EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6921
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A New Approach in Solving a Class of Control
Problems with Fractional Objectives

Tareq Saeed1, Savin Treanţă2,3,4,∗

1 Financial Mathematics and Actuarial Science (FMAS), Department of Mathematics,
Faculty of Sciences, King Abdulaziz University, 21589 Jeddah, Saudi Arabia
2 Department of Applied Mathematics, National University of Science and Technology
Politehnica Bucharest, 060042 Bucharest, Romania
3 Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, Romania
4 Fundamental Sciences Applied in Engineering - Research Center, National University of
Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania

Abstract. In this paper, we introduce and investigate a pair of symmetric multi-dimensional
variational fractional control problems. To this end, first, we formulate an updated concept of
pseudoinvexity associated with multiple integral type functionals. Further, we establish a very
important connection between the objective functionals of the studied symmetric models.
2020 Mathematics Subject Classifications: 49K20, 49K35, 49N15
Key Words and Phrases: Fractional control problems, properly efficient solution, pseudoinvex-
ity

1. Introduction

Optimization theory has recently undergone significant development. Many researchers
have managed to formulate optimality criteria and conditions that have advanced the cur-
rent level of knowledge. In this regard, Abdulaleem and Treanţă [1], under the framework
of E‑differentiability, established sufficient optimality conditions -notably KKT conditions
for vector optimization problems where the objective and constraint functions satisfy (gen-
eralized) V‑E‑type I properties. Ahmad [2] extended symmetric duality theory –a form
of duality where primal and dual problems have symmetric structures– to the domain of
multiobjective fractional variational problems. Specifically, he formulated various duality
results under assumptions of generalized invexity. Additionally, he explored the intrinsic
relationships between these variational problems and their corresponding multiobjective
fractional symmetric dual problems. Antczak et al. [3] studied a type of optimal control

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6921

Email addresses: tsalmalki@kau.edu.sa (T. Saeed),
savin.treanta@upb.ro (S. Treanţă)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 2 of 14

framework where the objective comprises multiple fractional terms that are not necessar-
ily convex or differentiable. Bagri et al. [4] tackled a multi-dimensional vector fractional
variational control problem while explicitly incorporating data uncertainty into the model.
The study’s core lies in formulating a Wolfe-type dual problem for the uncertain primal
control problem and establishing robust duality results, particularly under convexity as-
sumptions on the involved functionals. Bector and Husain [5] extended classical duality
theory into the realm of multiobjective variational problems, framing variational calculus
techniques within a multiobjective optimization context. Chandra et al. [6] introduced a
pair of symmetric dual fractional programming problems, extending classical duality the-
ory into the realm of fractional programming. The authors established appropriate duality
theorems for these problems, contributing to the development of duality theory in opti-
mization. In his works, Chen [7, 8] stated a pair of multiple-objective variational mixed
integer models over cones and establishes duality theorems under separability and partial-
invexity assumptions. The study provided weak, strong, converse, and self-duality results
related to efficient solutions. Craven [9] explored the relationship between Lagrange mul-
tipliers and quasiduality in optimization problems. He provided conditions under which
Lagrange multipliers exist and satisfy certain optimality conditions, contributing to the
understanding of quasiduality in optimization theory. Dantzig et al. [10] introduced a sym-
metric duality framework for nonlinear programming problems, extending classical duality
concepts to a broader class of problems. Das et al. [11] extended classical duality theo-
ries to set-valued fractional minimax programming problems by employing second-order
contingent epi-derivatives. Dorn [12] addressed duality theory specifically for quadratic
programming problems, which involve optimizing a quadratic objective function subject
to linear constraints. Gulati et al. [13] extended the concept of symmetric duality to
minimax variational problems, providing a dynamic generalization of symmetric and self-
duality theorems in nonlinear mixed integer programming. Guo et al. [14, 15] introduced
symmetric gH-derivative and formulated a duality theory in interval optimization. Jayswal
et al. [16] delved into the duality theory of multi-dimensional variational control problems
under data uncertainty. The authors formulated robust dual models, including Wolfe-type
and Mond-Weir-type dual problems, and established duality results under generalized con-
vexity assumptions. The study provided a comprehensive analysis of robust duality in the
context of variational control problems with data uncertainty. Kim and Lee [17] extended
the classical duality theory to multiobjective variational problems by introducing a sym-
metric duality framework under invexity conditions. The authors formulated dual pairs
for vector problems and proved duality results under generalized invexity assumptions.
These results provided a unified approach to analyzing multiobjective variational prob-
lems and contributed to the development of duality theory in optimization. Mond and
Hanson [18] derived duality theorems under certain convexity and concavity conditions,
providing necessary and sufficient optimality conditions for the primal and dual problems.
Recently, Prasad et al. [19] formulated optimality conditions for an interval-valued vector
problem. Saeed and Treanţă [20] employed the concept of convex multiple integral func-
tionals, and the notion of robust weak efficient solutions to develop a new mathematical
context for stating and proving the duality theorems. Schaible [21–23] presented a unified



