<s>
Simulated	B-Algorithm
annealing	I-Algorithm
(	O
SA	O
)	O
is	O
a	O
probabilistic	B-General_Concept
technique	I-General_Concept
for	O
approximating	O
the	O
global	O
optimum	O
of	O
a	O
given	O
function	O
.	O
</s>
<s>
Specifically	O
,	O
it	O
is	O
a	O
metaheuristic	B-Algorithm
to	O
approximate	O
global	O
optimization	O
in	O
a	O
large	O
search	O
space	O
for	O
an	O
optimization	O
problem	O
.	O
</s>
<s>
It	O
is	O
often	O
used	O
when	O
the	O
search	O
space	O
is	O
discrete	O
(	O
for	O
example	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
,	O
the	O
boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
,	O
protein	O
structure	O
prediction	O
,	O
and	O
job-shop	B-Algorithm
scheduling	I-Algorithm
)	O
.	O
</s>
<s>
For	O
problems	O
where	O
finding	O
an	O
approximate	O
global	O
optimum	O
is	O
more	O
important	O
than	O
finding	O
a	O
precise	O
local	O
optimum	O
in	O
a	O
fixed	O
amount	O
of	O
time	O
,	O
simulated	B-Algorithm
annealing	I-Algorithm
may	O
be	O
preferable	O
to	O
exact	O
algorithms	O
such	O
as	O
gradient	B-Algorithm
descent	I-Algorithm
or	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
.	O
</s>
<s>
Simulated	B-Algorithm
annealing	I-Algorithm
can	O
be	O
used	O
for	O
very	O
hard	O
computational	O
optimization	O
problems	O
where	O
exact	O
algorithms	O
fail	O
;	O
even	O
though	O
it	O
usually	O
achieves	O
an	O
approximate	O
solution	O
to	O
the	O
global	O
minimum	O
,	O
it	O
could	O
be	O
enough	O
for	O
many	O
practical	O
problems	O
.	O
</s>
<s>
In	O
1983	O
,	O
this	O
approach	O
was	O
used	O
by	O
Kirkpatrick	O
,	O
Gelatt	O
Jr.	O
,	O
Vecchi	O
,	O
for	O
a	O
solution	O
of	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
They	O
also	O
proposed	O
its	O
current	O
name	O
,	O
simulated	B-Algorithm
annealing	I-Algorithm
.	O
</s>
<s>
This	O
notion	O
of	O
slow	O
cooling	O
implemented	O
in	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
is	O
interpreted	O
as	O
a	O
slow	O
decrease	O
in	O
the	O
probability	O
of	O
accepting	O
worse	O
solutions	O
as	O
the	O
solution	O
space	O
is	O
explored	O
.	O
</s>
<s>
In	O
general	O
,	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithms	I-Algorithm
work	O
as	O
follows	O
.	O
</s>
<s>
The	O
method	O
is	O
an	O
adaptation	O
of	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
,	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
to	O
generate	O
sample	O
states	O
of	O
a	O
thermodynamic	O
system	O
,	O
published	O
by	O
N	O
.	O
Metropolis	O
et	O
al	O
.	O
</s>
<s>
At	O
each	O
step	O
,	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
heuristic	B-Algorithm
considers	O
some	O
neighboring	O
state	O
s*	O
of	O
the	O
current	O
state	O
s	O
,	O
and	O
probabilistically	O
decides	O
between	O
moving	O
the	O
system	O
to	O
state	O
s*	O
or	O
staying	O
in-state	O
s	O
.	O
These	O
probabilities	O
ultimately	O
lead	O
the	O
system	O
to	O
move	O
to	O
states	O
of	O
lower	O
energy	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
each	O
state	O
is	O
typically	O
defined	O
as	O
a	O
permutation	B-Algorithm
of	O
the	O
cities	O
