<s>
In	O
mathematical	O
optimization	O
,	O
the	O
cutting-plane	B-Algorithm
method	I-Algorithm
is	O
any	O
of	O
a	O
variety	O
of	O
optimization	O
methods	O
that	O
iteratively	O
refine	O
a	O
feasible	O
set	O
or	O
objective	O
function	O
by	O
means	O
of	O
linear	O
inequalities	O
,	O
termed	O
cuts	O
.	O
</s>
<s>
Such	O
procedures	O
are	O
commonly	O
used	O
to	O
find	O
integer	O
solutions	O
to	O
mixed	O
integer	O
linear	B-Algorithm
programming	I-Algorithm
(	O
MILP	O
)	O
problems	O
,	O
as	O
well	O
as	O
to	O
solve	O
general	O
,	O
not	O
necessarily	O
differentiable	O
convex	O
optimization	O
problems	O
.	O
</s>
<s>
The	O
use	O
of	O
cutting	B-Algorithm
planes	I-Algorithm
to	O
solve	O
MILP	O
was	O
introduced	O
by	O
Ralph	O
E	O
.	O
Gomory	O
.	O
</s>
<s>
Cutting	B-Algorithm
plane	I-Algorithm
methods	I-Algorithm
for	O
MILP	O
work	O
by	O
solving	O
a	O
non-integer	O
linear	B-Algorithm
program	I-Algorithm
,	O
the	O
linear	B-Algorithm
relaxation	I-Algorithm
of	O
the	O
given	O
integer	O
program	O
.	O
</s>
<s>
The	O
theory	O
of	O
Linear	B-Algorithm
Programming	I-Algorithm
dictates	O
that	O
under	O
mild	O
assumptions	O
(	O
if	O
the	O
linear	B-Algorithm
program	I-Algorithm
has	O
an	O
optimal	O
solution	O
,	O
and	O
if	O
the	O
feasible	O
region	O
does	O
not	O
contain	O
a	O
line	O
)	O
,	O
one	O
can	O
always	O
find	O
an	O
extreme	O
point	O
or	O
a	O
corner	O
point	O
that	O
is	O
optimal	O
.	O
</s>
<s>
A	O
cut	O
can	O
be	O
added	O
to	O
the	O
relaxed	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Cutting-plane	B-Algorithm
methods	I-Algorithm
for	O
general	O
convex	O
continuous	O
optimization	O
and	O
variants	O
are	O
known	O
under	O
various	O
names	O
:	O
Kelley	O
's	O
method	O
,	O
Kelley	O
–	O
Cheney	O
–	O
Goldstein	O
method	O
,	O
and	O
bundle	O
methods	O
.	O
</s>
<s>
Another	O
common	O
situation	O
is	O
the	O
application	O
of	O
the	O
Dantzig	B-Algorithm
–	I-Algorithm
Wolfe	I-Algorithm
decomposition	I-Algorithm
to	O
a	O
structured	O
optimization	O
problem	O
in	O
which	O
formulations	O
with	O
an	O
exponential	O
number	O
of	O
variables	O
are	O
obtained	O
.	O
</s>
<s>
Generating	O
these	O
variables	O
on	O
demand	O
by	O
means	O
of	O
delayed	B-Algorithm
column	I-Algorithm
generation	I-Algorithm
is	O
identical	O
to	O
performing	O
a	O
cutting	B-Algorithm
plane	I-Algorithm
on	O
the	O
respective	O
dual	O
problem	O
.	O
</s>
<s>
Cutting	B-Algorithm
planes	I-Algorithm
were	O
proposed	O
by	O
Ralph	O
Gomory	O
in	O
the	O
1950s	O
as	O
a	O
method	O
for	O
solving	O
integer	O
programming	O
and	O
mixed-integer	O
programming	O
problems	O
.	O
</s>
<s>
Things	O
turned	O
around	O
when	O
in	O
the	O
mid-1990s	O
Gérard	O
Cornuéjols	O
and	O
co-workers	O
showed	O
them	O
to	O
be	O
very	O
effective	O
in	O
combination	O
with	O
branch-and-bound	B-Algorithm
(	O
called	O
branch-and-cut	B-Algorithm
)	O
and	O
ways	O
to	O
overcome	O
numerical	O
instabilities	O
.	O
</s>
<s>
The	O
method	O
proceeds	O
by	O
first	O
dropping	O
the	O
requirement	O
that	O
the	O
xi	O
be	O
integers	O
and	O
solving	O
the	O
associated	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
to	O
obtain	O
a	O
basic	O
feasible	O
solution	O
.	O
</s>
<s>
This	O
is	O
then	O
added	O
as	O
an	O
additional	O
linear	O
constraint	O
to	O
exclude	O
the	O
vertex	O
found	O
,	O
creating	O
a	O
modified	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Cutting	B-Algorithm
plane	I-Algorithm
methods	I-Algorithm
are	O
also	O
applicable	O
in	O
nonlinear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
underlying	O
principle	O
is	O
to	O
approximate	O
the	O
feasible	O
region	O
of	O
a	O
nonlinear	O
(	O
convex	O
)	O
program	O
by	O
a	O
finite	O
set	O
of	O
closed	O
half	O
spaces	O
and	O
to	O
solve	O
a	O
sequence	O
of	O
approximating	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
Nonlinear	B-Algorithm
Programming	I-Algorithm
:	O
Analysis	O
and	O
Methods	O
.	O
</s>
<s>
Valid	O
Inequalities	O
for	O
Mixed	O
Integer	O
Linear	B-Algorithm
Programs	I-Algorithm
.	O
</s>