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 3 of 14

method for obtaining duality results for concave-convex fractional programs by trans-
forming the original nonconvex programming problem into an equivalent convex program.
The papers related known results by several authors and proved additional duality theo-
rems, including converse duality theorems for nondifferentiable and quadratic fractional
programs. Shi et al. [24] investigated a Lagrange-type dual theory associated with a
fuzzy optimization model involving mixed constraints. Smart and Mond [25] investigated
multiobjective variational problems -optimization problems defined over functions- where
multiple objective functionals are considered simultaneously. This multiobjective nature
required optimization under a partial ordering (e.g., Pareto optimality). Sun et al. [26] in-
troduced a mixed-type robust dual problem for optimization models subject to uncertainty
in both objective functions and constraints. This framework generalized classical duality
to the robust setting. Treanţă et al. [27, 28] introduced a new class of constrained robust
nonlinear optimization problems characterized by: (1) objective functionals defined via
path-independent curvilinear integrals, derived from controlled second-order Lagrangians
with uncertain data, (2) mixed constraints involving second-order partial derivatives and
embedded uncertainty. The paper’s core lied in formulating and analyzing three types of
robust dual optimization models -Wolfe-type, Mond–Weir-type, and mixed-type- within
this novel robust optimization context. Upadhyay et al. [29–31] investigated semi-infinite
programming problems on Hadamard manifolds. Yang et al. [32] addressed the symmetric
duality framework for a class of nondifferentiable multiobjective fractional programming
problems (optimization problems where multiple fractional objectives are optimized with-
out assuming differentiability).

In this paper, the authors introduce and investigate a pair of symmetric multidimen-
sional variational fractional control problems. To this end, first, we formulate an updated
concept of pseudoinvexity associated with multiple integral type functionals. Further, we
establish a very important connection (see Theorem 1) between the objective functionals
of the studied symmetric models. The main contributions associated with this article
are: (a) the presence of two state and control variables in the considered functionals; (b)
employing the multiple integrals as objective functionals; (c) building of suitable sym-
metric models and constraints; (d) introduction of pseudoinvexity notion for controlled
functionals driven by multiple integrals.

The rest of the paper is structured as follows. Next section includes some prelim-
inary ingredients which are useful is establishing the main results. Section 3 contains
the main contributions of this paper. The symmetric multi-dimensional multiobjective
dual programs are introduced and studied. Last section formulates the conclusions of this
study.

2. Preliminaries

Let K = [x1, x2] = [x11, x
1
2] × · · · × [xp1, x

p
2] be a multi-dimensional interval, with x1 =

(x11, · · · , x
p
1), x2 = (x12, · · · , x

p
n) two arbitrary points in Rp. Consider the functions

χδ(x, λ(x), λγ(x), π(x), ω(x), ωξ(x), ζ(x)),



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 4 of 14

Υδ(x, λ(x), λγ(x), π(x), ω(x), ωξ(x), ζ(x)),

where λ : K → Rn, π : K → Rs, ω : K → Rm, ζ : K → Rl (see λγ := ∂λ
∂xγ , γ = 1, p and

ωξ :=
∂ω
∂xξ , ξ = 1, p, as the partial derivatives for λ and ω with respect to the multi-variable

x = (x1, · · · , xp) ∈ K ⊂ Rp), are functionals of C2-class, for δ ∈ {1, 2, . . . , q}. In the paper,
χδ
λ, χ

δ
λγ
, χδ

π, χ
δ
ω, χ

δ
ωξ

and χδ
ζ represent the gradients associated with the functional

χδ(x, λ(x), λγ(x), π(x), ω(x), ωξ(x), ζ(x))

with respect to λ, λγ , π, ω, ωξ and ζ, that is

χδ
λ =

(
∂χδ

∂λ1
, . . . ,

∂χδ

∂λn

)T

, χδ
λγ

=

(
∂χδ

∂λ1γ
, . . . ,

∂χδ

∂λnγ

)T

,

χδ
ω =

(
∂χδ

∂ω1
, . . . ,

∂χδ

∂ωm

)T

, χδ
ωξ

=

(
∂χδ

∂ω1
ξ

, . . . ,
∂χδ

∂ωm
ξ

)T

,

χδ
π =

(
∂χδ

∂π1
, . . . ,

∂χδ

∂πs

)T

, χδ
ζ =

(
∂χδ

∂ζ1
, . . . ,

∂χδ

∂ζ l

)T

,

for δ ∈ {1, 2, . . . , q}. Similarly, Υδ
λ,Υ

δ
λγ
,Υδ

π,Υ
δ
ω,Υ

δ
ωξ

and Υδ
ζ denote the gradient vectors

of Υδ(x, λ(x), λγ(x), π(x), ω(x), ωξ(x), ζ(x)) with respect to λ, λγ , π, ω, ωξ and ζ.
Let C1 (K,Rn) and C1 (K,Rm) denote the families of functions λ and ω (state vari-

ables, continuously differentiable functions), respectively, having the associated norms
∥λ∥ = ∥λ∥∞ + ∥λγ∥∞ and ∥ω∥ = ∥ω∥∞ + ∥ωξ∥∞, respectively. Also, let C0 (K,Rs) and
C0
(
K,Rl

