conditions. In [17], Khazafi et al. introducedthe classes of (B, ρ)-type I functions and generalized (B, ρ)-type I functions andderived a series of sufficient optimality conditions and mixed-type duality resultsfor multiobjective variational problems. Nahak and Behera [26] used (generalized)ρ-(η, θ)-B-type I functions to establish sufficient optimality conditions and dualityresults for multiobjective variational problems. In [8], Arana-Jimenéz et al. provided

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Bulletin of the Iranian Mathematical Society (2019) 45:267–282 269

new pseudoinvexity conditions on the functionals involved in the considered multi-objective variational problem and they proved that all vector Kuhn–Tucker or FritzJohn points of a multiobjective variational problem are weakly efficient solutions ifand only if these conditions are fulfilled. Recently, Antczak [4] extended the conceptof (�, ρ)-invexity to the continuous case and, under (�, ρ)-invexity hypotheses, heproved sufficient optimality conditions and duality results for a class of nonconvexmultiobjective variational problems. Recently, Antczak andArana-Jiménez [5] provedsufficient optimality conditions and duality results in the sense of Mond–Weir for theconsidered multiobjective variational control problem under B-(p, r)-invexity.

In this paper, we use themodified objective functionmethod introduced byAntczak[3] in the case of differentiable vector optimization problems for solving a noncon-vex multiobjective variational problem involving univex functions. In this method,for the considered multiobjective variational problem, its associated multiobjectivevariational problem with the modified objective function is constructed at an arbitrarybut fixed feasible solution. This construction depends heavily on results proved inthis paper, which connects an efficient (weakly efficient) solution of the original mul-tiobjective variational problem to an efficient solution (weakly efficient solution) ofthe modified multiobjective variational problem constructed in the modified objectivefunction method. In general, the multiobjective variational problem with the modi-fied objective function is simpler to solve than the original multiobjective variationalproblem. Further, there exist also such cases in which the multiobjective variationalproblem with the modified objective function is convex although the original mul-tiobjective variational problem is nonconvex. These properties are important from apractical point of view. They have been illustrated in the paper by suitable examplesof nonconvex multiobjective variational problems.

2 Preliminaries and Notations

The following convention for equalities and inequalities will be used in the paper.For any x = (x1, x2, ..., xn)T , y = (y1, y2, ..., yn)T , where ()T denotes for the

transpose, we define:

(i) x = y if and only if xi = yi for all i = 1, 2, ..., n;(ii) x < y if and only if xi < yi for all i = 1, 2, ..., n;(iii) x � y if and only if xi � yi for all i = 1, 2, ..., n;(iv) x ≤ y if and only if x � y and x �= y.

All vectors will be taken as column vectors.Let I = [a, b] be a real interval, A = {1, 2, ..., p} and J = {1, 2, ...,m}.In this paper, we assume that x(t) is an n-dimensional piecewise smooth function

of t , and·x(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 → Rn withnorm ‖x‖ = ‖x‖∞ + ‖Dx‖∞, where the differentiation operator D is given by

z = Dx ⇐⇒ x(t) = x (a) +t∫

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

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Therefore, ddt ≡ D except at discontinuities. For notational simplicity, we write x(t)

and·x(t) as x and

·x , respectively.

Let f : I × Rn × Rn → Rp be a p-dimensional function and each of its componentis a continuously differentiable real scalar function, g : I × Rn × Rn → Rm be acontinuously differentiablem-dimensional function. Here t is the independent variable

and x(t) is the state variable. To consider f i(t, x (t) ,

·x (t)

), where x : I → Rn

is differentiable with derivative·x , denote the partial derivatives of f i , i ∈ A, with

respect to t, x and·x , respectively, by f it , f

ix , f

i·xsuch that f ix =

(∂ f i

∂x1, . . . ,

∂ f i

∂xn

)and

f i·x

=(

∂ f i

∂·x1

, . . . ,∂ f i

∂·xn

)

. Similarly, the partial derivatives of the vector function g can

be written, using matrices with m rows instead of one. Further, we denote by diag za p × p-dimensional matrix as follows:

diag z =

⎡

⎢⎢⎢⎢⎣

z1 0 ... 0

0 z2 ......

