<s>
In	O
numerical	B-General_Concept
optimization	O
,	O
the	O
Broyden	O
–	O
Fletcher	O
–	O
Goldfarb	O
–	O
Shanno	O
(	O
BFGS	B-Algorithm
)	O
algorithm	O
is	O
an	O
iterative	B-Algorithm
method	I-Algorithm
for	O
solving	O
unconstrained	O
nonlinear	B-Algorithm
optimization	I-Algorithm
problems	O
.	O
</s>
<s>
Like	O
the	O
related	O
Davidon	B-Algorithm
–	I-Algorithm
Fletcher	I-Algorithm
–	I-Algorithm
Powell	I-Algorithm
method	I-Algorithm
,	O
BFGS	B-Algorithm
determines	O
the	O
descent	O
direction	O
by	O
preconditioning	O
the	O
gradient	O
with	O
curvature	O
information	O
.	O
</s>
<s>
Since	O
the	O
updates	O
of	O
the	O
BFGS	B-Algorithm
curvature	O
matrix	O
do	O
not	O
require	O
matrix	O
inversion	O
,	O
its	O
computational	O
complexity	O
is	O
only	O
,	O
compared	O
to	O
in	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Also	O
in	O
common	O
use	O
is	O
L-BFGS	B-Algorithm
,	O
which	O
is	O
a	O
limited-memory	O
version	O
of	O
BFGS	B-Algorithm
that	O
is	O
particularly	O
suited	O
to	O
problems	O
with	O
very	O
large	O
numbers	O
of	O
variables	O
(	O
e.g.	O
,	O
>1000	O
)	O
.	O
</s>
<s>
The	O
BFGS-B	O
variant	O
handles	O
simple	O
box	O
constraints	O
.	O
</s>
<s>
by	O
finding	O
a	O
point	O
xk+1	O
satisfying	O
the	O
Wolfe	O
conditions	O
,	O
which	O
entail	O
the	O
curvature	O
condition	O
,	O
using	O
line	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Both	O
and	O
are	O
symmetric	B-Algorithm
rank-one	I-Algorithm
matrices	O
,	O
but	O
their	O
sum	O
is	O
a	O
rank-two	O
update	O
matrix	O
.	O
</s>
<s>
BFGS	B-Algorithm
and	O
DFP	B-Algorithm
updating	O
matrix	O
both	O
differ	O
from	O
its	O
predecessor	O
by	O
a	O
rank-two	O
matrix	O
.	O
</s>
<s>
Another	O
simpler	O
rank-one	O
method	O
is	O
known	O
as	O
symmetric	B-Algorithm
rank-one	I-Algorithm
method	O
,	O
which	O
does	O
not	O
guarantee	O
the	O
positive	O
definiteness	O
.	O
</s>
<s>
Perform	O
a	O
one-dimensional	O
optimization	O
(	O
line	B-Algorithm
search	I-Algorithm
)	O
to	O
find	O
an	O
acceptable	O
stepsize	O
in	O
the	O
direction	O
found	O
in	O
the	O
first	O
step	O
.	O
</s>
<s>
If	O
an	O
exact	O
line	B-Algorithm
search	I-Algorithm
is	O
performed	O
,	O
then	O
.	O
</s>
<s>
In	O
practice	O
,	O
an	O
inexact	O
line	B-Algorithm
search	I-Algorithm
usually	O
suffices	O
,	O
with	O
an	O
acceptable	O
satisfying	O
Wolfe	O
conditions	O
.	O
</s>
<s>
If	O
is	O
initialized	O
with	O
,	O
the	O
first	O
step	O
will	O
be	O
equivalent	O
to	O
a	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
but	O
further	O
steps	O
are	O
more	O
and	O
more	O
refined	O
by	O
,	O
the	O
approximation	O
to	O
the	O
Hessian	O
.	O
</s>
<s>
Perform	O
a	O
