Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 47, No. 1 (2025) pp. 205-229

Ras Bihari Soni¹,

DUALITY CRITERIA INVOLVING

(p, φ ,d)-INVEXITY AND (p, φ, d)

Dharamender Singh²

and

-PSEUODINVEXITY IN INTERVAL

-VALUED MULTIOBJECTIVE

OPTIMIZATION PROBLEMS

Kailash Chand

Sharma³

Abstract: In this present research work, study of duality associated with a

special class of multiobjective optimization that include the interval valued

components is delt. We define (ρ,φ,d)-Invexity and (ρ,φ,d-Pseuodinvexity,

which are connected with an interval valued multiple integral functional.

For such class of variational problems, we write dual problem associated

with primal problem. We prove weak, strong and converse duality theorems

for this type of variational problems. A brief comparison with existed

methods have been done to show the importance of this research work.

Additionally, numerical examples have been displayed at the appropriate

places to support the results which shows the significance of our Study.

Keywords: Multiobjective Optimization, (ρ, φ, d)-Invexity and (ρ, φ, d)-

Pseuodinvexity, Duality.

Mathematics Subject Classification (2010) No.: 58E17, 65K05, 90C46,

90C29, 26B25, 49K20,

49N15.

1. Introduction and Literature Review

Mathematical optimization problems involving multiple objective functions

that must be optimized simultaneously fall under the purview of multi-objective

optimization, also known as Pareto optimization or multi-objective programming,

vector optimization, multicriteria optimization, or multiattribute optimization. Several

scientific domains, such as engineering, economics, and logistics, have used multi-

206

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

objective vector optimization to make optimal judgments when there are trade-offs

between two or more competing objectives. Multi-objective optimization issues with

two or three objectives include things like optimizing performance while limiting fuel

consumption and vehicle emissions, and minimizing cost while maximizing comfort

while purchasing an automobile. There may be more than three objectives in practical

tasks.

The application of duality theory to more general classes of functions has

grown as a result of its success in mathematical programming. Kumar et al. [1] have

considered multiobjective semi-infinite variational problem (MSVP) and generalised

the concept of inveity. Kumar et al. [2] defined certain conditions on the functionals

of multi-objective fractional variational problem in order that it becomes F-Kuhn

Tucker pseudo invex or F-Fritz John pseudo invex. Bhardwaj and Ram [3]

established the relationships between a class of interval-valued vector optimization

problems and interval-valued vector variational-like inequality problems of both

Stampacchia and Minty kinds in terms of convexificators.

Upadhyay et al. [4] dealt with a certain class of multiobjective semi-infinite

programming problems with switching constraints (in short, MSIPSC) in the

framework of Hadamard manifolds. Sahay and Bhatia [5] introduced new classes of

higher order generalized strong invex functions under non-differentiable settings.

Soni et al. [6] discussed optimization problems with multiobjective functions

and their applications in engineering field.

Zalmai [7] established global semiparametric sufficient efficiency results

under various generalized (,b, , ,)-univexity assumptions for a multiobjective

fractional subset programming problem. Hachimi and Aghezzaf [8] generalized a fairly

large number of sufficient optimality conditions and duality results previously

obtained for multiobjective variational problems. Treanţă [9] introduced

necessary efficiency conditions for a class of multi-time vector fractional variational

problems with nonlinear equality and inequality constraints involving higher-order

partial derivatives. Treanţă [10] introduced a generalised condition on the

functionals involved in a multidimensional vector control problem and prove

that a (strongly) b-V-KT-pseudoinvex multidimensional control problem is

characterized so that all Kuhn-Tucker points are efficient solutions. Kim [11]

formulated duality for nondifferentiable multiobjective variational problems and

established the weak, strong, and converse duality theorems under generalized

(F, )-convexity assumptions. Gulati et al. [12] obtained Fritz John and Kuhn-

Tucker type necessary optimality conditions for a Pareto optimal (efficient) solution

of a multiobjective control problem are by first reducing the multiobjective control

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 207

problem to a system of single objective control problems, and then using already

established optimality conditions. Nahak and Nanda [13] presented the sufficient

optimality criteria for a class of multiobjective variational control problems under the

V-invexity assumption. They also proved duality results under a variety of V-invexity

assumptions.