.... . . 0

0 ... 0 z p

⎤

⎥⎥⎥⎥⎦

.

In the past few years, extensive literature relative to the other families of moregeneral functions to substitute the convex functions in the optimization theory hasgrown immensely. One of such classes of generalized convex functions is the class ofunivex functions, introduced by Bector et al. [7] for scalar optimization problems.

Let f : I ×Rn×Rn → Rp be a continuously differentiable functional with respectto each of their arguments and x ∈ X . The following definition introduces the conceptof univexity for the functional f .

Definition 2.1 If there exist functions � f = (� f 1 , ..., � f p

) : Rp → Rp, b f =(b f 1 , ..., b f p

) : X × X → Rp+, where b f i : X × X → R+\{0}, i = 1, ..., p,

η : I × Rn × Rn → Rn such that the inequality

diag b f (x, x) � f

⎛

⎝b∫

a

f(t, x,

·x)dt −

b∫

a

f

(

t, x,·x

)

dt

⎞

⎠

� (>)

b∫

a

{

η (t, x, x)T fx

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f ·x

(

t, x,·x

)}

dt

(2.1)

holds for all x ∈ X , (x �= x), then the functional f is said to be (strictly) univex at xwith respect to �, b, η. If (1) is satisfied for every x ∈ X , then f is said to be (strictly)univex on X .Equivalently, the inequality (1), defining the concept of univexity, can be written asfollows:

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b f i (x, x)� f i

⎛

⎝b∫

a

f i(t, x,

·x)dt −

b∫

a

f i(

t, x,·x

)

dt

⎞

⎠

� (>)

b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i·x

(

t, x,·x

)}

dt , i ∈ A.

(2.2)

Remark 2.2 Note that the concept of univexity generalizes some generalized convexitynotions previously defined in the literature for multiobjective variational problems.Indeed, in the case when � f i = I dRp , where I dRp is the identity in Rp, then weobtain the definition of a b-invex function in a continuous case (see [1]). Moreover, ifb f i (x, x) ≡ 1, i = 1, ..., p, for all x, x ∈ X , then we obtain the definition of an invexfunction in a continuous case (see, for example, [1]).

In the multiobjective variational control problem, under given conditions, the statevector x(t) is brought from specified initial state x(a) = α to some specified finalstate x(b) = β in such a way to minimize a given functional. A more precise mathe-matical formulation is given in the following multiobjective variational programmingproblem:

V -Minimize

b∫

a

f(t, x (t) ,

·x (t)

)dt (MVP)

=⎛

⎝b∫

a

f 1(t, x (t) ,

·x (t)

)dt, . . . ,

b∫

a

f p(t, x (t) ,

·x (t)

)dt

⎞

⎠

subject to g(t, x (t) ,

·x (t)

)� 0, t ∈ I ,

x (a) = α , (b) = β,

where f = (f 1, ..., f p

) : I×Rn×Rn → Rp and g = (g1, ..., gm

) : I×Rn×Rn →Rm are continuously differentiable functions with respect to each of their arguments.

Let S denote the set of all feasible solutions for the considered multiobjectivevariational programming problem (MVP), i.e.,

S ={x ∈ Rn : g

(t, x (t) ,

·x (t)

)� 0, ∀t ∈ I , x (a) = α, x (b) = β

}.

Definition 2.3 A feasible solution x of the considered multiobjective variational pro-gramming problem (MVP) is said to be weakly efficient of (MVP) if there exists noother x ∈ S such that

b∫

a

f(t, x,

·x)dt <

b∫

a

f

(

t, x,·x

)

d,

123


that is, there exists no other x ∈ S such that

b∫

a

f i(t, x,

·x)dt <

b∫

a

f i(

t, x,·x

)

dt, ∀i ∈ A.

