Presentation of Electromagnetism-like Mechanism Algorithm Technical - - PDF document

Presentation of Electromagnetism-like Mechanism Algorithm Technical Report

12

An Electromagnetism-likeMechanism for Global Optimization

Field of study: Artificial Intelligence and Robotics Lesson: Special topics in artificial intelligence Instructor: Dr. Azgomi edited by: Arman Daliri "November 2019"

(EM)

Contact : armandaliri@shafagh.ac.ir - daliriarman111@gmail.com - daliriarman111@outlook.com

This method was first proposed by Birbil and Feng in 2003, and like the genetic algorithm, it is a population-based method. In this method, each point of a charged particle in space is assumed, and its charge value is determined based on the value of its objective function. After determining the load of each point in the population, the result of the force applied to the points and to move them in each iteration is determined . Like electromagnetic forces, the result of the force acting on any point is obtained by summing all the forces acting on it. In some cases, this method differs from electromagnetic theory in that it describes the steps of the algorithm.

Electromagnetism-likeMechanism

Introduction

The electromagnetic algorithm is designed to solve problems with real and limited variables. The general picture of such issues is confronted.

𝑁𝑗𝑜 𝑔(𝑦) 𝑇. 𝑢, 𝑦 ∈ [𝑚, 𝑣] [𝑚, 𝑣]: = {𝑦 ∈ 𝑆𝑜 | 𝑚𝑙 < 𝑦𝑙 < 𝑣𝑙, 𝑙 = 1, … 𝑜}

This algorithm includes four main phases of setup, local search, computing the force applied to each particle, and moving the force vector, each of which is discussed in more detail below.

Electromagnetism-likeMechanism

Classification of algorithms

• ALGORITHM1. 𝐹𝑁
• ALGORITHM2. 𝐽𝑜𝑗𝑢𝑗𝑏𝑚𝑗𝑨𝑓 ()
• ALGORITHM3. 𝑀𝑝𝑑𝑏𝑚(𝑀𝑇𝐽𝑈 𝐹𝑆, 𝜀)
• ALGORITHM4. 𝐷𝑏𝑚𝑑𝐺(): 𝐺
• ALGORITHM5. 𝑁𝑝𝑤𝑓(𝐺)

Electromagnetism-likeMechanism

Setting up

At the beginning of the electromagnetic algorithm, similar to other population- based methods, m is randomly selected from the problem space. In order to produce a random point, it is assumed that each component is a random variable with a uniform distribution function between its lower and upper limits, then a random value is selected for each component. After creating this set, the value of the target function is calculated for each point and the best point is shown as xbest.

Electromagnetism-likeMechanism

General layout for EM

1: Initialize() 2: iteration←1 3: while iteration < MAXITER do 4: Local (LSITER, δ) 5: F←CalcF( ) 6: Move(F) 7: iteration←iteration+1 8: end while

• ALGORITHM1. EM

Electromagnetism-likeMechanism

Initialization

1: 𝑔𝑝𝑠 𝑗 = 1 𝑢𝑝 𝑛 𝑒𝑝 2: 𝑔𝑝𝑠 𝑙 = 1 𝑢𝑝 𝑜 𝑒𝑝 3: 𝜇 ← 𝑉( 0,1) 4: 𝑦 𝑗𝑙 ← 𝑚𝑙 + 𝜇( 𝑣𝑙 – 𝑚𝑙 ) 5: 𝑓𝑜𝑒 𝑔𝑝𝑠 6: 𝐷𝑏𝑚𝑑𝑣𝑚𝑏𝑢𝑓 𝑔(𝑦 𝑗) 7: 𝑓𝑜𝑒 𝑔𝑝𝑠 8: 𝑦𝑐𝑓𝑡𝑢 ← 𝑏𝑠𝑕𝑛𝑗𝑜 { 𝑔(𝑦𝑗) , ∀𝐽 }

The Initialize method is randomly used to sample m points. It is assumed that each coordinate is evenly distributed from a single point between the corresponding upper and lower boundary. After taking a point from space, the value of the calculated objective function is calculated using the function indicator f (x). This method ends with the identification of m points, and the point with the best value of performance is stored in xbest.

• ALGORITHM2. Initialize )(

Electromagnetism-likeMechanism

Local search

This step is done to examine the space around each of the created points. Birbil and Feng (2003) provided a simple algorithm for local search. In the method presented by them, for each of the dimensions of the created points, a random step is taken to increase or decrease the desired dimension. This step is randomly selected. Also, this step can be removed as much as possible depending on the step to the top or bottom of the desired dimension. After performing this step, if the new point created provides a better target function, it is replaced with the previous point, the search process for the next dimension is followed, otherwise, the initial point of retention and search for the next dimension is repeated. Repeat the search process to the maximum number of LSITERs for each dimension.

Electromagnetism-likeMechanism

Calculate the total force vector

According to electromagnetic theory, the force exerted by two charged particles is directly proportional to the distance between them and the amount of charge in each of them. In this algorithm, the points are determined based on the following relation in each iteration of the load.

