<s>
In	O
mathematical	O
optimization	O
,	O
a	O
trust	B-Algorithm
region	I-Algorithm
is	O
the	O
subset	O
of	O
the	O
region	O
of	O
the	O
objective	O
function	O
that	O
is	O
approximated	O
using	O
a	O
model	O
function	O
(	O
often	O
a	O
quadratic	O
)	O
.	O
</s>
<s>
If	O
an	O
adequate	O
model	O
of	O
the	O
objective	O
function	O
is	O
found	O
within	O
the	O
trust	B-Algorithm
region	I-Algorithm
,	O
then	O
the	O
region	O
is	O
expanded	O
;	O
conversely	O
,	O
if	O
the	O
approximation	O
is	O
poor	O
,	O
then	O
the	O
region	O
is	O
contracted	O
.	O
</s>
<s>
Trust-region	O
methods	O
are	O
in	O
some	O
sense	O
dual	O
to	O
line-search	B-Algorithm
methods	O
:	O
trust-region	O
methods	O
first	O
choose	O
a	O
step	O
size	O
(	O
the	O
size	O
of	O
the	O
trust	B-Algorithm
region	I-Algorithm
)	O
and	O
then	O
a	O
step	O
direction	O
,	O
while	O
line-search	B-Algorithm
methods	O
first	O
choose	O
a	O
step	O
direction	O
and	O
then	O
a	O
step	O
size	O
.	O
</s>
<s>
The	O
general	O
idea	O
behind	O
trust	B-Algorithm
region	I-Algorithm
methods	I-Algorithm
is	O
known	O
by	O
many	O
names	O
;	O
the	O
earliest	O
use	O
of	O
the	O
term	O
seems	O
to	O
be	O
by	O
Sorensen	O
(	O
1982	O
)	O
.	O
</s>
<s>
Additionally	O
,	O
in	O
an	O
early	O
foundational	O
work	O
on	O
the	O
method	O
,	O
Goldfeld	O
,	O
Quandt	O
,	O
and	O
Trotter	O
(	O
1966	O
)	O
refer	O
to	O
it	O
as	O
quadratic	B-Algorithm
hill-climbing	I-Algorithm
.	O
</s>
<s>
Conceptually	O
,	O
in	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
,	O
the	O
objective	O
function	O
is	O
iteratively	O
approximated	O
by	O
a	O
quadratic	O
surface	O
,	O
then	O
using	O
a	O
linear	O
solver	O
,	O
the	O
estimate	O
is	O
updated	O
.	O
</s>
<s>
If	O
the	O
ratio	O
is	O
more	O
than	O
0.5	O
,	O
then	O
we	O
are	O
damping	O
the	O
step	O
too	O
much	O
,	O
so	O
expand	O
the	O
trust	B-Algorithm
region	I-Algorithm
(	O
decrease	O
λ	O
)	O
and	O
iterate	O
.	O
</s>
<s>
If	O
the	O
ratio	O
is	O
smaller	O
than	O
0.25	O
,	O
then	O
the	O
true	O
function	O
is	O
diverging	O
"	O
too	O
much	O
"	O
from	O
the	O
trust-region	O
approximation	O
,	O
so	O
shrink	O
the	O
trust	B-Algorithm
region	I-Algorithm
(	O
increase	O
λ	O
)	O
and	O
try	O
again	O
.	O
</s>
