<s>
Benders	B-Algorithm
decomposition	I-Algorithm
(	O
or	O
Benders	B-Algorithm
 '	I-Algorithm
decomposition	I-Algorithm
)	O
is	O
a	O
technique	O
in	O
mathematical	O
programming	O
that	O
allows	O
the	O
solution	O
of	O
very	O
large	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
that	O
have	O
a	O
special	O
block	B-Algorithm
structure	I-Algorithm
.	O
</s>
<s>
This	O
block	B-Algorithm
structure	I-Algorithm
often	O
occurs	O
in	O
applications	O
such	O
as	O
stochastic	B-Algorithm
programming	I-Algorithm
as	O
the	O
uncertainty	O
is	O
usually	O
represented	O
with	O
scenarios	O
.	O
</s>
<s>
The	O
strategy	O
behind	O
Benders	B-Algorithm
decomposition	I-Algorithm
can	O
be	O
summarized	O
as	O
divide-and-conquer	O
.	O
</s>
<s>
That	O
is	O
,	O
in	O
Benders	B-Algorithm
decomposition	I-Algorithm
,	O
the	O
variables	O
of	O
the	O
original	O
problem	O
are	O
divided	O
into	O
two	O
subsets	O
so	O
that	O
a	O
first-stage	O
master	O
problem	O
is	O
solved	O
over	O
the	O
first	O
set	O
of	O
variables	O
,	O
and	O
the	O
values	O
for	O
the	O
second	O
set	O
of	O
variables	O
are	O
determined	O
in	O
a	O
second-stage	O
subproblem	O
for	O
a	O
given	O
first-stage	O
solution	O
.	O
</s>
<s>
Since	O
Benders	B-Algorithm
decomposition	I-Algorithm
adds	O
new	O
constraints	O
as	O
it	O
progresses	O
towards	O
a	O
solution	O
,	O
the	O
approach	O
is	O
called	O
"	O
row	O
generation	O
"	O
.	O
</s>
<s>
In	O
contrast	O
,	O
Dantzig	B-Algorithm
–	I-Algorithm
Wolfe	I-Algorithm
decomposition	I-Algorithm
uses	O
"	O
column	B-Algorithm
generation	I-Algorithm
"	O
.	O
</s>
<s>
Benders	B-Algorithm
Decomposition	I-Algorithm
is	O
applicable	O
to	O
problems	O
with	O
a	O
largely	O
block-diagonal	O
structure	O
.	O
</s>
<s>
Benders	B-Algorithm
decomposition	I-Algorithm
relies	O
on	O
an	O
iterative	O
procedure	O
that	O
chooses	O
successive	O
values	O
of	O
without	O
considering	O
the	O
inner	O
problem	O
except	O
through	O
a	O
set	O
of	O
cut	O
constraints	O
that	O
are	O
created	O
through	O
a	O
pass-back	O
mechanism	O
from	O
the	O
maximization	O
problem	O
.	O
</s>
<s>
If	O
for	O
some	O
the	O
objective	O
value	O
of	O
the	O
master	O
problem	O
is	O
equal	O
to	O
the	O
value	O
of	O
the	O
optimal	O
value	O
of	O
the	O
inner	O
problem	O
,	O
then	O
by	O
duality	B-Algorithm
theory	I-Algorithm
the	O
solution	O
is	O
optimal	O
.	O
</s>
<s>
By	O
duality	B-Algorithm
theory	I-Algorithm
,	O
when	O
a	O
dual	O
problem	O
has	O
unbounded	O
objective	O
the	O
corresponding	O
primal	O
problem	O
is	O
infeasible	O
.	O
</s>
<s>
By	O
duality	B-Algorithm
theory	I-Algorithm
for	O
linear	B-Algorithm
programs	I-Algorithm
,	O
the	O
optimal	O
value	O
of	O
the	O
subproblem	O
is	O
equal	O
to	O
the	O
optimal	O
value	O
of	O
the	O
original	O
problem	O
constrained	O
on	O
the	O
choice	O
of	O
.	O
</s>