Antczak and Jiménez [14] generalized the notion of B-(p, r)-invexity and

proved sufficient optimality conditions under the assumptions that the functions

constituting them are B-(p, r)-invex. Antczak [15] extended the notions of

(Φ, ρ)-invexity and generalized (Φ, ρ)-invexity to the continuous case and we use

these concepts to establish sufficient optimality conditions for the considered class of

nonconvex multiobjective variational control problems and established several mixed

duality results are under (Φ, ρ)-invexity. Khazafi et al. [16] introduced the classes of

(B, ρ)-type I and generalized (B, ρ)-type I, and derived various sufficient optimality

conditions and mixed type duality results for multiobjective control problems under

(B, ρ)-type I and generalized (B, ρ)-type I assumptions. Zhang et al. [17] extended the

vector-valued G-invex functions to multiobjective variational control problems, by

using this concept, a number of sufficient optimality results and Mond-Weir type

duality results were obtained for multiobjective variational control programming

problem. Treanţă and Arana [18] defined a Kuhn-Tucker (KT)-pseudoinvex

multidimensional control problem and introduced a new condition on the functions,

which were involved in a multidimensional control problem proved that a

KT-pseudoinvex multidimensional control problem is characterized such that a KT

point is an optimal solution. Mititelu [19] established necessary conditions for normal

efficient solutions of a class of multiobjective fractional variational problem (MFP)

with nonlinear equality and inequality constraints using a parametric approach to

relate efficient solutions of fractional problems and a non-fractional problem and

established the sufficiency of these conditions for efficiency solutions in problem

(MFP) using the (ρ, b)-quasiinvexity notion.

Mititelu and Treanţă [20] formulated and proved necessary and sufficient

optimality conditions in multiobjective control problems which involve multiple

integral and under (ρ, b)-quasiinvexity assumptions, sufficient efficiency conditions

for a feasible solution were derived. Treanţă and Mititelu [21] introduced several

results of duality for a class of multiobjective fractional control problems involving

multiple integrals and under (ρ, b)-quasiinvexity assumptions, they formulated and

prove weak, strong and converse duality results. Treanţă [22] formulated and proved

efficiency conditions for the considered uncertain variational control problem and

established sufficiency of Karush-Kuhn-Tucker conditions under some invexity and

(ρ, b)-quasiinvexity assumptions of the involved functionals. Treanţă [23] formulated

and proved weak, strong, and converse duality results for the considered class of

variational control problems by using the new notion of (ρ,ψ,d)-quasiinvexity

208

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

associated with an interval-valued multiple-integral functional. Treanţă [24]

investigated some connections between an LU-optimal solution of a variational

control problem governed by interval-valued multiple integral functional and a

saddle-point associated with an LU-Lagrange functional corresponding to a modified

interval-valued variational control problem.

In contrast to earlier studies, the current work addresses the duality study

related to a novel class of multiobjective optimization problems that involve interval-

valued ratio vector components. When taken into account simultaneously, these three

emphasized components are completely novel in the relevant literature. Additionally,

numerical example is given to show how useful the conclusions drawn in the study

are.

The following table compares our study with the available literature in this field

Invexity

and

Pseudoinvexity

Generalised

Invexity

Generalised

Approximate

Invexity

Inverval

Valued

Components

Mutliobjective

Optimization

Duality

Criteria

Research Article

Kumar et al. [1]

Yes

No

Bhardwaj et al. [3]

Yes

Upadhyay et al. [4]

Yes

No

Yes

Hachimi and

Aghezzaf [8]

No

Kim [11]

Gulati et al. [12]

Yes

No

Yes

Nahak and Nanda [13]

Yes

V-Invexity

No

Yes

Antczak and Jiménez

[14]

Yes

B-(p, r)-Invexity

No

Yes

Antczak [15]

Khazafi et al. [16]

Yes

No

Yes

No

Yes

No

Mititelu [19]

Yes

No

Treanţă and

Mititelu [21]

Yes

(ρ, b)-Q uasiinvexity

No

Yes

(ρ, ψ, d)-

Quasiinvexity

(p, b, d)-Invexity

Treanţă [23]

Treanţă [24]

Yes

No

Both (ρ,φ,d)-

Invexity and

(ρ, φ,d)-

Our Proposed Paper

Yes

Pseuodinvexity

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 209

In the field of multiobjective optimization, somewhere invexity or

pseudoinvexity were discussed, somewhere mutliobjective optimization with interval

valued components were discussed, while somewhere duality results were discussed.

To the best of our knowledge, all four components simultaneously with

(ρ,φ,d)-Invexity and (ρ, φ, d)-Pseuodinvexity were not discussed, so there was a

research gap in this field.

The structure of the paper is as follows: The problem formulation,

preliminary mathematical tools, and notations are included in Section 2 of this article.

The key findings are presented in Section 3 of this document. Results for Mond-Weir

weak, strong, and converse duality are developed and demonstrated for the recently

introduced category of multiobjective optimization problems. The paper is finally

concluded in Section 4.

2. The formulation of Problem and Notations

This part presents the definitions, notations, and preliminary findings that

will be utilized in the follow-up. Given this, we take into account:

Let us assume Ω be a compact domain which is a subset of Euclidean space

^mand a point in this compact domain Ω is represented by t  (t^) where

  1, 2.m

.

Now, following continuous differentiable functions are defined

X  (X_ⁱ) :   ⁿ ^k ^mnwhere i  1, 2..n and   1, 2.m

Y  (Y₁,Y₂..Y_q)  (Y_) :   ⁿ ^k ^qwhere   1, 2..q

It is assumed that the functions that are continuously differentiable

X_ (X_ⁱ) :   ⁿ ^k ^mnwhere i  1, 2..n and   1, 2.m

Satisfy the complete integrability conditions (closeness conditions)

D_X_ⁱ D_X_ⁱwhere   , ,  1, 2 m and i  1, 2 n

Where D_represent the total derivative operator.

