<s>
The	O
Bregman	B-Algorithm
method	I-Algorithm
is	O
an	O
iterative	B-Algorithm
algorithm	I-Algorithm
to	O
solve	O
certain	O
convex	O
optimization	O
problems	O
involving	O
regularization	O
.	O
</s>
<s>
The	O
algorithm	O
is	O
a	O
row-action	O
method	O
accessing	O
constraint	B-Application
functions	I-Application
one	O
by	O
one	O
and	O
the	O
method	O
is	O
particularly	O
suited	O
for	O
large	O
optimization	O
problems	O
where	O
constraints	O
can	O
be	O
efficiently	O
enumerated	O
.	O
</s>
<s>
In	O
order	O
to	O
be	O
able	O
to	O
use	O
the	O
Bregman	B-Algorithm
method	I-Algorithm
,	O
one	O
must	O
frame	O
the	O
problem	O
of	O
interest	O
as	O
finding	O
,	O
where	O
is	O
a	O
regularizing	O
function	O
such	O
as	O
.	O
</s>
<s>
The	O
Bregman	B-Algorithm
distance	I-Algorithm
is	O
defined	O
as	O
where	O
belongs	O
to	O
the	O
subdifferential	O
of	O
at	O
(	O
which	O
we	O
denoted	O
)	O
.	O
</s>
<s>
Then	O
,	O
for	O
each	O
constraint	B-Application
a	O
generalized	B-Algorithm
projection	I-Algorithm
onto	O
its	O
feasible	O
set	O
is	O
performed	O
,	O
updating	O
both	O
the	O
constraint	B-Application
's	O
dual	O
variable	O
and	O
all	O
primal	O
variables	O
for	O
which	O
there	O
are	O
non-zero	O
coefficients	O
in	O
the	O
constraint	B-Application
functions	I-Application
gradient	O
.	O
</s>
<s>
In	O
case	O
the	O
objective	O
is	O
strictly	O
convex	O
and	O
all	O
constraint	B-Application
functions	I-Application
are	O
convex	O
,	O
the	O
limit	O
of	O
this	O
iterative	O
projection	O
converges	O
to	O
the	O
optimal	O
primal	O
dual	O
pair	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
basis	O
pursuit-type	O
problem	O
,	O
the	O
Bregman	B-Algorithm
method	I-Algorithm
is	O
equivalent	O
to	O
ordinary	O
gradient	B-Algorithm
descent	I-Algorithm
on	O
the	O
dual	O
problem	O
.	O
</s>
<s>
The	O
Bregman	B-Algorithm
method	I-Algorithm
or	O
its	O
generalizations	O
can	O
be	O
applied	O
to	O
:	O
</s>
<s>
In	O
case	O
this	O
can	O
not	O
be	O
ensured	O
,	O
as	O
for	O
linear	B-Algorithm
programs	I-Algorithm
or	O
non-strictly	O
convex	O
quadratic	O
programs	O
,	O
additional	O
methods	O
such	O
as	O
proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
have	O
been	O
developed	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
Rudin-Osher-Fatemi	O
model	O
of	O
image	O
denoising	O
,	O
the	O
Bregman	B-Algorithm
method	I-Algorithm
provably	O
converges	O
.	O
</s>
<s>
Some	O
generalizations	O
of	O
the	O
Bregman	B-Algorithm
method	I-Algorithm
include	O
:	O
</s>
<s>
In	O
the	O
Linearized	O
Bregman	B-Algorithm
method	I-Algorithm
,	O
one	O
linearizes	O
the	O
intermediate	O
objective	O
functions	O
by	O
replacing	O
the	O
second	O
term	O
with	O
(	O
which	O
approximates	O
the	O
second	O
term	O
near	O
)	O
and	O
adding	O
the	O
penalty	O
term	O
for	O
a	O
constant	O
.	O
</s>
<s>
Sometimes	O
,	O
when	O
running	O
the	O
Linearized	O
Bregman	B-Algorithm
method	I-Algorithm
,	O
there	O
are	O
periods	O
of	O
"	O
stagnation	O
"	O
where	O
the	O
residual	O
is	O
almost	O
constant	O
.	O
</s>
<s>
To	O
alleviate	O
this	O
issue	O
,	O
one	O
can	O
use	O
the	O
Linearized	O
Bregman	B-Algorithm
method	I-Algorithm
with	O
kicking	O
,	O
where	O
one	O
essentially	O
detects	O
the	O
beginning	O
of	O
a	O
stagnation	O
period	O
,	O
then	O
predicts	O
and	O
skips	O
to	O
the	O
end	O
of	O
it	O
.	O
</s>
<s>
Since	O
Linearized	O
Bregman	O
is	O
mathematically	O
equivalent	O
to	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
it	O
can	O
be	O
accelerated	O
with	O
methods	O
to	O
accelerate	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
such	O
as	O
line	O
search	O
,	O
L-BGFS	B-Algorithm
,	O
Barzilai-Borwein	O
steps	O
,	O
or	O
the	O
Nesterov	O
method	O
;	O
the	O
last	O
has	O
been	O
proposed	O
as	O
the	O
accelerated	O
linearized	O
Bregman	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
Split	O
Bregman	B-Algorithm
method	I-Algorithm
solves	O
problems	O
of	O
the	O
form	O
,	O
where	O
and	O
are	O
both	O
convex	O
,	O
particularly	O
problems	O
of	O
the	O
form	O
.	O
</s>
<s>
The	O
Split	O
Bregman	B-Algorithm
method	I-Algorithm
has	O
been	O
generalized	O
to	O
optimization	O
over	O
complex	O
numbers	O
using	O
Wirtinger	O
derivatives	O
.	O
</s>