)
denote the classes of functions π and ζ (control variables, continuous functions),

respectively, with the corresponding uniform norm, as well.
In accordance with Bector and Husain [5], we state the multi-cost variational opti-

mization problem:

(Problem) min
(λ,π)

(∫
K
F 1(x, λ(x), π(x))dw, . . . ,

∫
K
F q(x, λ(x), π(x))dw

)
subject to

λ(x1) = α = given, λ(x2) = β = given, (or λ|∂K = given)

h(x, λ(x), π(x)) ≦ 0, x ∈ K,

ψτ,γ(x, λ(x), π(x))− λτγ(x) = 0, x ∈ K,

where F δ : C1 (K,Rn) × C0 (K,Rs) → R, δ = 1, q, hι : C1 (K,Rn) × C0 (K,Rs) → R, ι =
1, r and ψτ,γ : C1 (K,Rn) × C0 (K,Rs) → R, τ = 1, n, γ = 1, p, are continuously differen-
tiable functionals, dw := dx1 · · · dxp.

Let G be the feasible solution set of (Problem),

G =
{
(λ, π) ∈ C1 (K,Rn)× C0 (K,Rs) | λ(x1) = α, λ(x2) = β, h(x, λ(x), π(x)) ≦ 0,



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 5 of 14

ψτ,γ(x, λ(x), π(x))− λτγ(x) = 0, x ∈ K
}
.

In accordance with Geoffrion [33], we establish the next useful definitions.

Definition 1. We say that (λ0, π0) ∈ G is called efficient solution for (Problem) if
the relation ∫

K
F
(
x, λ0(x), π0(x)

)
dw ≤

∫
K
F (x, λ(x), π(x))dw

holds, for all (λ, π) ∈ G.

Definition 2. The efficient solution (λ0, π0) ∈ G is called properly efficient solution
for (Problem) if (∃) k > 0 fulfilling∫

K
F δ
(
x, λ0(x), π0(x)

)
dw −

∫
K
F δ(x, λ(x), π(x))dw

≤ k

(∫
K
F i(x, λ(x), π(x))dw −

∫
K
F i
(
x, λ0(x), π0(x)

)
dw

)
for δ ∈ {1, 2, . . . , q} and some i, with∫

K
F i(x, λ(x), π(x))dw >

∫
K
F i
(
x, λ0(x), π0(x)

)
dw

for (λ, π) ∈ G, and∫
K
F δ(x, λ(x), π(x))dw <

∫
K
F δ
(
x, λ0(x), π0(x)

)
dw.

Definition 3. The pair (λ0, π0) ∈ G is called improperly efficient solution for
(Problem) if for every sufficiently large k > 0, there exist (λ, π) ∈ G and δ ∈ {1, 2, . . . , q}
fulfilling ∫

K
F δ(x, λ(x), π(x))dw <

∫
K
F δ
(
x, λ0(x), π0(x)

)
dw

and ∫
K
F δ
(
x, λ0(x), π0(x)

)
dw −

∫
K
F δ(x, λ(x), π(x))dw

> k

(∫
K
F i(x, λ(x), π(x))dw −

∫
K
F i
(
x, λ0(x), π0(x)

)
dw

)
for i ∈ {1, 2, . . . , q} verifying∫

K
F i(x, λ(x), π(x))dw >

∫
K
F i
(
x, λ0(x), π0(x)

)
dw.



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 6 of 14

Definition 4. The pair (λ0, π0) ∈ G is called weak efficient solution for (Problem) if
there is no (λ, π) ∈ G satisfying∫

K
F δ
(
x, λ0(x), π0(x)

)
dw >

∫
K
F δ(x, λ(x), π(x))dw, for all δ ∈ {1, 2, . . . , q}.

Remark 1. We can remark that if (λ0, π0) ∈ G is an efficient solution for (Problem),
then it is a weak efficient solution for (Problem).

Definition 5. We say that the multiple integral type functional

E(λ̄, ω) =

∫
K
χ(x, λ̄(x), λ̄γ(x), π̄(x), ω(x), ωξ(x), ζ(x))dw

is called pseudoinvex at λ, λγ and π if there exist z ∈ Rn, with

z(x, λ, λγ , π, λ, λγ , π) = z|x=x1,x=x2 = 0, (vanishes on every face of ∂K)

and µ ∈ Rs, with
µ(x, λ, λγ , π, λ, λγ , π) = 0,

such that, for each ω, ωξ and ζ, we have∫
K

[
zTχλ̄(x, λ, λγ , π, ω, ωξ, ζ) + µTχπ̄(x, λ, λγ , π, ω, ωξ, ζ)

+
dzT

dxγ
χλ̄γ

(x, λ, λγ , π, ω, ωξ, ζ)
]
dw ≥ 0

⇒
∫
K
χ(x, λ̄, λ̄γ , π̄, ω, ωξ, ζ)dw ≥

∫
K
χ(x, λ, λγ , π, ω, ωξ, ζ)dw,

for all λ̄, λ̄γ and π̄.

