International Journal of Analysis and Applications

Volume 17, Number 2 (2019), 234-243

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-234

MULTI-OBJECTIVE OPTIMIZATION USING LOCAL FRACTIONAL DIFFERENTIAL
OPERATOR

RABHA W. IBRAHIM1,∗, MASLINA DARUS2

1Cloud computing center, University of Malaya, Malaysia

2Centre of Modelling and Data Sciences, Faculty of Science and Technology,

Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

∗Corresponding author: rabhaibrahim@yahoo.com

Abstract. In this effort, we aim to generalize the concept of Univex functions by utilizing a local fractional

differential-difference operator, based on different types of local fractional calculus (fractal calculus). This

study leads to a new class of these functions in some optimal problems by illustrating conditions on the

generalized functions. We call it the class of local fractional Univex functions. Strong, weak, converse,

and strict converse duality theorems are given. Multi-objective optimal problem involves the new process

is solved (local optimal problem). The main tool employed in the analysis is based on the local fractional

derivative operators.

1. Introduction

The notion of local fractional calculus (also labeled fractal calculus), which was first suggested by Kol-

wankar and Gangal [1] using the Riemann-Liouville fractional derivative [2]. It was employed to deal with

non-differentiable issues from science and engineering [3]- [5]. Local fractional derivative of φ(χ) of order

0 < α ≤ 1 is specified by

Received 2018-09-24; accepted 2018-11-20; published 2019-03-01.

2010 Mathematics Subject Classification. 44A45.

Key words and phrases. fractional calculus; fractional operator; fractional differential equation; univex function.
c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

234

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-234


Int. J. Anal. Appl. 17 (2) (2019) 235

Dα φ(χ) =
dα φ(χ)

χα

∣∣∣
χ=χ0

= lim
χ→χ0

dα[φ(χ) −φ(χ0)]
[d(χ−χ0)]α

, (1.1)

where the expression dα[φ(χ) −φ(χ0)]/[d(χ−χ0)]α is the Riemann-Liouville fractional derivative given by

dαφ(χ)

dχα
=

1

Γ(1 −α)
d

dχ

∫ χ
0

φ(t)

(χ− t)α
dt;

corresponding to the integral operator

(Iαφ)(χ) =
1

Γ(α)

∫ χ
0

(χ− t)α−1φ(t) dt.

This operator is well-defined and it is represented to the classical fractional calculus.

The local fractional derivative utilizing the fractal geometry is defined by the formula [5]

Dα φ(χ) =
dα φ(χ)

χα

∣∣∣
χ=χ0

= lim
χ→χ0

∆α[φ(χ) −φ(χ0)]
[(χ−χ0)]α

, (1.2)

where

∆α[φ(χ) −φ(χ0)] ∼= Γ(α + 1)[φ(χ) −φ(χ0)].

Dunkl operator (see [6,7]) is a structure for a diff-difference operator

Λκφ(χ) =
d

dχ
φ(χ) + κ

(φ(χ) −φ(−χ)
2χ

)
, κ ≥ 0. (1.3)

It generalized some special functions and integral transforms in several variables connected with reflection

groups. This class of operators has developed many other operators. It applied in the analysis of quantum

many body systems. Recently, this operator is given in term of fractional calculus [8]. By employing the

local fractional differential operator in (1.1) or (1.2), we introduce a generalization of (1.3) as follows:

Λακφ(χ) = D
αφ(χ) + κ

(φ(χ) −φ(−χ)
2χ

)
, (1.4)

(
κ ≥ 0,α ∈ (0, 1],χ 6= 0 ∈ R

)

In this study, we aim to generalize the concept of Univex functions by utilizing a local fractional differential-

difference operator (1.4). This study leads to a new class of these functions in some optimal problems by

illustrating conditions on the generalized functions. We call it the class of local fractional Univex functions.

Strong, weak, converse, and strict converse duality theorems are given, with examples in the sequel.



