<s>
In	O
optimization	O
problems	O
in	O
applied	O
mathematics	O
,	O
the	O
duality	B-Algorithm
gap	I-Algorithm
is	O
the	O
difference	O
between	O
the	O
primal	O
and	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
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
general	O
given	O
two	O
dual	B-Algorithm
pairs	I-Algorithm
separated	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
and	O
.	O
</s>
<s>
Then	O
let	O
be	O
a	O
perturbation	B-Algorithm
function	I-Algorithm
such	O
that	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	O
problem	O
.	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	O
problem	O
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	O
problem	O
:	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>
