<s>
The	O
dual	O
of	O
a	O
given	O
linear	B-Algorithm
program	I-Algorithm
(	O
LP	O
)	O
is	O
another	O
LP	O
that	O
is	O
derived	O
from	O
the	O
original	O
(	O
the	O
primal	O
)	O
LP	O
in	O
the	O
following	O
schematic	O
way	O
:	O
</s>
<s>
The	O
weak	O
duality	B-Algorithm
theorem	O
states	O
that	O
the	O
objective	O
value	O
of	O
the	O
dual	O
LP	O
at	O
any	O
feasible	O
solution	O
is	O
always	O
a	O
bound	O
on	O
the	O
objective	O
of	O
the	O
primal	O
LP	O
at	O
any	O
feasible	O
solution	O
(	O
upper	O
or	O
lower	O
bound	O
,	O
depending	O
on	O
whether	O
it	O
is	O
a	O
maximization	O
or	O
minimization	O
problem	O
)	O
.	O
</s>
<s>
The	O
strong	O
duality	B-Algorithm
theorem	O
states	O
that	O
,	O
moreover	O
,	O
if	O
the	O
primal	O
has	O
an	O
optimal	O
solution	O
then	O
the	O
dual	O
has	O
an	O
optimal	O
solution	O
too	O
,	O
and	O
the	O
two	O
optima	O
are	O
equal	O
.	O
</s>
<s>
These	O
theorems	O
belong	O
to	O
a	O
larger	O
class	O
of	O
duality	B-Algorithm
theorems	I-Algorithm
in	I-Algorithm
optimization	I-Algorithm
.	O
</s>
<s>
The	O
strong	O
duality	B-Algorithm
theorem	O
is	O
one	O
of	O
the	O
cases	O
in	O
which	O
the	O
duality	B-Algorithm
gap	I-Algorithm
(	O
the	O
gap	O
between	O
the	O
optimum	O
of	O
the	O
primal	O
and	O
the	O
optimum	O
of	O
the	O
dual	O
)	O
is	O
0	O
.	O
</s>
<s>
Suppose	O
we	O
have	O
the	O
linear	B-Algorithm
program	I-Algorithm
:	O
Maximize	O
cTx	O
subject	O
to	O
Ax	O
≤	O
b	O
,	O
x	O
≥	O
0.We	O
would	O
like	O
to	O
construct	O
an	O
upper	O
bound	O
on	O
the	O
solution	O
.	O
</s>
<s>
The	O
duality	B-Algorithm
theorem	O
has	O
an	O
economic	O
interpretation	O
.	O
</s>
<s>
If	O
we	O
interpret	O
the	O
primal	O
LP	O
as	O
a	O
classical	O
"	O
resource	B-Algorithm
allocation	I-Algorithm
"	O
problem	O
,	O
its	O
dual	O
LP	O
can	O
be	O
interpreted	O
as	O
a	O
"	O
resource	O
valuation	O
"	O
problem	O
.	O
</s>
<s>
Then	O
,	O
the	O
second	O
factory	O
's	O
optimization	O
problem	O
is	O
the	O
dual	O
LP:Minimize	O
bTy	O
subject	O
to	O
ATy	O
≥	O
c	O
,	O
y	O
≥	O
0The	O
duality	B-Algorithm
theorem	O
states	O
that	O
the	O
duality	B-Algorithm
gap	I-Algorithm
between	O
the	O
two	O
LP	B-Algorithm
problems	I-Algorithm
is	O
at	O
least	O
zero	O
.	O
</s>
<s>
The	O
strong	O
duality	B-Algorithm
theorem	O
further	O
states	O
that	O
the	O
duality	B-Algorithm
gap	I-Algorithm
is	O
zero	O
.	O
</s>
<s>
With	O
strong	O
duality	B-Algorithm
,	O
the	O
dual	O
solution	O
is	O
,	O
economically	O
speaking	O
,	O
the	O
"	O
equilibrium	O
price	O
"	O
(	O
see	O
shadow	O
price	O
)	O
for	O
the	O
raw	O
material	O
that	O
a	O
factory	O
with	O
production	O
matrix	O
and	O
raw	O
material	O
stock	O
would	O
accept	O
for	O
raw	O
material	O
,	O
given	O
the	O
market	O
price	O
for	O
finished	O
goods	O
.	O
</s>
<s>
The	O
duality	B-Algorithm
theorem	O
has	O
a	O
physical	O
interpretation	O
too	O
.	O
</s>
<s>
The	O
weak	O
duality	B-Algorithm
theorem	O
says	O
that	O
,	O
for	O
each	O
feasible	O
solution	O
x	O
of	O
the	O
primal	O
and	O
each	O
feasible	O
solution	O
y	O
of	O
the	O
dual	O
:	O
cTx	O
≤	O
bTy	O
.	O
</s>
<s>
Weak	O
duality	B-Algorithm
implies:maxx	O
cTx	O
≤	O
miny	O
bTyIn	O
particular	O
,	O
if	O
the	O
primal	O
is	O
unbounded	O
(	O
from	O
above	O
)	O
then	O
the	O
dual	O
has	O
no	O
feasible	O
solution	O
,	O
and	O
if	O
the	O
dual	O
is	O
unbounded	O
(	O
from	O
below	O
)	O
then	O
the	O
primal	O
has	O
no	O
feasible	O
solution	O
.	O
</s>
<s>
The	O
strong	O
duality	B-Algorithm