Definition 2.4 A feasible solution x of the considered multiobjective variational pro-gramming problem (MVP) is said to be efficient of (MVP) if there exists no otherx ∈ S such that

b∫

a

f(t, x,

·x)dt ≤

b∫

a

f

(

t, x,·x

)

dt,

that is, there exists no other x ∈ S such that

b∫

a

f i(t, x,

·x)dt �

b∫

a

f i(

t, x,·x

)

dt, ∀i ∈ A,

b∫

a

f i∗ (

t, x,·x)dt <

b∫

a

f i∗(

t, x,·x

)

dt for at least one i∗ ∈ A.

In order to prove sufficient optimality conditions for the considered multiobjectivevariational programming problem (MVP), we give the Karush–Kuhn–Tucker neces-sary optimality conditions for such vector optimization problems (see, for instance,[5,6,12,19,26]).

Theorem 2.5 Let x ∈ S be a normal efficient solution of the considered multiobjec-tive variational programming problem (MVP) at which the Kuhn–Tucker constraintqualification be satisfied. Then there exist λ ∈ Rp and the piecewise smooth functionξ (·) : I → Rm such that

λTfx

(

t, x,·x

)

+ ξ (t)T gx

(

t, x,·x

)

= d

dt

[

λTf ·x

(

t, x,·x

)

+ ξ (t)T g ·x

(

t, x,·x

)]

, t ∈ I , (2.3)

ξ (t)T g

(

t, x,·x

)

dt = 0, t ∈ I , (2.4)

λ ≥ 0, λTe = 1, ξ (t) � 0. (2.5)

For notational convenience, we use ξ for ξ (t).

123


Remark 2.6 [12] We shall use the following property to prove the main results in thepaper.

Let h : I × Rn × Rn → R be a continuously differentiable function with respectto each of its arguments and let η : I × Rn × Rn → Rn satisfy the conditionη (t, x, x) = 0. Let x, u : I → Rn be differentiable functions with x (a) = u (a) = α

and x (b) = u (b) = β. Then,

b∫

a

d

dt(η (t, x, u)) h ·

x

(t, u,

·u)dt = −

b∫

a

η (t, x, u)

(d

dth ·x

(t, u,

·u))

dt . (2.6)

3 Multiobjective Variational Problemwith theModified ObjectiveFunction

Let x be an arbitrary feasible solution of the considered multiobjective variationalprogrammingproblem (MVP) and thevector-valued functionη : I×Rn×Rn → Rn begiven. We now use the modified objective function method for solving the considerednonconvex multiobjective variational programming problem (MVP). Therefore, atthe given feasible point x , we construct the multiobjective variational programmingproblem (MVPη (x)) with the modified objective function associated with the originalmultiobjective variational programming problem (MVP) as follows:

V -Minimize⎛

⎝b∫

a

f 1(

t, x (t) ,·x (t)

)

dt +b∫

a

{

η (t, x (t) , x (t))T f 1x

(

t, x (t) ,·x (t)

)

+ d

dt

(η (t, x (t) , x (t))T

)f 1·x

(

t, x (t) ,·x (t)

)}

dt, . . . ,

b∫

a

f p(

t, x (t) ,·x (t)

)

dt +b∫

a

{

η (t, x (t) , x (t))T f px

(

t, x (t) ,·x (t)

)

+ d

dt

(η (t, x (t) , x (t))T

)f p·x

(

t, x (t) ,·x (t)

)}

dt

⎞

⎠

subject to g(t, x (t) ,

·x (t)

)� 0, t ∈ I (MV Pη (x))

x (a) = α , x (b) = β,

where f = (f 1, ..., f p

) : I ×Rn×Rn → Rp and g = (g1, ..., gm) : I ×Rn×Rn →Rm are continuously differentiable functions with respect to each of their arguments.

Now, we prove the equivalence between an efficient solution (weakly efficientsolution) of the original multiobjective variational problem and an efficient solution

123


(weakly efficient solution) of its associated modified multiobjective variational prob-lem constructed in the modified objective function method.

