<s>
In	O
numerical	O
optimization	O
,	O
the	O
nonlinear	B-Algorithm
conjugate	I-Algorithm
gradient	I-Algorithm
method	I-Algorithm
generalizes	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
to	O
nonlinear	B-Algorithm
optimization	I-Algorithm
.	O
</s>
<s>
One	O
simply	O
starts	O
in	O
the	O
opposite	O
(	O
steepest	B-Algorithm
descent	I-Algorithm
)	O
direction	O
:	O
</s>
<s>
with	O
an	O
adjustable	O
step	O
length	O
and	O
performs	O
a	O
line	B-Algorithm
search	I-Algorithm
in	O
this	O
direction	O
until	O
it	O
reaches	O
the	O
minimum	O
of	O
:	O
</s>
<s>
Perform	O
a	O
line	B-Algorithm
search	I-Algorithm
:	O
optimize	O
,	O
</s>
<s>
Subsequent	O
search	O
directions	O
lose	O
conjugacy	O
requiring	O
the	O
search	O
direction	O
to	O
be	O
reset	O
to	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
direction	O
at	O
least	O
every	O
N	O
iterations	O
,	O
or	O
sooner	O
if	O
progress	O
stops	O
.	O
</s>
<s>
However	O
,	O
resetting	O
every	O
iteration	O
turns	O
the	O
method	O
into	O
steepest	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
in	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
direction	O
)	O
,	O
or	O
when	O
some	O
tolerance	O
criterion	O
is	O
reached	O
.	O
</s>
<s>
linear	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
but	O
have	O
been	O
obtained	O
with	O
line	B-Algorithm
searches	I-Algorithm
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
can	O
follow	O
narrow	O
(	O
ill-conditioned	B-Algorithm
)	O
valleys	O
,	O
where	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
method	O
slows	O
down	O
and	O
follows	O
a	O
criss-cross	O
pattern	O
.	O
</s>
<s>
Fletcher	B-Algorithm
–	I-Algorithm
Reeves	I-Algorithm
:	O
</s>
<s>
These	O
formulas	O
are	O
equivalent	O
for	O
a	O
quadratic	O
function	O
,	O
but	O
for	O
nonlinear	B-Algorithm
optimization	I-Algorithm
the	O
preferred	O
formula	O
is	O
a	O
matter	O
of	O
heuristics	O
or	O
taste	O
.	O
</s>
<s>
There	O
,	O
both	O
step	O
direction	O
and	O
length	O
are	O
computed	O
from	O
the	O
gradient	O
as	O
the	O
solution	O
of	O
a	O
linear	O
system	O
of	O
equations	O
,	O
with	O
the	O
coefficient	O
matrix	O
being	O
the	O
exact	O
Hessian	O
matrix	O
(	O
for	O
Newton	O
's	O
method	O
proper	O
)	O
or	O
an	O
estimate	O
thereof	O
(	O
in	O
the	O
quasi-Newton	B-Algorithm
methods	I-Algorithm
,	O
where	O
the	O
observed	O
change	O
in	O
the	O
gradient	O
during	O
the	O
iterations	O
is	O
used	O
to	O
update	O
the	O
Hessian	O
estimate	O
)	O
.	O
</s>
<s>
For	O
high-dimensional	O
problems	O
,	O
the	O
exact	O
computation	O
of	O
the	O
Hessian	O
is	O
usually	O
prohibitively	O
expensive	O
,	O
and	O
even	O
its	O
storage	O
can	O
be	O
problematic	O
,	O
requiring	O
memory	O
(	O
but	O
see	O
the	O
limited-memory	O
L-BFGS	B-Algorithm
quasi-Newton	B-Algorithm
method	I-Algorithm
)	O
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
can	O
also	O
be	O
derived	O
using	O
optimal	O
control	O
theory	O
.	O
</s>
<s>
In	O
this	O
accelerated	O
optimization	O
theory	O
,	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
falls	O
out	O
as	O
a	O
nonlinear	O
optimal	O
feedback	O
controller	O
,	O
</s>