Definition 6. We say that the multiple integral type functional

−
∫
K
χ(x, λ(x), λγ(x), π(x), ω̄(x), ω̄ξ(x), ζ̄(x))dw

is called pseudoinvex at ω, ωξ and ζ if there exist η ∈ Rm, with

η(x, ω, ωξ, ζ, ω, ωξ, ζ) = η|x=x1,x=x2 = 0, (vanishes on every face of ∂K)

and ν ∈ Rl, with
ν(x, ω, ωξ, ζ, ω, ωξ, ζ) = 0,

such that, for each λ, λγ and π, we have∫
K

[
ηTχω̄(x, λ, λγ , π, ω, ωξ, ζ) + νTχζ̄(x, λ, λγ , π, ω, ωξ, ζ)



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 7 of 14

+
dηT

dxξ
χω̄ξ

(x, λ, λγ , π, ω, ωξ, ζ)
]
dw ≤ 0

⇒
∫
K
χ(x, λ, λγ , π, ω̄, ω̄ξ, ζ̄)dw ≤

∫
K
χ(x, λ, λγ , π, ω, ωξ, ζ)dw,

for all ω̄, ω̄ξ and ζ̄.
Remark 2. In the following, we briefly write z(x, λ, ω) for z(x, λ, λγ , π, ω, ωξ, ζ) and

η(x, λ, ω) for η(x, λ, λγ , π, ω, ωξ, ζ). In addition, we use (λ, ω) instead of (λ, π, ω, ζ).

3. Main result

In this section, according to Weir [34], we introduce the following symmetric multi-
dimensional multiobjective dual programs:

(P) min
(λ,ω)


∫
K
χ1(x, λ, λγ , π, ω, ωξ, ζ)dw∫

K
Υ1(x, λ, λγ , π, ω, ωξ, ζ)dw

, . . . ,

∫
K
χq(x, λ, λγ , π, ω, ωξ, ζ)dw∫

K
Υq(x, λ, λγ , π, ω, ωξ, ζ)dw


subject to

λ(x1) = 0 = λ(x2), ω(x1) = 0 = ω(x2),

λγ(x1) = 0 = λγ(x2), ωξ(x1) = 0 = ωξ(x2),

q∑
δ=1

Ωδ

{
Iδ(λ, ω)

(
χδ
ω − d

dxξ
χδ
ωξ

)
− Eδ(λ, ω)

(
Υδ

ω − d

dxξ
Υδ

ωξ

)}
≦ 0, x ∈ K,

q∑
δ=1

Ωδ
{
Iδ(λ, ω)

(
χδ
ζ − 0

)
− Eδ(λ, ω)

(
Υδ

ζ − 0
)}

≦ 0, x ∈ K,

ωT
q∑

δ=1

Ωδ

{
Iδ(λ, ω)

(
χδ
ω − d

dxξ
χδ
ωξ

)
− Eδ(λ, ω)

(
Υδ

ω − d

dxξ
Υδ

ωξ

)}
≥ 0, x ∈ K,

ζT
q∑

δ=1

Ωδ
{
Iδ(λ, ω)

(
χδ
ζ − 0

)
− Eδ(λ, ω)

(
Υδ

ζ − 0
)}

≥ 0, x ∈ K,

Ω > 0,

and

(D) max
(b,v)


∫
K
χ1(x, b, bγ , ρ, v, vξ, ϱ)dw∫

K
Υ1(x, b, bγ , ρ, v, vξ, ϱ)dw

, . . . ,

∫
K
χq(x, b, bγ , ρ, v, vξ, ϱ)dw∫

K
Υq(x, b, bγ , ρ, v, vξ, ϱ)dw


subject to

b(x1) = 0 = b(x2), v(x1) = 0 = v(x2),



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 8 of 14

bγ(x1) = 0 = bγ(x2), vξ(x1) = 0 = vξ(x2),

q∑
δ=1

Ωδ

{
Iδ(b, v)

(
χδ
λ − d

dxγ
χδ
λγ

)
− Eδ(b, v)

(
Υδ

λ − d

dxγ
Υδ

λγ

)}
≧ 0, x ∈ K,

q∑
δ=1

Ωδ
{
Iδ(b, v)

(
χδ
π − 0

)
− Eδ(b, v)

(
Υδ

π − 0
)}

≧ 0, x ∈ K,

bT
q∑

δ=1

Ωδ

{
Iδ(b, v)

