改进差分进化算法的定向变异操作
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"A directional mutation operator for differential evolution algorithms"
在优化领域，差分进化（Differential Evolution，简称DE）是一种广泛应用的全局数值优化算法。DE以其简单性和强大的全局搜索能力而受到广泛关注。本研究论文深入探讨了DE算法中的一个关键组件——变异算子，并提出了一种定向变异算子来提升DE的性能。
传统的DE变异操作中，随机变化是通过取两个随机选择的不同个体之间的差异来生成的。这种差异向量在DE的进化过程中起着核心作用。然而，DE在搜索过程中的一个问题在于，即使已经找到了迄今为止的最佳适应度解，算法也可能困于局部最优，无法有效地探索解决方案空间的其他区域。
作者Xin Zhanga和Shiu Yin Yuenb在研究中指出，定向变异算子的引入旨在解决这一问题。与传统的随机变异不同，定向变异算子考虑了种群中已知的最优解方向，使得变异操作更具有导向性，从而能更有效地引导搜索过程走向全局最优解。这种方法可以增强DE对局部极小值的逃逸能力，提高其全局优化性能。
论文详尽分析了定向变异算子的原理、实现方式以及与通用变异算子的对比。实验部分展示了在一系列基准测试函数上的应用效果，证明了定向变异算子在解决复杂优化问题时的优越性。关键词包括：差分进化、定向变异、通用变异算子、全局数值优化。这些关键词表明该研究主要关注DE算法的改进，特别是针对变异策略的创新，以及这些改进对全局优化任务的影响。
这篇论文为DE算法的优化提供了一个新的视角，定向变异算子的提出有助于提升DE在处理高维度、非线性或多模态优化问题时的表现，对于理解和改进进化计算方法具有重要意义。
532
X.
Zhang,
S.Y.
Yuen
/
Applied
Soft
Computing
30
(2015)
529–548
a
statistically
meaningful
way.
Brest
et
al.
[15]
proposed
another
self-adaptation
scheme,
called
jDE.
Both
F
and
Cr
are
encoded
into
the
individuals
and
updated,
relying
on
two
new
parameters
1
and
2
.
Each
individual
is
extended
by
two
dimensions.
As
claimed
by
the
authors,
the
better
parameter
values
lead
to
better
indi-
viduals
which,
in
turn,
are
more
likely
to
survive
and
propagate
offspring.
Brest
et
al.
[16]
reported
a
method
(called
dynNP-DE)
for
adapting
the
population
size.
Mallipeddi
et
al.
[17]
proposed
a
multi-population
DE
algorithm.
Ghosh
et
al.
proposed
a
fitness-
based
adaptive
DE,
called
FiADE
[18].
FiADE
tries
to
control
the
two
parameters
F
and
Cr
based
on
fitness
values
of
individuals.
2.2.2.
Variants
on
the
mutation
operator
Das
et
al.
[19]
proposed
a
neighbor-based
mutation
operator
which
is
an
extension
of
(5).
Zhang
et
al.
[20]
proposed
a
DE/target-
to-pbest/1/bin
mutation
operator
which
is
also
a
generalization
of
(5)
.
Epitropakis
et
al.
[21]
proposed
a
proximity
based
mutation
operator
framework.
It
is
based
on
the
observation
that
a
clustering
structure
of
the
population
can
arise
during
the
evolutionary
pro-
cess
of
DE.
Actually,
this
framework
provides
a
way
for
individual
selection,
hence
it
can
be
applied
to
any
mutation
strategy.
Its
effi-
ciency
has
been
experimentally
demonstrated
by
combining
with
six
DE
mutation
variants
and
eight
state
of
the
art
DE
algorithms.
However,
it
shows
only
a
little
improvement
for
(4),
and
does
not
show
improvement
for
(5).
It
spends
additional
computational
cost
in
calculating
and
storing
the
affinity
matrix
and
probability
matrix.
This
cost
can
be
substantial
when
the
cost
of
evaluation
is
trivial
[21].
Motivated
from
structured
EAs,
Dorronsoro
et
al.
[22]
study
several
DE
variants
using
seven
different
population
structures
based
on
complex
network
models.
A
new
mutation
operator,
called
GPBX-˛,
is
proposed.
The
authors
design
a
heterogeneous
distributed
algorithm
with
two
islands,
called
HdDE.
Islam
et
al.
[23]
proposed
modified
DE
with
p-best
crossover
(MDE
pBX).
A
new
mutant
variant
called
DE/current-to-
gr
best/1/bin
is
proposed.
It
randomly
selects
q%
individuals
of
the
total
population
to
form
a
group,
then
the
best
member
of
this
group
is
used
to
perturb
the
target
vector.
2.2.3.
Control
of
both
parameters
and
strategies
Qin
et
al.
[13]
proposed
a
self-adaptive
variant
of
DE,
called
(SaDE).
They
try
to
control
both
trial
vector
generation
strate-
gies
and
the
parameters
F
and
Cr.
Zhao
et
al.
[24]
proposed
an
enhancement
of
SaDE
by
incorporating
the
JADE
mutation
strat-
egy
with
archive
and
hybridizing
with
modified
multi-trajectory
search
(SaDE-MMTS).
Wang
et
al.
[25]
proposed
composite
DE
(CoDE).
CoDE
selects
three
trial
vector
generation
strategies
and
three
control
param-
eter
settings
based
on
the
experimental
results
reported
in
the
literature.