to	O
be	O
visited	O
,	O
and	O
the	O
neighbors	O
of	O
any	O
state	O
are	O
the	O
set	O
of	O
permutations	B-Algorithm
produced	O
by	O
swapping	O
any	O
two	O
of	O
these	O
cities	O
.	O
</s>
<s>
These	O
moves	O
usually	O
result	O
in	O
minimal	O
alterations	O
of	O
the	O
last	O
state	O
,	O
in	O
an	O
attempt	O
to	O
progressively	O
improve	O
the	O
solution	O
through	O
iteratively	O
improving	O
its	O
parts	O
(	O
such	O
as	O
the	O
city	O
connections	O
in	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
)	O
.	O
</s>
<s>
Simple	O
heuristics	B-Algorithm
like	O
hill	B-Algorithm
climbing	I-Algorithm
,	O
which	O
move	O
by	O
finding	O
better	O
neighbour	O
after	O
better	O
neighbour	O
and	O
stop	O
when	O
they	O
have	O
reached	O
a	O
solution	O
which	O
has	O
no	O
neighbours	O
that	O
are	O
better	O
solutions	O
,	O
cannot	O
guarantee	O
to	O
lead	O
to	O
any	O
of	O
the	O
existing	O
better	O
solutions	O
their	O
outcome	O
may	O
easily	O
be	O
just	O
a	O
local	O
optimum	O
,	O
while	O
the	O
actual	O
best	O
solution	O
would	O
be	O
a	O
global	O
optimum	O
that	O
could	O
be	O
different	O
.	O
</s>
<s>
Metaheuristics	B-Algorithm
use	O
the	O
neighbours	O
of	O
a	O
solution	O
as	O
a	O
way	O
to	O
explore	O
the	O
solution	O
space	O
,	O
and	O
although	O
they	O
prefer	O
better	O
neighbours	O
,	O
they	O
also	O
accept	O
worse	O
neighbours	O
in	O
order	O
to	O
avoid	O
getting	O
stuck	O
in	O
local	O
optima	O
;	O
they	O
can	O
find	O
the	O
global	O
optimum	O
if	O
run	O
for	O
a	O
long	O
enough	O
amount	O
of	O
time	O
.	O
</s>
<s>
The	O
probability	O
of	O
making	O
the	O
transition	B-Application
from	O
the	O
current	O
state	O
to	O
a	O
candidate	O
new	O
state	O
is	O
specified	O
by	O
an	O
acceptance	O
probability	O
function	O
,	O
that	O
depends	O
on	O
the	O
energies	O
and	O
of	O
the	O
two	O
states	O
,	O
and	O
on	O
a	O
global	O
time-varying	O
parameter	O
called	O
the	O
temperature	O
.	O
</s>
<s>
With	O
the	O
procedure	O
reduces	O
to	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
makes	O
only	O
the	O
downhill	O
transitions	O
.	O
</s>
<s>
In	O
the	O
original	O
description	O
of	O
simulated	B-Algorithm
annealing	I-Algorithm
,	O
the	O
probability	O
was	O
equal	O
to	O
1	O
when	O
—	O
i.e.	O
,	O
the	O
procedure	O
always	O
moved	O
downhill	O
when	O
it	O
found	O
a	O
way	O
to	O
do	O
so	O
,	O
irrespective	O
of	O
the	O
temperature	O
.	O
</s>
<s>
Many	O
descriptions	O
and	O
implementations	O
of	O
simulated	B-Algorithm
annealing	I-Algorithm
still	O
take	O
this	O
condition	O
as	O
part	O
of	O
the	O
method	O
's	O
definition	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
the	O
system	O
is	O
expected	O
to	O
wander	O
initially	O
towards	O
a	O
broad	O
region	O
of	O
the	O
search	O
space	O
containing	O
good	O
solutions	O
,	O
ignoring	O
small	O
features	O
of	O
the	O
energy	O
function	O
;	O
then	O
drift	O
towards	O
low-energy	O
regions	O
that	O
become	O
narrower	O
and	O
narrower	O
,	O
and	O
finally	O
move	O
downhill	O
according	O
to	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
heuristic	B-Algorithm
.	O
