Enhanced Flower Pollination Approach Applied to Electromagnetic

Optimization

Carlos Eduardo Klein

, Emerson Hochsteiner de Vasconcelos Segundo

Viviana Cocco Mariani

2,3

and

Leandro dos Santos Coelho

1,3

Pontifical Catholic University of Parana, Industrial and Systems Eng. Graduate Program (PPGEPS), Curitiba, PR, Brazil

Pontifical Catholic University of Parana, Mechanical Engineering Graduate Program (PPGEM), Curitiba, PR, Brazil

Federal University of Parana (UFPR), Department of Electrical Engineering (DEE/PPGEE), Curitiba, PR, Brazil

Keywords: Metaheuristics, Swarm Intelligence, Flower Pollination Algorithm, Electromagnetic Optimization.

Abstract: It is difficult to use the deterministic mathematical tools such as a gradient method to solve global

optimization problems. Flower pollination algorithm (FPA) is a new nature-inspired algorithm of the swarm

intelligence field to global optimization applications, based on the characteristics of flowering plants. To

enhance the performance of the standard FPA, an enhanced FPA (EFPA) approach based on beta

probability distribution was proposed in this paper. In order to verify the performance of the proposed

EFPA, five benchmark functions are chosen from the literature as the test suit. Furthermore, tests using

Loney’s solenoid benchmark, a classical problem in the electromagnetics area, are realized to evaluate the

effectiveness of the FPA and the proposed EFPA. Simulation results and comparisons with the FPA

demonstrated that the performance of the EFPA approach is promising in electromagnetics optimization.

1 INTRODUCTION

Swarm intelligence is the collective behaviour of a

decentralized, self-organized system, and it is able to

distribute the functionality of an overall big system

among smaller, less-expensive and cooperative

agents (Weng et al., 2014). Swarm behavior is one

of the main features of many species in the nature. In

this context, swarm intelligence originated from the

study of colonies or swarms of social organisms

(Engelbrecht, 2007).

Nature-inspired algorithms of the swarm

intelligence field perform powerfully and efficiently

in solving global optimization problems. Recent

research studies in optimization field have led to the

development of new approaches that exhibit certain

advantages over more traditional techniques in

various aspects. Inspired by nature, these

metaheuristic algorithms have obtained promising

performance over continuous domains of

optimization problems, such as ant colony (Dorigo

and Stützle, 2004), artificial bee colony (Karaboga,

2005), krill heard (Gandomi and Alavi, 2012), bat

algorithm (Gandomi and Yang, 2014), cuckoo

search algorithm (Coelho et al., 2013), bat algorithm

(Yang, 2010), and firefly algorithm (Yang, 2009).

In the nature, many floral traits are related to the

pollination and fertilization processes, i.e. floral

traits can be adjusted by selection to ensure pollen

transfer, the subsequent growth of pollen tubes

through the pistil, and finally ovule fertilization

(Fernández et al., 2009). Pollination is a process of

transfer of pollen from male parts of flower called

anther to the female part called stigma of a flower.

Pollination of flowers can be inspiration to generate

new optimization algorithms. Examples of

algorithms based on pollination are presented in

Kasinger and Bauer (2006) and Kaur and Singh

(2012).

Recently, the flower pollination algorithm

(FPA), developed by Xin-She Yang (Yang, 2012),

was proposed. FPA is a swarm intelligence method

based on the features of flowering plants. Being a

stochastic search process, FPA is not free from false

and/or premature convergence, especially over

multimodal fitness landscapes.

The aim of this paper is to improve the FPA to

achieve a better exploration/exploitation trade-off

when applied to continuous optimization problems.

The proposed enhanced FPA (EFPA) is based on

beta probability distribution. To demonstrate the

effectiveness of the proposed EFPA framework, a

223

Klein C., Segundo E., Mariani V. and Coelho L..

Enhanced Flower Pollination Approach Applied to Electromagnetic Optimization.