Int. J. Anal. Appl. 17 (2) (2019) 236

2. Univex function

In this section, we generalize the concept of the Univex function, by using the local fractional Dunkl

operator. Define the following functions η : [a,b] × [a,b] → R\{0}, φ : [a,b] → R and Φ : R → R.

Definition 1. A differential function φ is said to be a local fractional univex function of order α ∈ (0, 1] in

the direction of ξ ∈ J := [a,b] if for all χ ∈ J, we have

Λακ φ(χ) ≤
Φ
(
φ(χ) −φ(ξ)

)
η(χ,ξ)

.

Note that, this concept is one of significant tool for optimization. Also, we confirm that there are many

other techniques for optimization which are generalized by fractional formal operators (see [9]- [12]). The

advantage of using the fractional Dunkl operator, is that can be acted on multi-dimensional Euclidean spaces.

Therefor, it can be employed in non-linear multi-objective problem

Minimize Ψ(χ) =
(
ψ1(χ), ...,ψn(χ)

)
subjectto Θ(χ) ≤ 0,

(2.1)

where Ψ : J → Rn and Θ : J → Rn and 0 is the zero vector in Rn. The function Ψ(χ) has many applications

in various studies. It may represent a multi-agent function in cloud computing systems.

Definition 2. A point ξ ∈ J := {χ ∈ J : Θ(χ) ≤ 0} is said to be an efficient outcome of (2.1), if there is

no point χ ∈ J, with Ψ(χ) ≤ Ψ(ξ). Moreover, it is known as a weak efficient outcome when Ψ(χ) < Ψ(ξ).

Definition 3. The couple (Ψ, Θ) is called a local fractional univex of order α, if for all χ ∈ J we have

η1(χ,ξ).D
α
κ Ψ(χ) ≤ φ1

(
Ψ(χ) − Ψ(ξ)

)
and

η2(χ,ξ).D
α Θ(χ) ≤ −φ2

(
Θ(χ) − Θ(ξ)

)
,

where

η1 : J ×J → Rn, η2 : J ×J → Rn, φ1 : Rn → R, φ2 : Rn → R

and

DακΨ(χ) =
(
Λακψ1(χ), ..., Λ

α
κψn(χ)

)
.

This class of local fractional univex functions is denoted by α−type univex.



Int. J. Anal. Appl. 17 (2) (2019) 237

3. Results

In this section, we investigate some sufficient optimality conditions for a point to be an efficient solution

of (1.3) under the generalized (α,ρ,η,ϑ)-type Univex.

Theorem 3.1. Let ξ be an initial solution of the multi-objective problem (1.3) and c1 and c2 be two non-

negative constants such that

(A) Θ(ξ) = 0;

(B) c1
(
η1(x,ξ).D

α Ψ(x)
)

+ c2
(
η2(x,ξ).D

α Θ(x)
)
≥ 0;

(C) The couple (Ψ, Θ) is a strong (or weak ) pseudo-quasi (α,ρ,η,ϑ)-type univex at ξ ∈ Ω;

(D) u ≤ 0 ∈ Rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ Rp ⇒ φ2(v) ≥ 0;

(E) c1ρ1 + c2ρ2 ≥ 0.

Then ξ is an efficient solution of (1.3).

Proof. Suppose that ξ is not an efficient solution of (1.3), then there exists x ∈ Λ such that Ψ(x) ≤ Ψ(ξ).