</s>
<s>
For	O
any	O
given	O
finite	O
problem	O
,	O
the	O
probability	O
that	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
terminates	O
with	O
a	O
global	O
optimal	O
solution	O
approaches	O
1	O
as	O
the	O
annealing	O
schedule	O
is	O
extended	O
.	O
</s>
<s>
This	O
theoretical	O
result	O
,	O
however	O
,	O
is	O
not	O
particularly	O
helpful	O
,	O
since	O
the	O
time	O
required	O
to	O
ensure	O
a	O
significant	O
probability	O
of	O
success	O
will	O
usually	O
exceed	O
the	O
time	O
required	O
for	O
a	O
complete	B-Algorithm
search	I-Algorithm
of	O
the	O
solution	O
space	O
.	O
</s>
<s>
The	O
following	O
pseudocode	O
presents	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
heuristic	B-Algorithm
as	O
described	O
above	O
.	O
</s>
<s>
In	O
order	O
to	O
apply	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
method	O
to	O
a	O
specific	O
problem	O
,	O
one	O
must	O
specify	O
the	O
following	O
parameters	O
:	O
the	O
state	O
space	O
,	O
the	O
energy	O
(	O
goal	O
)	O
function	O
,	O
the	O
candidate	O
generator	O
procedure	O
,	O
the	O
acceptance	O
probability	O
function	O
,	O
and	O
the	O
annealing	O
schedule	O
AND	O
initial	O
temperature	O
.	O
</s>
<s>
Simulated	B-Algorithm
annealing	I-Algorithm
may	O
be	O
modeled	O
as	O
a	O
random	O
walk	O
on	O
a	O
search	O
graph	O
,	O
whose	O
vertices	O
are	O
all	O
possible	O
states	O
,	O
and	O
whose	O
edges	O
are	O
the	O
candidate	O
moves	O
.	O
</s>
<s>
To	O
investigate	O
the	O
behavior	O
of	O
simulated	B-Algorithm
annealing	I-Algorithm
on	O
a	O
particular	O
problem	O
,	O
it	O
can	O
be	O
useful	O
to	O
consider	O
the	O
transition	B-Application
probabilities	O
that	O
result	O
from	O
the	O
various	O
design	O
choices	O
made	O
in	O
the	O
implementation	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
For	O
each	O
edge	O
of	O
the	O
search	O
graph	O
,	O
the	O
transition	B-Application
probability	O
is	O
defined	O
as	O
the	O
probability	O
that	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
will	O
move	O
to	O
state	O
when	O
its	O
current	O
state	O
is	O
.	O
</s>
<s>
(	O
Note	O
that	O
the	O
transition	B-Application
probability	O
is	O
not	O
simply	O
,	O
because	O
the	O
candidates	O
are	O
tested	O
serially	O
.	O
)	O
</s>
<s>
This	O
formula	O
was	O
superficially	O
justified	O
by	O
analogy	O
with	O
the	O
transitions	O
of	O
a	O
physical	O
system	O
;	O
it	O
corresponds	O
to	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
,	O
in	O
the	O
case	O
where	O
T	O
=	O
1	O
and	O
the	O
proposal	O
distribution	O
of	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
is	O
symmetric	O
.	O
</s>
<s>
However	O
,	O
this	O
acceptance	O
probability	O
is	O
often	O
used	O
for	O
simulated	B-Algorithm
annealing	I-Algorithm
even	O
when	O
the	O
function	O
,	O
which	O
is	O
analogous	O
to	O
the	O
proposal	O
distribution	O
in	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
,	O
is	O
not	O
symmetric	O
,	O
or	O
not	O
probabilistic	O
at	O
all	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
transition	B-Application
probabilities	O
