radius particle swarm optimization

8/10/2019 Radius Particle Swarm Optimization

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Radius Particle Swarm Optimization

Mana AnantathanavitFaculty of Information Science and Technology

Mahanakorn University of Technology

Bangkok, Thailand

[email protected]

Mud-Armeen MunlinFaculty of Information Science and Technology

Mahanakorn University of Technology

Bangkok, Thailand

[email protected]

Abstract — Particle Swarm Optimization (PSO) is a swarm

intelligence based and stochastic algorithm to solve the

optimization problem. Nevertheless, the traditional PSO has

disadvantage from the premature convergence when finding the

global optimization. To prevent from falling into the local

optimum, we propose the Radius Particle Swarm Optimization

(R-PSO) which extends the Particle Swarm Optimization by

regrouping the agent particles within the given radius of the

circle. It initializes the group of particles, calculates the fitness

function, and finds the best particle in that group. The R-PSO

employs the group-swarm to keep the swarm diversity and

evolution by sharing information from the agent particles which

successfully maintain the balance between the global exploration

and the local exploitation. Therefore the agent particle guides the

neighbour particles to jump out of the local optimum and achieve

the global best. The proposed method is tested against the well-

known benchmark dataset. The results show that the R-PSO

performs better than the traditional PSO in solving the

multimodal complex problems.

Keywords—Particle Swarm Optimization(PSO); Local

Optimum; Global Optimum; Radius Particle SwarmOptimization(R-PSO)

I. I NTRODUCTION

Particle Swarm Optimization (PSO) was designed by

Kennedy and Eberhart [1]. This method imitates a swarm

behavior such as fish schooling and bird flocking. Bird canfind food and share information to others. Therefore, birds

flock into a promise position or region for food and this area is

shared and other birds will likely move into these share

location. The PSO imitates birds behavior by using a

population called swarm. The member in the group is a

particle. Each particle finds a solution of the problem. Thus,

position sharing of experience or information takes place andthe particles update position and search the solution

themselves.

PSO has been used in many applications [2, 3]. It is simple

to execute and consists of limited parameters. However, the

performance and efficiency of the PSO depend on the iteration

and size of particle. Therefore, the complex problem may

increase the iteration or size of particle to make the result

becomes stable and effective.

PSO has disadvantage for the premature convergence when

solving the complex problems. The PSO with particle

migration has been given by Ma gang et al. [4]. It is based on

the migratory behavior in the nature that the population is

randomly portioned into several sub-swarms and used the

tournament selection to employ the particle migration. Zhang

and Ding [5] propose the PSO on the four sub-swarms which

apply fitness adaptive strategy to update the inertia weight and

share the information to update current records adaptively andcooperatively.

In this paper, we present the Radius Particle SwarmOptimization (R-PSO) to solve the well-known complex

multimodal problem. We extend the traditional PSO algorithm

by using particles within the same radius and group them into

a new particle agent and iteratively find the best solution

under the given objective functions.

II. PARTICLE SWARM OPTIMIZATION(PSO)

The PSO is a population-based stochastic algorithm. It

initializes a population with random positions and search for

the best position under the given fitness function. At each

iteration, the velocity and the position particle in the swarmare updated following to its previous best position ( pbest i,j)

and the global best position ( gbest i,j). In PSO, there are two

update functions: velocity (v) and position ( x) and the inertia

weight (w) as given in equation (1), (2), and (3) respectively.

Where i is the index of particle in the swarm ( i = 1,…,n); n

is the population size, j is the index of position in the particle

( j = 1,…,m); m is the dimension size, t represents the iteration

number, vi,j is the velocity of the ith particle, xi,j is the position.

Note that R1 and R2 are random numbers between 0 and 1, c1

and c2 are the acceleration coefficients, and w is the positiveinertia weight.

2013 International Computer Science and Engineering Conference (ICSEC): ICSEC 2013 English Track Full Papers

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In the swarm population, there are two topologies [6]:

gbest and lbest as shown in Fig.1. In the gbest topology, each

particle is integrally united; the trajectories oscillate handinga fusion of the best solution that its particle has found, and an

attraction to the best solution that each particle in their

neighbourhood has found. The lbest topology allows each

individual to be influenced by some number of adjacent

numbers of the particle position. There are two lbest

topologies: a von Neumann method that all particles inswarm are connected to its four neighbor particles and a ring

topology in which where each particle is connected to its two

neighbour particles.

Fig.1 Particle Swarm Topologies: The gbest topology (left) and the lbesttopology (center and right)

In traditional PSO, each particle in the swarm moves to its

personal best and the global best position [7, 8]. Although,

the engine of this activation can result in a rapid convergence

rate, it becomes from the premature convergence problem,

where it may be certainly trapped into local minimum when

solving multimodal complex problems. When particles fly the

direction of gbest topology, the diversity between particles

gradually decreases. The solution position is contracted to a

small area containing these same particles. The efficiency of

PSO is concerned with diversity of particles, especially when

efforts are made to avoid premature convergence and to

escape from local optimum. Thus, in order to adjust the

performance of PSO, different type of lbest topologies suchas ring lattice and Von Neumann have been proposed [9]. In

this paper, we proposed another lbest topology using a circle

to find the agent particle based on particles distance or radius.