By the assumptions (A) and (D), we have

φ1(Ψ(x) − Ψ(ξ)) ≤ 0, and φ2(Θ(ξ)) ≥ 0. (3.1)

In view of the assumption (C), we get

c1
(
η1(x,ξ).D

α Ψ(x)
)
< −c1ρ1‖ϑ(x,ξ)‖2 (3.2)

and

c2
(
η2(x,ξ).D

α Θ(x)
)
≤−c2ρ2‖ϑ(x,ξ)‖2. (3.3)

Summing the above inequalities and utilizing (E), we conclude that

c1
(
η1(x,ξ).D

α Ψ(x)
)

+ c2
(
η2(x,ξ).D

α Θ(x)
)
< −

(
c1ρ1 + c2ρ2

)
‖ϑ(x,ξ)‖2

≤ 0,

which contradicts the assumption (B). Hence, ξ is an efficient solution of (1.3). This completes the proof. �

Theorem 3.2. If the following conditions are satisfied:

(A) ξ is a weakly efficient solution of (1.3);

(B) Θ is continuous in ξ;

(C) The functions Ψ and Θ are fractional Univex functions of order α ∈ (0, 1) in the direction of ξ ∈ Λ.

Moreover, for some x̄ ∈ Λ, we have Θ(x̄) < 0.



Int. J. Anal. Appl. 17 (2) (2019) 238

Then there are two constants c1 ≥ 0 and c2 ≥ 0 such that

c1
(
η1(x,ξ).D

α Ψ(x)
)

+ c2
(
η2(x,ξ).D

α Θ(x)
)
≥ 0,

(
x ∈ Ω, c2Θ(ξ) = 0, η1 : Ω × Ω → Rm, η2 : Ω × Ω → Rp

)
.

Proof. Our aim is to show that the system

η1(x,ξ).D
α Ψ(x) < 0, η2(x,ξ).D

α Θ(x) < 0,

has no solution for x ∈ Ω. Let the system has a solution y ∈ Ω. By the assumption (A), we have

Ψ(ξ + �1y) < Ψ(ξ) and Θ(ξ + �2y) < Θ(ξ),

for sufficient small arbitrary constants �1, �2 > 0. Now, we let x̄ := ξ +�2y; which implies that x̄ ∈ Λ∩N�2 (ξ)

thus by (B) and (C), we have Θ(ξ + �2y) = Θ(x̄) < 0; which contradicts (A), where ξ is a weak solution.

Therefore, the above inequalities are non-negative. Hence, in view of (C) these are two constants c1 and c2

satisfy the inequality

c1
(
η1(x,ξ).D

α Ψ(x)
)

+ c2
(
η2(x,ξ).D

α Θ(x)
)
≥ 0,

with the property c2Θ(ξ) = 0. This completes the proof. �

Next, we consider the dual problem of (1.3) as follows:

Max Ψ(χ) =
(
ψ1(χ), ...,ψm(χ)

)
subjectto c1

(
η1(x,χ).D

α Ψ(x)
)

+ c2
(
η2(x,χ).D

α Θ(x)
)
≥ 0,

c2Θ(χ) ≥ 0,

(3.4)

where χ ∈ Ω, c1 and c2 be two non negative constants.

Theorem 3.3. Let x,χ be initial solutions of the multi-objective problems (1.3) and (3.2) respectively. If

(A) The couple (Ψ, Θ) is a strong (or weak ) pseudo-quasi (α,ρ,η,ϑ)-type univex at ξ ∈ Ω;

(B) u ≤ 0 ∈ Rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ Rp ⇒ φ2(v) ≥ 0;

(C) c1ρ1 + ρ2 ≥ 0;

then Ψ(x) � Ψ(χ).

Proof. Suppose that Ψ(x) ≤ Ψ(χ). Since c1ρ1 + ρ2 ≥ 0 then by (B), we obtain

φ1(Ψ(x) − Ψ(χ)) ≤ 0

φ2(Θ(χ)) ≥ 0.



Int. J. Anal. Appl. 17 (2) (2019) 239

In virtue of the assumption (A) the above inequalities yield(
η1(x,χ).D

α Ψ(χ)
)
< −ρ1‖ϑ(x,χ)‖2(

η2(x,χ).D
α Θ(χ)

)
≤−ρ2‖ϑ(x,χ)‖2,

consequently, we obtain

c1
(
η1(x,ξ).D

α Ψ(x)
)
< −c1ρ1‖ϑ(x,χ)‖2

and

c2
(
η2(x,χ).D

α Θ(x)
)
≤−ρ2‖ϑ(x,χ)‖2.

