<s>
In	O
mathematical	O
optimization	O
,	O
the	O
revised	B-Algorithm
simplex	I-Algorithm
method	I-Algorithm
is	O
a	O
variant	O
of	O
George	O
Dantzig	O
's	O
simplex	B-Algorithm
method	I-Algorithm
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
revised	B-Algorithm
simplex	I-Algorithm
method	I-Algorithm
is	O
mathematically	O
equivalent	O
to	O
the	O
standard	O
simplex	B-Algorithm
method	I-Algorithm
but	O
differs	O
in	O
implementation	O
.	O
</s>
<s>
Instead	O
of	O
maintaining	O
a	O
tableau	O
which	O
explicitly	O
represents	O
the	O
constraints	O
adjusted	O
to	O
a	O
set	O
of	O
basic	O
variables	O
,	O
it	O
maintains	O
a	O
representation	O
of	O
a	O
basis	O
of	O
the	O
matrix	B-Architecture
representing	O
the	O
constraints	O
.	O
</s>
<s>
The	O
matrix-oriented	O
approach	O
allows	O
for	O
greater	O
computational	O
efficiency	O
by	O
enabling	O
sparse	O
matrix	B-Architecture
operations	O
.	O
</s>
<s>
For	O
the	O
rest	O
of	O
the	O
discussion	O
,	O
it	O
is	O
assumed	O
that	O
a	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
has	O
been	O
converted	O
into	O
the	O
following	O
standard	O
form	O
:	O
</s>
<s>
Without	O
loss	O
of	O
generality	O
,	O
it	O
is	O
assumed	O
that	O
the	O
constraint	O
matrix	B-Architecture
has	O
full	O
row	O
rank	O
and	O
that	O
the	O
problem	O
is	O
feasible	O
,	O
i.e.	O
,	O
there	O
is	O
at	O
least	O
one	O
such	O
that	O
.	O
</s>
<s>
For	O
linear	B-Algorithm
programming	I-Algorithm
,	O
the	O
Karush	O
–	O
Kuhn	O
–	O
Tucker	O
conditions	O
are	O
both	O
necessary	O
and	O
sufficient	O
for	O
optimality	O
.	O
</s>
<s>
By	O
what	O
is	O
sometimes	O
known	O
as	O
the	O
fundamental	O
theorem	O
of	O
linear	B-Algorithm
programming	I-Algorithm
,	O
a	O
vertex	O
of	O
the	O
feasible	O
polytope	O
can	O
be	O
identified	O
by	O
being	O
a	O
basis	O
of	O
chosen	O
from	O
the	O
latter	O
's	O
columns	O
.	O
</s>
<s>
Therefore	O
,	O
if	O
the	O
problem	O
is	O
bounded	O
,	O
the	O
revised	B-Algorithm
simplex	I-Algorithm
method	I-Algorithm
must	O
terminate	O
at	O
an	O
optimal	O
vertex	O
after	O
repeated	O
pivot	O
operations	O
because	O
there	O
are	O
only	O
a	O
finite	O
number	O
of	O
vertices	O
.	O
</s>
<s>
Because	O
the	O
revised	B-Algorithm
simplex	I-Algorithm
method	I-Algorithm
is	O
mathematically	O
equivalent	O
to	O
the	O
simplex	B-Algorithm
method	I-Algorithm
,	O
it	O
also	O
suffers	O
from	O
degeneracy	O
,	O
where	O
a	O
pivot	O
operation	O
does	O
not	O
result	O
in	O
a	O
decrease	O
in	O
,	O
and	O
a	O
chain	O
of	O
pivot	O
operations	O
causes	O
the	O
basis	O
to	O
cycle	O
.	O
</s>
<s>
Two	O
types	O
of	O
linear	O
systems	O
involving	O
are	O
present	O
in	O
the	O
revised	B-Algorithm
simplex	I-Algorithm
method	I-Algorithm
:	O
</s>
