<s>
In	O
the	O
theory	O
of	O
linear	B-Algorithm
programming	I-Algorithm
,	O
a	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
(	O
BFS	O
)	O
is	O
a	O
solution	O
with	O
a	O
minimal	O
set	O
of	O
non-zero	O
variables	O
.	O
</s>
<s>
This	O
fact	O
is	O
used	O
by	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
essentially	O
travels	O
from	O
some	O
BFS	O
to	O
another	O
until	O
an	O
optimal	O
one	O
is	O
found	O
.	O
</s>
<s>
For	O
the	O
definitions	O
below	O
,	O
we	O
first	O
present	O
the	O
linear	B-Algorithm
program	I-Algorithm
in	O
the	O
so-called	O
equational	O
form	O
:	O
</s>
<s>
Any	O
linear	B-Algorithm
program	I-Algorithm
can	O
be	O
converted	O
into	O
an	O
equational	O
form	O
by	O
adding	O
slack	B-Algorithm
variables	I-Algorithm
.	O
</s>
<s>
Given	O
a	O
basis	O
B	O
,	O
we	O
say	O
that	O
a	O
feasible	O
solution	O
is	O
a	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
with	O
basis	O
B	O
if	O
all	O
its	O
non-zero	O
variables	O
are	O
indexed	O
by	O
B	O
,	O
that	O
is	O
,	O
for	O
all	O
.	O
</s>
<s>
If	O
a	O
linear	B-Algorithm
program	I-Algorithm
has	O
an	O
optimal	O
solution	O
(	O
i.e.	O
,	O
it	O
has	O
a	O
feasible	O
solution	O
,	O
and	O
the	O
set	O
of	O
feasible	O
solutions	O
is	O
bounded	O
)	O
,	O
then	O
it	O
has	O
an	O
optimal	O
BFS	O
.	O
</s>
<s>
This	O
is	O
a	O
consequence	O
of	O
the	O
Bauer	O
maximum	O
principle	O
:	O
the	O
objective	O
of	O
a	O
linear	B-Algorithm
program	I-Algorithm
is	O
convex	O
;	O
the	O
set	O
of	O
feasible	O
solutions	O
is	O
convex	O
(	O
it	O
is	O
an	O
intersection	O
of	O
hyperspaces	O
)	O
;	O
therefore	O
the	O
objective	O
attains	O
its	O
maximum	O
in	O
an	O
extreme	O
point	O
of	O
the	O
set	O
of	O
feasible	O
solutions	O
.	O
</s>
<s>
This	O
is	O
not	O
the	O
most	O
efficient	O
way	O
to	O
solve	O
an	O
LP	O
;	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
examines	O
the	O
BFS-s	O
in	O
a	O
much	O
more	O
efficient	O
way	O
.	O
</s>
<s>
Consider	O
a	O
linear	B-Algorithm
program	I-Algorithm
with	O
the	O
following	O
constraints	O
:	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
every	O
basis	O
B	O
defines	O
a	O
unique	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
.	O
</s>
<s>
In	O
a	O
similar	O
way	O
,	O
each	O
basis	O
defines	O
a	O
solution	O
to	O
the	O
dual	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
:	O
</s>
<s>
In	O
practice	O
,	O
the	O
easiest	O
way	O
to	O
find	O
an	O
optimal	O
BFS	O
is	O
to	O
use	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
tableau	O
is	O
a	O
representation	O
of	O
the	O
linear	B-Algorithm
program	I-Algorithm
where	O
the	O
basic	O
variables	O
are	O
expressed	O
in	O
terms	O
of	O
the	O
non-basic	O
ones:where	O
is	O
the	O
vector	O
of	O
m	O
basic	O
variables	O
,	O
is	O
the	O
vector	O
of	O
n	O
non-basic	O
variables	O
,	O
and	O
is	O
the	O
maximization	O
objective	O
.	O
</s>
<s>
In	O
the	O
worst	O
case	O
,	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
may	O
require	O
exponentially	O
many	O
steps	O
to	O
complete	O
.	O
</s>
<s>
There	O
are	O
algorithms	O
for	O
solving	O
an	O
LP	O
in	O
weakly-polynomial	O
time	O
,	O
such	O
as	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
;	O
however	O
,	O
they	O
usually	O
return	O
optimal	O
solutions	O
that	O
are	O
not	O
basic	O
.	O
</s>
<s>
A	O
basis	O
B	O
of	O
the	O
LP	O
is	O
called	O
dual-optimal	O
if	O
the	O
solution	O
is	O
an	O
optimal	O
solution	O
to	O
the	O
dual	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
,	O
that	O
is	O
,	O
it	O
minimizes	O
.	O
</s>
<s>
Every	O
LP	O
with	O
an	O
optimal	O
solution	O
has	O
a	O
PD-optimal	O
basis	O
,	O
and	O
it	O
is	O
found	O
by	O
the	O
Simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
If	O
there	O
exists	O
a	O
strongly	O
polynomial	O
time	O
algorithm	O
that	O
inputs	O
an	O
optimal	O
solution	O
to	O
only	O
the	O
primal	O
LP	O
(	O
or	O
only	O
the	O
dual	O
LP	O
)	O
and	O
returns	O
an	O
optimal	O
basis	O
,	O
then	O
there	O
exists	O
a	O
strongly-polynomial	O
time	O
algorithm	O
for	O
solving	O
any	O
linear	B-Algorithm
program	I-Algorithm
(	O
the	O
latter	O
is	O
a	O
famous	O
open	O
problem	O
)	O
.	O
</s>
<s>
Megiddo	O
's	O
algorithms	O
can	O
be	O
executed	O
using	O
a	O
tableau	O
,	O
just	O
like	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
