Particle Swarms with Dynamic Topologies and Conservation of

Function Evaluations

Carlos M. Fernandes

1

, Juan L. J. Laredo

2

, Juan Julian Merelo

3

, Carlos Cotta

4

and Agostinho Rosa

1

1

LARSyS: Laboratory for Robotics and Systems in Engineering and Science, University of Lisbon,

Av. Rovisco Pais, 1, Lisbon, Portugal

2

Faculty of Sciences, Technology and Communications, University of Luxembourg,

6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Luxembourg

3

Departamento de Arquitectura y Tecnología de Computadores, University of Granada,

C/ Daniel Saucedo Aranda, s/n, 18071, Granada, Spain

4

Lenguages y Ciencias de la Computacion, Universidad de Malaga, ETSI Informática (3.2.49),

Universidad de Málaga, Campus de Teatinos, 29071, Malaga, Spain

Keywords: Particle Swarm Optimization, Population Structure, Dynamic Topologies, Swarm Intelligence.

Abstract: This paper proposes a general framework for structuring dynamic Particle Swarm populations and uses a

conservation of function evaluations strategy to increase the convergence speed. The population structure is

constructed by placing the particles on a 2-dimensional grid of nodes, where they interact and move

according to simple rules. During the running time of the algorithm, the von Neumann neighborhood is used

to decide which particles influence each other when updating their velocity and position. Each particle is

updated in each time-step but they are evaluated only if there are other particles in their neighborhood. A set

of experiments demonstrates that the dynamics imposed by the structure provides a more consistent and

stable behavior throughout the test set when compared to standard topologies, while the conservation of

evaluations significantly reduces the convergence speed of the algorithm. Furthermore, the working

mechanisms of the proposed structure are very simple and, except for the size of the grid, they do not

require parameters and tuning.

1 INTRODUCTION

Kennedy and Eberhart (1995) proposed the Particle

Swarm Optimization (PSO) algorithm for binary and

real-valued function optimization, a method inspired

by the swarming and social behavior of bird flocks

and fish schools. Since then, PSO has been applied

with success to a wide range of problems but the

proper balance between exploration (global search)

and exploitation (local search) is still an open

problem that motives several lines research on the

various mechanisms that control the algorithm’s

performance.

Population topology is one of PSO’s components

that affect the balance between exploration and

exploitation and the convergence speed and

accuracy of the algorithm. In the context of particle

swarms, topology is the structure that defines the

connections between the particles and consequently

the flow of information through the population. The

reason why particles are interconnected is the core

of the algorithm: the particles communicate so that

they acquire information on the regions explored by

other particles. In fact, it has been claimed that the

uniqueness of the PSO algorithm lies in the

interactions of the particles (Kennedy and Mendes,

2002). The population can be structured on any

possible topology, from sparse to dense (or even

fully connected) graphs), with different degrees of

connectivity and clustering. The classical and most

used population structures are the lbest (which

connects the individuals to a local neighborhood)

and the gbest (in which each particle is connected to

every other individual). These topologies are well-

studied and the major conclusions are that gbest is

fast but is frequently trapped in local optima, while

lbest is slower but converges more often to the

neighborhood of the global optima.

Since the first experiments on lbest and gbest

structures, researchers have tried to design networks

that hold the best traits given by each structure

86

Fernandes C., Laredo J., Merelo J., Cotta C. and Rosa A..

Particle Swarms with Dynamic Topologies and Conservation of Function Evaluations.

DOI: 10.5220/0005087900860094

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2014), pages 86-94

ISBN: 978-989-758-052-9

Copyright

c

2014 SCITEPRESS (Science and Technology Publications, Lda.)

(Parsopoulos and Vrahatis, 2004). Some studies also

try to understand what makes a good structure: for

instance, Kennedy and Mendes (2002) investigate

several types of topologies and recommend the use

of a lattice with von Neumann neighborhood (which

results in a connectivity degree between that of lbest

and gbest).