(
χδ
λ − d

dxγ
χδ
λγ

)
− Eδ(b, v)

(
Υδ

λ − d

dxγ
Υδ

λγ

)}
≤ 0, x ∈ K,

ρT
q∑

δ=1

Ωδ
{
Iδ(b, v)

(
χδ
π − 0

)
− Eδ(b, v)

(
Υδ

π − 0
)}

≤ 0, x ∈ K,

Ω > 0,

where, for δ = 1, 2, . . . , q, χδ : K × R2n × Rs × R2m × Rl → R+, and Υδ : K × R2n × Rs ×
R2m × Rl → R+\{0} are functionals of C2-class and

Eδ(λ, ω) =

∫
K
χδ(x, λ, λγ , π, ω, ωξ, ζ)dw, I

δ(λ, ω) =

∫
K
Υδ(x, λ, λγ , π, ω, ωξ, ζ)dw.

Remark 3. If we consider Iδ(λ, ω) = 1, δ = 1, 2, . . . , q, and remove control functions,
the considered symmetric programs (P) and (D) are converted into the optimization mod-
els investigated by Gulati et al. [35]. Also, for q = 1 and removing the control variables,
the symmetric problems (P) and (D) are transformed in the variational problems con-
sidered by Gulati et al. [36]. If K is a classical real interval, then the present study is
investigated by Treanţă et al. [37]. In addition, let us note that we do not impose (see
Kim et al. [38]) the constraint ΩT e = 1, e = (1, 1, · · · , 1) ∈ Rq, in (P) and (D) since it
does not matter in establishing the main result (see Theorem 1).

Further, we reformulate the considered symmetric programs (P) and (D) as follows:

(P’) min
(λ,ω)

(
Y 1, Y 2, . . . , Y q

)
subject to

λ(x1) = 0 = λ(x2), ω(x1) = 0 = ω(x2), (1)

λγ(x1) = 0 = λγ(x2), ωξ(x1) = 0 = ωξ(x2), (2)∫
K
χδ(x, λ, λγ , π, ω, ωξ, ζ)dw − Y δ

∫
K
Υδ(x, λ, λγ , π, ω, ωξ, ζ)dw = 0, δ = 1, 2, . . . , q, (3)

q∑
δ=1

Ωδ

{(
χδ
ω − d

dxξ
χδ
ωξ

)
− Y δ

(
Υδ

ω − d

dxξ
Υδ

ωξ

)}
≦ 0, for almost every x ∈ K, (4)



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 9 of 14

q∑
δ=1

Ωδ
{(
χδ
ζ − 0

)
− Y δ

(
Υδ

ζ − 0
)}

≦ 0, for almost every x ∈ K, (4′)

ωT
q∑

δ=1

Ωδ

{(
χδ
ω − d

dxξ
χδ
ωξ

)
− Y δ

(
Υδ

ω − d

dxξ
Υδ

ωξ

)}
≥ 0, for almost every x ∈ K, (5)

ζT
q∑

δ=1

Ωδ
{(
χδ
ζ − 0

)
− Y δ

(
Υδ

ζ − 0
)}

≥ 0, for almost every x ∈ K, (5′)

Ω > 0, (6)

and
(D’) max

(b,v)

(
X1, X2, . . . , Xq

)
subject to

b(x1) = 0 = b(x2), v(x1) = 0 = v(x2), (7)

bγ(x1) = 0 = bγ(x2), vξ(x1) = 0 = vξ(x2), (8)∫
K
χδ(x, b, bγ , ρ, v, vξ, ϱ)dw −Xδ

∫
K
Υδ(x, b, bγ , ρ, v, vξ, ϱ)dw = 0, δ = 1, 2, . . . , q, (9)

q∑
δ=1

Ωδ

{(
χδ
λ − d

dxγ
χδ
λγ

)
−Xδ

(
Υδ

λ − d

dxγ
Υδ

λγ

)}
≧ 0, for almost every x ∈ K, (10)

q∑
δ=1

Ωδ
{(
χδ
π − 0

)
−Xδ

(
Υδ

π − 0
)}

≧ 0, for almost every x ∈ K, (10′)

bT
q∑

δ=1

Ωδ

{(
χδ
λ − d

dxγ
χδ
λγ

)
−Xδ

(
Υδ

λ − d

dxγ
Υδ

λγ

)}
≤ 0, for almost every x ∈ K, (11)

ρT
q∑

δ=1

Ωδ
{(
χδ
π − 0

)
−Xδ

(
Υδ

π − 0
)}

≤ 0, for almost every x ∈ K, (11′)

Ω > 0. (12)

Next, denote by A and B the set of feasible solutions associated with the symmetric
models (P) and (D), respectively.

The main result given in the following is formulated in terms of (P’) and (D’). Of
course, this result is equally valid to (P) and (D).