Theorem 3.1 Let x be a feasible solution of the considered multiobjective variationalprogrammingproblem (MVP)and theKarush–Kuhn–Tucker necessary optimality con-ditions (2.3)–(2.5) be satisfied at this point with λ ∈ Rp and the piecewise smoothfunction ξ (·) : I → Rm. Further, assume that ξ (t)T g (t, ·, ·) is univex at x on S withrespect to �g, bg, η, where a � 0 �⇒ �g (a) � 0 and η (t, x, x) = 0. If λ > 0, thenx is an efficient solution of the multiobjective variational programming problem withthe modified objective function (MVPη (x)) associated with the original multiobjectivevariational problem (MVP).

Proof Suppose, contrary to the result, that x is not an efficient solution of the multi-objective variational programming problem (MVPη (x)) with the modified objectivefunction associated with the original multiobjective variational problem (MVP). Then,there exists x ∈ S such that

b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i·x

(

t, x,·x

)}

dt �

b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i·x

(

t, x,·x

)}

dt, ∀i ∈ A,

(3.1)

b∫

a

{

η (t, x, x)T f i∗

x

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i

∗·x

(

t, x,·x

)}

dt <

b∫

a

{


x

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i

∗·x

(

t, x,·x

)}

dt

for at least one i∗ ∈ A. (3.2)

Since η (t, x (t) , x (t)) = 0, inequalities (3.1) and (3.2) yield, respectively,

b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i·x

(

t, x,·x

)}

dt � 0, ∀i ∈ A,

(3.3)

b∫

a

{


x

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i

∗·x

(

t, x,·x

)}

dt < 0

for at least one i∗ ∈ A. (3.4)

123


By assumption, λ > 0. Hence, (3.3) and (3.4) imply

b∫

a

{

η (t, x, x)T λTfx

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)λTf ·x

(

t, x,·x

)}

dt < 0.

(3.5)By (2.6), inequality (3.5) gives

b∫

a

{

η (t, x, x)T[

λTfx

(

t, x,·x

)

− λT d

dt

(

f ·x

(

t, x,·x

))]}

dt < 0. (3.6)

Now, using x , x ∈ S together with the necessary optimality conditions (2.4) and (2.5),we get

b∫

a

ξ (t)T g

(

t, x,·x

)

dt −b∫

a

ξ (t)T g

(

t, x,·x

)

dt � 0.

Hence, by hypothesis a � 0 �⇒ �g (a) � 0, the above inequality implies

�g

⎛

⎝b∫

a

ξ (t)T g

(

t, x,·x

)

dt −b∫

a

ξ (t)T g

(

t, x,·x

)

dt

⎞

⎠ � 0. (3.7)

Since bg (x, x) > 0, (3.7) gives

bg (x, x)�g

⎛

⎝b∫

a

ξ (t)T g

(

t, x,·x

)

dt −b∫

a

ξ (t)T g

(

t, x,·x

)

dt

⎞

⎠ � 0. (3.8)

Hence, by Definition 2.1, (3.8) implies

b∫

a

{

η (t, x, x)T ξ (t)T gx

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)ξ (t)T g ·

x

(

t, x,·x

)}

dt � 0.

(3.9)By (2.6), the above inequality (3.9) gives

b∫

a

{

η (t, x, x)T ξ (t)T gx

(

t, x,·x

)

− η (t, x, x)Td

dt

(

ξ (t)T g ·x

(

t, x,·x

))}

dt � 0.

(3.10)

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Combining (3.6) and (3.10), we obtain that the following inequality:

b∫

a

η (t, x, x)T[

λTfx

(

t, x,·x

)

+ ξ (t)T gx

(

t, x,·x

)

− ddt

(

λTf ·x

(

t, x,·x

)

+ ξ (t)T g ·x

(

t, x,·x

))]

dt < 0

holds, which contradicts the necessary optimality conditions (2.3). This completes theproof of this theorem. ��Corollary 3.2 Let x be an efficient solution of the consideredmultiobjective variationalprogrammingproblem (MVP)and theKarush–Kuhn–Tucker necessary optimality con-ditions (2.3)–(2.5) be satisfied at this point with λ ∈ Rp and the piecewise smoothfunction ξ (·) : I → Rm. Further, assume that ξ (t)T g (t, ·, ·) is univex at x on S withrespect to �g, bg, η, where a � 0 �⇒ �g (a) � 0 and η (t, x, x) = 0. If λ > 0,then x is an efficient solution of the multiobjective variational programming problem(MVP η (x)) with the modified objective function associated with the original varia-tional programming problem (MVP).

