MULTI-OBJECTIVE OPTIMIZATION USING LOCAL FRACTIONAL DIFFERENTIAL OPERATOR

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.


Introduction
The notion of local fractional calculus (also labeled fractal calculus), which was first suggested by Kolwankar 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 where the expression d α [φ(χ) − φ(χ 0 )]/[d(χ − χ 0 )] α is the Riemann-Liouville fractional derivative given by corresponding to the integral operator 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] where Dunkl operator (see [6,7]) is a structure for a diff-difference operator 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: In this study, we aim to generalize the concept of Univex functions by utilizing a local fractional differentialdifference 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.

Univex function
In this section, we generalize the concept of the Univex function, by using the local fractional Dunkl 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 where Ψ : J → R n and Θ : J → R n and 0 is the zero vector in R n .The function Ψ(χ) has many applications in various studies.It may represent a multi-agent function in cloud computing systems.
Definition 3. The couple (Ψ, Θ) is called a local fractional univex of order α, if for all χ ∈ J we have and where This class of local fractional univex functions is denoted by α−type univex.

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 c 1 and c 2 be two nonnegative constants such that Then ξ is an efficient solution of (1.3).
By the assumptions (A) and (D), we have In view of the assumption (C), we get and Summing the above inequalities and utilizing (E), we conclude that which contradicts the assumption (B).Hence, ξ is an efficient solution of (1.3).This completes the proof.
Then there are two constants c 1 ≥ 0 and c 2 ≥ 0 such that Proof.Our aim is to show that the system has no solution for x ∈ Ω.Let the system has a solution y ∈ Ω.By the assumption (A), we have for sufficient small arbitrary constants 1 , 2 > 0. Now, we let x := ξ + 2 y; which implies that x ∈ Λ ∩ N 2 (ξ) thus by (B) and (C), we have Θ(ξ + 2 y) = Θ(x) < 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 c 1 and c 2 satisfy the inequality with the property c 2 Θ(ξ) = 0.This completes the proof.
Next, we consider the dual problem of (1.3) as follows: where χ ∈ Ω, c 1 and c 2 be two non negative constants.

Simulation
In this section, we illustrate a simulation to show how the fractional calculus is effected on the multiobjective functions.
Let Ψ, Θ : R → R 2 such that 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 k v for v = 1.
The fractional Dunkl operator acts on the functions Ψ and Θ as follows:

Now, by letting
we have .
To apply the conditions of Theorem 3.1, we assume that c 1 = c 2 = 1; thus, we have 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 φ 1 (Y ) = 3Y and φ 2 (Y ) = −3Y, the couple (Ψ, Θ) is not To evaluate the fractional Dunkl operator, a calculation implies that Therefore, one can attain Table 2 shows the evaluation of the fractional multi-objective functions for different values of α.
Fractional multi-objective function, k v = ConclusionThis 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 R n .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 the cost function and the entropy function.One can replace the Riemann-Liouville fractional differential operator of any type of fractional calculus.

Table 2 .
Fractional multi-objective function, k v = 1Thus, we conclude that the conditions of Theorem 3.1 are satisfied when c 1