Recently, dynamic structures have been tested in

order to improve the algorithm’s adaptability to

different fitness landscapes and overcome the

rigidity of static structures, like in (Liang et al.,

2006). Fernandes et al. (2003) try a different

approach and propose a dynamic and partially

connected von Neumann structure with Brownian

motion. This paper uses this model but introduces a

strategy for the conservation of function evaluations

(Majercik, 2013) with the aim of taking advantage of

the underlying structure and reduce convergence

speed. Furthermore, a formal description of the

dynamic network is given here, opening the way for

more sophisticated dynamics.

In the proposed topology, particles are placed in a

2-dimensional m-nodes grid where . Every

time-step, each individual checks its von Neumann

neighborhood and, as in the standard PSO, updates

its velocity and position using the information given

by the neighbors. However, while the connectivity

degree (number of numbers, considering the particle

itself) of the von Neumann topology is 5, the

degree of the proposed topology is variable: 5.

Furthermore, the structure is dynamic: in each time-

step, every particle updates its position on the grid

(which is a different concept from the position of the

particle on the fitness landscape) according to a pre-

defined rule that selects the destination node. The

movement rule, which is implemented locally and

without any knowledge on the global state of the

system, can be based on stigmergy (Grassé, 1959) or

Brownian motion.

As stated above, the connectivity degree of each

particle in each time-step is variable and lies in the

range 15. Depending on the size of the grid,

there are, in each time-step, a number of particles

with 1. These particles without neighbors

(except the particle itself) do not learn from any

local neighborhood at that specific iteration.

Therefore, it is expected that they continue to follow

their previous trajectory in the fitness landscape.

Taking into account these premises, the algorithm

proposed in this study does evaluate the position of

the particles when 1. Regardless of the loss of

informant intrinsic to a conservation of evaluations

policy, we hypothesize that the strategy is

particularly suited for the proposed dynamic

topology (in which the particles are sometimes

isolated from the flow of information) and the

number of function evaluations required for meeting

the stop criteria can be significantly reduced.

Furthermore, it is the structure of the population and

the position of the particles at a specific time-step

that decides the application of the conservation rule

and not any extra parameter or pre-defined decision

rule.

A classical PSO experimental setup is used for the

tests and the results demonstrate that the proposed

algorithm consistently improves the speed of

convergence of the standard von Neumann structure

without degrading the quality of solutions. The

experiments also demonstrate that the introduction

of the conservation strategy reduces significantly the

convergence speed without affecting the quality of

the final solutions.

The remaining of the paper is organized as follows.

Section 2 describes PSO and gives an overview on

population structures for PSOs. Section 3 gives a

formal description of the proposed structure. Section

4 describes the experiments and discusses the results

and, finally, Section 5 concludes the paper and

outlines future research.

2 BACKGROUND REVIEW

PSO is described by a simple set of equations that

define the velocity and position of each particle. The

position of the i-th particle is given by

,

,

,

,…

,

), where is the dimension of the

search space. The velocity is given by

,

,

,

,…

,

). The particles are evaluated with a

fitness function

and then their positions and

velocities are updated by:

,

,

1

,

,

1

,

,

1

(1)

,

,

1

,

(2)

were

is the best solution found so far by particle

and

is the best solution found so far by the

neighborhood. Parameters

and

are random

numbers uniformly distributed in the range 0,1] and

and

are acceleration coefficients that tune the

relative influence of each term of the formula. The

first term is known as the cognitive part, since it

relies on the particle’s own experience. The last term

ParticleSwarmswithDynamicTopologiesandConservationofFunctionEvaluations

87

is the social part, since it describes the influence of

the community in the trajectory of the particle.

In order to prevent particles from stepping out of the

limits of the search space, the positions

,

are

limited by constants that, in general, correspond to

the domain of the problem:

,

∊

,

. Velocity may also be

limited within a range in order to prevent the

explosion of the velocity vector:

,

∊

,

.

