<s>
In	O
mathematical	O
optimization	O
theory	O
,	O
duality	B-Algorithm
or	O
the	O
duality	B-Algorithm
principle	I-Algorithm
is	O
the	O
principle	O
that	O
optimization	O
problems	O
may	O
be	O
viewed	O
from	O
either	O
of	O
two	O
perspectives	O
,	O
the	O
primal	B-Algorithm
problem	I-Algorithm
or	O
the	O
dual	O
problem	O
.	O
</s>
<s>
This	O
fact	O
is	O
called	O
weak	B-Algorithm
duality	I-Algorithm
.	O
</s>
<s>
Their	O
difference	O
is	O
called	O
the	O
duality	B-Algorithm
gap	I-Algorithm
.	O
</s>
<s>
For	O
convex	O
optimization	O
problems	O
,	O
the	O
duality	B-Algorithm
gap	I-Algorithm
is	O
zero	O
under	O
a	O
constraint	B-Application
qualification	O
condition	O
.	O
</s>
<s>
This	O
fact	O
is	O
called	O
strong	B-Algorithm
duality	I-Algorithm
.	O
</s>
<s>
The	O
Lagrangian	O
dual	O
problem	O
is	O
obtained	O
by	O
forming	O
the	O
Lagrangian	O
of	O
a	O
minimization	O
problem	O
by	O
using	O
nonnegative	O
Lagrange	O
multipliers	O
to	O
add	O
the	O
constraints	B-Application
to	O
the	O
objective	O
function	O
,	O
and	O
then	O
solving	O
for	O
the	O
primal	O
variable	O
values	O
that	O
minimize	O
the	O
original	O
objective	O
function	O
.	O
</s>
<s>
This	O
solution	O
gives	O
the	O
primal	O
variables	O
as	O
functions	O
of	O
the	O
Lagrange	O
multipliers	O
,	O
which	O
are	O
called	O
dual	O
variables	O
,	O
so	O
that	O
the	O
new	O
problem	O
is	O
to	O
maximize	O
the	O
objective	O
function	O
with	O
respect	O
to	O
the	O
dual	O
variables	O
under	O
the	O
derived	O
constraints	B-Application
on	O
the	O
dual	O
variables	O
(	O
including	O
at	O
least	O
the	O
nonnegativity	O
constraints	B-Application
)	O
.	O
</s>
<s>
If	O
there	O
are	O
constraint	B-Application
conditions	O
,	O
these	O
can	O
be	O
built	O
into	O
the	O
function	O
by	O
letting	O
where	O
is	O
a	O
suitable	O
function	O
on	O
that	O
has	O
a	O
minimum	O
0	O
on	O
the	O
constraints	B-Application
,	O
and	O
for	O
which	O
one	O
can	O
prove	O
that	O
.	O
</s>
<s>
for	O
satisfying	O
the	O
constraints	B-Application
and	O
otherwise	O
)	O
.	O
</s>
<s>
Then	O
extend	O
to	O
a	O
perturbation	B-Algorithm
function	I-Algorithm
such	O
that	O
.	O
</s>
<s>
The	O
duality	B-Algorithm
gap	I-Algorithm
is	O
the	O
difference	O
between	O
the	O
values	O
of	O
any	O
primal	O
solutions	O
and	O
any	O
dual	O
solutions	O
.	O
</s>
<s>
If	O
is	O
the	O
optimal	O
dual	O
value	O
and	O
is	O
the	O
optimal	O
primal	O
value	O
,	O
then	O
the	O
duality	B-Algorithm
gap	I-Algorithm
is	O
equal	O
to	O
.	O
</s>
<s>
The	O
duality	B-Algorithm
gap	I-Algorithm
is	O
zero	O
if	O
and	O
only	O
if	O
strong	B-Algorithm
duality	I-Algorithm
holds	O
.	O
</s>
<s>
Otherwise	O
the	O
gap	O
is	O
strictly	O
positive	O
and	O
weak	B-Algorithm
duality	I-Algorithm
holds	O
.	O
</s>
<s>
In	O
computational	O
optimization	O
,	O
another	O
"	O
duality	B-Algorithm
gap	I-Algorithm
"	O
is	O
often	O
reported	O
,	O
which	O
is	O
the	O
difference	O
in	O
value	O
between	O
any	O
dual	O
solution	O
and	O
the	O
value	O
of	O
a	O