DOI: 10.5220/0005074502230227

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2014), pages 223-227

ISBN: 978-989-758-052-9

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

set of two benchmark functions and a numerically

ill-conditioned inverse problem in the

electromagnetic field called Loney’s solenoid

benchmark problem (Di Barba and Savini, 1995;

Ciuprina et al., 2002) are solved. Loney’s solenoid

benchmark problem is a significant testbed of the

rough objective function surface typical of many

electromagnetic problems. Such problem are ideally

suited for stochastic techniques which escape from

local minima. Optimization results and convergence

performance are compared with the classical FPA.

The remainder of this paper is organized as

follows. Section 2 covers background information

on the FPA and EFPA. Section 3 provides the

description of the Loney’s solenoid benchmark.

Section 4 presents the simulation results and

discussions. Finally, we present concluding remarks

on this work in Section 5.

2 FPA AND EFPA ALGORITHMS

Swarm intelligence is based on self-organized

individuals, generally called agents, whose actions

and interaction add up to intelligent global behavior.

In particular, agents communicate to one another

merely by modifying their local environment. These

local interactions finally yield to the global self-

properties that make a system self-managing.

This section describes the classical FPA and the

proposed EFPA, both swarm intelligence

approaches. First, the fundamentals of the FPA are

introduced, and finally the mechanisms of the

proposed EFPA are mentioned.

2.1 FPA

FPA is a swarm intelligence inspired paradigm by

the flow pollination process of flowering plants.

FPA is useful to solve multimodal continuous

optimization problems. For simplicity, we use the

following four rules in the FPA, all rules suggest by

Yang (2012):

1. Biotic and cross-pollination can be considered

as a process of global pollination process, and

pollen-carrying pollinators move in a way which

obeys Lévy flights (Rule 1).

2. For local pollination, abiotic and self-

pollination are used (Rule 2).

3. Pollinators such as insects can develop flower

constancy, which is equivalent to a reproduction

probability that is proportional to the similarity of

two flowers involved (Rule 3).

4. The interaction or switching of local

pollination and global pollination can be controlled

by a switch probability p in the range [0, 1], with a

slight bias towards local pollination (Rule 4).

In reality, each plant can have multiple flowers,

and each flower patch often release millions and

even billions of pollen gametes. However, for

simplicity, we assume that each plant only has one

flower, and each flower only produce one pollen

gamete (Yang, 2012).

In order to formulate updating formulas, we have

to convert the aforementioned rules into updating

equations. For example, in the global pollination

step, flower pollen gametes are carried by

pollinators such as insects, and pollen can travel

over a long distance because insects can often fly

and move in a much longer range (Yang, 2012).

Therefore, Rule 1 and flower constancy can be

represented mathematically as:

])([)()()1( BtxL txtx

iii















(1)

where

)(tx

is the pollen i or solution vector

x at

iteration t, and B is the current best solution found

among all solutions at the current

generation/iteration. Here γ is a scaling factor to

control the step size. In addition, L(λ) is the

parameter that corresponds to the strength of the

pollination, which essentially is also the step size.

Since insects may move over a long distance with

various distance steps, we can use a Lévy flight to

imitate this characteristic efficiently.

The Lévy distribution, named for the French

mathematician Paul Lévy (Lévy, 1925), is important

in the study of Brownian motion. Lévy stable

distribution (Nolan, 2010) is a rich class of

probability distributions. It is worthy of noting that

the well-known Gaussian and Cauchy distributions

are its special cases.

A Lévy flight is a random walk in which the

step-lengths have a probability distribution that is

heavy-tailed. That is, we draw L>0 from a Lévy

distribution:









)0( ,

12/sin













(2)

Here, Γ(λ) is the standard gamma function, and

this distribution is valid for large steps S > 0. Then,

to model the local pollination, both Rule 2 and Rule

3 can be represented as

])()([)()1( txtx Utxtx

kjii











(3)

where )(tx

and )(tx

are pollen from different

flowers of the same plant species. This essentially

imitates the flower constancy in a limited

ECTA2014-InternationalConferenceonEvolutionaryComputationTheoryandApplications

224

neighborhood (Abdel-Raouf et al., 2014).