For achieving a better balancing between local and

global search, Shi an Eberhart (1998) added the

inertia weight as a multiplying factor of the first

term of equation 1. This paper uses PSOs with

inertia weight.

The neighborhood of the particle defines the value

of

and is a key factor in the performance of PSO.

Most of the PSOs use one of two simple sociometric

principles for defining the neighborhood network.

One connects all the members of the swarm to one

another, and it is called gbest, were g stands for

global. The degree of connectivity of gbest is ,

where is the number of particles. Since all the

particles are connected to every other and

information spreads easily through the network, the

gbest topology is known to converge fast but

unreliably (it often converges to local optima).

The other standard configuration, called lbest (where

l stands for local), creates a neighborhood that

comprises the particle itself and its nearest

neighbors. The most common lbest topology is the

ring structure, in which the particles are arranged in

a ring structure (resulting in a degree of connectivity

3, including the particle). The lbest converges

slower than the gbest structure because information

spreads slower through the network but for the same

reason it is less prone to converge prematurely to

local optima. In-between the ring structure with

3 and the gbest with there are several

types of structure, each one with its advantages on a

certain type of fitness landscapes. Choosing a proper

structure depends on the target problem and also on

the objectives or tolerance of the optimization

process.

Kennedy and Mendes (2002) published an

exhaustive study on population structures for PSOs.

They tested several types of structures, including the

lbest, gbest and von Neumann configuration with

radius 1 (also kown as 5 neighborhood). They also

tested populations arranged in randomly generated

graphs. The authors conclude that when the

configurations are ranked by the performance the

structures with k = 5 (like the 5) perform better,

but when ranked according to the number of

iterations needed to meet the criteria, configurations

with higher degree of connectivity perform better.

These results are consistent with the premise that

low connectivity favors robustness, while higher

connectivity favors convergence speed (at the

expense of reliability). Amongst the large set of

graphs tested in (Kennedy and Mendes, 2002), the

Von Neumann with radius 1 configuration

performed more consistently and the authors

recommend its use.

Alternative topologies that combine standard

structures’ characteristics or introduce some kind of

dynamics in the connections have been also

proposed. Parsopoulos and Vrahatis (2004) describe

the unified PSO (UPSO), which combines the gbest

and lbest configurations. Equation 1 is modified in

order to include a term with

and a term with

and a parameter balances the weight of each term.

The authors argue that the proposed scheme exploits

the good properties of gbest and lbest. Peram et al.

(2003) proposed the fitness–distance-ratio-based

PSO (FDR-PSO), which defines the “neighborhood”

of a particle as its closest particles in the

population (measured by the Euclidean distance). A

selective scheme is also included: the particle selects

nearby particles that have also visited a position of

higher fitness. The algorithm is compared to a

standard PSO and the authors claim that FDR-PSO

performs better on several test functions. However,

the FDR-PSO is compared only to a gbest

configuration, which is known to converge

frequently to local optima in the majority of the

functions of the test set. More recently, a

comprehensive-learning PSO (CLPSO) was

proposed (Liang et al. 2006). Its learning strategy

abandons the global best information and introduces

a complex and dynamic scheme that uses all other

particles’ past best information. CLPSO can

significantly improve the performance of the

original PSO on multimodal problems. Finally,

Hseigh et al. (2009) use a PSO with varying swarm

size and solution-sharing that, like in (Liang et al.

2006), uses the past best information from every

particle.

A different approach is given in (Fernandes et al.,

2013). The authors describe a structure that is based

on a grid of nodes (with ) on which the

particles move and interact. The von Neumann

neighborhood is checked for chosing the informants

of the cognitive part of equation 1. Since , the

number of neighbors lies in the range 15 and

ECTA2014-InternationalConferenceonEvolutionaryComputationTheoryandApplications

88

the distribution of the particles on the grid at a given

time-step defines the PSO structure at that precise

iteration. The results demonstrate that the proposed

structure performs consistently throughout the test

set, improving the performance of other topologies

in the majority of the scenarios and under different

performance evaluation criteria. Furthermore, the

structure is very simple and only the grid size needs

to be set before the run (the authors suggest 1:2 ratio

between the swarm size and the number of nodes).

