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8/11/2019 Velocity Self Adaptation
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2008 IEEE Swann Intelligence Symposium
St. Louis MO USA, September21-23, 2008
Velocity Self-Adaptation
ade
Particle Swarm Optimization
aster
Guangming Lin, Lishan Kang, Yongsheng Liang, Yuping Chen
bstract The lognormal self-adaptation has been used
extensively in evolutionary programming EP) and evolution
strategies ES) to adjust the search step size for each objective
variable. The Particle Swarm Optimization PSO) relies on two
kinds of factors: velocity and position of particles to generate
better particles. In this paper, we propose Self-Adaptive Velocity
PSO SAVPSO) in which we firstly introduce lognormal
self-adaptation strategies to efficiently control the velocity of
PSO. Extensive empirical studies have been carried out to
evaluate the performance of SAVPSO, standard PSO and some
other improved versions of PSO. From the experimental results
on 7 widely used test functions, we can show that
SA
VPSO
outperforms standard PSO.
I INTRODUCTION
volutionary algorithms EAs) have been applied to lnany
optimization problems successfully in recent years. They
are population-based search algorithms with the
generation-and-test feature [1, 2]. Newoffspring are generated
by perturbations and tested to detennine the acceptable
individuals for the next generation. For large search spaces,
the methods
of
EAs are more sufficient than the classical
exhaustive methods; they are stochastic algorithms whose
search methods model some natural phenomena: genetic
inheritance and Darwinian strife for survival. The best known
techniques in the class
of
EAs are Genetic Algorithlns GA),
Evolution Strategies ES), Evolution Programming EP), and
Genetic Programlning GP). The Particle SwannOptimization
PSO) is also a stochastic search algorithln first proposed by
Kennedy and Eherhart [8,9], which developed out of work
simulating themovement
of
flocks
of
birds. PSO shares many
features with EAs. It has shown to be an efficient, robust and
simple optimization algorithln. PSO has been applied
successfully to many different kinds
of
problems [18, 19]
Optimization using EAs and PSO can be explained by two
lnajor steps:
1
Generate the solutions in the current population, and
Manuscript received June 16, 2008. This work was supportedin part by the
National Natural Science Foundation
of
China No. 60473081).
Guangming Lin is with Shenzhen Institute
of
Information Technology.
No.1068 West Niguang Road, Shenzhen 518029, China corresponding
author, phone: 86-755-25859105; e-mail: lingm@sziit.com.cn)
Lishan Kang is with School
of
Computer Science, China University of
Geoscience, Wuhan, China
Yongsheng Liang is with Shenzhen Institute
of
InformationTechnology.
Yuping Chen is with School of Computer Science, China University of
Geoscience, Wuhan, China
2 Select the next generation from the generated and the
current solutions.
These two steps can be regarded as a population-base
version of the classical generate-and-test method, where we
use mutation or velocity and position in PSO) to generated
new solutions and selection is used to test which of the newly
generated solutions should survive to the next generation.
Fonnulating EAs as a special case
of
the generate-and-test
method establishes a bridge between PSO and other search
algorithms, such as EP, ES, GA, simulated annealing SA),
tabu search TS), and others, and thus facilitates
cross-fertilization amongst different research areas.
Standard PSO perfonns well in the early iterations, but
has problelns reaching a near optimal solution in some
of
the
multi-modal optitnization problelns [8]. PSO could often
easily fall into local optima, because the particle could quickly
get closer to the best particle. Both Eberhart [8] and Angeline
[10] conclude that hybrid models of the EAs and the PSO,
could lead to further advances. Some researchers have been
done to tackle this problem [18, 19] In [18, 19], a method
hybrid Fast EP and PSO to fonn a Fast PSO, which is uses
Cauchy mutation operator to mutate the best position
of
particles gbest, It is to hope that the long jump from Cauchy
mutation could get the best position out
of
the local optitna
where it has fallen. FPSO focus on the best position
of
particle
gbest. Actually in PSO procedure, there is another important
factor is the velocity
of
particle. During PSO search, theglobal
best position gbest and the current best position of particles
pbest indicate the search direction. The velocity
of
particle is
the search step size. In [2] we analyzed the importanceof steps
size affect the perfonnance
of
EAs. In this paper we focus on
velocity the search step size of PSO. We first introduce the
10gnonna1
self-adaptation strategy to control the velocity
of
PSO. According to the global optimization search strategy, in
the early stages, we should increase the step size to enhance
the global search ability, and in the final stages, we should
decrease the step size to enhance the local search ability. The
characteristics of lognonnal function fit this search strategy
very well. We proposed a new self-adaptive velocity PSO
SAVPSO) algorithm to efficiently control the global and
local search
of
PSO. Weuse a suite
of
7 functions to test PSO
and SAVPSO. We can see SAVPSOsignificantly outperfonns
PSO in all the test functions.
The rest
of
thepaper is organized as following: Section 2
fonnulates the global optimization problem considered in this
paper and describes the implementation
of
EP, FEP and PSO.
