<s>
Powell	B-Algorithm
's	I-Algorithm
dog	I-Algorithm
leg	I-Algorithm
method	I-Algorithm
,	O
also	O
called	O
Powell	O
's	O
hybrid	O
method	O
,	O
is	O
an	O
iterative	B-Algorithm
optimisation	O
algorithm	O
for	O
the	O
solution	O
of	O
non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
problems	O
,	O
introduced	O
in	O
1970	O
by	O
Michael	O
J	O
.	O
D	O
.	O
Powell	O
.	O
</s>
<s>
Similarly	O
to	O
the	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
,	O
it	O
combines	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
with	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
but	O
it	O
uses	O
an	O
explicit	O
trust	B-Algorithm
region	I-Algorithm
.	O
</s>
<s>
At	O
each	O
iteration	O
,	O
if	O
the	O
step	O
from	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
is	O
within	O
the	O
trust	B-Algorithm
region	I-Algorithm
,	O
it	O
is	O
used	O
to	O
update	O
the	O
current	O
solution	O
.	O
</s>
<s>
If	O
not	O
,	O
the	O
algorithm	O
searches	O
for	O
the	O
minimum	O
of	O
the	O
objective	O
function	O
along	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
direction	O
,	O
known	O
as	O
Cauchy	O
point	O
.	O
</s>
<s>
If	O
the	O
Cauchy	O
point	O
is	O
outside	O
of	O
the	O
trust	B-Algorithm
region	I-Algorithm
,	O
it	O
is	O
truncated	O
to	O
the	O
boundary	O
of	O
the	O
latter	O
and	O
it	O
is	O
taken	O
as	O
the	O
new	O
solution	O
.	O
</s>
<s>
If	O
the	O
Cauchy	O
point	O
is	O
inside	O
the	O
trust	B-Algorithm
region	I-Algorithm
,	O
the	O
new	O
solution	O
is	O
taken	O
at	O
the	O
intersection	O
between	O
the	O
trust	B-Algorithm
region	I-Algorithm
boundary	O
and	O
the	O
line	O
joining	O
the	O
Cauchy	O
point	O
and	O
the	O
Gauss-Newton	B-Algorithm
step	O
(	O
dog	O
leg	O
step	O
)	O
.	O
</s>
<s>
The	O
name	O
of	O
the	O
method	O
derives	O
from	O
the	O
resemblance	O
between	O
the	O
construction	O
of	O
the	O
dog	O
leg	O
step	O
and	O
the	O
shape	O
of	O
a	O
dogleg	O
hole	O
in	O
golf	B-Application
.	O
</s>
<s>
with	O
,	O
Powell	B-Algorithm
's	I-Algorithm
dog	I-Algorithm
leg	I-Algorithm
method	I-Algorithm
finds	O
the	O
optimal	O
point	O
by	O
constructing	O
a	O
sequence	O
that	O
converges	O
to	O
.	O
</s>
<s>
Given	O
a	O
trust	B-Algorithm
region	I-Algorithm
of	O
radius	O
,	O
Powell	B-Algorithm
's	I-Algorithm
dog	I-Algorithm
leg	I-Algorithm
method	I-Algorithm
selects	O
the	O
update	O
step	O
as	O
equal	O
to	O
:	O
</s>
<s>
,	O
if	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
step	O
is	O
within	O
the	O
trust	B-Algorithm
region	I-Algorithm
(	O
)	O
;	O
</s>
<s>
if	O
both	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
and	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
steps	O
are	O
outside	O
the	O
trust	B-Algorithm
region	I-Algorithm
(	O
)	O
;	O
</s>
<s>
with	O
such	O
that	O
,	O
if	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
step	O
is	O
outside	O
the	O
trust	B-Algorithm
region	I-Algorithm
but	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
step	O
is	O
inside	O
(	O
dog	O
leg	O
step	O
)	O
.	O
</s>