However, the structure itself and the distribution of

the particles on the grid suggest that there is still

room for improvement, namely of convergence

speed, which is a critical aspect when optimizing

real-world functions.

In the proposed structure, a particle, at a given time-

step, may have no neighbors except itself. The

isolated particles will continue to follow its previous

trajectory, based on their current information, until

they find another particle in the neighborhood.

Therefore, we intend to investigate if the loss of

information caused by not evaluating these particles

is overcome by the payoff in the convergence speed.

Common ways of addressing the computational cost

of evaluating solutions in hard real-world problems

are function approximation (Landa-Becerra et al.,

2008), fitness inheritance (Reyes-Sierra and Coello

Coello, 2007) and conservation of evaluations

(Majercik, 2013). Due to the underlying structure of

the proposed algorithm, we have tested a

conservation policy similar to the GREEN-PSO

proposed by Majercik (2013). However, in our

algorithm the decision on evaluating or not is made

by the position of the particle in the grid (isolated

particles are not evaluated) while in the GREEN-

PSO the decision is probabilistic and the likelihood

of conserving a solution is controlled by a

parameter.

The following section gives a formal description of

the proposed network and presents the transition

rules that define the model for dynamic population

structures.

3 PARTIALLY CONNECTED

STRUCTURES

Let us consider a rectangular grid of size

, where is the size of the population of any

population-based metaheuristics or model. Each

node

of the grid is a tuple

〈

,

〉

, where

∈

1,…,

∪

•

and

∈

∪

•

for

some domain . The value

indicates the index

of the individual that occupies the position

〈

,

〉

in

the grid. If

• then the corresponding position

is empty. However, that same position may still have

information, namely a mark (or clue)

. If

= •

then the position is empty and unmarked. Please

note that when , the topology is a static 2-

dimensional lattice and when and

the topology is the standard square grid graph.

In the case of a PSO, the marks are placed by

particles that occupied that position in the past and

they consist of information about those particles,

like their fitness

or position in the fitness

landscape, as well as a time stamp

that indicates

the iteration in which the mark was placed. The

marks have a lifespan of iterations, after which

they are deleted.

Initially,

•,• for all

〈

,

〉

. Then, the

particles are placed randomly on the grid (only one

particle per node). Afterwards, all particles are

subject to a movement phase (or grid position

update), followed by a PSO phase. The process

(position update and PSO phase) repeats until a stop

criterion is met.

The PSO phase is the standard iteration of a PSO,

comprising position and velocity update. The only

difference to a static structure is that in this case a

particle may find empty nodes in its neighborhood.

In the position update phase, each individual moves

to an adjacent empty node. Adjacency is defined by

the Moore neighborhood of radius , so an

individual at

〈

,

〉

can move to an empty

node

〈

,

〉

for which

〈

,

〉

,

〈

,

〉

. If

empty positions are unavailable, the individual stays

in the same node. Otherwise, it picks a neighboring

empty node according to the marks on them. If there

are no marks, the destination is chosen randomly

amongst the free nodes.

With this framework, there are two possibilities for

the position update phase: stimergic, whereby the

individual looks for a mark that is similar to itself;

and Brownian, whereby the individual selects an

empty neighbor regardless of the marks. For the first

option, let

〈

,

〉

〈

,

〉

,…,

〈

,

〉

be

the collection of empty neighboring nodes and let

be the individual to move. Then, the individual

attempts to move to a node whose mark is as close

as possible to its own corresponding trait (fitness or

position in the fitness landscape, for instance) or to

an adjacent cell picked at random if there are no

marks in the neighborhood. In the alternative

Brownian policy, the individual moves to an

ParticleSwarmswithDynamicTopologiesandConservationofFunctionEvaluations

89

adjacent empty position picked at random. In either

case, the process is repeated for the whole

population.

