<s>
In	O
mathematics	O
,	O
nonlinear	B-Algorithm
programming	I-Algorithm
(	O
NLP	O
)	O
is	O
the	O
process	O
of	O
solving	O
an	O
optimization	O
problem	O
where	O
some	O
of	O
the	O
constraints	B-Application
or	O
the	O
objective	O
function	O
are	O
nonlinear	O
.	O
</s>
<s>
An	O
optimization	O
problem	O
is	O
one	O
of	O
calculation	O
of	O
the	O
extrema	O
(	O
maxima	O
,	O
minima	O
or	O
stationary	O
points	O
)	O
of	O
an	O
objective	O
function	O
over	O
a	O
set	O
of	O
unknown	O
real	O
variables	O
and	O
conditional	O
to	O
the	O
satisfaction	O
of	O
a	O
system	O
of	O
equalities	O
and	O
inequalities	O
,	O
collectively	O
termed	O
constraints	B-Application
.	O
</s>
<s>
A	O
typical	O
non-convex	O
problem	O
is	O
that	O
of	O
optimizing	O
transportation	O
costs	O
by	O
selection	O
from	O
a	O
set	O
of	O
transportation	O
methods	O
,	O
one	O
or	O
more	O
of	O
which	O
exhibit	O
economies	O
of	O
scale	O
,	O
with	O
various	O
connectivities	O
and	O
capacity	O
constraints	B-Application
.	O
</s>
<s>
One	O
tries	O
to	O
find	O
a	O
best	B-Algorithm
fit	I-Algorithm
numerically	O
.	O
</s>
<s>
In	O
this	O
case	O
one	O
often	O
wants	O
a	O
measure	O
of	O
the	O
precision	O
of	O
the	O
result	O
,	O
as	O
well	O
as	O
the	O
best	B-Algorithm
fit	I-Algorithm
itself	O
.	O
</s>
<s>
There	O
are	O
several	O
possibilities	O
for	O
the	O
nature	O
of	O
the	O
constraint	B-Application
set	O
,	O
also	O
known	O
as	O
the	O
feasible	O
set	O
or	O
feasible	O
region	O
.	O
</s>
<s>
An	O
infeasible	O
problem	O
is	O
one	O
for	O
which	O
no	O
set	O
of	O
values	O
for	O
the	O
choice	O
variables	O
satisfies	O
all	O
the	O
constraints	B-Application
.	O
</s>
<s>
That	O
is	O
,	O
the	O
constraints	B-Application
are	O
mutually	O
contradictory	O
,	O
and	O
no	O
solution	O
exists	O
;	O
the	O
feasible	O
set	O
is	O
the	O
empty	O
set	O
.	O
</s>
<s>
A	O
feasible	O
problem	O
is	O
one	O
for	O
which	O
there	O
exists	O
at	O
least	O
one	O
set	O
of	O
values	O
for	O
the	O
choice	O
variables	O
satisfying	O
all	O
the	O
constraints	B-Application
.	O
</s>
<s>
If	O
the	O
objective	O
function	O
is	O
concave	O
(	O
maximization	O
problem	O
)	O
,	O
or	O
convex	O
(	O
minimization	O
problem	O
)	O
and	O
the	O
constraint	B-Application
set	O
is	O
convex	O
,	O
then	O
the	O
program	O
is	O
called	O
convex	O
and	O
general	O
methods	O
from	O
convex	O
optimization	O
can	O
be	O
used	O
in	O
most	O
cases	O
.	O
</s>
<s>
If	O
the	O
objective	O
function	O
is	O
quadratic	O
and	O
the	O
constraints	B-Application
are	O
linear	O
,	O
quadratic	B-Algorithm
programming	I-Algorithm
techniques	O
are	O
used	O
.	O
</s>
<s>
If	O
the	O
objective	O
function	O
is	O
a	O
ratio	O
of	O
a	O
concave	O
and	O
a	O
convex	O
function	O
(	O
in	O
the	O
maximization	O
case	O
)	O
and	O
the	O
constraints	B-Application
are	O
convex	O
,	O
then	O
the	O
problem	O
can	O
be	O
transformed	O
to	O
a	O
convex	O
optimization	O
problem	O
using	O
fractional	B-Algorithm
programming	I-Algorithm
techniques	O
.	O
</s>
<s>
One	O
approach	O
is	O
to	O
use	O
special	O
formulations	O
of	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Another	O
method	O
involves	O
the	O
use	O
of	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
techniques	O
,	O
where	O
the	O
program	O
is	O
divided	O
into	O
subclasses	O
to	O
be	O
solved	O
with	O
convex	O
(	O
minimization	O
problem	O
)	O
or	O
linear	O
approximations	O
that	O
form	O
a	O
lower	O
bound	O
on	O
the	O
overall	O
cost	O
within	O
the	O
subdivision	O
.	O
</s>
<s>
Under	O
differentiability	O
and	O
constraint	B-Application
qualifications	O
,	O
the	O
Karush	O
–	O
Kuhn	O
–	O
Tucker	O
(	O
KKT	O
)	O
conditions	O
provide	O
necessary	O
conditions	O
for	O
a	O
solution	O
to	O
be	O
optimal	O
.	O
</s>