one-dimensional	O
optimization	O
(	O
line	B-Algorithm
search	I-Algorithm
)	O
to	O
find	O
an	O
acceptable	O
stepsize	O
in	O
the	O
direction	O
found	O
in	O
the	O
first	O
step	O
.	O
</s>
<s>
If	O
an	O
exact	O
line	B-Algorithm
search	I-Algorithm
is	O
performed	O
,	O
then	O
.	O
</s>
<s>
In	O
practice	O
,	O
an	O
inexact	O
line	B-Algorithm
search	I-Algorithm
usually	O
suffices	O
,	O
with	O
an	O
acceptable	O
satisfying	O
Wolfe	O
conditions	O
.	O
</s>
<s>
In	O
statistical	O
estimation	O
problems	O
(	O
such	O
as	O
maximum	O
likelihood	O
or	O
Bayesian	O
inference	O
)	O
,	O
credible	B-General_Concept
intervals	I-General_Concept
or	O
confidence	O
intervals	O
for	O
the	O
solution	O
can	O
be	O
estimated	O
from	O
the	O
inverse	O
of	O
the	O
final	O
Hessian	O
matrix	O
.	O
</s>
<s>
However	O
,	O
these	O
quantities	O
are	O
technically	O
defined	O
by	O
the	O
true	O
Hessian	O
matrix	O
,	O
and	O
the	O
BFGS	B-Algorithm
approximation	O
may	O
not	O
converge	O
to	O
the	O
true	O
Hessian	O
matrix	O
.	O
</s>
<s>
GNU	B-Language
Octave	I-Language
uses	O
a	O
form	O
of	O
BFGS	B-Algorithm
in	O
its	O
fsolve	O
function	O
,	O
with	O
trust	B-Algorithm
region	I-Algorithm
extensions	O
.	O
</s>
<s>
The	O
GSL	B-Application
implements	O
BFGS	B-Algorithm
as	O
gsl_multimin_fdfminimizer_vector_bfgs2	O
.	O
</s>
<s>
In	O
R	B-Language
,	O
the	O
BFGS	B-Algorithm
algorithm	O
(	O
and	O
the	O
L-BFGS-B	O
version	O
that	O
allows	O
box	O
constraints	O
)	O
is	O
implemented	O
as	O
an	O
option	O
of	O
the	O
base	O
function	O
optim( )	O
.	O
</s>
<s>
In	O
SciPy	B-Application
,	O
the	O
scipy.optimize.fmin_bfgs	O
function	O
implements	O
BFGS	B-Algorithm
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
run	O
BFGS	B-Algorithm
using	O
any	O
of	O
the	O
L-BFGS	B-Algorithm
algorithms	O
by	O
setting	O
the	O
parameter	O
L	O
to	O
a	O
very	O
large	O
number	O
.	O
</s>
<s>
In	O
Julia	B-Application
,	O
the	O
package	O
implements	O
BFGS	B-Algorithm
and	O
L-BFGS	B-Algorithm
as	O
a	O
solver	O
option	O
to	O
the	O
optimize( )	O
function	O
(	O
among	O
other	O
options	O
)	O
.	O
</s>
<s>
The	O
large	O
scale	O
nonlinear	B-Algorithm
optimization	I-Algorithm
software	O
Artelys	B-Application
Knitro	I-Application
implements	O
,	O
among	O
others	O
,	O
both	O
BFGS	B-Algorithm
and	O
L-BFGS	B-Algorithm
algorithms	O
.	O
</s>
<s>
In	O
the	O
MATLAB	O
Optimization	B-Application
Toolbox	I-Application
,	O
the	O
fminunc	O
function	O
uses	O
BFGS	B-Algorithm
with	O
cubic	O
line	B-Algorithm
search	I-Algorithm
when	O
the	O
problem	O
size	O
is	O
set	O
to	O
"	O
medium	O
scale.	O
"	O
</s>
<s>
Mathematica	B-Language
includes	O
BFGS	B-Algorithm
.	O
</s>