of	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
do	O
not	O
correspond	O
to	O
the	O
transitions	O
of	O
the	O
analogous	O
physical	O
system	O
,	O
and	O
the	O
long-term	O
distribution	O
of	O
states	O
at	O
a	O
constant	O
temperature	O
need	O
not	O
bear	O
any	O
resemblance	O
to	O
the	O
thermodynamic	O
equilibrium	O
distribution	O
over	O
states	O
of	O
that	O
physical	O
system	O
,	O
at	O
any	O
temperature	O
.	O
</s>
<s>
Nevertheless	O
,	O
most	O
descriptions	O
of	O
simulated	B-Algorithm
annealing	I-Algorithm
assume	O
the	O
original	O
acceptance	O
function	O
,	O
which	O
is	O
probably	O
hard-coded	O
in	O
many	O
implementations	O
of	O
SA	O
.	O
</s>
<s>
Moscato	O
and	O
Fontanari	O
conclude	O
from	O
observing	O
the	O
analogous	O
of	O
the	O
"	O
specific	O
heat	O
"	O
curve	O
of	O
the	O
"	O
threshold	O
updating	O
"	O
annealing	O
originating	O
from	O
their	O
study	O
that	O
"	O
the	O
stochasticity	O
of	O
the	O
Metropolis	O
updating	O
in	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
does	O
not	O
play	O
a	O
major	O
role	O
in	O
the	O
search	O
of	O
near-optimal	O
minima	O
"	O
.	O
</s>
<s>
When	O
choosing	O
the	O
candidate	O
generator	O
,	O
one	O
must	O
consider	O
that	O
after	O
a	O
few	O
iterations	O
of	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
,	O
the	O
current	O
state	O
is	O
expected	O
to	O
have	O
much	O
lower	O
energy	O
than	O
a	O
random	O
state	O
.	O
</s>
<s>
This	O
heuristic	B-Algorithm
(	O
which	O
is	O
the	O
main	O
principle	O
of	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
)	O
tends	O
to	O
exclude	O
"	O
very	O
good	O
"	O
candidate	O
moves	O
as	O
well	O
as	O
"	O
very	O
bad	O
"	O
ones	O
;	O
however	O
,	O
the	O
former	O
are	O
usually	O
much	O
less	O
common	O
than	O
the	O
latter	O
,	O
so	O
the	O
heuristic	B-Algorithm
is	O
generally	O
quite	O
effective	O
.	O
</s>
<s>
In	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
above	O
,	O
for	O
example	O
,	O
swapping	O
two	O
consecutive	O
cities	O
in	O
a	O
low-energy	O
tour	O
is	O
expected	O
to	O
have	O
a	O
modest	O
effect	O
on	O
its	O
energy	O
(	O
length	O
)	O
;	O
whereas	O
swapping	O
two	O
arbitrary	O
cities	O
is	O
far	O
more	O
likely	O
to	O
increase	O
its	O
length	O
than	O
to	O
decrease	O
it	O
.	O
</s>
<s>
A	O
more	O
precise	O
statement	O
of	O
the	O
heuristic	B-Algorithm
is	O
that	O
one	O
should	O
try	O
first	O
candidate	O
states	O
for	O
which	O
is	O
large	O
.	O
</s>
<s>
Such	O
"	O
closed	O
catchment	O
basins	O
"	O
of	O
the	O
energy	O
function	O
may	O
trap	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
with	O
high	O
probability	O
(	O
roughly	O
proportional	O
to	O
the	O
number	O
of	O
states	O
in	O
the	O
basin	O
)	O
and	O
for	O
a	O
very	O
long	O
time	O
(	O
roughly	O
exponential	O
on	O
the	O
energy	O
difference	O
between	O
the	O
surrounding	O
states	O
and	O
the	O
bottom	O
of	O
the	O
basin	O
)	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
one	O
can	O
often	O
vastly	O
improve	O
the	O
efficiency	O
of	O
simulated	B-Algorithm
annealing	I-Algorithm
by	O
relatively	O
simple	O
