<s>
In	O
mathematical	O
optimization	O
,	O
affine	B-Algorithm
scaling	I-Algorithm
is	O
an	O
algorithm	O
for	O
solving	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Specifically	O
,	O
it	O
is	O
an	O
interior	B-Algorithm
point	I-Algorithm
method	I-Algorithm
,	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>
Affine	B-Algorithm
scaling	I-Algorithm
has	O
a	O
history	O
of	O
multiple	O
discovery	O
.	O
</s>
<s>
Dikin	O
's	O
work	O
went	O
largely	O
unnoticed	O
until	O
the	O
1984	O
discovery	O
of	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
the	O
first	O
practical	O
polynomial	O
time	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Several	O
groups	O
then	O
independently	O
came	O
up	O
with	O
a	O
variant	O
of	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
E	O
.	O
R	O
.	O
Barnes	O
at	O
IBM	O
,	O
a	O
team	O
led	O
by	O
R	O
.	O
J	O
.	O
Vanderbei	O
at	O
AT&T	O
,	O
and	O
several	O
others	O
replaced	O
the	O
projective	O
transformations	O
that	O
Karmarkar	O
used	O
by	O
affine	B-Algorithm
ones	O
.	O
</s>
<s>
After	O
a	O
few	O
years	O
,	O
it	O
was	O
realized	O
that	O
the	O
"	O
new	O
"	O
affine	B-Algorithm
scaling	I-Algorithm
algorithms	O
were	O
in	O
fact	O
reinventions	O
of	O
the	O
decades-old	O
results	O
of	O
Dikin	O
.	O
</s>
<s>
managed	O
to	O
produce	O
an	O
analysis	O
of	O
affine	B-Algorithm
scaling	I-Algorithm
's	O
convergence	O
properties	O
.	O
</s>
<s>
Karmarkar	O
,	O
who	O
had	O
also	O
came	O
with	O
affine	B-Algorithm
scaling	I-Algorithm
in	O
this	O
timeframe	O
,	O
mistakenly	O
believed	O
that	O
it	O
converged	O
as	O
quickly	O
as	O
his	O
own	O
algorithm	O
.	O
</s>
<s>
Affine	B-Algorithm
scaling	I-Algorithm
works	O
in	O
two	O
phases	O
,	O
the	O
first	O
of	O
which	O
finds	O
a	O
feasible	O
point	O
from	O
which	O
to	O
start	O
optimizing	O
,	O
while	O
the	O
second	O
does	O
the	O
actual	O
optimization	O
while	O
staying	O
strictly	O
inside	O
the	O
feasible	O
region	O
.	O
</s>
<s>
Both	O
phases	O
solve	O
linear	B-Algorithm
programs	I-Algorithm
in	O
equality	O
form	O
,	O
viz	O
.	O
</s>
<s>
These	O
problems	O
are	O
solved	O
using	O
an	O
iterative	B-Algorithm
method	I-Algorithm
,	O
which	O
conceptually	O
proceeds	O
by	O
plotting	O
a	O
trajectory	O
of	O
points	O
strictly	O
inside	O
the	O
feasible	O
region	O
of	O
a	O
problem	O
,	O
computing	O
projected	O
gradient	B-Algorithm
descent	I-Algorithm
steps	O
in	O
a	O
re-scaled	O
version	O
of	O
the	O
problem	O
,	O
then	O
scaling	O
the	O
step	O
back	O
to	O
the	O
original	O
problem	O
.	O
</s>
<s>
Formally	O
,	O
the	O
iterative	B-Algorithm
method	I-Algorithm
at	O
the	O
heart	O
of	O
affine	B-Algorithm
scaling	I-Algorithm
takes	O
as	O
inputs	O
,	O
,	O
,	O
an	O
initial	O
guess	O
that	O
is	O
strictly	O
feasible	O
(	O
i.e.	O
,	O
)	O
,	O
a	O
tolerance	O
and	O
a	O
stepsize	O
.	O
</s>
<s>
Let	O
be	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
on	O
its	O
diagonal	O
.	O
</s>
<s>
Compute	O
a	O
vector	O
of	O
reduced	O
costs	O
,	O
which	O
measure	O
the	O
slackness	B-Algorithm
of	O
inequality	O
constraints	O
in	O
the	O
dual	O
:	O
</s>
<s>
While	O
easy	O
to	O
state	O
,	O
affine	B-Algorithm
scaling	I-Algorithm
was	O
found	O
hard	O
to	O
analyze	O
.	O
</s>
<s>
For	O
step	O
sizes	O
,	O
Vanderbei	O
's	O
variant	O
of	O
affine	B-Algorithm
scaling	I-Algorithm
has	O
been	O
proven	O
to	O
converge	O
,	O
while	O
for	O
,	O
an	O
example	O
problem	O
is	O
known	O
that	O
converges	O
to	O
a	O
suboptimal	O
value	O
.	O
</s>
