<s>
In	O
mathematical	O
optimization	O
,	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
is	O
an	O
iterative	B-Algorithm
method	I-Algorithm
for	O
minimizing	O
convex	O
functions	O
.	O
</s>
<s>
When	O
specialized	O
to	O
solving	O
feasible	O
linear	B-Algorithm
optimization	I-Algorithm
problems	O
with	O
rational	O
data	O
,	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
is	O
an	O
algorithm	O
which	O
finds	O
an	O
optimal	O
solution	O
in	O
a	O
number	O
of	O
steps	O
that	O
is	O
polynomial	O
in	O
the	O
input	O
size	O
.	O
</s>
<s>
The	O
ellipsoid	B-Algorithm
method	I-Algorithm
generates	O
a	O
sequence	O
of	O
ellipsoids	O
whose	O
volume	O
uniformly	O
decreases	O
at	O
every	O
step	O
,	O
thus	O
enclosing	O
a	O
minimizer	O
of	O
a	O
convex	O
function	O
.	O
</s>
<s>
The	O
ellipsoid	B-Algorithm
method	I-Algorithm
has	O
a	O
long	O
history	O
.	O
</s>
<s>
As	O
an	O
iterative	B-Algorithm
method	I-Algorithm
,	O
a	O
preliminary	O
version	O
was	O
introduced	O
by	O
Naum	O
Z	O
.	O
Shor	O
.	O
</s>
<s>
In	O
1972	O
,	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
for	O
real	O
convex	O
minimization	O
was	O
studied	O
by	O
Arkadi	O
Nemirovski	O
and	O
David	O
B	O
.	O
Yudin	O
(	O
Judin	O
)	O
.	O
</s>
<s>
As	O
an	O
algorithm	O
for	O
solving	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
with	O
rational	O
data	O
,	O
the	O
ellipsoid	B-Algorithm
algorithm	I-Algorithm
was	O
studied	O
by	O
Leonid	O
Khachiyan	O
;	O
Khachiyan	O
's	O
achievement	O
was	O
to	O
prove	O
the	O
polynomial-time	O
solvability	O
of	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
This	O
was	O
a	O
notable	O
step	O
from	O
a	O
theoretical	O
perspective	O
:	O
The	O
standard	O
algorithm	O
for	O
solving	O
linear	B-Algorithm
problems	I-Algorithm
at	O
the	O
time	O
was	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
has	O
a	O
run	O
time	O
that	O
typically	O
is	O
linear	O
in	O
the	O
size	O
of	O
the	O
problem	O
,	O
but	O
for	O
which	O
examples	O
exist	O
for	O
which	O
it	O
is	O
exponential	O
in	O
the	O
size	O
of	O
the	O
problem	O
.	O
</s>
<s>
Khachiyan	O
's	O
work	O
showed	O
,	O
for	O
the	O
first	O
time	O
,	O
that	O
there	O
can	O
be	O
algorithms	O
for	O
solving	O
linear	B-Algorithm
programs	I-Algorithm
whose	O
runtime	O
can	O
be	O
proven	O
to	O
be	O
polynomial	O
.	O
</s>
<s>
Specifically	O
,	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
an	O
interior-point	B-Algorithm
method	I-Algorithm
,	O
is	O
much	O
faster	O
than	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
in	O
practice	O
.	O
</s>
<s>
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
also	O
faster	O
in	O
the	O
worst	O
case	O
.	O
</s>
<s>
The	O
ellipsoidal	B-Algorithm
algorithm	I-Algorithm
allows	O
complexity	O
theorists	O
to	O
achieve	O
(	O
worst-case	O
)	O
bounds	O
that	O
depend	O
on	O
the	O
dimension	O
of	O
the	O
problem	O
and	O
on	O
the	O
size	O
of	O
the	O
data	O
,	O
but	O
not	O
on	O
the	O
number	O
of	O
rows	O
,	O
so	O
it	O
remained	O
important	O
in	O
combinatorial	O
optimization	O
theory	O
for	O
many	O
years	O
.	O
</s>
<s>
Only	O
in	O
the	O
21st	O
century	O
have	O
interior-point	B-Algorithm
algorithms	O
with	O
similar	O
complexity	O
properties	O
appeared	O
.	O
</s>
<s>
The	O
output	O
of	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
is	O
either	O
:	O
</s>
<s>
It	O
turns	O
out	O
that	O
any	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
can	O
be	O
reduced	O
to	O
a	O
linear	O
feasibility	O
problem	O
(	O
e.g.	O
</s>
<s>
One	O
way	O
to	O
do	O
this	O
is	O
by	O
combining	O
the	O
primal	O
and	O
dual	O
linear	B-Algorithm
programs	I-Algorithm
together	O
into	O
one	O
program	O
,	O
and	O
adding	O
the	O
additional	O
(	O
linear	O
)	O
constraint	O
that	O
the	O
value	O
of	O
the	O
primal	O
solution	O
is	O
no	B-Algorithm
worse	I-Algorithm
than	I-Algorithm
the	O
value	O
of	O
the	O
dual	O
solution	O
.	O
</s>
<s>
Another	O
way	O
is	O
to	O
treat	O
the	O
objective	O
of	O
the	O
linear	B-Algorithm
program	I-Algorithm
as	O
an	O
additional	O
constraint	O
,	O
and	O
use	O
binary	O
search	O
to	O
find	O
the	O
optimum	O
value	O
.	O
</s>
<s>
The	O
ellipsoid	B-Algorithm
method	I-Algorithm
is	O
used	O
on	O
low-dimensional	O
problems	O
,	O
such	O
as	O
planar	O
location	O
problems	O
,	O
where	O
it	O
is	O
numerically	B-Algorithm
stable	I-Algorithm
.	O
</s>
<s>
On	O
even	O
"	O
small	O
"	O
-sized	O
problems	O
,	O
it	O
suffers	O
from	O
numerical	B-Algorithm
instability	I-Algorithm
and	O
poor	O
performance	O
in	O
practice	O
.	O
</s>
<s>
However	O
,	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
is	O
an	O
important	O
theoretical	O
technique	O
in	O
combinatorial	O
optimization	O
.	O
</s>
<s>
In	O
computational	O
complexity	O
theory	O
,	O
the	O
ellipsoid	B-Algorithm
algorithm	I-Algorithm
is	O
attractive	O
because	O
its	O
complexity	O
depends	O
on	O
the	O
number	O
of	O
columns	O
and	O
the	O
digital	O
size	O
of	O
the	O
coefficients	O
,	O
but	O
not	O
on	O
the	O
number	O
of	O
rows	O
.	O
</s>
<s>
In	O
the	O
21st	O
century	O
,	O
interior-point	B-Algorithm
algorithms	O
with	O
similar	O
properties	O
have	O
appeared	O
.	O
</s>