𝑟𝑗 = (−𝑜

𝑔 𝑦𝑗 −𝑔(𝑦

) 𝑐𝑓𝑡𝑢

𝑙=1

𝑛

𝑔(𝑦𝑙)−𝑔(𝑦𝑐𝑓𝑡𝑢) )

Electromagnetism-likeMechanism

How the formula worked

The electric charge of each point determines the power of absorption or repulsion of that point, the points that have a better target function value will have more charge. Therefore, xbest will be most likely to be repeated in each iteration. Depending on the formula used, the load of a point may vary in two different iterations. It is noteworthy that, unlike electric charges, the charge of points in this method has no sign, for the force applied to each pair of points after determining the value of their

• bjective function.

Between each pair of points, the point that has a better target function absorbs the

• ther point, and the point that has a worse target function repels the other point,

and if the objective functions are equal, the force points will not enter each other. Therefore, at each repetition of the point that has the best value of the objective function, it absorbs the other points and the point that has the worst value of the

• bjective function repels the other points.

Electromagnetism-likeMechanism

The amount of force exerted by point j on point i It is calculated as follows:

𝑗≠𝑘

𝑛

𝑦𝑘 − 𝑦𝑗

𝑟𝑗𝑟𝑘 𝑦𝑘− 𝑦𝑗 2

𝑗𝑔 𝑔 𝑦𝑘 < 𝑔 𝑦𝑗 𝑦𝑗 − 𝑦𝑘

𝑟𝑗𝑟𝑘 𝑦𝑘− 𝑦𝑗 2

𝑗𝑔 𝑔 𝑦𝑘 ≥ 𝑔 𝑦𝑗 ฀ } , ∀i

Electromagnetism-likeMechanism

Explain the algorithm

As can be seen in algorithm 4 (lines 7-8), between two points, the point with the best performance value absorbs the other point. Conversely, a point that is worth performing worse will repel another (lines 9-10). Because xbest has the least amount of

• bjective performance, it acts as an absolute point of attraction,

meaning it attracts all other parts of the population.

• ALGORITHM4. CalcF():F

1: for i =1 tom do 2: 𝑟_𝑗 = (−𝑜 (𝑔(𝑦^𝑗 ) − 𝑔(𝑦^(𝑐𝑓𝑡𝑢)))/( _(𝑙 = 1)^𝑛▒〖(𝑔(𝑦^𝑙 〗)

− 𝑔(𝑦^𝑐𝑓𝑡𝑢))) )

3: Fi ← 0 4: end for 5: for i =1 tom do 6: for j =1 tom do 7: if f(xj) < f(x i) then 8: 𝑔𝑗 ← 𝑦𝑘 − 𝑦𝑗

𝑟𝑗𝑟𝑘 𝑦𝑘− 𝑦𝑗 2 {Attraction}

9: else 10: 𝑔𝑘 ← 𝑦𝑗 − 𝑦𝑘

𝑟𝑗𝑟𝑘 𝑦𝑘− 𝑦𝑗 2 {Repulsion}

11: end if 12: end for 13: end for

Electromagnetism-likeMechanism

Move according to the total force

After evaluating the total force vector Fi, the point (i) moves in a random direction along the equation. Here the length of the random step, λ is assumed to be evenly distributed between 0 and 1. Obviously, there are many other distributions that can be used to calculate the length of this step. But for ease of calculation, we have applied uniform distribution.

Electromagnetism-likeMechanism

RNG

It is the vector whose components indicate the permissible movement towards the upper limit, 𝑣𝑙, or lower limit, 𝑚𝑙, for the corresponding dimension. Is equal to :

𝑦𝑗 = 𝑦𝑗 + 𝜇 𝑔𝑗 𝑔𝑗 RNG i = 1,2, … , m

Electromagnetism-likeMechanism

Motion Algorithm

Quasi-code gives the method of movement. Note that the best point, xbest, has not been moved and will be moved to the next iterations (line 2). This suggests that we may avoid calculating the total force at the best current point in algorithm 4 (but computational effort to calculate the total force at the best current point is negligible).

• ALGORITHM5. Move(F)

1: for i =1 to m do 2: if i ≠ best then 3: λ ← U( 0,1) 4: Fi ←

𝑔𝑗 𝑔𝑗

5: for k =1 ton do 6: if Fi k > 0 then 7: xi k ← xi k +λFi k(uk −xi k) 8: else 9: xi k ← xi k +λFi k(xi k −lk) 10: end if 11: end for 12: end if 13: end for

Electromagnetism-likeMechanism

References

• Birbil Şİ, Fang SC. An electromagnetism-like mechanism for global optimization.

Journal of global optimization. 2003 Mar 1;25(3):263-82.

• SADEGHI MH, TAVAKKOLI MR. A HYBRID ELECTROMAGNETISM-LIKE ALGORITHM

FOR A MULTI-MODE RESOURCE-CONSTRAINED PROJECT SCHEDULING PROBLEM.

Contact : armandaliri@shafagh.ac.ir - daliriarman111@gmail.com - daliriarman111@outlook.com