![](bg3.png)
where
rand
j
is a random index chosen from [1, D] to
ensure at least one component is different from
G
i
U
and
G
i
X
, and the parameter Cr is within the range [0,
1], indicating the crossover rate of the generation. If
any component of the trial vectors is beyond the
search space, they will be reinitialized randomly and
uniformly within the search space.
2.4 Selection
In this phase, we determine which vector is going
into the next generation and which should be
deleted. The procedure is done following rule for the
function minimization:
otherwiseU
UfXfifX
X
G
i
G
i
G
i
G
i
G
i
,
))()((,
1
(4)
Every trial vector is only compared with its target
vector, and the one with better fitness is kept. Hence,
all the individuals of the next generation are going to
get better or remain the same, thus the whole
population evolves.
3 REPLICATOR DYNAMIC
INSPIRED DE ALGORITHM
Being a crucial factor of the DE algorithm, control
parameters selection determines the performance of
the algorithm directly. Hence, a good deal of
research on the parameters selection of DE has been
done. Storn (1995) suggested that F within the range
[0.5,1], Cr in [0.8,1] and NP = 5D or 10D. Gämperle
et al. (2002) suggested that NP be between 3D and
8D, F= 0.6, and Cr between [0.3,0.9]. At the same
time, several adaptive and self-adaptive mechanisms
have been proposed to dynamically change the value
of the parameters. Zaharie (2003) used a
multipopulation method for the parameter adaptation
(ADE). Omran et al. (2005) proposed a mechanism
to self-adapt the scaling factor F (SDE). Later on,
Brest et al. (2006) encoded F and Cr into individuals
and modulate them by two parameters. In the same
year, Teo (2006) proposed a DE algorithm with a
dynamic population sizing strategy based on self-
adaptation (DESAP). Lately, Qin et al. (2009)
proposed SaDE, in which both generation strategy
and the parameters are adapted.
In our paper, we focus on the adaptation of Cr
during the evolution, as Cr is an especially
significant parameter. The suitable choice of Cr can
lead to good result while an improper one may result
in the failure of the algorithm (Price et al., 2005).
3.1 Inspired by Replicator Dynamic
The main idea of this paper is to self-adapt the
probability distribution of the crossover rate, so that
the parameter could be more suitable to various
kinds of problems. At the same time, different
distributions of Cr may perform better at different
generations for a certain problem, so the distribution
of Cr is expected to be fit for every moment of the
evolution as well. To achieve this, a mechanism of
multiple evolutions is proposed: the first evolution
refers to DE algorithm itself, and the second one
means that the probability distribution of Cr value is
evolving independently with the idea of evolutionary
game theory.
We build a candidate set (CRSet), containing
several possible values of Cr. Whenever the
crossover operation is executed, each individual
choose one value from the set via a particular
probability distribution. The value of Cr is a real
number within the range [0, 1], and the set is
expected to cover the range uniformly. In our
proposal, we let
CRSet
,,,{
321
CRCRCR },
54
CRCR
,
where
i
CR
is set to (0.2×i-0.1). For each
i
CR
, a
i
P
is assigned to indicate the probability to choose it,
the distribution of
i
P
is
. At each generation,
every individual choose a Cr from the CRSet via the
distribution of
i
P
, and the distribution
is evolving
according to the fitness of each
i
CR
of the current
and previous generations with replicator dynamic.
Probability distribution to choose values for Cr is
very similar to mixed strategy equilibrium of a game
theory, and a definite value of Cr corresponds with a
pure strategy. Our attention is on the dynamically
changing of the distribution, thus a method of
evolutionary game theory is introduced. We assume
that a new population of plentiful individuals is
generated to seek a reasonable probability
distribution for
i
CR
with the idea of evolution. Any
individual in the population is called replicator,
choosing a certain value in the CRSet and passing its
choice to the descendants without modification. Let
tn
i
be the number of individuals choosing
i
CR
at
time point t, then the total population size is
tntN
ii
5
1
, and the proportion of individuals to
choose
i
CR
is
tNtntpcr
ii
/
. The population
state is the distribution of
tpcr
i
, i.e.,
,(
1
tpcrtP
cr
.
),,,
5432
tpcrtpcrtpcrtpcr
, Let
and
be the
ReplicatorDynamicInspiredDifferentialEvolutionAlgorithmforGlobalOptimization
135