<s>
In	O
mathematics	O
,	O
a	O
constraint	B-Application
is	O
a	O
condition	O
of	O
an	O
optimization	O
problem	O
that	O
the	O
solution	O
must	O
satisfy	O
.	O
</s>
<s>
There	O
are	O
several	O
types	O
of	O
constraintsprimarily	O
equality	O
constraints	O
,	O
inequality	B-Application
constraints	I-Application
,	O
and	O
integer	B-Algorithm
constraints	I-Algorithm
.	O
</s>
<s>
The	O
second	O
and	O
third	O
lines	O
define	O
two	O
constraints	O
,	O
the	O
first	O
of	O
which	O
is	O
an	O
inequality	B-Application
constraint	I-Application
and	O
the	O
second	O
of	O
which	O
is	O
an	O
equality	O
constraint	B-Application
.	O
</s>
<s>
These	O
two	O
constraints	O
are	O
hard	B-Application
constraints	I-Application
,	O
meaning	O
that	O
it	O
is	O
required	O
that	O
they	O
be	O
satisfied	O
;	O
they	O
define	O
the	O
feasible	O
set	O
of	O
candidate	O
solutions	O
.	O
</s>
<s>
The	O
solution	O
of	O
the	O
constrained	B-Application
optimization	I-Application
problem	I-Application
stated	O
above	O
is	O
,	O
which	O
is	O
the	O
point	O
with	O
the	O
smallest	O
value	O
of	O
that	O
satisfies	O
the	O
two	O
constraints	O
.	O
</s>
<s>
If	O
an	O
inequality	B-Application
constraint	I-Application
holds	O
with	O
equality	O
at	O
the	O
optimal	O
point	O
,	O
the	O
constraint	B-Application
is	O
said	O
to	O
be	O
,	O
as	O
the	O
point	O
cannot	O
be	O
varied	O
in	O
the	O
direction	O
of	O
the	O
constraint	B-Application
even	O
though	O
doing	O
so	O
would	O
improve	O
the	O
value	O
of	O
the	O
objective	O
function	O
.	O
</s>
<s>
If	O
an	O
inequality	B-Application
constraint	I-Application
holds	O
as	O
a	O
strict	O
inequality	O
at	O
the	O
optimal	O
point	O
(	O
that	O
is	O
,	O
does	O
not	O
hold	O
with	O
equality	O
)	O
,	O
the	O
constraint	B-Application
is	O
said	O
to	O
be	O
,	O
as	O
the	O
point	O
could	O
be	O
varied	O
in	O
the	O
direction	O
of	O
the	O
constraint	B-Application
,	O
although	O
it	O
would	O
not	O
be	O
optimal	O
to	O
do	O
so	O
.	O
</s>
<s>
Under	O
certain	O
conditions	O
,	O
as	O
for	O
example	O
in	O
convex	O
optimization	O
,	O
if	O
a	O
constraint	B-Application
is	O
non-binding	O
,	O
the	O
optimization	O
problem	O
would	O
have	O
the	O
same	O
solution	O
even	O
in	O
the	O
absence	O
of	O
that	O
constraint	B-Application
.	O
</s>
<s>
If	O
a	O
constraint	B-Application
is	O
not	O
satisfied	O
at	O
a	O
given	O
point	O
,	O
the	O
point	O
is	O
said	O
to	O
be	O
infeasible	O
.	O
</s>
<s>
If	O
the	O
problem	O
mandates	O
that	O
the	O
constraints	O
be	O
satisfied	O
,	O
as	O
in	O
the	O
above	O
discussion	O
,	O
the	O
constraints	O
are	O
sometimes	O
referred	O
to	O
as	O
hard	B-Application
constraints	I-Application
.	O
</s>
<s>
However	O
,	O
in	O
some	O
problems	O
,	O
called	O
flexible	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
,	O
it	O
is	O
preferred	O
but	O
not	O
required	O
that	O
certain	O
constraints	O
be	O
satisfied	O
;	O
such	O
non-mandatory	O
constraints	O
are	O
known	O
as	O
soft	B-Application
constraints	I-Application
.	O
</s>
<s>
Soft	B-Application
constraints	I-Application
arise	O
in	O
,	O
for	O
example	O
,	O
preference-based	B-Application
planning	I-Application
.	O
</s>
<s>
Some	O
of	O
them	O
,	O
such	O
as	O
the	O
constraint	B-Application
,	O
can	O
be	O
rewritten	O
as	O
a	O
conjunction	O
of	O
atomic	O
constraints	O
in	O
a	O
simpler	O
language	O
:	O
the	O
alldifferent	O
constraint	B-Application
holds	O
on	O
n	O
variables	O
,	O
and	O
is	O
satisfied	O
if	O
the	O
variables	O
take	O
values	O
which	O
are	O
pairwise	O
different	O
.	O
</s>
<s>
Other	O
global	O
constraints	O
extend	O
the	O
expressivity	O
of	O
the	O
constraint	B-Application
framework	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
regular	B-Application
constraint	I-Application
expresses	O
that	O
a	O
sequence	O
of	O
variables	O
is	O
accepted	O
by	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
Global	O
constraints	O
are	O
used	O
to	O
simplify	O
the	O
modeling	O
of	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
,	O
to	O
extend	O
the	O
expressivity	O
of	O
constraint	B-Application
languages	O
,	O
and	O
also	O
to	O
improve	O
the	O
constraint	B-Application
resolution	I-Application
:	O
indeed	O
,	O
by	O
considering	O
the	O
variables	O
altogether	O
,	O
infeasible	O
situations	O
can	O
be	O
seen	O
earlier	O
in	O
the	O
solving	O
process	O
.	O
</s>