changes	O
to	O
the	O
generator	O
.	O
</s>
<s>
In	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
,	O
for	O
instance	O
,	O
it	O
is	O
not	O
hard	O
to	O
exhibit	O
two	O
tours	O
,	O
,	O
with	O
nearly	O
equal	O
lengths	O
,	O
such	O
that	O
(	O
1	O
)	O
is	O
optimal	O
,	O
(	O
2	O
)	O
every	O
sequence	O
of	O
city-pair	O
swaps	O
that	O
converts	O
to	O
goes	O
through	O
tours	O
that	O
are	O
much	O
longer	O
than	O
both	O
,	O
and	O
(	O
3	O
)	O
can	O
be	O
transformed	O
into	O
by	O
flipping	O
(	O
reversing	O
the	O
order	O
of	O
)	O
a	O
set	O
of	O
consecutive	O
cities	O
.	O
</s>
<s>
The	O
physical	O
analogy	O
that	O
is	O
used	O
to	O
justify	O
simulated	B-Algorithm
annealing	I-Algorithm
assumes	O
that	O
the	O
cooling	O
rate	O
is	O
low	O
enough	O
for	O
the	O
probability	O
distribution	O
of	O
the	O
current	O
state	O
to	O
be	O
near	O
thermodynamic	O
equilibrium	O
at	O
all	O
times	O
.	O
</s>
<s>
In	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
,	O
the	O
relaxation	O
time	O
also	O
depends	O
on	O
the	O
candidate	O
generator	O
,	O
in	O
a	O
very	O
complicated	O
way	O
.	O
</s>
<s>
Note	O
that	O
all	O
these	O
parameters	O
are	O
usually	O
provided	O
as	O
black	O
box	O
functions	O
to	O
the	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithm	O
.	O
</s>
<s>
Adaptive	B-Algorithm
simulated	I-Algorithm
annealing	I-Algorithm
algorithms	O
address	O
this	O
problem	O
by	O
connecting	O
the	O
cooling	O
schedule	O
to	O
the	O
search	O
progress	O
.	O
</s>
<s>
Other	O
adaptive	O
approach	O
as	O
Thermodynamic	O
Simulated	B-Algorithm
Annealing	I-Algorithm
,	O
automatically	O
adjusts	O
the	O
temperature	O
at	O
each	O
step	O
based	O
on	O
the	O
energy	O
difference	O
between	O
the	O
two	O
states	O
,	O
according	O
to	O
the	O
laws	O
of	O
thermodynamics	O
.	O
</s>
<s>
This	O
process	O
is	O
called	O
restarting	O
of	O
simulated	B-Algorithm
annealing	I-Algorithm
.	O
</s>
<s>
Interacting	B-Algorithm
Metropolis	I-Algorithm
–	I-Algorithm
Hasting	I-Algorithm
algorithms	I-Algorithm
(	O
a.k.a.	O
</s>
<s>
sequential	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
)	O
combines	O
simulated	B-Algorithm
annealing	I-Algorithm
moves	O
with	O
an	O
acceptance-rejection	O
of	O
the	O
best	O
fitted	O
individuals	O
equipped	O
with	O
an	O
interacting	O
recycling	O
mechanism	O
.	O
</s>
<s>
Quantum	B-Algorithm
annealing	I-Algorithm
uses	O
"	O
quantum	O
fluctuations	O
"	O
instead	O
of	O
thermal	O
fluctuations	O
to	O
get	O
through	O
high	O
but	O
thin	O
barriers	O
in	O
the	O
target	O
function	O
.	O
</s>
<s>
Stochastic	O
tunneling	O
attempts	O
to	O
overcome	O
the	O
increasing	O
difficulty	O
simulated	B-Algorithm
annealing	I-Algorithm
runs	O
have	O
in	O
escaping	O
from	O
local	O
minima	O
as	O
the	O
temperature	O
decreases	O
,	O
by	O
'	O
tunneling	O
 '	O
through	O
barriers	O
.	O
</s>
<s>
Tabu	B-Algorithm
search	I-Algorithm
normally	O
moves	O
to	O
neighbouring	O
states	O
of	O
lower	O
energy	O
,	O
but	O
will	O
take	O
uphill	O
moves	O
when	O
it	O
finds	O
itself	O
stuck	O
in	O
a	O
local	O
minimum	O
;	O
and	O
avoids	O
