<s>
Interior-point	B-Algorithm
methods	I-Algorithm
(	O
also	O
referred	O
to	O
as	O
barrier	B-Algorithm
methods	I-Algorithm
or	O
IPMs	O
)	O
are	O
a	O
certain	O
class	O
of	O
algorithms	O
that	O
solve	O
linear	O
and	O
nonlinear	O
convex	O
optimization	O
problems	O
.	O
</s>
<s>
An	O
interior	B-Algorithm
point	I-Algorithm
method	I-Algorithm
was	O
discovered	O
by	O
Soviet	O
mathematician	O
I	O
.	O
I	O
.	O
Dikin	O
in	O
1967	O
and	O
reinvented	O
in	O
the	O
U.S.	O
in	O
the	O
mid-1980s	O
.	O
</s>
<s>
In	O
1984	O
,	O
Narendra	O
Karmarkar	O
developed	O
a	O
method	O
for	O
linear	B-Algorithm
programming	I-Algorithm
called	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
runs	O
in	O
provably	O
polynomial	O
time	O
and	O
is	O
also	O
very	O
efficient	O
in	O
practice	O
.	O
</s>
<s>
It	O
enabled	O
solutions	O
of	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
that	O
were	O
beyond	O
the	O
capabilities	O
of	O
the	O
simplex	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
simplex	B-Algorithm
method	I-Algorithm
,	O
it	O
reaches	O
a	O
best	O
solution	O
by	O
traversing	O
the	O
interior	O
of	O
the	O
feasible	O
region	O
.	O
</s>
<s>
The	O
method	O
can	O
be	O
generalized	O
to	O
convex	O
programming	O
based	O
on	O
a	O
self-concordant	B-Algorithm
barrier	O
function	O
used	O
to	O
encode	O
the	O
convex	O
set	O
.	O
</s>
<s>
The	O
idea	O
of	O
encoding	O
the	O
feasible	O
set	O
using	O
a	O
barrier	O
and	O
designing	O
barrier	B-Algorithm
methods	I-Algorithm
was	O
studied	O
by	O
Anthony	O
V	O
.	O
Fiacco	O
,	O
Garth	O
P	O
.	O
McCormick	O
,	O
and	O
others	O
in	O
the	O
early	O
1960s	O
.	O
</s>
<s>
These	O
ideas	O
were	O
mainly	O
developed	O
for	O
general	O
nonlinear	B-Algorithm
programming	I-Algorithm
,	O
but	O
they	O
were	O
later	O
abandoned	O
due	O
to	O
the	O
presence	O
of	O
more	O
competitive	O
methods	O
for	O
this	O
class	O
of	O
problems	O
(	O
e.g.	O
</s>
<s>
sequential	B-Algorithm
quadratic	I-Algorithm
programming	I-Algorithm
)	O
.	O
</s>
<s>
They	O
guarantee	O
that	O
the	O
number	O
of	O
iterations	B-Algorithm
of	O
the	O
algorithm	O
is	O
bounded	O
by	O
a	O
polynomial	O
in	O
the	O
dimension	O
and	O
accuracy	O
of	O
the	O
solution	O
.	O
</s>
<s>
Karmarkar	O
's	O
breakthrough	O
revitalized	O
the	O
study	O
of	O
interior-point	B-Algorithm
methods	I-Algorithm
and	O
barrier	O
problems	O
,	O
showing	O
that	O
it	O
was	O
possible	O
to	O
create	O
an	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
characterized	O
by	O
polynomial	O
complexity	O
and	O
,	O
moreover	O
,	O
that	O
was	O
competitive	O
with	O
the	O
simplex	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Already	O
Khachiyan	O
's	O
ellipsoid	B-Algorithm
method	I-Algorithm
was	O
a	O
polynomial-time	O
algorithm	O
;	O
however	O
,	O
it	O
was	O
too	O
slow	O
to	O
be	O
of	O
practical	O
interest	O
.	O
</s>
<s>
The	O
class	O
of	O
primal-dual	O
path-following	O
interior-point	B-Algorithm
methods	I-Algorithm
is	O
considered	O
the	O
most	O
successful	O
.	O
</s>
<s>
Mehrotra	B-Algorithm
's	I-Algorithm
predictor	I-Algorithm
–	I-Algorithm
corrector	I-Algorithm
algorithm	I-Algorithm
provides	O
the	O
basis	O
for	O
most	O
implementations	O
of	O
this	O
class	O
of	O
methods	O
.	O
</s>
<s>
The	O
primal-dual	O
method	O
's	O
idea	O
is	O
easy	O
to	O
demonstrate	O
for	O
constrained	O
nonlinear	B-Algorithm
optimization	I-Algorithm
.	O
</s>
<s>
For	O
simplicity	O
,	O
consider	O
the	O
following	O
nonlinear	B-Algorithm
optimization	I-Algorithm
problem	O
with	O
inequality	O
constraints	O
:	O
</s>
<s>
where	O
is	O
the	O
Hessian	O
matrix	O
of	O
,	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
,	O
and	O
is	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
.	O
</s>