210

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

If we consider any two vectors d  (d₁,d₂.....d_s) and e  (e₁,e₂.....e_s) in



^s, then following partial ordering is used

d  e  d_r e_r

,

d  e  d_r e_r

,

d  e  d_r e_r

,

d  e  d_r e_r,d_r e_r, r  1, 2.........s

Now let us assume that

K

is the set of all closed and bounded real intervals,

we represent a closed and bounded interval by F  [f ^L, f^U], where f ^Land f^U

are the lower and upper bounds of

F

, respectively. The interval operations covered

in this paper can be carried out in the following ways:

(1)

(2)

F  G  f ^L g^Land f^U g^U

;

if f ^L f^U f then F  [f, f ]  f

;

(3)

(4)

F  G  [f^l g^L, f^U g^U]

;

F   [f ^L, f^U]  [ f ^L,  f^U];

(5)

(6)

(7)

(8)

(9)

For any h  R, h  F  {h  f ^L, h  f^U]

;

For any h  R and h  0, hF  [hf ^L, hf^U]

;

For any h  R and h  0, hF  [hf^U, hf ^L]

;

F  G  [f ^L g^L, f^U g^U]

;

F /G  [f ^L/g^L, f^U/g^U], where g^L, g^U 0

.

Now we have some following definitions

Definition 1: If

F

and

G

are two closed and bounded real intervals, i.e.

F,G  K , then we have

F  G  f ^L g^Uand f^U g^U

Definition 2: If

F

and

G

are two closed and bounded real intervals, i.e.

F,G  K , then we have

F  G  f ^L g^Uand f^U g^U

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 211

Definition 3: Interval valued Functions

If we define

a

function

f

from

  ⁿ ^k

to

K

,

i.e.

f    ⁿ ^k K such that

f(t,b(t),c(t))  [f ^L(t,b(t),c(t)), f^U(t,b(t),c(t))], where t  

Where both f ^L(t,b(t),c(t)) and f^U(t,b(t),c(t)) are real valued functions

and the condition f ^L(t,b(t),c(t))  f^U(t,b(t),c(t))t   is satisfied , then

f

is

said to be an interval valued function.

The following (per Mititelu and Treantă [19], and Treantă [21]) was used to

formulate and demonstrate the primary findings of this work, now we are going to

introduce (ρ, φ, d)-Invexity and (ρ, φ, d)-Pseuodinvexity with the help of functional

which is interval valued multiple integral.

For this first we consider an interval-valued function which is continuously

differentiable

h :   Rⁿ R^mn R^k K such that

h  h(t,b(t),b_(t),c(t))  [h^L(t,b(t),b_(t),c(t)), h^U(t,b(t),b_(t),c(t))]

Where b_(t) represents partial derivative of b(t) with respect to t^i.e.

b

t^

b_(t) 

(t).

Now for any b  B and c  C , we define following interval-valued

multiple integral functional:

H : B  C  K such that

_^.

H(b,c) 

h(t,b(t),b_(t),c(t))dt

.

_

_^.

 [

h^L(t,b(t),b_(t),c(t))dt,

h^U(t,b(t),b_(t),c(t))dt]

If



is a real number and  : B  C  B  C  [0, ) be a positive

functional and (d(b,c), (b⁰,c⁰)) is a real valued function defined on(B  C)²

.

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R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

Definition 4: (ρ, φ,d)-Invexity and (ρ, φ,d)-Pseudoinvexity

(i) Now if there exists a functional such that

 :   ⁿ ^k ⁿ ^k ⁿsuch that

  (t,b(t),c(t),b⁰(t),c⁰(t))  (_i(t,b(t),c(t),b⁰(t),c⁰(t))) where i  1, 2..n,

of the C¹class functional with (t,b(t),c(t),b⁰(t),c⁰(t))  0, t  ,  _ 0,

and another functional such that

 :   ⁿ ^k ⁿ ^k ^Ksuch that

  (t,b(t),c(t),b⁰(t),c⁰(t))  (_j(t,b(t),c(t),b⁰(t),c⁰(t))) where j  1, 2..k,

of the C ⁰class function with (t,b(t),c(t),b⁰(t),c⁰(t))  0, t  ,  _ 0

such that for each (b,c)  B  C

:

H(b,c)  H(b⁰,c⁰)

_^.

 (b,c,b⁰,c⁰) [h ^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U(t,b⁰(t),b_⁰(t),c⁰(t))]dt

b

_^.

(b,c,b⁰,c⁰) [h ^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U(t,b⁰(t),b_⁰(t),c⁰(t))]D_dt

b



_^.

(b,c,b⁰,c⁰) [h_c^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U_c(t,b⁰(t),b_⁰(t),c⁰(t))]dt

 (b,c,b⁰,c⁰)d²((b,c),(b⁰,c⁰))  0

Or in other words

_^.