Summing the above inequalities and utilizing (C), we conclude that

c1
(
η1(x,χ).D

α Ψ(χ)
)

+ c2
(
η2(x,χ).D

α Θ(χ)
)
< −

(
c1ρ1 + ρ2

)
‖ϑ(x,χ)‖2

≤ 0,

which contradicts the assumption (C). This completes the proof. �

Theorem 3.4. Let x0 and χ0 be initial solution for the problems (1.3) and (3.2) respectively. If Ψ(x0) =

Ψ(χ0) then the (weak or strong) duality problems (1.3) and (3.2) has efficient solutions x0 and χ0 respectively.

Proof. Suppose that x0 is not efficient for (1.3), then for some x ∈ Λ

Ψ(x) ≤ Ψ(x0) = Ψ(χ0),

which contradicts weak (strong) duality theorems as χ0 is initial solution for (3.2). Therefore, x0 is efficient

for (1.3). Similarly χ0 is efficient solution for (3.2). Hence the proof. �

Theorem 3.5. Let χ0 be an initial solution of the multi-objective problem (3.2) and c1 and c2 be two non

negative constants such that

(A) The couple (Ψ, Θ) is a strong (or weak ) pseudo-quasi (α,ρ,η,ϑ)-type univex at ξ ∈ Ω;

(B) u ≤ 0 ∈ Rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ Rp ⇒ φ2(v) ≥ 0;

(C) c1ρ1 + ρ2 ≥ 0.

Then χ0 is an efficient solution of (3.2).

Proof. Suppose that χ0 is not an efficient solution of (3.2), then there exists x0 ∈ Λ such that Ψ(x0) ≤ Ψ(χ0).

Now going on as in Theorem 3.3, we have a contradiction. Hence, χ0 is an efficient solution of (3.2). �

Theorem 3.6. Let x0,χ0 be initial solutions of the multi-objective problems (1.3) and(3.2) respectively. If

(A) Ψ(x0) ≤ Ψ(χ0);

(B) The couple (Ψ, Θ) is a strong (or weak ) pseudo-quasi (α,ρ,η,ϑ)-type univex at ξ ∈ Ω;

(C) u ≤ 0 ∈ Rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ Rp ⇒ φ2(v) ≥ 0;



Int. J. Anal. Appl. 17 (2) (2019) 240

(D) c1ρ1 + ρ2 ≥ 0;

then x0 = χ0.

Proof. Suppose that x0 6= χ0. Since χ0 is an initial solution for (3.2) then by (A) and (C), we have

φ1(Ψ(x0) − Ψ(χ0)) ≤ 0

φ2(Θ(χ0)) ≥ 0.

In virtue of the assumption (B) the above inequalities imply that(
η1(x0,χ0).D

α Ψ(χ0)
)
< −ρ1‖ϑ(x0,χ0)‖2(

η2(x0,χ0).D
α Θ(χ0)

)
≤−ρ2‖ϑ(x0,χ0)‖2,

which on summing yields

c1
(
η1(x0,χ0).D

α Ψ(χ0)
)

+ c2
(
η2(x0,χ0).D

α Θ(χ0)
)
< −

(
c1ρ1 + ρ2

)
‖ϑ(x0,χ0)‖2

≤ 0,

which contradicts to initially of χ0. Then we obtain x0 = χ0. This completes the proof. �

4. Simulation

In this section, we illustrate a simulation to show how the fractional calculus is effected on the multi-

objective functions.

Let Ψ, Θ : R → R2 such that

Ψ(x) =
(
x2,x3

)
; Θ(x) =

(
x,x2

)
.

Our aim is to show that the couple (Ψ, Θ) is (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. To determine the fractional

Dunkl operator on these functions, we shall introduce three cases depending on the value of kv for v = 1.