In order to imitate this, we can effectively use the

switch probability like in Rule 4 or the proximity

probability p to switch between common global

pollination to intensive local pollination.

The procedure for implementing the FPA can be

summarized by the following steps:

Step 1: Initialization of a population of flo-

wers/pollen gametes: Initialize a vectors population

(floating-point representation) of flowers/pollen

gametes in the n-dimensional problem space using

uniform probability distribution function. The

counter of generations t is initialized too.

Step 2: Evaluation of population’s fitness:

Evaluate each flower’s fitness value.

Step 3: Determine the best solution in the

population: Find the best solution B in the initial

population.

Step 4: Apply local or global pollination:

Randomly with a switch probability p if a global

pollination using Lévy distribution is applied or a

local pollination using a uniform probability

distribution function. This procedure is applied to all

flowers in the population. Update t = t + 1;

Step 5: Repeating the evolutionary cycle: Return

to Step 2 until a stopping criterion is met. In this

paper, evolutionary process is performed predefined

maximum number of generations (adopted as

stopping criterion) t

max

is reached.

2.2 Proposed EFPA

In spite of the prominent merits, sometimes FPA

shows the premature convergence and slowing down

of convergence as the region of global optimum is

approached. In this context, a trade-off between

exploration and exploitation actions must be

developed. Exploration is the process of visiting

entirely new regions of a search space, whilst

exploitation is the process of visiting those regions

of a search space within the neighborhood of

previously visited points.

In the classical FPA, the proximity probability p

is constant during the optimization process and p

takes values in the range [0, 1]. However, no optimal

choice of p was proposed in Yang (2012). This

means p is strongly problem-dependent and the user

should choose p carefully after some trial and error

tests.

In this paper, the proposed EFPA presents three

modifications in Step 4 in relation to the classical

FPA using beta probability distribution. The use of

the beta probability distribution (Ali, 2007) can be

useful to preserve diversity and helps to explore

hidden areas in the search space. The modifications

are the following:

1. The EFPA incorporates the tuning of p during the

evolutionary cycle given by:





)1.0,(5.0)

max

/(5.0 rrttp 









(4)

where

)1.0,( rr





is a beta distribution probability, r

is a random number generated with uniform

distribution in range [0,1], r and

r1.0 are beat

distribution parameters (see script betarnd in

MatLab environment).

2. The EFPA uses beta distribution probability

instead of Lévy distribution (see equation (1)) to

global pollination given by:

])([)1.0,()()1( Bt

xrr t

x 













(5)

where



is a scale parameter (adopted



=1.6 in

this paper).

3. The EFPA uses beta distribution probability

instead of uniform probability distribution (see

equation (3)) to local pollination given by:

if ra > 0.7 then





else





5.0)

max

/(5.0









end

])()([)1.0,()()1( t

xrrt

x 













where ra is a random number generated with

uniform distribution in range [0,1] and



is a scale

parameter.

3 LONEY'S SOLENOID DESIGN

Loney’s solenoid design problem is to find the

dimensions called position (l) and size (s) of two

coils to generate possibly uniform magnetic field on

the segment (-z

, z

). This is a minimization problem

with non-analytical objective function. The box

constraints are

cm 200



and

.cm 200  l

The

upper half plane of the axial cross-section of the

system is presented in Fig. 1.

Figure 1: Axial cross-section of Loney’s solenoid (upper

half-plane).

EnhancedFlowerPollinationApproachAppliedtoElectromagneticOptimization

225

4 OPTIMIZATION RESULTS

In the following sub-sections, the optimization

results are presented and discussed.

