<s>
In	O
an	O
optimization	O
problem	O
,	O
a	O
slack	B-Algorithm
variable	I-Algorithm
is	O
a	O
variable	O
that	O
is	O
added	O
to	O
an	O
inequality	B-Application
constraint	I-Application
to	O
transform	O
it	O
into	O
an	O
equality	O
.	O
</s>
<s>
Introducing	O
a	O
slack	B-Algorithm
variable	I-Algorithm
replaces	O
an	O
inequality	B-Application
constraint	I-Application
with	O
an	O
equality	O
constraint	B-Application
and	O
a	O
non-negativity	O
constraint	B-Application
on	O
the	O
slack	B-Algorithm
variable	I-Algorithm
.	O
</s>
<s>
Slack	B-Algorithm
variables	I-Algorithm
are	O
used	O
in	O
particular	O
in	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
As	O
with	O
the	O
other	O
variables	O
in	O
the	O
augmented	O
constraints	O
,	O
the	O
slack	B-Algorithm
variable	I-Algorithm
cannot	O
take	O
on	O
negative	O
values	O
,	O
as	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
requires	O
them	O
to	O
be	O
positive	O
or	O
zero	O
.	O
</s>
<s>
If	O
a	O
slack	B-Algorithm
variable	I-Algorithm
associated	O
with	O
a	O
constraint	B-Application
is	O
zero	O
at	O
a	O
particular	O
candidate	O
solution	O
,	O
the	O
constraint	B-Application
is	O
binding	O
there	O
,	O
as	O
the	O
constraint	B-Application
restricts	O
the	O
possible	O
changes	O
from	O
that	O
point	O
.	O
</s>
<s>
If	O
a	O
slack	B-Algorithm
variable	I-Algorithm
is	O
positive	O
at	O
a	O
particular	O
candidate	O
solution	O
,	O
the	O
constraint	B-Application
is	O
non-binding	O
there	O
,	O
as	O
the	O
constraint	B-Application
does	O
not	O
restrict	O
the	O
possible	O
changes	O
from	O
that	O
point	O
.	O
</s>
<s>
If	O
a	O
slack	B-Algorithm
variable	I-Algorithm
is	O
negative	O
at	O
some	O
point	O
,	O
the	O
point	O
is	O
infeasible	O
(	O
not	O
allowed	O
)	O
,	O
as	O
it	O
does	O
not	O
satisfy	O
the	O
constraint	B-Application
.	O
</s>
<s>
Slack	B-Algorithm
variables	I-Algorithm
are	O
also	O
used	O
in	O
the	O
Big	B-Algorithm
M	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Slack	B-Algorithm
variables	I-Algorithm
give	O
an	O
embedding	O
of	O
a	O
polytope	O
into	O
the	O
standard	O
f-orthant	O
,	O
where	O
is	O
the	O
number	O
of	O
constraints	O
(	O
facets	O
of	O
the	O
polytope	O
)	O
.	O
</s>
<s>
This	O
map	O
is	O
one-to-one	O
(	O
slack	B-Algorithm
variables	I-Algorithm
are	O
uniquely	O
determined	O
)	O
but	O
not	O
onto	O
(	O
not	O
all	O
combinations	O
can	O
be	O
realized	O
)	O
,	O
and	O
is	O
expressed	O
in	O
terms	O
of	O
the	O
constraints	O
(	O
linear	O
functionals	O
,	O
covectors	O
)	O
.	O
</s>
<s>
Slack	B-Algorithm
variables	I-Algorithm
are	O
dual	B-Algorithm
to	O
generalized	O
barycentric	O
coordinates	O
,	O
and	O
,	O
dually	O
to	O
generalized	O
barycentric	O
coordinates	O
(	O
which	O
are	O
not	O
unique	O
but	O
can	O
all	O
be	O
realized	O
)	O
,	O
are	O
uniquely	O
determined	O
,	O
but	O
cannot	O
all	O
be	O
realized	O
.	O
</s>
<s>
Dually	O
,	O
generalized	O
barycentric	O
coordinates	O
express	O
a	O
polytope	O
with	O
vertices	O
(	O
dual	B-Algorithm
to	O
facets	O
)	O
,	O
regardless	O
of	O
dimension	O
,	O
as	O
the	O
image	O
of	O
the	O
standard	O
-simplex	O
,	O
which	O
has	O
vertices	O
–	O
the	O
map	O
is	O
onto	O
:	O
and	O
expresses	O
points	O
in	O
terms	O
of	O
the	O
vertices	O
(	O
points	O
,	O
vectors	O
)	O
.	O
</s>