4.1. Case (i) kv = 0. The fractional Dunkl operator acts on the functions Ψ and Θ as follows:

DαΨ(x) =
( Γ(3)

Γ(3 −α)
x2−α,

Γ(4)

Γ(4 −α)
x3−α

)
; DαΘ(x) =

( Γ(2)
Γ(2 −α)

x1−α,
Γ(3)

Γ(3 −α)
x2−α

)
.

Now, by letting

η1,2(x,ξ) =
(x− ξ

2
,
x− ξ

2

)
, ξ = 0,

we have

η1(x,ξ).D
αΨ(x) =

x3−α

Γ(3 −α)
+

3x4−α

Γ(4 −α)
; η2(x,ξ).D

αΘ(x) =
x2−α

2Γ(2 −α)
+

x3−α

Γ(3 −α)
.

Consider ρ1 = ρ2 = 1, x ∈ [0, 1] and ϑ(x,ξ) = x2 − ξ, therefore, we obtain

‖ϑ(x,ξ)‖2 = x4, ξ = 0.



Int. J. Anal. Appl. 17 (2) (2019) 241

Table 1. Fractional multi-objective function, kv = 0

(α) Eq. (3.3) Eq.(3.4)

0.25 1.6 1.9

0.5 2.6 2.4

0.75 3.1 3.2

It is clear that

Ψ(ξ) = Ψ(0) = (0, 0); Θ(ξ) = Θ(0) = (0, 0),

then by assuming

φ1

(
Ψ(x) − Ψ(ξ)

)
= 5x, φ2

(
Θ(x) − Θ(ξ)

)
= −5x, x ∈ [0, 1],

we conclude that

η1(x,ξ).D
α Ψ(x) + ρ1‖ϑ(x,ξ)‖2 =

x3−α

Γ(3 −α)
+

3x4−α

Γ(4 −α)
+ x4

< 5x, x ∈ [0, 1]

= φ1

(
Ψ(x) − Ψ(ξ)

) (4.1)

and

η2(x,ξ).D
α Θ(x) + ρ2‖ϑ(x,ξ)‖2 =

x2−α

2Γ(2 −α)
+

x3−α

Γ(3 −α)
+ x4

< 5x, x ∈ [0, 1]

= −φ2
(

Θ(x) − Θ(ξ)
) (4.2)

Hence, the couple (Ψ, Θ) is (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. Table 1 shows that for various values of

α ∈ (0, 1), the outcomes yield the fractional univexty of the couple (Ψ, Θ).

To apply the conditions of Theorem 3.1, we assume that c1 = c2 = 1; thus, we have c1ρ1 + c2ρ2 = 2 > 0

with the inequalities (3.3) and (3.4). This leads to all the conditions of Theorem 3.1 are achieved and hence,

ξ = 0 is an efficient solution. Note that if we let φ1(Y ) = 3Y and φ2(Y ) = −3Y, the couple (Ψ, Θ) is not

(α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1].

4.2. Case (ii) kv = 1. To evaluate the fractional Dunkl operator, a calculation implies that

σx2 = x
2 − 2

v.x2

v.v
= −x2, σx3 = −x3.

Therefore, one can attain

η1(x,ξ).D
αΨ(x) =

x3−α

Γ(3 −α)
+
x(2x2)α

2
+

3x4−α

Γ(4 −α)
+
x(2x3)α

2



Int. J. Anal. Appl. 17 (2) (2019) 242

and

η2(x,ξ).D
αΘ(x) =

x2−α

2Γ(2 −α)
+
x(2x)α

2
+

x3−α

Γ(3 −α)
+
x(2x2)α

2
.

Table 2 shows the evaluation of the fractional multi-objective functions for different values of α.

Table 2. Fractional multi-objective function, kv = 1

(α) Eq.(3.3) Eq. (3.4)

0.25 2.7 2.9

0.5 5 3.8

0.75 4.7 4.8

Thus, we conclude that the conditions of Theorem 3.1 are satisfied when c1 = c2 = 1; such that c1ρ1 +c2ρ2 =

2 > 0 with the inequalities (3.3) and (3.4). Consequently, we obtain ξ = 0 is an efficient solution.