(b,c,b⁰,c⁰) [h ^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U(t,b⁰(t),b_⁰(t),c⁰(t))]dt

b

_^.

(b,c,b⁰,c⁰) [h ^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U(t,b⁰(t),b_⁰(t),c⁰(t))]D_dt

b



_^.

(b,c,b⁰,c⁰) [h_c^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U_c(t,b⁰(t),b_⁰(t),c⁰(t))]dt

 (b,c,b⁰,c⁰)d²((b,c), (b⁰,c⁰))  0  H(b,c)  H(b⁰,c⁰)

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 213

In this case

H

is called as (ρ, φ, d)-Invex at point (b⁰,c⁰)B  C with

respect to



and



.

(ii) Now if there exists a functional such that

 :   ⁿ ^k ⁿ ^k ⁿsuch that

  (t,b(t),c(t),b⁰(t),c⁰(t))  (_i(t,b(t),c(t),b⁰(t),c⁰(t))) where i  1, 2..n,

of the C¹class functional with(t,b(t),c(t),b⁰(t),c⁰(t))  0, t  ,   _ 0 ,

and another functional such that

 :   ⁿ ^k ⁿ ^k ^Ksuch that

  (t,b(t),c(t),b⁰(t),c⁰(t))  (_j(t,b(t),c(t),b⁰(t),c⁰(t))) where j  1, 2..k,

of the C ⁰class function with (t,b(t),c(t),b⁰(t),c⁰(t))  0, t  ,  _ 0

such that for each (b,c)  (b⁰,c⁰)  B  C

:

H(b,c)  H(b⁰,c⁰)

_^.

 (b,c,b⁰,c⁰) [h ^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U(t,b⁰(t),b_⁰(t),c⁰(t))]dt

b

_^.

(b,c,b⁰,c⁰) [h ^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U(t,b⁰(t),b_⁰(t),c⁰(t))]D_dt

b



_^.

(b,c,b⁰,c⁰) [h_c^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U_c(t,b⁰(t),b_⁰(t),c⁰(t))]dt

 (b,c,b⁰,c⁰)d²((b,c),(b⁰,c⁰))  0

Or in other words

_^.

(b,c,b⁰,c⁰) [h ^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U(t,b⁰(t),b_⁰(t),c⁰(t))]dt

b

_^.

(b,c,b⁰,c⁰) [h ^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U(t,b⁰(t),b_⁰(t),c⁰(t))]D_dt

b



_^.

(b,c,b⁰,c⁰) [h_c^L(t,b⁰(t),b_⁰(t),c⁰(t)), h^U_c(t,b⁰(t),b_⁰(t),c⁰(t))]dt

 (b,c,b⁰,c⁰)d²((b,c), (b⁰,c⁰))  0  H(b,c)  H(b⁰,c⁰)

214

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

In this case

H

is called as (ρ, φ, d)-pseudoinvex at point (b⁰,c⁰)B  C

with respect to



and



.

Definition 5: Now if we consider a vector valued continuously differentiable

function such that

h

h :   ⁿ ^mn ^k  ^psuch that

h  h(t,b(t),b_(t),c(t)) where r  1, 2 p

 ([h ^L(t,b(t),b_(t),c(t)), h^U(t,b(t),b_(t),c(t))]

1

................[h ^L_p(t,b(t),b_(t),c(t)), h^U_p(t,b(t),b_(t),c(t))])

Now we define vector multiple integral functional

continuously differentiable function

H

with the help of above

H : B  C  K ^psuch that

_^.

H(b,c) 

h(t,b(t),b_(t),c(t))dt

.

^{ }_



_^.





_h^L(t,b(t),b_(t),c(t))dt,

h^U(t,b(t),b_(t),c(t))dt ,

1



 

^_^.

_^.

 

 

_h_p^L(t,b(t),b_(t),c(t))dt,

h^U_p(t,b(t),b_(t),c(t))dt







Now this vector valued multiple integral functional

H

is said to be (ρ, φ,d)-

Invex or (ρ, φ,d)-Pseudoinvex at point (b⁰,c⁰)B  C with respect to



and



if

each of the interval valued component of the vector is (ρ, φ,d)-Invex or (ρ, φ, d)-

Pseudoinvex respectively at point (b⁰,c⁰)B  C with respect to



and



.

Now consider a vector valued continuous differentiable function g such that

g  (g₁, g₂.......g_p)

where g_r:   Rⁿ R^k K ^p

,

r  1, 2........p

We may now design a new class of multiobjective variational control

problems with interval-valued components that we refer to as Primal Problems

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 215

(abbreviated PP for short)

.











min_(b,c)G(b,c) 

g₁(t,b(t),c(t)dt,........

g (t,b(t),c(t)dt

p



__

_



subject to

i

b

(t)  X_ⁱ(t,b(t),c(t), i  1, 2.....n and   1, 2........m and t  

(1)



t

Y(t,b(t),c(t))  0, t  

(2)

(3)

b(t)_ (t)  given

Now for r  1, 2........p we have

.