4.1 Benchmark Functions Results

The performance of the EFPA and the classical FPA

has been analyzed over five benchmarks functions

(minimization problems) applied to dimension equal

to 10. The optimization methods, EFPA and FPA,

were employed using the following parameter

settings: population size equal to 50 flowers and the

stopping criterion is 2,000 generations.

Fifty independent runs of each algorithm on each

problem are taken. The minimum, maximum, mean

and the standard deviation of the best objective

function values (50 runs) are presented in Tables 1-

5. Best entries have been marked in boldface in

Tables 1-5. A closed look in Tables 1-5 reveals that

the EFPA has the ability to avoid local optima and it

presents superior performance when compared with

the FPA. However, FPA presents a promising

performance in the Griewank case.

Table 1: Results of f

(Rastrigin function).

Index FPA EFPA

Maximum

(Worst)

35.81 8.19×10

-12

Mean 19.44

2.22×10

-13

Minimum

(Best)

3.98

Standard

Deviation

7.61 1.17×10

-12

Table 2: Results of f

(Ackley function).

Index FPA EFPA

Maximum

(Worst)

6.28 8.99×10

-13

Mean 3.31

1.07×10

-13

Minimum

(Best)

4.93×10

-12

7.99×10

-15

Standard

Deviation

1.44 1.71×10

-13

4.2 Loney’s Solenoid Results

We used the following parametric setup for tested

FPA and EFPA to optimize the Loney’s solenoid

benchmark problem (with dimension equal to 2):

population size equal to 20 flowers and the stopping

criterion is 150 generations. In particular, three

Table 3: Results of f

(Sphere function).

Index FPA EFPA

Maximum

(Worst)

100 0

Mean 140

Minimum

(Best)

0 0

Standard

Deviation

35.05 0

Table 4: Results of f

(Griewank function).

Index FPA EFPA

Maximum

(Worst)

6.65×10

-1

7.40×10

-3

Mean 2.19×10

-1

6.57×10

-4

Minimum

(Best)

3.94×10

-2

Standard

Deviation

1.42×10

-1

2.05×10

-3

Table 5: Results of f

(Rosenbrock function).

Index FPA EFPA

Maximum

(Worst)

408.49 1.29

Mean 31.22

6.18×10

-2

Minimum

(Best)

1.86×10

-2

6.15×10

-6

Standard

Deviation

69.57 1.99×10

-1

different basins of attraction of local minima can be

recognized in the domain of f with values of f>4·10

-8

(high level region), 3·10

-8

< f < 4·10

-8

(low level

region), and f < 3·10

-8

(very low level region –

global minimum region) (Coelho and Alotto, 2007).

Table 6 summarizes the optimization results of

FPA and EFPA. A result with boldface means the

best values in terms of minimum and mean values in

f found in Table 6.

As seen from Table 6, EFPA outperforms FPA

clearly. The best result (minimum) using EFPA

presented f = 2.0666·10

-8

with s = 11.6013 cm and

l=1.5110 cm. On the other hand, the best f using

FPA was with s = 12.3459 cm and l = 2.0691 cm.

Table 6: Results of Loney’s solenoid problem.

F(s, l)·10

-8

Approach

Maximum

(Worst)

Mean

Minimum

(Best)

Standard

Deviation

FPA 3.9526 3.4920 2.2524 0.5259

EPFA 3.9952

3.0206 2.0666

0.6098

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5 CONCLUSION

Although during the last years, research on and with

swarm intelligence has reached an impressive state,

there are still many open problems, and new

application areas are continually emerging for the

optimization paradigms.

We undertook a comparative study of EFPA with

classical FPA over a test-suite comprising 5 well-

known numerical benchmarks and the Loney’s

solenoid problem. Our simulation results indicate

that the EFPA remains always better than FPA. In

near future, we are planning to compare the EFPA

with good performing algorithms available in

literature, such differential evolution and covariance

matrix adaptation evolution strategy.

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