<s>
In	O
mathematical	O
optimization	O
,	O
the	O
perturbation	B-Algorithm
function	I-Algorithm
is	O
any	O
function	O
which	O
relates	O
to	O
primal	O
and	O
dual	O
problems	O
.	O
</s>
<s>
In	O
some	O
texts	O
the	O
value	B-Algorithm
function	I-Algorithm
is	O
called	O
the	O
perturbation	B-Algorithm
function	I-Algorithm
,	O
and	O
the	O
perturbation	B-Algorithm
function	I-Algorithm
is	O
called	O
the	O
bifunction	B-Algorithm
.	O
</s>
<s>
Given	O
two	O
dual	B-Algorithm
pairs	I-Algorithm
of	O
separated	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
and	O
.	O
</s>
<s>
Then	O
is	O
a	O
perturbation	B-Algorithm
function	I-Algorithm
if	O
and	O
only	O
if	O
.	O
</s>
<s>
For	O
any	O
choice	O
of	O
perturbation	B-Algorithm
function	I-Algorithm
F	O
weak	B-Algorithm
duality	I-Algorithm
holds	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
conditions	O
which	O
if	O
satisfied	O
imply	O
strong	B-Algorithm
duality	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
F	O
is	O
proper	O
,	O
jointly	O
convex	O
,	O
lower	O
semi-continuous	O
with	O
(	O
where	O
is	O
the	O
algebraic	O
interior	O
and	O
is	O
the	O
projection	O
onto	O
Y	O
defined	O
by	O
)	O
and	O
X	O
,	O
Y	O
are	O
Fréchet	B-Algorithm
spaces	I-Algorithm
then	O
strong	B-Algorithm
duality	I-Algorithm
holds	O
.	O
</s>
<s>
Let	O
and	O
be	O
dual	B-Algorithm
pairs	I-Algorithm
.	O
</s>
<s>
Given	O
a	O
primal	O
problem	O
(	O
minimize	O
f(x )	O
)	O
and	O
a	O
related	O
perturbation	B-Algorithm
function	I-Algorithm
(F(x , y )	O
)	O
then	O
the	O
Lagrangian	O
is	O
the	O
negative	O
conjugate	O
of	O
F	O
with	O
respect	O
to	O
y	O
(	O
i.e.	O
</s>
<s>
Let	O
and	O
be	O
dual	B-Algorithm
pairs	I-Algorithm
.	O
</s>
<s>
Assume	O
there	O
exists	O
a	O
linear	B-Architecture
map	I-Architecture
with	O
adjoint	O
operator	O
.	O
</s>
<s>
In	O
particular	O
if	O
the	O
primal	O
objective	O
is	O
then	O
the	O
perturbation	B-Algorithm
function	I-Algorithm
is	O
given	O
by	O
,	O
which	O
is	O
the	O
traditional	O
definition	O
of	O
Fenchel	O
duality	O
.	O
</s>