Theorem 1. If (λ(x), ω(x),Ω, Y ) ∈ A and (b(x), v(x),Ω, X) ∈ B are some feasible
solutions for the considered symmetric models (P’) and (D’), respectively, and the next
assumptions are satisfied:



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 10 of 14

(i)
q∑

δ=1

Ωδ

∫
K

{
χδ(x, (·), (·), (·), v, vξ, ϱ)−XδΥδ(x, (·), (·), (·), v, vξ, ϱ)

}
dw is pseudoin-

vex at b, bγ and ρ, with z(x, λ, b) + b(x) ≧ 0, µ(x, λ, b) + ρ(x) ≧ 0, x ∈ K;

(ii) −
q∑

δ=1

Ωδ

∫
K

{
χδ(x, b, bγ , ρ, (·), (·), (·))− Y δΥδ(x, b, bγ , ρ, (·), (·), (·))}dw is pseudoin-

vex at v, vξ and ϱ, with η(x, v, ω) + ω(x) ≧ 0, ν(x, v, ω) + ζ(x) ≧ 0, x ∈ K;
then the connection Y ̸≤ X is true between the corresponding objective functionals of

(P’) and (D’).
Proof. By considering (10) and (10’), together with z(x, λ, b) + b(x) ≧ 0, µ(x, λ, b) +

ρ(x) ≧ 0, x ∈ K, we get

[z(x, λ, b) + b(x)]T
[
Ω

{(
χλ − d

dxγ
χλγ

)
−X

(
Υλ − d

dxγ
Υλγ

)}]
≥ 0,

[µ(x, λ, b) + ρ(x)]T [Ω {(χπ − 0)−X (Υπ − 0)}] ≥ 0,

and, by using (11) and (11’), we get

(z(x, λ, b))T
[
Ω

{(
χλ − d

dxγ
χλγ

)
−X

(
Υλ − d

dxγ
Υλγ

)}]
≥ 0, x ∈ K,

(µ(x, λ, b))T [Ω {(χπ − 0)−X (Υπ − 0)}] ≥ 0, x ∈ K,

which imply

0 ≤
∫
K
z(x, λ, b)T

[
Ω

{
(χλ −XΥλ)−

d

dxγ
(
χλγ −XΥλγ

)}]
dw

=

∫
K

[
z(x, λ, b)TΩ(χλ −XΥλ) +

dz(x, λ, b)T

dxγ
Ω
(
χλγ −XΥλγ

) ]
dw

− z(x, λ, b)TΩ
(
χλγ −XΥλγ

)∣∣x=x2

x=x1
,

and ∫
K

[
µ(x, λ, b)TΩ(χπ −XΥπ) +

dµ(x, λ, b)T

dxγ
Ω(0−X · 0)

]
dw ≥ 0.

Since z(x, λ, b) = 0, at x = x1 and x = x2, it follows∫
K

[
z(x, λ, b)TΩ(χλ −XΥλ) +

dz(x, λ, b)T

dxγ
Ω
(
χλγ −XΥλγ

) ]
dw ≥ 0

and ∫
K

[
µ(x, λ, b)TΩ(χπ −XΥπ) +

dµ(x, λ, b)T

dxγ
Ω(0−X · 0)

]
dw ≥ 0,

involving ∫
K

[
z(x, λ, b)TΩ(χλ −XΥλ) + µ(x, λ, b)TΩ(χπ −XΥπ)



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 11 of 14

+
dz(x, λ, b)T

dxγ
Ω
(
χλγ −XΥλγ

) ]
dw ≥ 0.

Now, by using the pseudoinvexity assumption given in (i), we obtain

Ω

∫
K
{χ(x, λ, λγ , π, v, vξ, ϱ)−XΥ(x, λ, λγ , π, v, vξ, ϱ)} dw

≥ Ω

∫
K
{χ(x, b, bγ , ρ, v, vξ, ϱ)−XΥ(x, b, bγ , ρ, v, vξ, ϱ)} dw.

By (9), the inequality given above involves

Ω

∫
K
{χ(x, λ, λγ , π, v, vξ, ϱ)−XΥ(x, λ, λγ , π, v, vξ, ϱ)} dw ≥ 0. (13)

On the other hand, relations given in (4) and (4’), together with η(x, v, ω) + ω(x) ≧
0, ν(x, v, ω) + ζ(x) ≧ 0, x ∈ K, implies