_^.

_

g_r(t,b(t),c(t)dt  [

g_r^L(t,b(t),c(t)dt, g^U_r(t,b(t),c(t)dt]

or

G_r(b,c)  [G_r^L(b,c),G^U_r(b,c)]

The set of all feasible solutions in primal problem is defined by

D  {(b,c) b  B and c  C} satisfying equations (1), (2) and (3).

Definition 6: A feasible solution (b⁰,c⁰)  D in primal problem is said to

be an LU-optimal solution if there does not exist any(b,c)  D

such that

G(b,c)  G(b⁰,c⁰)

.

Constrained by certain qualification assumptions, if (b⁰,c⁰)  D is an LU-

Optimal solution of the variational control, then Treantă [21] and Mititelu and

Treantă [19] can be considered. According to this there exists piecewise

smooth functions ,  and



, with (t)  (^L(t),^U(t)), (t)  (^(t)) and

(t)  ^_i(t) such that

g_r^l

bⁱ

X_ⁱ

bⁱ

 _l^r

(t,b⁰(t),c⁰(t))  ^_i(t)

(t,b⁰(t),c⁰(t))

216

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

_i^

t^

Y_

bⁱ

 ^(t)

(t,b⁰(t),c⁰(t)) 

(t)  0.

(4)

Where i  1, 2.........n and l  L,U

.

g_r^l

c^j

X_ⁱ

c^j

 _l^r

(t,b⁰(t),c⁰(t))  ^_i(t)

(t,b⁰(t),c⁰(t))

Y_

c^j

 ^(t)

(t,b⁰(t),c⁰(t))  0.

(5)

(6)

Where j  1, 2.........k and l  L,U

.

And ^(t)Y_(t,b⁰(t),c⁰(t))  0 (no summation) (t), (t)  0

for all t   except at the point of discontinuities.

Definition 7: For the primal problem an LU-Optimal solution (b⁰,c⁰)  D

is called an normal LU-optimal solution if above necessary LU-optimality conditions

in equation (4) to (6) are satisfied.

3. Dual problem associated with Primal problem

Suppose that the set P  {1, 2......q} is partitioned into the set

{P , P , ........P }, where s  q . Using the same notations as in Section 2, we relate

1

2

s

the next multiobjective variational control problem with interval-valued vector

components, known as the Dual Problem (DP), to the above primal problem for

(a, u)  B  C

:

.











min_(a,u)G(a, u) 

g₁(t,a(t), u(t)dt, ........ g (t,a(t), u(t)dt

p



__

_



subject to

g_r^l

aⁱ

X_ⁱ

aⁱ

 _l^r

(t,a(t), u(t))  ^_i(t)

(t,a(t), u(t))

_i^

t^

Y_

aⁱ

 ^(t)

(t,a(t), u(t)) 

(t)  0.

(7)

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 217

Where i  1, 2.........n and l  L,U

g_r^l

u^j

X_ⁱ

u^j

 _l^r

(t,a(t), u(t))  ^_i(t)

(t,a(t), u(t))

Y_

u^j

 ^(t)

(t,a(t), u(t))  0.

(8)

Where j  1, 2.........k and l  L,U

.

bⁱ

t^





^(t) X_ⁱ(t,a(t), u(t)0 

(t)  0

.

(9)





i





P



And  (t)Y_P(t,a(t), u(t)0  0 where   1, 2........s

(10)



Where

l  L,U

  ( _l^r)  0, (t)  (^(t))  0 ,

a(t) _ (t)  given

(11)

.

P



And the expression is  (t)Y_P(t,a(t), u(t)) is



P

^(t)Y_(t,a(t), u(t))



 (t)Y_P(t,a(t), u(t)) 

_P



In this section, we prove that, under (ρ, φ, d)-Invexity hypotheses, the

multiobjective optimization problems with interval-valued components, Primal

Problem and Dual Problem, are a Mond-Weir dual pair. Moreover, keep in mind that



is the collection of all feasible solutions related to dual problem.

Now we formulate and establish the initial duality result, which is also

known as weak duality.

Weak Duality theorem-For any multiobjective variational problem with

interval-valued components (Primal Problem), let (b,c)  D be a feasible solution;

similarly, let (a, u,, , )   be a feasible solution for the multiobjective

variational problem with interval-valued components (Dual Problem). Furthermore,

keep in mind that the following prerequisites are satisfied:

218

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

(i)

For each

r

, the functional

_^.

a,u

g

(b,c) 

g_r(t,b(t),c(t)dt

r  1, 2......p

and l  L,U

is

r,l

(ꢀ¹, φ, d)-Invex at (a, u) with regard to



and



or in other words, each

interval-valued multiple-integral functional

r  1, 2......p is (ꢀ¹, φ, d)-Invex at

a,u

r,L

a,u

r,U

g

^a,u(b,c)  [g

(b,c),g

(b,c)]

,

r

(a, u) with regard to



and



.

bⁱ

t^





_^.