For this paper, the investigation is restricted to the

Brownian structure. The algorithm is referred in the

remaining of the paper has PSO-B, followed by the

grid size . An extension of the PSO-B is also

proposed by introducing a conservation of function

evaluations (cfe) strategy. If at a given time-step a

particle has no neighbors, then the particle is

updated but its position is not evaluated. This

version of the algorithm is referred to as PSO-Bcfe.

The following section describes the results attained

by the PSOs with dynamic structure and Brownian

movement, with and without conservation of

function evaluations and compares them to the

standard topologies.

4 EXPERIMENTS AND RESULTS

This section describes the experiments conducted for

evaluating the performance of the proposed stru-

cture. The algorithm is first compared to the version

without conservation of function evaluations.

Table 1: Benchmarks for the experiments. Dynamic range,

initialization range and stop criteria.

function

mathematical

representation

Rangeof

search/

Rangeof

initialization

stop

Sphere

f

1

100,100

(50,100

0.01

Rosenbrock

f

2

100

1

100,100

15,30

100

Rastrigin

f

3

10cos

2

10

10,10

2.56,5.12

100

Griewank

f

4

1

1

4000

cos

√

600,600

300,600

0.05

Schaffer

f5

0.5

sin

0.5

1.00.001

100,100

15,30

0.00001

Table 2: PSO-B and PSO-Bcfe. Best fitness values

averaged over 50 runs.

f

1

f

2

f

3

f

4

f

5

PSO-B

8×8

2.71e‐40 7.98e+00 6.57e+01 7.43e‐03 1.17e‐03

±4.21e‐40 ±1.11e+01 ±1.78e+01±9.02e‐03 ±3.19E‐03

PSO-Bcfe

8×8

3.20e‐40 1.12e+01 6.29e+01 7.44e‐03 1.94e‐04

±8.44e‐40 ±2.02e+01 ±1.35e+01±7.96e‐03±1.37E‐03

PSO-B

10×10

8.74e‐38 1.25e+01 6.23e+01 7.73e‐03 0.00e+00

±1.29e‐37 ±2.07e+01 ±1.88e+01±9.50e‐03±0.00E+00

PSO-Bcfe

10×10

2.43e‐39 1.09e+01 6.16e+01 5.61e‐03 0.00e+00

±4.38e‐39 ±1.67e+01 ±1.60e+01±7.47e‐03±0.00E+00

PSO-B

15×15

3.47e‐33 1.53e+01 6.05e+01 3.79e‐03 3.89e‐04

±4.31e‐33 ±2.65e+01 ±1.45e+01±6.22e‐03±1.92E‐03

PSO-Bcfe

15×15

2.47e‐45 1.31 e+01 6.62e+01 6.45e‐03 0.00e+00

±4.21e‐45 ±2.42e+01 ±2.05e+01±9.90e‐03±0.00E+00

Then, the topology with Brownian motion and

conservation of evaluations is compared to the lbest,

gbest, and standard square von Neumann topologies.

An experimental setup was constructed with five

benchmark unimodal and multimodal functions that

are commonly used for investigating the

performance of PSO − see (Kennedy and Mendes,

2002) and (Trelea, 2003) for instance). The

functions are described in Table 1. The optimum

(minimum) of all functions is located in the origin

with fitness 0. The dimension of the search space is

set to 30 (except Schaffer, with 2). In

order to set a square grid graph for the standard von

Neumann topology, the population size is set to 49

(which is within the typical range of PSO’s swarm

size). The acceleration coefficients were set to 1.494

and the inertia weight is 0.729, as in (Trelea, 2003).

is defined as usual by the domain’s upper

limit and . A total of 50 runs for

each experiment are conducted. Asymmetrical

initialization is used (the initialization range for each

function is given in Table 1).