[η(x, v, ω) + ω(x)]T
[
Ω

{(
χω − d

dxξ
χωξ

)
− Y

(
Υω − d

dxξ
Υωξ

)}]
≤ 0,

[ν(x, v, ω) + ζ(x)]T [Ω {(χζ − 0)− Y (Υζ − 0)}] ≤ 0,

and, by using (5) and (5’), we get

(η(x, v, ω))T
[
Ω

{(
χω − d

dxξ
χωξ

)
− Y

(
Υω − d

dxξ
Υωξ

)}]
≤ 0, x ∈ K,

(ν(x, v, ω))T [Ω {(χζ − 0)− Y (Υζ − 0)}] ≤ 0, x ∈ K,

which imply

0 ≥
∫
K
η(x, v, ω)T

[
Ω

{
(χω − YΥω)−

d

dxξ
(
χωξ

− YΥωξ

)}]
dw

=

∫
K

[
η(x, v, ω)TΩ(χω − YΥω) +

dη(x, v, ω)T

dxξ
Ω
(
χωξ

− YΥωξ

) ]
dw

− η(x, v, ω)TΩ
(
χωξ

− YΥωξ

)∣∣x=x2

x=x1
,

and ∫
K

[
ν(x, v, ω)TΩ(χζ − YΥζ) +

dν(x, v, ω)T

dxξ
Ω(0− Y · 0)

]
dw ≤ 0.

Since η(x, v, ω) = 0, at x = x1 and x = x2, it follows∫
K

[
η(x, v, ω)TΩ(χω − YΥω) +

dη(x, v, ω)T

dxξ
Ω
(
χωξ

− YΥωξ

) ]
dw ≤ 0

and ∫
K

[
ν(x, v, ω)TΩ(χζ − YΥζ) +

dν(x, v, ω)T

dxξ
Ω(0− Y · 0)

]
dw ≤ 0,



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 12 of 14

involving ∫
K

[
η(x, v, ω)TΩ(χω − YΥω) + ν(x, v, ω)TΩ(χζ − YΥζ)

+
dη(x, v, ω)T

dxξ
Ω
(
χωξ

− YΥωξ

) ]
dw ≤ 0.

Now, by using the pseudoinvexity assumption given in (ii), we obtain

Ω

∫
K
{χ(x, λ, λγ , π, v, vξ, ϱ)− YΥ(x, λ, λγ , π, v, vξ, ϱ)} dw

≤ Ω

∫
K
{χ(x, λ, λγ , π, ω, ωξ, ζ)− YΥ(x, λ, λγ , π, ω, ωξ, ζ)} dw.

By (3), the inequality given above involves

Ω

∫
K
{χ(x, λ, λγ , π, v, vξ, ϱ)− YΥ(x, λ, λγ , π, v, vξ, ϱ)} dw ≤ 0.

The above inequality along with (13) yields
q∑

δ=1

Ωδ
(
Y δ −Xδ

)∫
K
Υδ(x, λ, λγ , π, v, vξ, ϱ)dw ≥ 0. (14)

If, for some δ ∈ {1, 2, . . . , q}, we have Y δ < Xδ, and, for i ∈ {1, 2, . . . , q}, with i ̸= δ,
we have Y i ≤ Xi, then since

∫
K
Υδ(x, λ, λγ , π, v, vξ, ϱ)dw > 0 and Ω > 0, we are in

contradiction with (14). □

4. Conclusions

In this paper, we have introduced a pair of symmetric multi-dimensional variational
fractional control problems. By using the updated concept of pseudoinvexity associated
with multiple integral type functionals, we have established a very important connection
between the objective functionals of the studied symmetric models. This approach (duality
technique) is useful in solving complex optimization problems that arise in various research
and practical areas. Therefore, this paper offered a solid and original contribution to the
theory of variational fractional control problems and duality.

References

[1] N Abdulaleem and S Treanţă. Optimality conditions and duality for e-differentiable
multiobjective programming involving v-e-type i functions. Opsearch, 60:1824–1843,
2023.

[2] I Ahmad. Symmetric duality for multiobjective fractional variational problems with
generalized invexity. Inform. Sci., 176:2192–2207, 2006.



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 13 of 14

[3] T Antczak M Arana-Jimenéz and S Treanţă. On efficiency and duality for a class
of nonconvex nondifferentiable multiobjective fractional variational control problems.
Opuscula Math., 43:335–391, 2023.

[4] R Bagri S Treanţă D Agarwal and G Sachdev. Robust duality in multi-dimensional
vector fractional variational control problem. Opsearch, DOI: 10.1007/s12597-024-
00756-2:22 pages, 2024.

[5] C R Bector and I Husain. Duality for multiobjective variational problems. J. Math.
Anal. Appl., 166:214–229, 1992.

[6] S Chandra B D Craven and B Mond. Symmetric dual fractional programming. Z.
Oper. Res., 29:59–64, 1985.

[7] X Chen. Symmetric duality for the multiobjective fractional variational problem with
partial invexity. J. Math. Anal. Appl., 245:105–123, 2000.

[8] X Chen. Minimax and symmetric duality for a class of multiobjective variational
mixed integer programming problems. Eur. J. Oper. Res., 154:71–83, 2004.

[9] B D Craven. Langragean conditions and quasiduality. Bull. Austral. Math. Soc.,
16:325–339, 1977.

[10] G B Dantzig E Eisenberg and R W Cottle. Symmetric dual nonlinear programs. Pac.
J. Math., 15:809–812, 1965.