(ii)

(iii)

The functional X(b,c) 

^(t) X_ⁱ(t,b(t),c(t), 

(t) dt

is





i





(ꢀ²,φ, d)-Invex at (a, u) with regard to



and



.

Each functional

_^.

Q

  1, 2.......s is (_³,,d)

-



Y_(b,c) 

 (t)Y_Q_(t,b(t),c(t)dt

Invex at (a, u) with respect to



and



.

(iv)

(v)

With regard to ꢁ ꢂꢃꢄ ꢅ , at least one of the functionals provided in (i) to

(iii) is (ꢀ, φ, d)-Pseudoinvex at (a, u), where   _r¹, ²and

_³

.

For the given

_^s_1

 ^r_r¹ ²

_³

 0

where ¹_r, ²and  ³ 

.

l



Then, supremum of dual problem is less than or equal to the infimum of

primal problem.

Proof: The values of primal problem at (b,c)  D and dual problem at

(a, u,, , )   are denoted by (b,c) and (a, u,, , ) respectively. Contrast

to the result, if possible, suppose that (b,c)  (a, u,, , )

.

Now, take into consideration the following non-empty set for

r  1, 2......p,l  L,U and   1, 2,...........s:

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 219

S  {(b,c)  B  C g ^a,u(b,c)  g ^b,c(a,u), X(b,c)  X(a,u),Y_(b,c)  Y_(a,u)}

r,l

Now by using above (i) for (b,c)  S and r  1, 2 p and l  L,U

,

we have

a,u

r,l

b,c

g

(b,c)  g (a, u) 

r,l

_^.

(b,c,a, u) (g_r^l)_a(t,a(t), u(t)dt  (b,c,a, u) (g_r^l)_u(t,a(t), u(t)dt

 ¹_r(b,c,a, u)d²((b,c)(a, u))

Now we multiply this by   _l^r 0 where l  L,U and take summation

over r  1, 2 p , we get the following

.

_^.

(b,c,a, u)

^r(g_r^l)_a(t,a(t), u(t)dt  (b,c,a, u)

^r(g_r^l)_u(t,a(t), u(t)dt

l

_

 ^r¹_r(b,c,a, u)d²((b,c)(a, u)).

(12)

l

Now since for each (b,c)  S , the inequality X(b,c)  X(a, u) satisfies,

now according to (ii), we have the following

_^.

(b,c,a, u) [^_i(t)(X_ⁱ)_a(t,a(t),u(t))  ^(t)D_  ^_i(t)(X_ⁱ)_u(t,a(t), u(t))]dt

  ²(b,c,a, u)d²((b,c)(a, u)).

(13)

Similarly, for each (b,c)  S , the inequality Y_(b,c)  Y_(a, u) for

  1, 2s exists, now using (iii) , we have the following

.

Q



(b,c,a, u) [ (t)(Y_Q)_a(t,a(t),u(t))   (t)(Y_Q)_u(t,a(t),u(t))]dt

_



  _³(b,c,a, u)d²((b,c)(a, u)).

Now, taking the summation over   1, 2s , we have

220

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

_^.

(b,c,a, u) [^(t)(Y_)_a(t,a(t),u(t))  ^(t)(Y_)_u(t,a(t),u(t))]dt

s

 

³(b,c,a, u)d²((b,c)(a, u))

(14)





 1

Now, adding equations (12),(13) and (14) and taking condition (iv) under

consideration, we have

.

(b,c,a, u)

^r(g_r^l)_a(t,a(t), u(t))dt

l

_

_^.

(b,c,a, u) [^_i(t)(X_ⁱ)_a(t,a(t), u(t))  ^(t)(Y_)_a(t,a(t), u(t))]dt

.

(b,c,a, u)

^r(g_r^l)_u(t,a(t), u(t))dt

l

_

_^.

(b,c,a, u) [^_i(t)(X_ⁱ)_u(t,a(t), u(t))  ^(t)(Y_)_u(t,a(t), u(t))]dt

s





_^.

(b,c,a,u) [^(t)D_]dt    ^r¹_r ²

³ (b,c,a,u)d²((b,c),(a,u))



l







 1





Where l  L,U

.

Since, (b,c,a, u)  0, using this, we have the following

_^.

^r(g_r^l)_q(t,a(t), u(t))dt

l

.

 [^_i(t)(X_ⁱ)_a(t,a(t), u(t))  ^(t)(Y_)_a(t,a(t),u(t))]dt





.

^r(g_r^l)_u(t,a(t), u(t))dt

^_

l

.

 [^_i(t)(X_ⁱ)_u(t,a(t), u(t))  ^(t)(Y_)_u(t,a(t),u(t))]dt





s





.

 [^(t)D_]dt    ^r¹_r ²

³ (b,c,a,u)d²((b,c),(a,u))

.



l











 1





Where ꢆ = ꢇ, ꢈ.