Two experiments were conducted. Firstly, the

algorithms were run for a limited amount of function

evaluations (147000 for

and

, 49000 for

,

and

) and the fitness of the best solution found was

averaged over the 50 runs. In the second experiment

the algorithms were run for 980000 iterations

(corresponding to 20000 iterations of standard PSO

with 49) or until reaching the stop criterion.

The criteria were taken from (Kennedy and Mendes,

2002) and are given in Table 1. For each function

and each algorithm, the number of function

evaluations required to meet the criterion is recorded

and averaged over the 50 runs. A success measure is

defined as the number of runs in which an algorithm

ECTA2014-InternationalConferenceonEvolutionaryComputationTheoryandApplications

90

attains the fitness value established as the stop

criterion.

Table 3: PSO-B and PSO-Bcfe. Function evaluations

averaged over 50 runs.

f

1

f

2

f

3

f

4

f

5

PSO-B

8×8

21070.98 56472.50 13224.90 19959.66 14369.25

±1023.60

(50)

±49342.60

(50)

±3894.30

(48)

±1408.36

(50)

±20599.03

(46)

PSO-Bcfe

8×8

21157.7 96148.42 13427.12 19968.40 10275.69

±1092.81

(50)

±112675.6

(50)

±2259.73

(50)

±1103.20

(50)

±6952.548

(49)

PSO-B

10×10

22700.72 65769.76 15114.46 21574.70 10741.78

±906.39

(50)

±70232.15

(50)

±3939.74

(48)

±1107.56

(50)

±10658.16

(50)

PSO-Bcfe

10×10

21796.04 57704.16 13953.04 20430.34 11817.06

±832.54

(50)

±71260.77

(50)

±3341.41

(49)

±1176.32

(50)

±15647.63

(50)

PSO-B

15×15

26122.88 74321.24 20408.50 24626.42 11830.83

±950.08

(50)

±83535.59

(50)

±3692.10

(50)

±1406.53

(50)

±11576.37

(48)

PSO-Bcfe

15×15

19600.76 77348.06 16713.59 18734.94 10890.55

±730.62

(50)

±91374.00

(50)

±4387.78

(46)

±1029.26

(50)

±11624.59

(47)

Table 4: PSO-Bcfe and standard topologies. Best fitness

values averaged over 50 runs.

f

1

f

2

f

3

f

4

f

5

PSO

VN

4.26e‐36 1.39e+01 6.40e+01 5.61e‐03 0.00e+00

±1.32e‐35 ±2.39e+01 ±1.59e+01 ±8.78e‐03 ±0.00e+00

PSO

1.20e‐25 1.19e+01 1.11e+02 2.95e‐04 1.94e‐04

±1.30e‐25 ±2.48e+01 ±1.74e+01 ±2.09e‐03 ±1.37e‐03

PSO

3.80e+03 1.19e+01 9.86e+01 3.80e+01 1.36e‐03

±5.67e+03 ±1.50e+01 ±2.84e+01 ±5.80e+01 ±3.41e‐03

PSO-Bcfe

10×10

2.43e‐39 1.09e+01 6.16e+01 5.61e‐03 0.00e+00

±4.38e‐39 ±1.67e+01 ±1.60e+01 ±7.47e‐03 ±0.00E+00

Table 2 shows the average fitness values attained by

PSO-B and PSO-Bcfe with different grid sizes.

Table 3 shows the average number of function

evaluations required to meet the stop criteria as well

as the number of successful runs. The performance

according to the fitness values is very similar with

no significant differences between the algorithm in

every function except

(PSO-Bcfe is better). When

considering the number of function evaluations (i.e.,

convergence speed), PSO-Bcfe is significantly better

or statistically equivalent in every function. For the

statistical tests comparing two algorithms, non-

parametric Kolmogorov-Smirnov tests (with 0.05

level of significance) have been used.

The results confirm that PSO-Bcfe is able to

improve the convergence speed of PSO-B without

degrading the accuracy of the solutions. The loss of

information that results from conserving evaluations

is clearly overcome by the benefits of reducing the

computational cost per iteration.

