<s>
In	O
discrete	O
optimization	O
,	O
a	O
special	B-Algorithm
ordered	I-Algorithm
set	I-Algorithm
(	O
SOS	O
)	O
is	O
an	O
ordered	O
set	O
of	O
variables	O
used	O
as	O
an	O
additional	O
way	O
to	O
specify	O
integrality	O
conditions	O
in	O
an	O
optimization	O
model	O
.	O
</s>
<s>
Special	O
order	O
sets	O
are	O
basically	O
a	O
device	O
or	O
tool	O
used	O
in	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
methods	O
for	O
branching	O
on	O
sets	O
of	O
variables	O
,	O
rather	O
than	O
individual	O
variables	O
,	O
as	O
in	O
ordinary	O
mixed	O
integer	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Knowing	O
that	O
a	O
variable	O
is	O
part	O
of	O
a	O
set	O
and	O
that	O
it	O
is	O
ordered	O
gives	O
the	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
algorithm	I-Algorithm
a	O
more	O
intelligent	O
way	O
to	O
face	O
the	O
optimization	O
problem	O
,	O
helping	O
to	O
speed	O
up	O
the	O
search	O
procedure	O
.	O
</s>
<s>
The	O
members	O
of	O
a	O
special	B-Algorithm
ordered	I-Algorithm
set	I-Algorithm
individually	O
may	O
be	O
continuous	O
or	O
discrete	O
variables	O
in	O
any	O
combination	O
.	O
</s>
<s>
However	O
,	O
even	O
when	O
all	O
the	O
members	O
are	O
themselves	O
continuous	O
,	O
a	O
model	O
containing	O
one	O
or	O
more	O
special	B-Algorithm
ordered	I-Algorithm
sets	I-Algorithm
becomes	O
a	O
discrete	O
optimization	O
problem	O
requiring	O
a	O
mixed	O
integer	O
optimizer	O
for	O
its	O
solution	O
.	O
</s>
<s>
The	O
‘	O
only’	O
benefit	O
of	O
using	O
Special	B-Algorithm
Ordered	I-Algorithm
Sets	I-Algorithm
compared	O
with	O
using	O
only	O
constraints	O
is	O
that	O
the	O
search	O
procedure	O
will	O
generally	O
be	O
noticeably	O
faster	O
.	O
</s>
<s>
There	O
are	O
two	O
sorts	O
of	O
Special	B-Algorithm
Ordered	I-Algorithm
Sets	I-Algorithm
:	O
</s>
<s>
Special	B-Algorithm
Ordered	I-Algorithm
Sets	I-Algorithm
of	O
type	O
1	O
(	O
SOS1	O
or	O
S1	O
)	O
:	O
are	O
a	O
set	O
of	O
variables	O
,	O
at	O
most	O
one	O
of	O
which	O
can	O
take	O
a	O
non-zero	O
value	O
,	O
all	O
others	O
being	O
at	O
0	O
.	O
</s>
<s>
Special	B-Algorithm
Ordered	I-Algorithm
Sets	I-Algorithm
of	O
type	O
2	O
(	O
SOS2	O
or	O
S2	O
)	O
:	O
an	O
ordered	O
set	O
of	O
non-negative	O
variables	O
,	O
of	O
which	O
at	O
most	O
two	O
can	O
be	O
non-zero	O
,	O
and	O
if	O
two	O
are	O
non-zero	O
these	O
must	O
be	O
consecutive	O
in	O
their	O
ordering	O
.	O
</s>
<s>
Special	B-Algorithm
Ordered	I-Algorithm
Sets	I-Algorithm
of	O
type	O
2	O
are	O
typically	O
used	O
to	O
model	O
non-linear	O
functions	O
of	O
a	O
variable	O
in	O
a	O
linear	O
model	O
.	O
</s>
<s>
They	O
are	O
the	O
natural	O
extension	O
of	O
the	O
concepts	O
of	O
Separable	O
Programming	O
,	O
but	O
when	O
embedded	O
in	O
a	O
Branch	B-Algorithm
and	I-Algorithm
Bound	I-Algorithm
code	O
enable	O
truly	O
global	O
optima	O
to	O
be	O
found	O
,	O
and	O
not	O
just	O
local	O
optima	O
.	O
</s>
