<s>
In	O
computational	O
science	O
,	O
particle	B-Algorithm
swarm	I-Algorithm
optimization	I-Algorithm
(	O
PSO	O
)	O
is	O
a	O
computational	O
method	O
that	O
optimizes	O
a	O
problem	O
by	O
iteratively	B-Algorithm
trying	O
to	O
improve	O
a	O
candidate	O
solution	O
with	O
regard	O
to	O
a	O
given	O
measure	O
of	O
quality	O
.	O
</s>
<s>
PSO	O
is	O
originally	O
attributed	O
to	O
Kennedy	O
,	O
Eberhart	O
and	O
Shi	O
and	O
was	O
first	O
intended	O
for	O
simulating	B-Application
social	O
behaviour	O
,	O
as	O
a	O
stylized	O
representation	O
of	O
the	O
movement	O
of	O
organisms	O
in	O
a	O
bird	O
flock	O
or	O
fish	O
school	O
.	O
</s>
<s>
The	O
book	O
by	O
Kennedy	O
and	O
Eberhart	O
describes	O
many	O
philosophical	O
aspects	O
of	O
PSO	O
and	O
swarm	B-Architecture
intelligence	I-Architecture
.	O
</s>
<s>
PSO	O
is	O
a	O
metaheuristic	B-Algorithm
as	O
it	O
makes	O
few	O
or	O
no	O
assumptions	O
about	O
the	O
problem	O
being	O
optimized	O
and	O
can	O
search	O
very	O
large	O
spaces	O
of	O
candidate	O
solutions	O
.	O
</s>
<s>
Also	O
,	O
PSO	O
does	O
not	O
use	O
the	O
gradient	O
of	O
the	O
problem	O
being	O
optimized	O
,	O
which	O
means	O
PSO	O
does	O
not	O
require	O
that	O
the	O
optimization	O
problem	O
be	O
differentiable	O
as	O
is	O
required	O
by	O
classic	O
optimization	O
methods	O
such	O
as	O
gradient	B-Algorithm
descent	I-Algorithm
and	O
quasi-newton	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
However	O
,	O
metaheuristics	B-Algorithm
such	O
as	O
PSO	O
do	O
not	O
guarantee	O
an	O
optimal	O
solution	O
is	O
ever	O
found	O
.	O
</s>
<s>
The	O
two	O
other	O
parameters	O
can	O
be	O
then	O
derived	O
thanks	O
to	O
the	O
constriction	O
approach	O
,	O
or	O
freely	O
selected	O
,	O
but	O
the	O
analyses	O
suggest	O
convergence	B-Algorithm
domains	O
to	O
constrain	O
them	O
.	O
</s>
<s>
The	O
PSO	O
parameters	O
can	O
also	O
be	O
tuned	O
by	O
using	O
another	O
overlaying	O
optimizer	O
,	O
a	O
concept	O
known	O
as	O
meta-optimization	B-Algorithm
,	O
or	O
even	O
fine-tuned	O
during	O
the	O
optimization	O
,	O
e.g.	O
,	O
by	O
means	O
of	O
fuzzy	O
logic	O
.	O
</s>
<s>
This	O
school	O
of	O
thought	O
contends	O
that	O
the	O
PSO	O
algorithm	O
and	O
its	O
parameters	O
must	O
be	O
chosen	O
so	O
as	O
to	O
properly	O
balance	O
between	O
exploration	O
and	O
exploitation	O
to	O
avoid	O
premature	O
convergence	B-Algorithm
to	O
a	O
local	O
optimum	O
yet	O
still	O
ensure	O
a	O
good	O
rate	O
of	O
convergence	B-Algorithm
to	O
the	O
optimum	O
.	O
</s>
<s>
In	O
relation	O
to	O
PSO	O
the	O
word	O
convergence	B-Algorithm
typically	O
refers	O
to	O
two	O
different	O
definitions	O
:	O
</s>
<s>
Convergence	B-Algorithm
of	O
the	O
sequence	O
of	O
solutions	O
(	O
aka	O
,	O
stability	O
analysis	O
,	O
converging	B-Algorithm
)	O
in	O
which	O
all	O
particles	O
have	O
converged	O
to	O
a	O
point	O
in	O
the	O
search-space	O
,	O
which	O
may	O
or	O
may	O
not	O
be	O
the	O
optimum	O
,	O
</s>
<s>
Convergence	B-Algorithm
to	O
a	O
local	O
optimum	O
where	O
all	O
personal	O
bests	O
p	O
or	O
,	O
alternatively	O
,	O
the	O
swarm	O
's	O
best	O
known	O
position	O
g	O
,	O
approaches	O
a	O
local	O
optimum	O
of	O
the	O
problem	O
,	O
regardless	O
of	O
how	O
the	O
swarm	O
behaves	O
.	O
</s>
<s>
Convergence	B-Algorithm
of	O
the	O
sequence	O
of	O
solutions	O
has	O
been	O
investigated	O
for	O
PSO	O
.	O
</s>
<s>
These	O
analyses	O
have	O
resulted	O
in	O
guidelines	O
for	O
selecting	O
PSO	O
parameters	O
that	O
are	O
believed	O