4.3. Case (iii) kv = 2. By applying (1.2), we have

η1(x,ξ).D
αΨ(x) =

x3−α

Γ(3 −α)
+ x(2x2)α +

3x4−α

Γ(4 −α)
+ x(2x3)α

and

η2(x,ξ).D
αΘ(x) =

x2−α

2Γ(2 −α)
+ x(2x)α +

x3−α

Γ(3 −α)
+ x(2x2)α.

Table 3 shows the evaluation of the fractional multi-objective functions for different values of α. It is clear

that the couple (Ψ, Θ) is not (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. It is of (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1],

when α ∈ (0, 0.25]. Hence, Theorem 3.1 can be applied only for this value of α.

Table 3. Fractional multi-objective function, kv = 2

(α) Eq. (3.3) Eq. (3.4)

0.25 3.5 4.1

0.5 5.4 5.2

0.75 6.4 5.5

5. Conclusion

This effort is generalized, for the first time, two important concepts in science. The Dunkl operator

and the Univex function, by utilizing the Riemann-Liouville fractional differential operator. These two

generalizations are combined to deliver the fractional multi-objective problems. We studied the duality cases

by minimize and maximize the desired function in the Rn. Simulation is provided to apply the existing

solutions. It has been found that the fractional case converges to the ordinary case. These problems can be

employed in many studies not only in mathematics, but also in the economy; such as the utility function



Int. J. Anal. Appl. 17 (2) (2019) 243

the cost function and the entropy function. One can replace the Riemann-Liouville fractional differential

operator of any type of fractional calculus.

Acknowledgement

The work here is partially supported by UKM grant: GUP-2017-064.

References

[1] K.M. Kolwankar, A.D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos 6 (4)

(1996), 505-513.

[2] A.A. Kilbas, H.M. Srivastava, J.J. Trujiilo, Theory and Applications of Fractional Differential Equations. Amsterdam,

Netherlands: Elsevier, 2006.

[3] G. Jumarie, Maximum entropy, information without probability and complex fractals: classical and quantum approach

(Vol. 112). Springer Science & Business Media, 2013.

[4] X-Jun Yang, D. Baleanu and H. M. Srivastava, Local Fractional Integral Transforms and Their Applications, Elsevier Ltd,

2016.

[5] X.-Jun. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher Limited, Hong Kong,

2011.

[6] C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167–183.

[7] M.Rosler, M. Voit, Markov processes related with Dunkl operators. Adv. Appl. Math. 21(4)(1998), 575–643.

[8] R. W. Ibrahim, Optimality and duality defined by the concept of tempered fractional univex functions in multi-objective

optimization, Int. J. Anal. Appl. 15 (2017), 75-85.

[9] R. W. Ibrahim, Abdullah Gani, A new algorithm in cloud computing of multi-agent fractional differential economical

system, Computing 98 (11) (2016), 1061-1074.

[10] R. W. Ibrahim, Hamid A. Jalab, and Abdullah Gani, Perturbation of fractional multi-agent systems in cloud entropy

computing, Entropy 18 (2016), 31.

[11] R. W. Ibrahim, Abdullah Ghani, Hybrid cloud entropy systems based on Wiener process, Kybernetes 45 (7) (2016),

1072-1083.

[12] R. W. Ibrahim, and Abdullah Gani, A mathematical model of cloud computing in the economic fractional dynamic system,

Iran. J. Sci. Technol. Trans. A, Sci. 42 (2018), 65-72.


	1. Introduction
	2. Univex function
	3. Results
	4. Simulation
	4.1. Case (i) kv=0.
	4.2. Case (ii) kv=1.
	4.3. Case (iii) kv=2.

	5. Conclusion
	Acknowledgement
	References