cycles	O
by	O
keeping	O
a	O
"	O
taboo	O
list	O
"	O
of	O
solutions	O
already	O
seen	O
.	O
</s>
<s>
Dual-phase	B-Algorithm
evolution	I-Algorithm
is	O
a	O
family	O
of	O
algorithms	O
and	O
processes	O
(	O
to	O
which	O
simulated	B-Algorithm
annealing	I-Algorithm
belongs	O
)	O
that	O
mediate	O
between	O
local	O
and	O
global	O
search	O
by	O
exploiting	O
phase	O
changes	O
in	O
the	O
search	O
space	O
.	O
</s>
<s>
Reactive	B-Application
search	I-Application
optimization	I-Application
focuses	O
on	O
combining	O
machine	O
learning	O
with	O
optimization	O
,	O
by	O
adding	O
an	O
internal	O
feedback	O
loop	O
to	O
self-tune	O
the	O
free	O
parameters	O
of	O
an	O
algorithm	O
to	O
the	O
characteristics	O
of	O
the	O
problem	O
,	O
of	O
the	O
instance	O
,	O
and	O
of	O
the	O
local	O
situation	O
around	O
the	O
current	O
solution	O
.	O
</s>
<s>
Genetic	B-Algorithm
algorithms	I-Algorithm
maintain	O
a	O
pool	O
of	O
solutions	O
rather	O
than	O
just	O
one	O
.	O
</s>
<s>
Memetic	B-Algorithm
algorithms	I-Algorithm
search	O
for	O
solutions	O
by	O
employing	O
a	O
set	O
of	O
agents	O
that	O
both	O
cooperate	O
and	O
compete	O
in	O
the	O
process	O
;	O
sometimes	O
the	O
agents	O
 '	O
strategies	O
involve	O
simulated	B-Algorithm
annealing	I-Algorithm
procedures	O
for	O
obtaining	O
high	O
quality	O
solutions	O
before	O
recombining	O
them	O
.	O
</s>
<s>
Graduated	B-Algorithm
optimization	I-Algorithm
digressively	O
"	O
smooths	O
"	O
the	O
target	O
function	O
while	O
optimizing	O
.	O
</s>
<s>
Ant	B-Algorithm
colony	I-Algorithm
optimization	I-Algorithm
(	O
ACO	O
)	O
uses	O
many	O
ants	O
(	O
or	O
agents	O
)	O
to	O
traverse	O
the	O
solution	O
space	O
and	O
find	O
locally	O
productive	O
areas	O
.	O
</s>
<s>
The	O
cross-entropy	B-Algorithm
method	I-Algorithm
(	O
CE	O
)	O
generates	O
candidates	O
solutions	O
via	O
a	O
parameterized	O
probability	O
distribution	O
.	O
</s>
<s>
Stochastic	B-Algorithm
optimization	I-Algorithm
is	O
an	O
umbrella	O
set	O
of	O
methods	O
that	O
includes	O
simulated	B-Algorithm
annealing	I-Algorithm
and	O
numerous	O
other	O
approaches	O
.	O
</s>
<s>
Particle	B-Algorithm
swarm	I-Algorithm
optimization	I-Algorithm
is	O
an	O
algorithm	O
modelled	O
on	O
swarm	O
intelligence	O
that	O
finds	O
a	O
solution	O
to	O
an	O
optimization	O
problem	O
in	O
a	O
search	O
space	O
,	O
or	O
model	O
and	O
predict	O
social	O
behavior	O
in	O
the	O
presence	O
of	O
objectives	O
.	O
</s>
<s>
The	O
runner-root	O
algorithm	O
(	O
RRA	O
)	O
is	O
a	O
meta-heuristic	B-Algorithm
optimization	O
algorithm	O
for	O
solving	O
unimodal	O
and	O
multimodal	O
problems	O
inspired	O
by	O
the	O
runners	O
and	O
roots	O
of	O
plants	O
in	O
nature	O
.	O
</s>
<s>
Parallel	B-Algorithm
tempering	I-Algorithm
is	O
a	O
simulation	O
of	O
model	O
copies	O
at	O
different	O
temperatures	O
(	O
or	O
Hamiltonians	O
)	O
to	O
overcome	O
the	O
potential	O
barriers	O
.	O
</s>
<s>
Multi-objective	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithms	I-Algorithm
have	O
been	O
used	O
in	O
multi-objective	O
optimization	O
.	O
</s>
