<s>
The	O
"	O
second-order	O
cone	O
"	O
in	O
SOCP	B-Algorithm
arises	O
from	O
the	O
constraints	O
,	O
which	O
are	O
equivalent	O
to	O
requiring	O
the	O
affine	O
function	O
to	O
lie	O
in	O
the	O
second-order	O
cone	O
in	O
.	O
</s>
<s>
SOCPs	B-Algorithm
can	O
be	O
solved	O
by	O
interior	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
and	O
in	O
general	O
,	O
can	O
be	O
solved	O
more	O
efficiently	O
than	O
semidefinite	O
programming	O
(	O
SDP	O
)	O
problems	O
.	O
</s>
<s>
Some	O
engineering	O
applications	O
of	O
SOCP	B-Algorithm
include	O
filter	O
design	O
,	O
antenna	O
array	O
weight	O
design	O
,	O
truss	O
design	O
,	O
and	O
grasping	O
force	O
optimization	O
in	O
robotics	O
.	O
</s>
<s>
Applications	O
in	O
quantitative	O
finance	O
include	O
portfolio	O
optimization	O
;	O
some	O
market	O
impact	O
constraints	O
,	O
because	O
they	O
are	O
not	O
linear	O
,	O
cannot	O
be	O
solved	O
by	O
quadratic	B-Algorithm
programming	I-Algorithm
but	O
can	O
be	O
formulated	O
as	O
SOCP	B-Algorithm
problems	O
.	O
</s>
<s>
When	O
for	O
,	O
the	O
SOCP	B-Algorithm
reduces	O
to	O
a	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
When	O
for	O
,	O
the	O
SOCP	B-Algorithm
is	O
equivalent	O
to	O
a	O
convex	O
quadratically	O
constrained	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Convex	O
quadratically	O
constrained	O
quadratic	B-Algorithm
programs	I-Algorithm
can	O
also	O
be	O
formulated	O
as	O
SOCPs	B-Algorithm
by	O
reformulating	O
the	O
objective	O
function	O
as	O
a	O
constraint	O
.	O
</s>
<s>
Semidefinite	O
programming	O
subsumes	O
SOCPs	B-Algorithm
as	O
the	O
SOCP	B-Algorithm
constraints	O
can	O
be	O
written	O
as	O
linear	O
matrix	O
inequalities	O
(	O
LMI	O
)	O
and	O
can	O
be	O
reformulated	O
as	O
an	O
instance	O
of	O
semidefinite	O
program	O
.	O
</s>
<s>
In	O
fact	O
,	O
while	O
any	O
closed	O
convex	O
semialgebraic	O
set	O
in	O
the	O
plane	O
can	O
be	O
written	O
as	O
a	O
feasible	O
region	O
of	O
a	O
SOCP	B-Algorithm
,	O
it	O
is	O
known	O
that	O
there	O
exist	O
convex	O
semialgebraic	O
sets	O
that	O
are	O
not	O
representable	O
by	O
SDPs	O
,	O
that	O
is	O
,	O
there	O
exist	O
convex	O
semialgebraic	O
sets	O
that	O
can	O
not	O
be	O
written	O
as	O
a	O
feasible	O
region	O
of	O
a	O
SDP	O
.	O
</s>