[11] K Das S Treanţă and T Saeed. Mond-weir and wolfe duality of set-valued fractional
minimax problems in terms of contingent epi-derivative of second-order. Mathematics,
10:938, 2022.

[12] W S Dorn. A symmetric dual theorem for quadratic programs. J. Oper. Res. Soc.
Japan, 2:93–97, 1960.

[13] T R Gulati I Ahmad and I Husain. Symmetric duality for minimax variational
problems. Math. Meth. Oper. Res., 48:81–95, 1998.

[14] Y Guo G Ye W Liu D Zhao and S Treanţă. On symmetric gh-derivative applications
to dual interval-valued optimization problems. Chaos Solitons Fractals, 158:112068,
2022.

[15] Y Guo G Ye W Liu D Zhao and S Treanţă. Optimality conditions and duality for
a class of generalized convex interval-valued optimization problems. Mathematics,
9:2979, 2021.

[16] A Jayswal Preeti and S Treanţă. Multi-dimensional Control Problems - Robust Ap-
proach. Springer Nature Singapore Pte Ltd., Singapore, 2022.

[17] D S Kim and W J Lee. Symmetric duality for multiobjective variational problems
with invexity. J. Math. Anal. Appl., 218:34–48, 1998.

[18] B Mond and M A Hanson. Symmetric duality for variational problems. J. Math.
Anal. Appl., 23:161–172, 1968.

[19] A K Prasad J Khatri and I Ahmad. Optimality conditions for an interval-valued
vector problem. Kybernetika, 61:221–237, 2025.

[20] T Saeed and S Treanţă. Various duality models associated with some constrained
robust nonlinear optimal control problems. Int. J. Control, 98:544–554, 2025.

[21] S Schaible. Duality in fractional programming - a unified approach. Oper. Res.,
24:452–461, 1976.



T. Saeed, S. Treanţă / Eur. J. Pure Appl. Math, 18 (4) (2025), 6921 14 of 14

[22] S Schaible. Fractional programming i - duality. Manag. Sci., 22:858–867, 1976.
[23] S Schaible. Fractional Programming. Kluwer Academic, Dordrecht, 1995.
[24] F Shi G Ye W Liu D Zhao and S Treanţă. Lagrangian dual theory and stability

analysis for fuzzy optimization problems. Inform. Sci., 657:119953, 2024.
[25] I Smart and B Mond. Symmetric duality with invexity in variational problems. J.

Math. Anal. Appl., 152:536–545, 1990.
[26] X Sun K L Teo and L Tang. Dual approaches to characterize robust optimal solution

sets for a class of uncertain optimization problems. J. Optim. Theory Appl., 182:984–
1000, 2019.

[27] S Treanţă and Şt Mititelu. Duality with (ρ, ω)-quasiinvexity for multidimensional
vector fractional control problems. J. Inform. Optim. Sci., 40:1429–1445, 2019.

[28] S Treanţă and T Saeed. Duality results for a class of constrained robust nonlinear
optimization problems. Mathematics, 11:192, 2023.

[29] B B Upadhyay A Ghosh and S Treanţă. Optimality conditions and duality for non-
smooth multiobjective semi-infinite programming problems on hadamard manifolds.
Bull. Iran. Math. Soc., 49:45, 2023.

[30] B B Upadhyay A Ghosh and S Treanţă. Efficiency conditions and duality for mul-
tiobjective semi-infinite programming problems on hadamard manifolds. J. Global
Optim., 89:723–744, 2024.

[31] B B Upadhyay A Ghosh and S Treanţă. Optimality conditions and duality for nons-
mooth multiobjective semi-infinite programming problems with vanishing constraints
on hadamard manifolds. J. Math. Anal. Appl., 531:127785, 2024.

[32] X M Yang S Y Wang and X T Deng. Symmetric duality for multiobjective fractional
programming problems. J. Math. Anal. Appl., 274:279–295, 2002.

[33] A M Geoffrion. Proper efficiency and the theory of vector maximization. J. Math.
Anal. Appl., 22:613–630, 1968.

[34] T Weir. Symmetric dual multiobjective fractional programming. J. Austral. Math.
Soc. (Ser. A), 50:67–74, 1991.

[35] T R Gulati I Husain and A Ahmed. Symmetric duality for multiobjective variational
problems. J. Math. Anal. Appl., 210:22–38, 1997.

[36] T R Gulati I Husain and A Ahmed. Symmetric duality with invexity in static and
continuous fractional programming. Optimization, 40:41–56, 1997.

[37] G Ye and W Liu S Treanţă, V Ionică. On symmetric dual models associated with
multiple cost control problems. Arch. Control Sci., 35:265–288, 2025.

[38] D S Kim W J Lee and S Schaible. Symmetric duality for invex multiobjective frac-
tional variational problems. J. Math. Anal. Appl., 289:505–521, 2004.