theorem	O
says	O
that	O
if	O
one	O
of	O
the	O
two	O
problems	O
has	O
an	O
optimal	O
solution	O
,	O
so	O
does	O
the	O
other	O
one	O
and	O
that	O
the	O
bounds	O
given	O
by	O
the	O
weak	O
duality	B-Algorithm
theorem	O
are	O
tight	O
,	O
i.e.	O
</s>
<s>
:maxx	O
cTx	O
=	O
miny	O
bTyThe	O
strong	O
duality	B-Algorithm
theorem	O
is	O
harder	O
to	O
prove	O
;	O
the	O
proofs	O
usually	O
use	O
the	O
weak	O
duality	B-Algorithm
theorem	O
as	O
a	O
sub-routine	O
.	O
</s>
<s>
One	O
proof	O
uses	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
and	O
relies	O
on	O
the	O
proof	O
that	O
,	O
with	O
the	O
suitable	O
pivot	O
rule	O
,	O
it	O
provides	O
a	O
correct	O
solution	O
.	O
</s>
<s>
The	O
proof	O
establishes	O
that	O
,	O
once	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
finishes	O
with	O
a	O
solution	O
to	O
the	O
primal	O
LP	O
,	O
it	O
is	O
possible	O
to	O
read	O
from	O
the	O
final	O
tableau	O
,	O
a	O
solution	O
to	O
the	O
dual	O
LP	O
.	O
</s>
<s>
So	O
,	O
by	O
running	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
we	O
obtain	O
solutions	O
to	O
both	O
the	O
primal	O
and	O
the	O
dual	O
simultaneously	O
.	O
</s>
<s>
The	O
weak	O
duality	B-Algorithm
theorem	O
implies	O
that	O
finding	O
a	O
single	O
feasible	O
solution	O
is	O
as	O
hard	O
as	O
finding	O
an	O
optimal	O
feasible	O
solution	O
.	O
</s>
<s>
The	O
combined	O
LP	O
has	O
both	O
x	O
and	O
y	O
as	O
variables:Maximize	O
1	O
subject	O
to	O
Ax	O
≤	O
b	O
,	O
ATy	O
≥	O
c	O
,	O
cTx	O
≥	O
bTy	O
,	O
x	O
≥	O
0	O
,	O
y	O
≥	O
0If	O
the	O
combined	O
LP	O
has	O
a	O
feasible	O
solution	O
(	O
x	O
,	O
y	O
)	O
,	O
then	O
by	O
weak	O
duality	B-Algorithm
,	O
cTx	O
=	O
bTy	O
.	O
</s>
<s>
The	O
strong	O
duality	B-Algorithm
theorem	O
provides	O
a	O
"	O
good	O
characterization	O
"	O
of	O
the	O
optimal	O
value	O
of	O
an	O
LP	O
in	O
that	O
it	O
allows	O
us	O
to	O
easily	O
prove	O
that	O
some	O
value	O
t	O
is	O
the	O
optimum	O
of	O
some	O
LP	O
.	O
</s>
<s>
In	O
accordance	O
with	O
the	O
strong	O
duality	B-Algorithm
theorem	O
,	O
the	O
maximum	O
of	O
the	O
primal	O
equals	O
the	O
minimum	O
of	O
the	O
dual	O
.	O
</s>
<s>
We	O
use	O
this	O
example	O
to	O
illustrate	O
the	O
proof	O
of	O
the	O
weak	O
duality	B-Algorithm
theorem	O
.	O
</s>
<s>
The	O
primal	B-Algorithm
problem	I-Algorithm
would	O
be	O
the	O
farmer	O
deciding	O
how	O
much	O
wheat	O
(	O
)	O
and	O
barley	O
(	O
)	O
to	O
grow	O
if	O
their	O
sell	O
prices	O
are	O
and	O
per	O
unit	O
.	O
</s>
<s>
The	O
primal	B-Algorithm
problem	I-Algorithm
deals	O
with	O
physical	O
quantities	O
.	O
</s>
<s>
Duality	B-Algorithm
theory	O
tells	O
us	O
that	O
:	O
</s>
<s>
The	O
max-flow	O
min-cut	O
theorem	O
is	O
a	O
special	O
case	O
of	O
the	O
strong	O
duality	B-Algorithm
theorem	O
:	O
flow-maximization	O
is	O
the	O
primal	O
LP	O
,	O
and	O
cut-minimization	O
is	O
the	O
dual	O
LP	O
.	O
</s>
<s>
Other	O
graph-related	O
theorems	O
can	O
be	O
proved	O
using	O
the	O
strong	O
duality	B-Algorithm
theorem	O
,	O
in	O
particular	O
,	O
Konig	O
's	O
theorem	O
.	O
</s>
<s>
The	O
Minimax	O
theorem	O
for	O
zero-sum	O
games	O
can	O
be	O
proved	O
using	O
the	O
strong-duality	O
theorem	O
.	O
</s>
<s>
Consider	O
the	O
following	O
linear	B-Algorithm
program	I-Algorithm
:	O
</s>
<s>
However	O
,	O
any	O
linear	B-Algorithm
program	I-Algorithm
may	O
be	O
transformed	O
to	O
standard	O
form	O
and	O
it	O
is	O
therefore	O
not	O
a	O
limiting	O
factor	O
.	O
</s>