In the case of

, PSO-Bcfe also significantly

improves the quality of the solutions, namely with

larger grids. The proposed scheme seems to be

particularly efficient in unimodal landscapes, but

further tests are required in order to confirm this

hypothesis and understand what mechanisms make

PSO-Bcfe so efficient in finding more precise

solutions for the sphere function.

Table 5: PSO-Bcfe and standard topologies. Function

evaluations averaged over 50 runs.

f

1

f

2

f

3

f

4

f

5

PSO

VN

23530.78 72707.18 18424.00 22015.70 17622.36

±954.74

(50)

±92916.33

(50)

±11082.75

(49)

±1304.60

(50)

±16056.6

8

(50)

PSO

32488.96 80547.18 233260.18 30200.66 26263.00

±921.45

(50)

±112067.67

(50)

±281453.62

(17)

±1703.88

1

(50)

±27266.8

6

(49)

PSO

16082.39 56681.24 9602.04 14856.07 13933.09

±2697.41

(33)

±88165.30

(50)

±3599.04

(25)

±2028.12

(27)

±21576.6

3

(43)

PSO-

Bcfe

15×15

21796.04 57704.16 13953.04 20430.34 11817.06

±832.54

(50)

±71260.77

(50)

±3341.41

(49)

±1176.32

(50)

±15647.63

(50)

Figure 1: Rank by Holm-Bonferroni test.

Tables 4 and 5 compare the PSO-Bcfe with standard

PSOs: square grid (77) with von Neumann (VN)

neighborhood, lbest with 3 (ring) and gbest.

As expected, gbest is fast but its solutions are clearly

worst than the other algorithms results (except for

, and

) and the success rates are lower than the

average; lbest is fairly accurate when compared to

other strategies but it clearly requires more

evaluations than the other PSOs in order to meet the

ParticleSwarmswithDynamicTopologiesandConservationofFunctionEvaluations

91

same criteria. The standard von Neumann strategy

and the PSO-Bcfe are more consistent throughout

the test set, but PSO-Bcfe is faster is most of the

functions and significantly faster in

,

, and

. In

addition, its solutions for

are significantly better

than the fitness values attained by the PSO with von

Neumann topology. The proposed strategy maintains

or improves the accuracy of the von Neumann

strategy throughout the test set — the consistency of

the von Neumann topology has been reported by

Kennedy and Mendes (2002) —, while increasing

significantly the convergence speed.

A statistical analysis of the algorithms (including

also PSO-B) on the entire test set was conducted.

First, a Friedman test showed that there are no

significant differences in the performance of the

algorithms considering the fitness values criteria.

When taking into account the function evaluations

and success rates the test reveals significant

differences between the algorithms.

After the Friedman test, a Holm-Bonferroni test was

conducted for ranking the algorithms according to

their convergence speed and reliability and detect

significant differences between the algorithms. PSO-

Bcfe ranks first, followed by PSO-B, PSO with von

Neumann topology, PSO with lbest and finally PSO

with gbest (see Figure 1). Considering 0.1,

PSO-Bcfe is significantly better than the von

Neumann, lbest and gbest topologies. Table 6 shows

the results of the Holm test.

Table 6: Holm test results. .

z‐statistic p‐value

⁄

PSO-B

0.800000 0.211855 0.10000

PSO VN

1.700000 0.044565 0.05000

PSO

2.600000 0.004661 0.33333

PSO

2.900000 0.001866 0.02500

The Holm test concludes that although PSO-Bcfe

attains a better average ranking than PSO-B, there

are no significant differences between the

algorithms. Please remember that the grid size

considered in the general comparisons is 10×10. I

f

we consider a 15×15 grid size we see that PSO-Bcfe

is faster in every function and significantly faster in

f_1, f_3 and f_4, while for grids with size 8×8 the

algorithms are statistically equivalent in ever

y

function. In general, the performance of PSO-Bcfe is

improved or preserved when the grid size grows,

while the performance of PSO-B degrades with the

grid size.

