Biomedical Engineering Reference
In-Depth Information
2.2.4.3 Archive Truncation Method
In.each.generation,.the.archive.of.the.next.generation.is.formed.by.copying.
all. nondominated. individuals. from. the. archive. and. the. population.. If. the.
number.of.all.those.nondominated.individuals.exceeds.the.archive.size. | |,.
the.archive.truncation.method.is.used.to.iteratively.remove.some.nondomi-
nated.individuals.until.the.number.reaches.|
A
.
An. individual
i
. is. selected. for. removal. if. ≤
i
j
. for. all.
j A
t
1
,. with. ≤
∈
i
j
.
d
+
d
defined.as
(
)
0
k A
t
|
|:
1
k
k
.or.
0
k A
|
|:
0
l
k
s.t.
l
l
and
k
k
∀
<
<
σ
= σ
∃
<
<
∀
<
<
σ = σ
σ
< σ
+
i
j
t
+
1
i
j
i
j
.
where.σ
i
k
.is.the.distance.of.the.individual
i
to.its.
k
th.nearest.neighbor.in.
A
t
+
,
.
1
and
t
is.the.counter.of.generations.
It.means.that.an.individual.having.the.minimum.distance.to.another.indi-
vidual.is.chosen.at.each.stage..If.there.are.several.individuals.with.the.mini-
mum.distance,.the.tie.is.broken.by.considering.the.second-smallest.distances.
and.so.forth.
This.archive.truncation.method.can.avoid.the.removal.of.boundary.non-
dominated.solutions,.which.is.more.favorable.for.MOPs.
2.2.5 Pareto archived evolution Strategy
The.Pareto.archived.evolution.strategy.(PAES).is.a.local.search-based.algo-
rithm. [47,48],. imitating. the. evolution. strategy.. Its. flowchart. is. shown. in.
Figure 2.10.
.
In. the. PAES,. the. only. genetic. operation,. mutation,. provides. a.
hill-climbing-like.strategy..An.archive.with.limited.size.is.used.to.store.the.
previously.found.nondominated.solutions.
The.PAES.has.three.versions:.(1 + 1)-PAES,.(1 + λ)-PAES,.and.(.μ + λ)-PAES..
The.first.one.means.that.a.single.parent.generates.one.offspring..The.second.
one.represents.a.single.parent.producing.λ.offspring..The.last.one.means.that.
a. population. of. μ. parents. generates. λ. offspring.. In. comparing. these. three.
versions,.(1 + 1)-PAES.has.the.lowest.computational.overhead.(i.e.,.it.is.a.faster.
algorithm).and.is.the.simplest.
The.unique.characteristic.of.the.PAES.is.the.adaptive.grid.scheme..Its.notion.
is. the. use. of. a. new. crowding. procedure. based. on. recursively. dividing. up.
the.
M
-dimensional.objective.space.to.trace.the.crowding.degrees.of.different.
regions.in.which.the.nondominated.solutions.in.the.archive.fall..Its.purpose.is.
to.diversify.those.nondominated.solutions.and.help.to.remove.excessive.non-
dominated.solutions.located.in.the.crowded.grids.(i.e.,.the.grids.with.a.high.
degree.of.crowding).if.the.number.of.those.solutions.exceeds.the.archive.size.
The.procedures.of.the.adaptive.grid.scheme.are.outlined.as.follows:.When.
each. solution. is. generated,. it. is. necessary. to. determine. its. grid. location. in.
the. objective. space.. Suppose. that. the. range. of. the. space. is. deined. in. each.
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