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 221

Now using the constraints (7) and (8) of dual problem, we have

.

_^.

 [D_^(t)dt  [^(t)D_]dt  0





s





    ^r¹_r ²

³ (b,c,a, u)d²((b,c),(a, u))



l







 1





Where ꢆ = ꢇ, ꢈ.

By direct formula of derivative, we know that

D_[^(t)]  ^(t)D_  D_^(t)

D_^(t)  D_[^(t)]  ^(t)D_

Now applying integral over the region Ω , we have

.

_^.



D ^(t)dt 

D_[^(t)]dt 

[^(t)D_]dt





Using the condition   _ 0 and applying the flow-divergence formula,

we get

_^.

_^._



D_[^(t)dt 

[^(t)nd  0



Where n  (n)_where   1, 2 m , is the unit normal vector to the

hyper surface  , now it follows that

_^.

D_^(t)dt 

[^(t)D_]dt

or

.

D_^(t)dt 

[^(t)D_]dt  0

.

^_





Therefore, we have

s





0     ^r¹_r ²

³ (b,c,a, u)d²((b,c), (a, u))

.



l







 1





222

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

Where ꢆ = ꢇ, ꢈ.

Now applying the condition (v) and d²((b,c),(a, u))  0, we get a

contradiction. Therefore, supremum of dual problem is less than or equal to the

infimum of primal problem.

The following outcome proves a strong duality between the two

multiobjective optimization problems with interval-valued components under

consideration.

Strong Duality theorem-If we consider the same (ρ, φ, d)-Invexity

hypotheses mentioned in above weak duality theorem, if (b⁰,c⁰)  D is a normal

LU-optimal solution of the given primal problem, then  ⁰, ⁰(t) and ⁰(t) such

that (b⁰,c⁰,⁰, ⁰, ⁰)   is an LU-optimal solution of the dual problem, and the

values of corresponding objective functions are equal.

Proof: Consider that (b⁰,c⁰)  D is a normal LU-optimal solution of the

primal problem, the necessary LU-optimality conditions mentioned in equations (4)

to (6) involve that  ⁰, ⁰(t) and ⁰(t) such that (b⁰,c⁰,⁰, ⁰, ⁰)   is an

feasible solution for dual problem.

b⁰ⁱ

t^

(t)  X_ⁱ(t,b⁰(t),c⁰(t)) for i  1, 2,......n

,

  1, 2......m t  

Now by equation (6)

^(t)Y_(t,b⁰,(t),c⁰(t)0  0 , (summation is taken over



)

and t  

.

Therefore, the value of objective function of dual problem has the same

value of objective function of primal problem. Hence by weak duality theorem

(b⁰,c⁰,⁰, ⁰, ⁰)   is an LU-optimal solution of dual problem.

A converse duality conclusion related to considered multiobjective

optimization problems with interval-valued components is formulated in the

following theorem.

CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 223

Converse Duality theorem-Assume that the LU-optimal solution of dual

problem is (b⁰,c⁰,⁰, ⁰, ⁰)   . Furthermore, presumptively the following

circumstances hold true:

(i)

(b, c)  D is a normal LU-optimal solution of the given primal problem.

(ii)

For (b⁰,c⁰,⁰, ⁰, ⁰), the hypotheses of weak duality theorem are met.

Consequently, the corresponding objective values are equal and

(b, c )  (b⁰,c⁰)

.

Proof: In contrast to the outcome, let's assume that (b, c)  (b⁰,c⁰)and

that (b⁰,c⁰) is not a normal LU-optimal solution of primal problem. According to

Treantă and Mititelu and Treantă, since (b, c)  D is a normal LU-optimal solution

of primal problem, there exist  , u(t) and (t), satisfying equations (4) to (6) and

definition of normal LU-optimal solution. Consequently

i



b



 ^_i(t) X_ⁱ(t,b (t), c(t) 

(t)  0,











t

Q



 (t)Y_Q_(t,b (t), c(t)  0

,

  1, 2.......s

where (b, c, , , )   as a result. Additionally, (b, c)  (b, c, , , )  

is present. We obtain

(b, c)  (b⁰,c⁰,⁰, ⁰, ⁰)

in accordance with weak

duality theorem, or

(b,c, , , )  (b⁰,c⁰,⁰, ⁰, ⁰). The maximal LU-

optimality of (b⁰,c⁰,⁰, ⁰, ⁰) is in conflict with this. As a result, the

corresponding objective values are identical and (b, c )  (b⁰,c⁰)

.

Illustrative instance: The following two-dimensional interval-valued

variational control problem is taken into consideration:

.

min_(b,c)

g(t,b(t),c(t),dt

__(0.3)

^_^.