In
[26],
Mallipeddi
et
al.
proposed
an
ensemble
of
parameters,
mutation
strategies
and
crossover
strategies
(EPSDE1).
Mallipeddi
et
al.
[27]
proposed
to
employ
an
ensemble
of
parameters
and
strategies
with
the
DE
(EPSDE2).
2.2.4.
Hybrids
with
other
methods
Rahnamayan
et
al.
[28]
proposed
an
opposition
based
DE
(ODE)
for
accelerating
the
convergence
rate
of
DE.
ODE
utilizes
opposite
point
in
both
population
initialization
and
generation
jumping.
DE
has
been
hybridized
with
other
direct
search
method
such
as
PSO
[29,30],
bacterial
foraging
[31],
and
biogeography-based
optimization
[32].
The
hybrid
DE
with
biogeography-based
opti-
mization
has
also
been
applied
to
economic
load
dispatch
problems
[33].
Sardar
et
al.
[34]
applied
gradient-based
repair
technique
to
handle
constraint
violations
for
constrained
optimization
prob-
lems.
The
gradient-based
repair
is
incorporated
with
(3).
2.2.5.
Memetic
variants
of
differential
evolution
The
DE
scheme
contains
a
limited
amount
of
search
moves,
and
some
researchers
try
to
increase
the
set
of
search
moves
to
enhance
the
performance
of
DE
[35].
Weber
et
al.
[36]
propose
a
scale
fac-
tor
inheritance
method
for
distributed
DE.
The
authors
have
also
studied
the
effect
of
different
scale
factor
values
in
distributed
DE
[37].
Mininno
et
al.
[38]
propose
compact
DE
(cDE).
cDE
belongs
to
the
class
of
estimation
of
distribution
algorithms
(EDAs).
Neri
et
al.
[39]
propose
disturbed
exploitation
compact
DE
(DEcDE).
DEcDE
employs
a
compact
structure
as
in
cDE.
It
periodically
perturbs
the
virtual
population
which
could
inhibit
premature
convergence.
Noman
et
al.
[40]
propose
to
accelerate
DE
using
adaptive
hill-
climbing
simplex
crossover
(DEahcSPX).
Caponio
et
al.
[41]
propose
super-fit
memetic
DE
(SFMDE).
Neri
et
al.
[42]
propose
scale
factor
local
search
DE
(SFLSDE).
2.2.6.
Convergence
analysis
Ghosh
et
al.
[43]
studied
the
convergence
analysis
of
DE.
Under
suitable
assumptions,
they
showed
that
the
dynamics
asymp-
totically
converges
around
the
global
optimum
of
the
objective
function,
assuming
the
shape
of
a
Dirac
delta
distribution.
2.2.7.
Application
of
differential
evolution
DE
finds
many
applications
in
solving
real
world
problems
like
automatic
clustering
[44],
design
of
fractional
order
controllers
[45],
Cartesian
robot
control
[46]
and
so
on.
3.
The
directional
mutation
operator
This
section
will
describe
the
proposed
directional
mutation
operator.
It
can
be
combined
with
any
other
DE
mutation
strat-
egy.
When
combined
with
CDE,
it
is
called
directional
mutation
differential
evolution
(DMDE).
An
empirical
analysis
is
reported
by
comparing
DMDE
with
CDE.
Then
the
computation
complexity
of
DMDE
is
analyzed
with
CDE
as
the
baseline.
3.1.
Description
of
the
directional
mutation
operator
First,
we
give
a
formal
definition
about
fitness
improvement
at
a
generation.
Definition
(Fitness
Improvement):
Without
loss
of
generality,
let
the
optimization
be
a
minimization.
Assume
in
a
DE
algorithm,
G
g
is
the
g
th
generation
with
a
population
of
individuals
G
g
=
{x
1,g
,
·
·
·
,
x
i,g
,
·
·
·
,
x
NP,g
}.
The
minimum
fitness
in
G
g
is
denoted
by
f
best
g
.
Through
mutation
and
crossover,
a
pool
of
NP
trial
vectors
U
=
{u
1,g
,
·
·
·
,
u
i,g
,
·
·
·
,
u
NP,g
}
is
obtained.
In
the
survivor
selection
step,
a
new
population
G
g+1
=
{x
1,g+1
,
·
·
·
,
x
i,g+1
,
·
·
·
,
x
NP,g+1
}
is
obtained
by
NP
parallel
(1
+
1)
competitions
between
u
i,g
and
x
i,g
,
i
=
1,
2,
·
·
·
,
NP.
If
there
exists
at
least
one
index
i,
such
that:
f
(u
i,g
)
≤
f
best
g
(8)
then
this
DE
algorithm
has
fitness
improvement
at
G
g
.
Suppose
for
one
index
i,
f
(u
i,g
)
≤
f
best
g
,
in
this
case
u
i,g
will
survive
and
become
x
i,g+1
.
Thus,
u
i,g
must
contain
some
better
components
than
x
i,g
.
The
difference
vector
(u
i,g
−
x
i,g
)
should
be
a
descent
direction
at
x
i,g
.
For
all
possible
index
i
(1≤i≤
NP),
a
pool
of
such
difference
vectors
can
be
calculated.
Denote
the
pool
of
dif-
ference
vectors
as
g+1
=
{ı
1,g+1
,
·
·
·
,
ı
k,g+1
,
·
·
·
,
ı
ND,g+1
},
where
ND
(1≤ND≤
NP)
is
the
size
of
the
difference
vector
pool.
This
pool
indi-
cates
the
possible
descent
directions
found
at
G
g
.
These
directions
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