Section 3 describes the implementation
of
the new SAVPSO
978-1-4244-2705-5/08/ 25.00 2008 IEEE
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8/11/2019 Velocity Self Adaptation
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(2.3)
algorithm. Section 4 lists benchmark functions use in the
experiments, and gives the experimental settings. Section 5
presents and discusses the experimental resul ts. F inally,
Section 6 concludes with a summary and a few remarks.
II. OPTIMIZATION USING EP, FEP AND PSO
A global minimization problem can be represented as a
pair S,j), where S
R
n is a bounded set on
R
n andf SH
R
is an n-dimensional real-valued function. The problem is to
find a point
x min E S
such that
j x min
is a global minitnum on
S. More specifically, it is required to find x min E S such that
\;fXE
S : f x m i n ~ f x ,
Wherej
does not need to
be
continuous but it must be bounded.
This paper considers only the unconstrained optimization
functions.
A Classical Evolutionary Programming
Fogel [4] and Back and Schwefel [6] have indicated that
CEP with self-adaptive mutation usually performs better than
CEP withou t se lf-adapt ive muta tion for the funct ion they
tested. For this reason, CEP with self-adaptive mutation will
be invest igated in this paper. As described by Back and
Schwefel [6], CEP implemented in this study is as follows:
1. Generate the initial population of Jl individuals, and
set
k=1.
Each individual is taken as a pair of
real-valued vectors, Xi,TJi),ViE {1,2,oo.,p} , where
Xi
s are var iables and
lli
s are standard deviations for
Gaussian mutations (also know as strategy parameters
in self-adaptive evolutionary algorithms).
2. Evaluate the fitness score for each
individual
Xi 1JJ,
Vi
E {1,2,oo.,p}, of
the population
based on the objective function
f x
i
) .
3. Each parent XpTJi) Vi E
{1,2,oo.,jl},
creates a single
offspring X i ,
i
) by: for}=1,2,
n,
x; j = xi J) + 11/J)N
j
(0, 1 (2.1)
17i j)=17i j) xexp T N O, l )+T N
j
O,l; 2.2)
Where
Xi
j),
Xi
j),
i
j and
i
j)
denote the j-th
component of the vecto rs
Xi Xi
1]i and 1]i
respectively.
N O 1
denote a normally distributed
one-d imens ional r andom number wi th mean and
standard deviation 1. Nj (0,1) indicates that a new random
number is generated for each value of}.
The
and
ar e commonly set to
Mr1and
[6].
4. Calculate
the fitness
of
each
offspring
x; , l; ), V iE
{1,2,
...
,
p}
5. Conduct pair WIse comparison over t he union of
parents
Xi
17
i
and
offspring
x
i
, 1]
), i i
E
{1,2,..., , l l} .
For each individual, q
opponents are chosen uniformly at random from all the
parents and offspring. For each comparison, if an
individual s fitness is better than its opponent, then it is
the winner.
6. Select
Jl
individuals , out of (
Xi
i
) and
( x; ,Tl; , 17 i E {1,2,... ll} those are winners, to be
parents in the next generation.
7. Stop if the halting cri terion is sat isfied; otherwise,
k=k+
1
and go to Step 3.
B The Standard Particle
Swarm
Optimization
Particle swarm optimization (PSO) algorithm is a recent
addition to the list of global search Inethods.
It
is a population
based stochast ic opti tnization technique developed by
Kennedy and Eberhart [8] in 1995, inspired
by
social behavior
of
organisms such as fish schooling, bird flocking and swarm
intelligence theory.
PSO
has been found to be robus t in
solving continuous nonlinear optimization problems. Recently,
PSO has been successfully employed to solve non-smooth
cOInplex optimization problems. In past several years,
PSO
has been widely appl ied in many research and applica tion
areas.
The Particle Swarm Optimization s imula tes social
behavior such as a school of flying birds in searching of food.
The behavior of each individual is impacted by the behaviors
of
neighborhoods and the swarm.
PSO
is initialized with a population
of
random solutions
of the objective function. It uses a population of individuals,
called particles, with an initial population distributed randomly
over
the search space.
It
searches for the optimal value
of
a
function by updat ing the popula tion through a number of
generations. Each
new
populat ion is generated from the old
population with a set of simple rules that
have
stochastic
elements.
Each part ic le searches the opt imum posi tion l ike the
behavior of a b irds search food that
it
flown through the
problem space by following the current optimal particles. The
position of each particle is updated by a new velocity
calculated through equations (2.3) and (2.4) which is based on
its previous velocity, the position at which the best solution so
far has been achieved by the particle (pbest or pb), and the
position at which the best solution so far has been achieved by
the global population (gbest
or
gb).
v i
+1 =
OJxv i)+c
1
Xli
x pb-x i))+
c
2
x r
x gb-x i))
x i + = x i) + v i +
(2.4)
In equation(
1
,