_^._(0.3)



(c²(t)  8c(t)  16)dt¹dt²,

(c²(t)dt¹dt²

,







(0.3)



224

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

Subject to

b

t¹

b

t²

(t) 

(t)  3  c(t)

where t  (t¹,t²)  _(0.3)

81  b²(t)  0

where t  (t¹,t²)  _(0.3)

b(0)  b(0, 0)  6

,

b(3)  b(3, 3)  8

where t₀ (t₀¹, t₀²)  (0, 0) and t₁ (t¹, t²)  33 in ²are the diagonally

1

 8 8

opposed points that fix the square b : 

(0.3)

 ., c : 

(0.3)

  ,

and





3 3





_(t₀_1,t₀₂₎ _(0.3)

.

Furthermore, we consider that in the examined variational control problem in

which affine state functions are the only ones that interest us. It is possible to

demonstrate by direct computation that the feasible point

1

3

8

3

b⁰(t) 

(t¹ t²)  6

,

c⁰(t) 

,

t  (t¹,t²)  _(0.3)

5

is a normal LU-optimal solution with   (¹, ²)  (1, ),  (^L,^U)  (1, 1)

3

and

  0 for the optimization problem under consideration. Moreover, the

(ꢀ, 1, 0)-invexity (with    ) of the functionals involved (refer to weak duality

theorem) at (b⁰,c⁰) with regard to



and



may be easily verified as follows: Given

by , : _(0.3) (  )² 

0



b(t)  b (t),

t  int(_(0.3)

)



(t,b(t),c(t),b⁰(t),c⁰(t) 



0,

t  _(0.3)





0





c(t)  c (t),

t  int(_(0.3))

(t,b(t),c(t),b⁰(t),c⁰(t) 



0,

t  _(0.3)





CRITERIA INVOLVING (p, φ ,d)-INVEXITY AND (p, φ, d)-PSEUODINVEXITY 225

where int(_(0.3)

)

and (_(0.3)

)

represent interior region and boundary of _(0.3)

respectively.

 1

8

5



Therefore, by strong duality theorem

(t¹ t²)  6, , (1, 1), (1, ), 0







3



will be an LU-optimal solution for the dual problem mentioned below

.

max_(a,u)

g(t,a(t), u(t),dt

__(0.3)

^_^.



_^._(0.3)



(u²(t)  8u(t)  16)dt¹dt²,

u²(t)dt¹dt²







(0.3)



subject to

¹

t¹

²

t²

2(t)a(t) 

(t) 

(t)  0

where t  (t¹,t²)  _(0.3)

2^Lu(t)  8^L 2^Uu(t)  ¹(t)  ²(t)  0

where t  (t¹,t²)  _(0.3)



a

t¹





b

t²



¹(t) 3  u(t) 

(t)  ²(t) 3  u(t) 

(t)  0

















where t  (t¹,t²)  _(0.3)

(t)(81  a²(t))  0

,

where t  (t¹,t²)  _(0.3)

  ^L,^U [0, 0]

,

(t)  0

,

a(0)  a(0, 0)  6

,

b(3)  b(3, 3)  8

.

and the values of objective of both primal and dual problem are equal.

4. Conclusions

In this paper we have formulated and proved Mond-Weir weak, strong, and

converse duality theorems for a completely new concept of multiobjective

optimization problems having interval-valued components, based on the completely

new notion of (ρ, φ, d)-Invexity and (ρ, φ, d)-Pseudoinvexity related with an interval-

valued multiple-integral functional. Considering the relevance of interval analysis

and duality theory to optimization and control, this work constitutes a significant

contribution for applied sciences researchers and engineers.

226

Future Scope

This paper can be extended from numerous points of view for additional

R. B. SONI, DHARAMENDER SINGH AND K. C. SHARMA

exploration. In this paper we have studied for one parameter t, which can be

generalised for two or three parameters. On the other hand, here, we have studied

Both (ρ, φ, d)-Invexity and (ρ, φ, d)-Pseuodinvexity for multiobjective optimization,

which can also be studied for fractional programming or Inverse optimization. So,

this study has great future scope.

Funding

There was no external funding for this study from any agency.

Conflict of Interest

All authors declare that they have no conflicts of interest.

Acknowledgement

All authors would like to express their gratitude to the concerned research

centre Department of Mathematics, Maharani Shree Jaya Govt. P.G. College and

Maharaja Surajmal Brij University, Bharatpur for their invaluable support in

completing this research work.

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1. Department of Mathematics,

Govt. Birla P.G. College,

(Received, September 10, 2024)

Bhawani Mandi, Jhalawar, Rajasthan

(Affiliated by University of Kota, Kota, Rajasthan), India

1. E-mail: rbsoni68@gmail.com,

Orcid: https://orcid.org/0000-0002-1718-2512)

2. Department of Mathematics,

Maharani Shri Jaya Govt. College, Bharatpur, Rajasthan

(Affiliated by Maharaja Surajmal Brij University, Bharatpur, Rajasthan), India

2. E-mail: dharamender.singh6@gmail.com,

Orcid: https://orcid.org/0000-0001-5601-7790

3. E-mail: ks.pragya@gmail.com ,

Orcid: https://orcid.org/0009-0008-6242-3873)