Figure 2: PSO-B and PSO-Bcfe. Function evaluations

required to meet stop criteria when using grids with

different sizes.

19000

21000

23000

25000

27000

8x8 10x10 15x15

functionevaluations

gridsize

spheref1

PSO‐B

PSO‐Bcfe

50000

60000

70000

80000

90000

100000

8x8 10x10 15x15

functionevaluations

gridsize

rosenbrockf2

PSO‐B

PSO‐Bcfe

12000

14000

16000

18000

20000

22000

8x8 10x10 15x15

functionevaluations

gridsize

rastriginf3

PSO‐B

PSO‐Bcfe

18000

20000

22000

24000

26000

8x8 10x10 15x15

functionevaluations

gridsize

griewankf4

VNRAND

G‐PSO

10000

12000

14000

16000

8x8 10x10 15x15

functionevaluations

gridsize

schafferf5

PSO‐B

PSO‐Bcfe

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This behavior may be explained by the fact that in

larger grids the particles are isolated more often and

for longer periods of time. During these periods, the

particles are not using information from the rest of

the swarm. PSO-Bcfe partially overcomes the loss of

communication and information by not evaluating

the particles, thus saving computational resources.

Figure 2 graphically displays the evaluations

required by each version of the algorithm in each

function for different grids. Except for

, PSO-B

number of function evaluations to meet the criteria

increase with the size of the grid, while PSO-Bcfe

scales better, namely in functions

,

and

.

These results show that it is possible to improve

standard PSO’s performance by structuring the

particles on a grid of nodes, let them move

according to simple rules and save computational

resources by letting them follow their current

trajectories without evaluation the new positions.

5 CONCLUSIONS

This paper proposes a general scheme for structuring

dynamic populations for the Particle Swarm

Optimization (PSO) algorithm. The particles are

placed on a grid of nodes where the number of nodes

is larger than the swarm size. The particles move on

the grid according to simple rules and the network of

information is defined by the particle’s position on

the grid and its neighborhood (von Neumann vicinity

is considered here). If isolated (i.e., no neighbors

except itself), a particle is updated but its position is

not evaluated. This strategy results in loss of

information but it decreases the number of

evaluations per generation. The results show that the

payoff in convergence speed overcomes the loss of

information: the number of function evaluations is

reduced in the entire test set, while the accuracy of

the algorithm (i.e., the averaged final fitness) is not

degraded by the conservation of evaluations

strategy.

The proposed algorithm is tested with a Brownian

motion rule and compared to standard static

topologies. Statistical tests and ranking according to

convergence speed and success rates shows that the

dynamic structure with conservation of function

evaluations ranks first and it is significantly better

than the von Neumann, ring and lbest topologies.

Furthermore, the conservation of evaluations

strategy results in a more stable performance when

varying the grid size, while removing this strategy

from the proposed dynamic structure results in a

drop off of the convergence speed when the size of

the grid increases in relation to the swarm size.

The present study is restricted to dynamic structures

based on particles with Brownian motion. However,

a self-organized behavior based on communication

via the grid (stigmergy) can be modeled by the

general framework proposed in this paper. Future

research will be focused on dynamic structures with

stigmergic behavior based on the fitness and position

of the particles.

ACKNOWLEDGEMENTS

The first author wishes to thank FCT, Ministério da

Ciência e Tecnologia, his Research Fellowship

SFRH/ BPD/66876/2009. The work was supported

by FCT PROJECT [PEst-OE/EEI/LA0009/2013],

Spanish Ministry of Science and Innovation projects

TIN2011-28627-C04-02 and TIN2011-28627-C04-

01, Andalusian Regional Government P08-TIC-

03903 and P10-TIC-6083, CEI-BioTIC UGR project

CEI2013-P-14, and UL-EvoPerf project.

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