III. R ADIUS PARTICLE SWARM OPTIMIZATION(R-PSO)

The main concept of the R-PSO is based on the lbest circle

topology to find the agent particle within the radius of circle as

shown in Fig.2. Each particle in the overlap radius can be in

multiple groups. Once, a group is defined, we find the best

particle in that the swarm group and assign to the agent

particle represented by abest i,j as shown in Fig.3. Finally, the

agent particle is the candidate for finding the optimal solution

or the gbest position as shown in Fig.4.

Fig. 2 The lbest circle topology and candidate agent particles.

Fig.3 The local neighborhood search for the best agent particle (abest ).

Fig.4 The global neighborhood search for the gbest.

Thus, the distance (d ) between particles in the radius-

neighborhood is computed in equation (4) and (5)

respectively.


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Where d is the distance, p is the particle in the radius-

neighborhood search ( p = 1,…,Ø ); Ø is the population size, j

is the index of position in the particle ( j = 1,…,m); m is thedimension size, r is the radius value, μ is the constant number

( μ = [0.1,0.9]), and vmax is a limit of each particle‘s velocity

to a specified maximum velocity. The maximum velocity is

calculated from the distance between the feasible bounds in

the benchmark function. In this paper, we discuss the

algorithm to search for the optimum solution using the agent particle abest i,j which is a radius local optimum as given in

equation (6).

Where f ( ) is the fitness function, ß is the index of particle in

the radius-neighborhood ( ß=1,…,Ø ). Therefore, the position is

applied to the entire velocity update equation:

The pseudo-code of the R-PSO algorithm is given below.

for each time step t to t max do

for each particle i in the swarm do

update position xi,j(t+1) with Eq.(2) and Eq.(7)

calculated particle’s fitness f(xi,j(t+1)

if particle’s fitness f( xi,j(t+1) better than pbest xi,j(t+1) then

update pbest xi,j(t+1)

end if

find agent particle

if agent particle’s fitness f (agent i,j(t+1)) better than

gbest then

update gbest

end if

end for

end for

Fig.5. R-PSO Algorithm

IV. EXPERIMENTS AND R ESULTS

The performance of R-PSO depends on the radius value.Therefore, experiments are carried out to find the most

suitable value of radius. The fine tuning is performed for the

five well-known test functions as given in Table I.

A. Benchmark function

The benchmark functions that are commonly used in

literature [4, 5, 11] are employed to test the performance of the

R-PSO. This benchmark is chosen for their variety-functions[12]; Sphere ( f1), Schwefel 1.2 ( f2) and Generalized

Rosenbrock ( f3) are simple unimodal problems, Generalize

Schwefel 2.6 ( f4) and Generalize Rastrigin ( f5) are highly

complex multimodal problem with few local minimum. Thefunction, the size of the variable and the solution are shown in

Table I and Table II. Table I provides characteristics of

function. The initialization bounds in Table II are an area

equal to one quarter of the search area that does not contain

the optimum solution.

TABLE I. BENCHMARK FUNCTIONS

F Common name Expression

f1 Sphere

f2 Schwefel 1.2

f3 Generalized

Rosenbrock

f4 Generalize

Schwefel 2.6

f5 Generalize

Rastrigin

TABLE II. OPTIMA AND INITIALIZATION RANGES FOR ALL FUNCTION

F Feasible

Bounds

Dimension Optimum Initialization

Bounds

f1 (-100,100) 30 0 (50,100)

f2 (-100,100) 30 0 (50,100)

f3 (-30,30) 30 0 (15,30)

f4 (-500,500) 30 0 (250,500)

f5 (-5.12,5.12) 30 0 (2.56,100)

B.

Parameter selectionWe test our R-PSO against traditional PSO algorithm. Each

algorithm runs on the all the benchmark function given in

Table I. The parameter w is set from the recommended value

by Shi and Eberchart [10] with a linear decreasing from 0.9 to

0.4. The acceleration constants c1, c2 are both 1.42694, xmax

and vmax are set to be equal, population size in the swarm is

60 and iteration number is 5000. Therefore, the number offunction evaluations is 300000. A total of 30 runs for each

experimental setting are conducted and average fitness of the

best solution throughout the run is recorded. In the case where

a gbest value is < 10-8

, it is rounded to 0.0.

The value of Radius may affect the performance of R-

PSO. Thus, improper values of Radius may potentially leadthe algorithm to be trapped in the local minimum. To select

the better value Radius, we define a μ parameter under variant

μ = [0.1,0.9], respectively.