In order to prove that a feasible solution x is weakly efficient in problem (MVP),the assumption λ > 0 can be omitted.

Corollary 3.3 Let x be a weakly efficient solution of the considered multiobjectivevariational programming problem (MVP) and the Karush–Kuhn–Tucker conditions(2.3)–(2.5) be satisfied at this point with λ ∈ Rp and the piecewise smooth functionξ (·) : I → Rm. Further, assume that ξ (t)T g (t, ·, ·) is univex at x on S with respect to�g, bg, η, where a � 0 �⇒ �g (a) � 0 and η (t, x, x) = 0. Then x is a weakly effi-cient solution of themultiobjective variational programming problem (MVPη (x)) withthe modified objective function associated with the original variational programmingproblem (MVP).

Remark 3.4 Note that we have established the results in Theorem 3.1 and Corollaries3.2 and 3.3 without any univexity assumption imposed on the objective function.

Now, we prove the converse results to those ones established above.

Theorem 3.5 Let x be an efficient solution of the multiobjective variational program-ming problem (MVPη (x)) with the modified objective function. Further, assume thatf i (t, ·, ·), i = 1, ..., p, is strictly univex at x on S with respect to � f i , b f i , η, wherea � 0 �⇒ � f i (a) � 0, i ∈ I , and, moreover, η (t, x, x) = 0. Then x is an efficientsolution of the considered multiobjective variational problem (MVP).

Proof Suppose, contrary to the result, that x is not a weakly efficient solution of theconsidered multiobjective variational programming problem (MVP). Then, there existx ∈ S and x �= x such that

b∫

a

f

(

t, x,·x

)

dt ≤b∫

a

f

(

t, x,·x

)

dt .

123


Thus,

b∫

a

f i(

t, x,·x

)

dt �b∫

a

f i(

t, x,·x

)

dt, i ∈ A, (3.11)

b∫

a

f i∗(

t, x,·x

)

dt <

b∫

a

f i∗(

t, x,·x

)

dt for at least on i∗ ∈ A. (3.12)

Then, by hypothesis imposed on� f , inequalities (3.11) and (3.12) yield, respectively,

� f i

⎛

⎝b∫

a

f i(

t, x,·x

)

dt −b∫

a

f i(

t, x,·x

)

dt

⎞

⎠ � 0, i ∈ A. (3.13)

Since each function f i (t, ·, ·), i = 1, ..., p, is strictly univex at x on S with respectto � f i , b f i , η, by Definition 2.1, the following inequalities

b f i (x, x)� f i

⎛

⎝b∫

a

f i(t, x,

·x)dt −

b∫

a

f i(

t, x,·x

)⎞

⎠ dt

>

b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i·x

(

t, x,·x

)}

dt , i ∈ A

hold for all x ∈ S, (x �= x). Therefore, the inequalities above are also satisfied forx = x ∈ S. Thus,

b f i (x, x)� f i

⎛

⎝b∫

a

f i(

t, x,·x

)

dt −b∫

a

f i(

t, x,·x

)⎞

⎠ dt

>

b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i·x

(

t, x,·x

)}

dt , i ∈ A.

(3.14)

Combining (3.13) and (3.14), we get

b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i·x

(

t, x,·x

)}

dt < 0, i ∈ A.

(3.15)

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By assumption, η (t, x, x) = 0. Hence, (3.15) implies that the following inequalities

b∫

a

f i(

t, x,·x

)

dt +b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)Tf i·x

(

t, x,·x

)}

dt

<

b∫

a

f i(

t, x,·x

)

dt +b∫

a

{

η (t, x, x)T f ix

(

t, x,·x

)

+ d

dt

(η (t, x, x)T

)f i·x

(

t, x,·x

)}

dt

hold, contradicting the efficiency of x for the multiobjective variational programmingproblem (MVPη (x)) with the modified objective function associated with the originalmultiobjective variational problem (MVP). This completes the proof of this theorem.

