Apical-dominant particle swarm optimization
Zhihua Cui
a,b,
*
, Xingjuan Cai
b
, Jianchao Zeng
b
, Guoji Sun
a
a
State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China
b
Division of System Simulation and Computer Application, Taiyuan University of Science and Technology, Taiyuan 030024, China
Received 4 March 2008; received in revised form 18 June 2008; accepted 26 June 2008
Abstract
Particle swarm optimization (PSO) is a new stochastic population-based search methodology by simulating the animal social behaviors
such as birds flocking and fish schooling. Many improvements have been proposed within the framework of this biological assumption.
However, in this paper, the search pattern of PSO is used to model the branch growth process of natural plants. It provides a different poten-
tial manner from artificial plant. To illustrate the effectiveness of this new model, apical dominance phenomenon is introduced to construct a
novel variant by emphasizing the influence of the phototaxis. In this improvement, the population is divided into three different kinds of buds
associated with their performances. Furthermore, a mutation strategy is applied to enhance the ability escaping from a local optimum. Sim-
ulation results demonstrate good performance of the new method when solving high-dimensional multi-modal problems.
Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in
China Press. All rights reserved.
Keywords: Apical-dominance phenomenon; Particle swarm optimization; Branch growth model; High-dimensional multi-modal benchmarks
1. Introduction
Particle swarm optimization (PSO) [1,2] is a population-
based, self-adaptive swarm intelligent technique inspired by
the simplified animal social behaviors such as fish school-
ing, birds flocking, and insects herding. Due to the fast con-
vergence speed and easy implementation, it has been
successfully applied in many areas such as neural network,
dynamic web organizing, fitness prediction, and mountain
clustering [3–5].
In the PSO algorithm, each individual (called particle)
represents a potential solution, and flies over the problem
space to seek the food (optimum point). The particles
adapt their search patterns with collaborative and compet-
itive manners. Therefore, the process of seeking optima in
the problem space is analogous to the food searching pro-
cess of birds in nature. In this paper, only the following
unconstrained optimization problems are considered:
Minf ðxÞ; x ¼½x
1
; ...; x
D
# R
D
ð1Þ
Suppose v
j
ðtÞ¼ðv
j
1
ðtÞ; v
j
2
ðtÞ; ...; v
j
D
ðtÞÞ is the velocity of
the jth particle, where v
j
k
ðtÞ represents the kth coordinate
value of the velocity vector. Inspired by the artificial Boid
model made by Reynolds [6], Eberhart and Kennedy [1,2]
proposed a velocity update equation
v
j
ðt þ 1Þ¼wv
j
ðtÞþc
1
r
1
ðp
j
ðtÞx
j
ðtÞÞ þ c
2
r
2
ðp
g
ðtÞx
j
ðtÞÞ
ð2Þ
where the vector x
j
ðtÞ¼ðx
j
1
ðtÞ; x
j
2
ðtÞ ; ...; x
j
D
ðtÞÞ represents
the position of particle j. In each iteration, it is updated
with the following manner
x
j
ðt þ 1Þ¼x
j
ðtÞþx
j
ðt þ 1Þð3Þ
where p
j
ðtÞ¼ðp
j
1
ðtÞ; p
j
2
ðtÞ; ...; p
j
D
ðtÞÞ is the best location
found by particle j, p
g
ðtÞ¼ðp
g
1
ðtÞ; p
g
2
ðtÞ; ...; p
g
D
ðtÞÞ is the
best location found by the entire swarm. Inertia weight w
1002-0071/$ - see front matter Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited
and Science in China Press. All rights reserved.
doi:10.1016/j.pnsc.2008.06.005
*
Corresponding author. Tel.: +86 03516962060; fax: +86 03516998027.
E-mail address: cuizhihua@gmail.com (Z. Cui).
www.elsevier.com/locate/pnsc
Available online at www.sciencedirect.com
Progress in Natural Science 18 (2008) 1577–1582