feasible	O
but	O
suboptimal	O
iterate	O
for	O
the	O
primal	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
This	O
alternative	O
"	O
duality	B-Algorithm
gap	I-Algorithm
"	O
quantifies	O
the	O
discrepancy	O
between	O
the	O
value	O
of	O
a	O
current	O
feasible	O
but	O
suboptimal	O
iterate	O
for	O
the	O
primal	B-Algorithm
problem	I-Algorithm
and	O
the	O
value	O
of	O
the	O
dual	O
problem	O
;	O
the	O
value	O
of	O
the	O
dual	O
problem	O
is	O
,	O
under	O
regularity	O
conditions	O
,	O
equal	O
to	O
the	O
value	O
of	O
the	O
convex	O
relaxation	O
of	O
the	O
primal	B-Algorithm
problem	I-Algorithm
:	O
The	O
convex	O
relaxation	O
is	O
the	O
problem	O
arising	O
replacing	O
a	O
non-convex	O
feasible	O
set	O
with	O
its	O
closed	O
convex	O
hull	O
and	O
with	O
replacing	O
a	O
non-convex	O
function	O
with	O
its	O
convex	O
closure	O
,	O
that	O
is	O
the	O
function	O
that	O
has	O
the	O
epigraph	O
that	O
is	O
the	O
closed	O
convex	O
hull	O
of	O
the	O
original	O
primal	O
objective	O
function	O
.	O
</s>
<s>
Linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
are	O
optimization	O
problems	O
in	O
which	O
the	O
objective	O
function	O
and	O
the	O
constraints	B-Application
are	O
all	O
linear	O
.	O
</s>
<s>
In	O
the	O
primal	B-Algorithm
problem	I-Algorithm
,	O
the	O
objective	O
function	O
is	O
a	O
linear	O
combination	O
of	O
n	O
variables	O
.	O
</s>
<s>
There	O
are	O
m	O
constraints	B-Application
,	O
each	O
of	O
which	O
places	O
an	O
upper	O
bound	O
on	O
a	O
linear	O
combination	O
of	O
the	O
n	O
variables	O
.	O
</s>
<s>
The	O
goal	O
is	O
to	O
maximize	O
the	O
value	O
of	O
the	O
objective	O
function	O
subject	O
to	O
the	O
constraints	B-Application
.	O
</s>
<s>
In	O
the	O
dual	O
problem	O
,	O
the	O
objective	O
function	O
is	O
a	O
linear	O
combination	O
of	O
the	O
m	O
values	O
that	O
are	O
the	O
limits	O
in	O
the	O
m	O
constraints	B-Application
from	O
the	O
primal	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
There	O
are	O
n	O
dual	O
constraints	B-Application
,	O
each	O
of	O
which	O
places	O
a	O
lower	O
bound	O
on	O
a	O
linear	O
combination	O
of	O
m	O
dual	O
variables	O
.	O
</s>
<s>
In	O
the	O
linear	O
case	O
,	O
in	O
the	O
primal	B-Algorithm
problem	I-Algorithm
,	O
from	O
each	O
sub-optimal	O
point	O
that	O
satisfies	O
all	O
the	O
constraints	B-Application
,	O
there	O
is	O
a	O
direction	O
or	O
subspace	O
of	O
directions	O
to	O
move	O
that	O
increases	O
the	O
objective	O
function	O
.	O
</s>
<s>
Moving	O
in	O
any	O
such	O
direction	O
is	O
said	O
to	O
remove	O
slack	O
between	O
the	O
candidate	O
solution	O
and	O
one	O
or	O
more	O
constraints	B-Application
.	O
</s>
<s>
An	O
infeasible	O
value	O
of	O
the	O
candidate	O
solution	O
is	O
one	O
that	O
exceeds	O
one	O
or	O
more	O
of	O
the	O
constraints	B-Application
.	O
</s>
<s>
In	O
the	O
dual	O
problem	O
,	O
the	O
dual	O
vector	O
multiplies	O
the	O
constraints	B-Application
that	O
determine	O
the	O
positions	O
of	O
the	O
constraints	B-Application