to	O
cause	O
convergence	B-Algorithm
to	O
a	O
point	O
and	O
prevent	O
divergence	O
of	O
the	O
swarm	O
's	O
particles	O
(	O
particles	O
do	O
not	O
move	O
unboundedly	O
and	O
will	O
converge	O
to	O
somewhere	O
)	O
.	O
</s>
<s>
Convergence	B-Algorithm
to	O
a	O
local	O
optimum	O
has	O
been	O
analyzed	O
for	O
PSO	O
in	O
and	O
.	O
</s>
<s>
This	O
means	O
that	O
determining	O
convergence	B-Algorithm
capabilities	O
of	O
different	O
PSO	O
algorithms	O
and	O
parameters	O
still	O
depends	O
on	O
empirical	O
results	O
.	O
</s>
<s>
One	O
attempt	O
at	O
addressing	O
this	O
issue	O
is	O
the	O
development	O
of	O
an	O
"	O
orthogonal	O
learning	O
"	O
strategy	O
for	O
an	O
improved	O
use	O
of	O
the	O
information	O
already	O
existing	O
in	O
the	O
relationship	O
between	O
p	O
and	O
g	O
,	O
so	O
as	O
to	O
form	O
a	O
leading	O
converging	B-Algorithm
exemplar	O
and	O
to	O
be	O
effective	O
with	O
any	O
PSO	O
topology	O
.	O
</s>
<s>
The	O
aims	O
are	O
to	O
improve	O
the	O
performance	O
of	O
PSO	O
overall	O
,	O
including	O
faster	O
global	O
convergence	B-Algorithm
,	O
higher	O
solution	O
quality	O
,	O
and	O
stronger	O
robustness	O
.	O
</s>
<s>
Without	O
the	O
need	O
for	O
a	O
trade-off	O
between	O
convergence	B-Algorithm
( 	O
 '	O
exploitation	O
 '	O
)	O
and	O
divergence	O
( 	O
 '	O
exploration	O
 '	O
)	O
,	O
an	O
adaptive	O
mechanism	O
can	O
be	O
introduced	O
.	O
</s>
<s>
Adaptive	O
particle	B-Algorithm
swarm	I-Algorithm
optimization	I-Algorithm
(	O
APSO	O
)	O
features	O
better	O
search	O
efficiency	O
than	O
standard	O
PSO	O
.	O
</s>
<s>
APSO	O
can	O
perform	O
global	O
search	O
over	O
the	O
entire	O
search	O
space	O
with	O
a	O
higher	O
convergence	B-Algorithm
speed	O
.	O
</s>
<s>
Another	O
research	O
trend	O
is	O
to	O
try	O
and	O
alleviate	O
premature	O
convergence	B-Algorithm
(	O
that	O
is	O
,	O
optimization	O
stagnation	O
)	O
,	O
e.g.	O
</s>
<s>
by	O
reversing	O
or	O
perturbing	O
the	O
movement	O
of	O
the	O
PSO	O
particles	O
,	O
another	O
approach	O
to	O
deal	O
with	O
premature	O
convergence	B-Algorithm
is	O
the	O
use	O
of	O
multiple	O
swarms	O
(	O
multi-swarm	B-Algorithm
optimization	I-Algorithm
)	O
.	O
</s>
<s>
Another	O
argument	O
in	O
favour	O
of	O
simplifying	O
PSO	O
is	O
that	O
metaheuristics	B-Algorithm
can	O
only	O
have	O
their	O
efficacy	O
demonstrated	O
empirically	O
by	O
doing	O
computational	O
experiments	O
on	O
a	O
finite	O
number	O
of	O
optimization	O
problems	O
.	O
</s>
<s>
This	O
means	O
a	O
metaheuristic	B-Algorithm
such	O
as	O
PSO	O
cannot	O
be	O
proven	O
correct	O
and	O
this	O
increases	O
the	O
risk	O
of	O
making	O
errors	O
in	O
its	O
description	O
and	O
implementation	O
.	O
</s>
<s>
A	O
good	O
example	O
of	O
this	O
presented	O
a	O
promising	O
variant	O
of	O
a	O
genetic	B-Algorithm
algorithm	I-Algorithm
(	O
another	O
popular	O
metaheuristic	B-Algorithm
)	O
but	O
it	O
was	O
later	O
found	O
to	O
be	O
defective	O
as	O
it	O
was	O
strongly	O
biased	O
in	O
its	O
optimization	O
search	O
towards	O
similar	O
values	O
for	O
different	O
dimensions	O
in	O
the	O
search	O
space	O
,	O
which	O
happened	O
to	O
be	O
the	O
optimum	O
of	O
the	O
benchmark	O
problems	O
considered	O
.	O
</s>
<s>
Another	O
simpler	O
variant	O
is	O
the	O
accelerated	O
particle	B-Algorithm
swarm	I-Algorithm
optimization	I-Algorithm
(	O
APSO	O
)	O
,	O
which	O
also	O
does	O
not	O
need	O
to	O
use	O
velocity	O
and	O
can	O
speed	O
up	O
the	O
convergence	B-Algorithm
in	O
many	O
applications	O
.	O
</s>