��

Remark 3.6 Note that we have established the result in Theorem 3.5 without anyunivexity assumption imposed on the constraint function.

If we assume weaker univexity hypothesis imposed on the objective function, thenwe are in a position to obtain the following result.

Theorem 3.7 Let x be a weakly efficient solution of the multiobjective variationalproblem (MVPη (x)) with the modified objective function. Further, assume that eachobjective function f i (t, ·, ·), i = 1, ..., p, is univex at x on S with respect to� f i , b f i ,η, where a < 0 �⇒ � f i (a) < 0, i ∈ I , and, moreover, η (t, x, x) = 0. Then x is aweakly efficient solution of the considered multiobjective variational problem (MVP).

By Corollary 3.2 and Theorem 3.5, it follows the equivalence between an efficientsolution of the original multiobjective variational problem and an efficient solution ofits associated modified multiobjective variational problem constructed in the modifiedobjective function method.

Theorem 3.8 Let all hypotheses of Theorems 3.1 and 3.5 be fulfilled. Then x ∈ S isan efficient solution of the original multiobjective variational problem (MVP) if andonly if it is an efficient solution of its associated modified multiobjective variationalproblem constructed in the modified objective function method.

The similar result is true by Corollary 3.3 and Theorem 3.7:

Theorem 3.9 Let all hypotheses of Corollary 3.3 and Theorem 3.7 be fulfilled. Thena feasible solution x is a weakly efficient solution of the original multiobjective vari-ational problem (MVP) if and only if it is a weakly efficient solution of its associatedmodified multiobjective variational problem constructed in the modified objectivefunction method.

Now, we give an example of a nonconvex multiobjective variational problem toillustrate the result established in the paper.

123


Example 3.10 Let I = [0, 1]. Consider the following nonconvex multiobjective vari-ational problem:

V -Minimize∫ 1

0f (t, x, x)dt =

( ∫ 1

0(1 + t + x − x2 + sin

2x)dt,

∫ 1

0

(t + xex

)dt

)

subject to g(t, x, x) = x2−x � 0, t ∈ I (MVP1)

x(0) = 0 , x(1) = 0.

The set of all feasible solutions of the considered multiobjective variational problem(MVP1) is given by S = {x ∈ R : x(0) = 0, x(1) = 0 and 0 � x (t) � 1 for eacht ∈ I } and, moreover, x(t) = 0 is feasible for the problem (MVP1). Further, note thatλ = (λ1, λ2) = ( 12 ,

12 ) and ξ (t) = 1 verify Theorem 2.5.

We set

� f1 (a) = ea − 1, � f2 (a) = a, �g(a) = a,

b f1(x, x) = 1, b f2(x, x) = 1, bg(x, x) = 1

and, moreover, η : I × R × R �→ R be defined as

η(t, x, x) = x − x2 − x − x2.

Hence, it can be established byDefinition 2.1 that f i (t, ·, ·), i = 1, 2, is strictly univexat x on S with respect to � f i , b f i , and η and ξ (t)T g (t, ·, ·) is univex at x on S withrespect to �g , bg and η. Using the approach analyzed in the paper, we construct thefollowingmultiobjective variational problem (MVP1η(x)) with themodified objectivefunction as follows:

V -Minimize( ∫ 1

0(1 + t+x − x2)dt ,

∫ 1

0(t+x − x2)dt

)

subject to g(t, x, x) = x2−x � 0, t ∈ I (MVP1η(x))

x(0) = 0 , x(1) = 0.

Since all hypotheses of Theorem 3.1 are fulfilled, we conclude that x(t) = 0 isan efficient solution of the multiobjective variational problem (MVP1η(x)) with themodified objective function. Further, all hypotheses of Theorem3.5 are also satisfied atx = 0. Therefore, x(t) = 0, being efficient in the multiobjective variational problem(MVP1η(x)), it is also efficient for the original multiobjective variational problem(MVP1). Note, moreover, that this result cannot be proved under convexity hypothesessince the functions constituting themultiobjective variational problem (MVP1) are notconvex on S.

