Differential evolution algorithm 263
because it biases the new trial solution in the direction where
the best one of three individuals is chosen for the mutation.
As an extension of the classic DE/current-to-best strategy,
Zhang and Sanderson (2009) presented a new mutation
strategy called DE/current-to-pbest/1 in their algorithm
JADE. This new strategy utilises the information of other
good solutions instead of only adopting the best individual.
As another well-known improved variant of DE/current-
to-best/1, Das et al. (2009) modified this strategy by
introducing a local neighbourhood model. In this model,
each target vector is mutated by using the best individual
solution found so far in its small neighbourhood. What’s
more, this local mutation model is combined with a global
mutation model through a weight factor. Along with the
idea of generating mutant vectors within neighbourhoods,
Qu et al. (2012) recently proposed a neighbourhood-based
mutation strategy to maintain the multiple optima found
during the evolution. In their neighbourhood mutation
strategy, difference vector is limited to a number of similar
individuals as measured by Euclidean distance. Epitropakis
et al. (2011) presented a proximity-based mutation operation by
incorporating information from neighbouring individuals.
Instead of focusing on the mutation strategy, Wang et al.
(2012a) studied limitations of conventional crossover
operations in DE, and presented an orthogonal crossover
operator which is based on orthogonal design for enhancing
the search ability of DE.
Liu and Lampinen (2005) proposed a fuzzy adaptive DE,
named FADE, which uses a fuzzy logic controller to set the
probability of mutation and crossover operations. Brest et al.
(2006) presented an adaptive DE (jDE) for automatically
adapting the parameters F and CR for each individual. Their
reported experiments show that jDE achieves better results
than the classical DE, classical evolutionary programming,
and fast evolutionary programming (Yao et al., 1999) on most
test functions. Recently, Ghosh et al. (2011) proposed a
fitness-based adaptation scheme for adaptive adjustment of
the parameter F in their algorithm FiADE. In addition to the
automatic adjustment of F and CR, adaptive or self-adaptive
mutation strategy also shows promising performance.
Qin and Suganthan (2005) presented a self-adaptive DE
(SaDE) for numerical optimisation. In SaDE, several mutation
strategies coexist rather than a single strategy as in the classic
DE. Recently, Gong et al. (2011) proposed an adaptive
strategy selection scheme in DE. A strategy pool is composed
of four distinct mutation strategies and the most suitable
strategy is picked out from this pool during the evolution.
Mallipeddi et al. (2011) introduced an ensemble of mutation
strategies and control parameters with DE (EPSDE), in which
a pool of distinct mutation strategies along with a pool of
values of each parameter coexists throughout the evolution
process and competes to produce offspring. The experimental
results showed that EPSDE outperforms SaDE, jDE, and
JADE on a majority of test problems. Similarly, Wang et al.
(2011a) choose three mutation strategies and three parameter
settings to construct a strategy candidate pool and parameter
candidate pool, respectively. Unlike the EPSDE, however,
the mutation strategy and parameter settings are combined
randomly to generate the trial vectors.
From the above brief review, it can be seen that many
works have been dedicated to adaptively adjust the mutation
strategy and parameters of F and CR. However, only few
works have been focused on how to adjust the parameter of
population size. Teo (2006) firstly attempted at self-adapting
the population size parameter in DE. He argued that an
absolute encoding methodology for self-adapting population
size in DE produced results with greater optimisation
reliability compared to a relative encoding methodology. As
an enhanced version of jDE, Brest and Maučec (2008)
extended jDE with a population size reduction scheme for
high-dimensional optimisation problems.
In this section, we only present a brief overview of some
related works with DE variants, and some good comprehensive
surveys can be referred to Das and Suganthan (2010) and Neri
and Tirronen (2010).
3 DE with dynamic population scheme
For complex multimodal problems, DE always suffers from
premature convergence because of quick losing of diversity.
To address this issue, a reasonable way is to add some new
individuals into the population for increment of dissimilarities
among individuals. Meanwhile, the search regions can also be
extended by doing this. However, the population size cannot
always be extended owing to the limited computational
resources and the running time burden. So in our approach a
predefined maximum population size
max
NP is given. In fact,
almost every approach of variable population size also has
this parameter. In addition to
max
NP , another two questions
should be tackled for the variable population size of DE: (1)
how many new individuals should be added if the population
is considered to be trapped into local minima, and (2) how to
generate these new individuals if the incremental number is
determined. As solutions to these two questions, in our
proposed DP scheme, the logistic model and OBL strategy
are employed, respectively.
3.1 Logistic model for controlling population size
The logistic model was first proposed by Verhulst in 1838
as a way of describing population growth in a limited
environment, which is a slight modification of Malthus’s
model. This model describes that the rate of reproduction is
proportional to both the existing population and the amount
of available resources, and it can be formalised by the
following differential equation (Storn and Price, 1997):
==1
Pt
Pt pt Pt
M
(8)