In order to investigate the scalability of radius, different μ

ranging from 0.1 to 0.9 is analyzed. μ is a scale factor that has

value between 0 and 1, where μ = 0 means random particle

selection and μ = 1 means traditional PSO particle selection.

We have done experiment with the aforementioned five

functions to calibrate the μ and obtain the best μ = 0.4 as


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shown in Table III. We will employ this number in our R-PSO

to measure the performance against traditional PSO.

TABLE III. SOLUTION TO EACH R ADIUS VALUE

μ f1 f2 f3 f4 f5

0.1 0.00 2830.79 90.47 5569.33 133.63

0.2 0.00 6998.51 100.25 5355.56 167.050.3 0.00 536.85 3040.54 6025.58 116.49

0.4 0.00 0.012 0.68 4051.17 18.00

0.5 0.00 5062.98 572.89 5123.18 197.71

0.6 0.00 279.98 225.78 5891.35 31.83

0.7 0.00 9032.17 25.22 5825.92 19.91

0.8 1000.00 25058.84 3040.65 4049.57 22.88

0.9 2000.00 10405.57 25.51 4445.10 80.66

C. Results

We compare the performance of R-PSO and PSO. Table IVand Figure 6-10 present the best fitness result of the algorithm.

The results are average over the 300,000 function evaluations.

TABLE IV. MEAN FITNESS AFTER 30 TRIALS OF 300000 EVALUATION

F Common Name Evaluation R-PSO PSO

f1 Sphere Best 0 0

Mean 0 0

Worst 0 0

Std. ±0.0 ±0.0

f2 Schwefel 1.2 Best 0.04 26532.42

Mean 0.12 26666.66

Worst 0.13 26714.23

Std. ±0.002 ±33.78

f3 Generalized

Rosenbrock

Best 0.65 6.82

Mean 0.68 8.13

Worst 0.78 8.39

Std. ±0.178 ±0.273

f4 Generalize

Schwefel 2.6

Best 4049.57 5180.10

Mean 4051.17 5121.67

Worst 4065.93 5201.65

Std. ±2.830 ±10.480

f5 Generalize

Rastrigin

Best 16.13 152.13

Mean 18.00 157.27

Worst 19.68 157.28

Std. ±0.461 ±0.922

Fig.6 Algorithm Convergence for Sphere ( f1)

Fig. 7 Algorithm Convergence for Schwefel 1.2 ( f2)

Fig. 8 Algorithm Convergence for Generalized Rosenbrock ( f3)

Fig. 9 Algorithm Convergence for Generalize Schwefel 2.6 ( f4)


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Fig. 10 Algorithm Convergence for Generalize Rastrigin 2.6 ( f5)

Table IV presents the mean fitness for minimization result

on the functions. The performance of R-PSO outperform the

PSO for functions f2 – f5. For function f1, it is the simplesphere model and the mean fitness result is the same.

However, the convergence of PSO is faster than R-PSO due to

a single lbest but R-PSO still tries different lbest which is

already a gbest as shown in Fig.6.

For function f2, the R-PSO is very close to the solution,

while the PSO is divergent without having close the the

solution as a result of the best local optimum (or gbest) as

shown in Fig .7. The PSO is trapped in the lbest while R-PSOcan escape and find the new lbest using the agent particle. The

R-PSO can practically prevent the premature convergence and

notably increase the convergence in the evolution procedure.

For function f3, a saddle point contains serval lbest,

therefore, it is hardly to find the optimum solution. PSO

eventually diverges while the R-PSO still can demonstrates a

better result and convergence precision as shown in fig.8.

Function f4 and f5 are multimodal functions where the local

optimum increases exponentially depending on the dimension

size. It is the complex problem and very hard to find the

solution. PSO can be trapped in a local optimum on its route to

the global optimum, while R-PSO eventually can still find a

solution as shown in Fig.9 and Fig.10. It has been shown that

our R-PSO is the better optimization method than the PSO for

such complex problems in term of both convergence and

speed. Thus, the proposed lbest circle topology using radius-

neighborhood search is very effective for most test functions.

V. CONCLUSION

We propose the R-PSO by regrouping the particles within a

given radius and determine the agent particle which is the best

particle of the group for each local optimum. The R-PSO is

able to maintain appropriate swarm diversity, jump out thelocal optimum using the agent particle to achieve the global

optimum. We prove the performance of the R-PSO using well-

known complex problems. The result shows that our proposedmethod can solve the complex problems more effectively and

efficiently than the traditional PSO.

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[4] M.Gang, Z.Wei and C. Xiaolin . “A novel particle swarm optimization base on particle migration,” Applied Mathematics and Computation.Vol. 218, pp. 6620–6636, 2012

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[10] Y. Shi, and R.C. Eberhart, “Empirical Study of Particle SwarmOptimization,” in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC), pp. 1945-1950, 1999

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[12] Available from : http://www2.denizyuret.com/pub/aitr1569/node19.html


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radius particle swarm optimization

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