in	O
the	O
primal	O
.	O
</s>
<s>
Varying	O
the	O
dual	O
vector	O
in	O
the	O
dual	O
problem	O
is	O
equivalent	O
to	O
revising	O
the	O
upper	O
bounds	O
in	O
the	O
primal	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
the	O
dual	O
vector	O
is	O
minimized	O
in	O
order	O
to	O
remove	O
slack	O
between	O
the	O
candidate	O
positions	O
of	O
the	O
constraints	B-Application
and	O
the	O
actual	O
optimum	O
.	O
</s>
<s>
It	O
sets	O
the	O
candidate	O
positions	O
of	O
one	O
or	O
more	O
of	O
the	O
constraints	B-Application
in	O
a	O
position	O
that	O
excludes	O
the	O
actual	O
optimum	O
.	O
</s>
<s>
This	O
intuition	O
is	O
made	O
formal	O
by	O
the	O
equations	O
in	O
Linear	B-Algorithm
programming	I-Algorithm
:	O
Duality	B-Algorithm
.	O
</s>
<s>
In	O
nonlinear	B-Algorithm
programming	I-Algorithm
,	O
the	O
constraints	B-Application
are	O
not	O
necessarily	O
linear	O
.	O
</s>
<s>
They	O
provide	O
necessary	O
conditions	O
for	O
identifying	O
local	O
optima	O
of	O
non-linear	B-Algorithm
programming	I-Algorithm
problems	O
.	O
</s>
<s>
There	O
are	O
additional	O
conditions	O
(	O
constraint	B-Application
qualifications	O
)	O
that	O
are	O
necessary	O
so	O
that	O
it	O
will	O
be	O
possible	O
to	O
define	O
the	O
direction	O
to	O
an	O
optimal	O
solution	O
.	O
</s>
<s>
The	O
dual	B-Algorithm
function	I-Algorithm
g	O
is	O
concave	O
,	O
even	O
when	O
the	O
initial	O
problem	O
is	O
not	O
convex	O
,	O
because	O
it	O
is	O
a	O
point-wise	O
infimum	O
of	O
affine	O
functions	O
.	O
</s>
<s>
The	O
dual	B-Algorithm
function	I-Algorithm
yields	O
lower	O
bounds	O
on	O
the	O
optimal	O
value	O
of	O
the	O
initial	O
problem	O
;	O
for	O
any	O
and	O
any	O
we	O
have	O
.	O
</s>
<s>
If	O
a	O
constraint	B-Application
qualification	O
such	O
as	O
Slater	O
's	O
condition	O
holds	O
and	O
the	O
original	O
problem	O
is	O
convex	O
,	O
then	O
we	O
have	O
strong	B-Algorithm
duality	I-Algorithm
,	O
i.e.	O
</s>
<s>
For	O
a	O
convex	O
minimization	O
problem	O
with	O
inequality	B-Application
constraints	I-Application
,	O
</s>
<s>
where	O
the	O
objective	O
function	O
is	O
the	O
Lagrange	O
dual	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Also	O
,	O
the	O
equality	O
constraint	B-Application
is	O
nonlinear	O
in	O
general	O
,	O
so	O
the	O
Wolfe	O
dual	O
problem	O
is	O
typically	O
a	O
nonconvex	O
optimization	O
problem	O
.	O
</s>
<s>
In	O
any	O
case	O
,	O
weak	B-Algorithm
duality	I-Algorithm
holds	O
.	O
</s>
<s>
According	O
to	O
George	O
Dantzig	O
,	O
the	O
duality	B-Algorithm
theorem	O
for	O
linear	B-Algorithm
optimization	I-Algorithm
was	O
conjectured	O
by	O
John	O
von	O
Neumann	O
immediately	O
after	O
Dantzig	O
presented	O
the	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Von	O
Neumann	O
noted	O
that	O
he	O
was	O
using	O
information	O
from	O
his	O
game	O
theory	O
,	O
and	O
conjectured	O
that	O
two	O
person	O
zero	O
sum	O
matrix	O
game	O
was	O
equivalent	O
to	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