Now, we give an example of a nonconvex multiobjective variational problem toillustrate one of the properties of the approach analyzed in the paper. Namely, for some

123


nonlinear nonconvex multiobjective variational problems, multiobjective variationalproblems with the modified objective function constructed in the analyzed approachmay be linear.

Example 3.11 Let I = [0, 1]. Consider the following nonconvex multiobjective vari-ational problem:

V -Minimize∫ 1

0f (t, x, x)dt =

( ∫ 1

0(t x3 + 1

2x)dt ,

∫ 1

0

(

sinx + x − 3

4x2

)

dt)

subject to g1(t, x, x) = −x � 0, t ∈ I

g2(t, x, x) = x − 1 � 0, t ∈ I ,

x(0) = 0 , x(1) = 0. (MVP2)

The set of all feasible solutions of problem (MVP2) is given by S = {x ∈ R : x(0) =0, x(1) = 0 and 0 � x (t) � 1 for each t ∈ I } and x(t) = 0. Further, note that

λ = (λ1, λ2) = ( 12 ,12 ) and ξ (t) =

(54 , 0

)verify Theorem 2.5.

We set

� f1 (a) = a, � f2 (a) = a , �g(a) = a,

b f1(x, x) = 1

2, b f2(x, x) = 2, bg(x, x) = 1

2

and, moreover, η : I × R × R �→ R be defined as

η(t, x, x) = 1

2(x − x) .

Hence, it can be established by Definition 2.1, that f i (t, ·, ·), i = 1, 2, are strictlyunivex at x on S with respect to � f i , b f i , η and ξ (t)T g (t, ·, ·) is univex at x on Swith respect to �g , bg , η. Using the approach analyzed in the paper, we construct thefollowingmultiobjective variational problem (MVP2η(x)) with themodified objectivefunction as follows:

V -Minimize( ∫ 1

0

1

4xdt ,

∫ 1

0xdt

)

subject to g(t, x, x) = −x � 0, t ∈ I

g2(t, x, x) = x − 1 � 0, t ∈ I ,

x(0) = 0 , x(1) = 0. (MVP2η(x))

Since all hypotheses of Theorem 3.1 are fulfilled, we conclude that x(t) = 0 is an effi-cient solution of themultiobjective variational problem (MVP2η(x)) with themodifiedobjective function. Further, since all hypotheses of Theorem 3.5 are satisfied at x = 0,which is an efficient solution of the multiobjective variational problem (MVP2η(x)), xis also efficient in the original multiobjective variational problem (MVP2). Although

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we consider the nonlinear nonconvex multiobjective variational problem (MVP2),then the multiobjective variational problem (MVP2 η(x)) with the modified objectivefunction constructed in the used approach is linear. Then, the multiobjective vari-ational problem (MVP2η(x)) with the modified objective function is less complexthan the original multiobjective variational problem (MVP2). Therefore, it is easier tosolve than the original one. This property of the modified objective function methodis important from a practical point of view.

4 Conclusion

In this paper, the class of nonconvexmultiobjective variational programming problemswith univex functions has been considered. To solve such nonconvex multiobjectivevariational programming problems, we have used the modified objective functionmethod. In this method, for the considered multiobjective variational programmingproblem, its associatedmultiobjective variational programmingproblemwith themod-ified objective function has been constructed at the given feasible solution. Underunivexity hypotheses, the equivalence between an (weakly efficient) efficient solutionof the considered multiobjective variational programming problem and an (weaklyefficient) efficient solution of its associated multiobjective variational programmingproblemwith themodified objective function has been established. It turned out that, ingeneral, the multiobjective variational programming problemwith themodified objec-tive function generated in the modified objective function method has simpler formthan the considered multiobjective variational programming problem. What is more,in some cases, the multiobjective variational programming problem with the modifiedobjective function generated in the modified objective function method is linear orconvex although the considered multiobjective variational programming problem isnonlinear and nonconvex. An example of such a nonlinear nonconvex multiobjectivevariational programming problem has been presented in the paper. The property men-tioned above is important from the practical point of view since we are in a position tosolve nonlinear nonconvex multiobjective variational programming problems by thehelp of linear or convex vector variational ones.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

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