<s>
Augmented	B-Algorithm
Lagrangian	I-Algorithm
methods	I-Algorithm
are	O
a	O
certain	O
class	O
of	O
algorithms	O
for	O
solving	O
constrained	B-Application
optimization	O
problems	O
.	O
</s>
<s>
They	O
have	O
similarities	O
to	O
penalty	B-Algorithm
methods	I-Algorithm
in	O
that	O
they	O
replace	O
a	O
constrained	B-Application
optimization	O
problem	O
by	O
a	O
series	O
of	O
unconstrained	O
problems	O
and	O
add	O
a	O
penalty	O
term	O
to	O
the	O
objective	O
;	O
the	O
difference	O
is	O
that	O
the	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
adds	O
yet	O
another	O
term	O
,	O
designed	O
to	O
mimic	O
a	O
Lagrange	O
multiplier	O
.	O
</s>
<s>
Viewed	O
differently	O
,	O
the	O
unconstrained	O
objective	O
is	O
the	O
Lagrangian	O
of	O
the	O
constrained	B-Application
problem	O
,	O
with	O
an	O
additional	O
penalty	O
term	O
(	O
the	O
augmentation	O
)	O
.	O
</s>
<s>
The	O
method	O
was	O
originally	O
known	O
as	O
the	O
method	O
of	O
multipliers	O
,	O
and	O
was	O
studied	O
much	O
in	O
the	O
1970	O
and	O
1980s	O
as	O
a	O
good	O
alternative	O
to	O
penalty	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
method	O
was	O
also	O
studied	O
by	O
Dimitri	O
Bertsekas	O
,	O
notably	O
in	O
his	O
1982	O
book	O
,	O
together	O
with	O
extensions	O
involving	O
nonquadratic	O
regularization	O
functions	O
,	O
such	O
as	O
entropic	B-Algorithm
regularization	I-Algorithm
,	O
which	O
gives	O
rise	O
to	O
the	O
"	O
exponential	O
method	O
of	O
multipliers	O
,	O
"	O
a	O
method	O
that	O
handles	O
inequality	B-Application
constraints	I-Application
with	O
a	O
twice	O
differentiable	O
augmented	O
Lagrangian	O
function	O
.	O
</s>
<s>
Since	O
the	O
1970s	O
,	O
sequential	B-Algorithm
quadratic	I-Algorithm
programming	I-Algorithm
(	O
SQP	O
)	O
and	O
interior	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
(	O
IPM	O
)	O
have	O
had	O
increasing	O
attention	O
,	O
in	O
part	O
because	O
they	O
more	O
easily	O
use	O
sparse	B-Algorithm
matrix	I-Algorithm
subroutines	O
from	O
numerical	O
software	O
libraries	O
,	O
and	O
in	O
part	O
because	O
IPMs	O
have	O
proven	O
complexity	O
results	O
via	O
the	O
theory	O
of	O
self-concordant	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
The	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
was	O
rejuvenated	O
by	O
the	O
optimization	O
systems	O
LANCELOT	B-Application
,	O
ALGENCAN	O
and	O
AMPL	B-Language
,	O
which	O
allowed	O
sparse	B-Algorithm
matrix	I-Algorithm
techniques	O
to	O
be	O
used	O
on	O
seemingly	O
dense	O
but	O
"	O
partially	O
separable	O
"	O
problems	O
.	O
</s>
<s>
Around	O
2007	O
,	O
there	O
was	O
a	O
resurgence	O
of	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
methods	I-Algorithm
in	O
fields	O
such	O
as	O
total-variation	B-Algorithm
denoising	I-Algorithm
and	O
compressed	O
sensing	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
variant	O
of	O
the	O
standard	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
that	O
uses	O
partial	O
updates	O
(	O
similar	O
to	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
for	O
solving	O
linear	O
equations	O
)	O
known	O
as	O
the	O
alternating	O
direction	O
method	O
of	O
multipliers	O
or	O
ADMM	B-Algorithm
gained	O
some	O
attention	O
.	O
</s>
<s>
Let	O
us	O
say	O
we	O
are	O
solving	O
the	O
following	O
constrained	B-Application
problem	O
:	O
</s>
<s>
For	O
reference	O
,	O
we	O
first	O
list	O
the	O
kth	O
step	O
of	O
the	O
penalty	B-Algorithm
method	I-Algorithm
approach	O
:	O
</s>
<s>
The	O
penalty	B-Algorithm
method	I-Algorithm
solves	O
this	O
problem	O
,	O
then	O
at	O
the	O
next	O
iteration	O
it	O
re-solves	O
the	O
problem	O
using	O
a	O
larger	O
value	O
of	O
(	O
and	O
using	O
the	O
old	O
solution	O
as	O
the	O
initial	O
guess	O
or	O
"	O
warm-start	O
"	O
)	O
.	O
</s>
<s>
The	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
uses	O
the	O
following	O
unconstrained	O
objective	O
:	O
</s>
<s>
The	O
major	O
advantage	O
of	O
the	O
method	O
is	O
that	O
unlike	O
the	O
penalty	B-Algorithm
method	I-Algorithm
,	O
it	O
is	O
not	O
necessary	O
to	O
take	O
in	O
order	O
to	O
solve	O
the	O
original	O
constrained	B-Application
problem	O
.	O
</s>
<s>
The	O
method	O
can	O
be	O
extended	O
to	O
handle	O
inequality	B-Application
constraints	I-Application
.	O
</s>
<s>
The	O
alternating	O
direction	O
method	O
of	O
multipliers	O
(	O
ADMM	B-Algorithm
)	O
is	O
a	O
variant	O
of	O
the	O
augmented	O
Lagrangian	O
scheme	O
that	O
uses	O
partial	O
updates	O
for	O
the	O
dual	O
variables	O
.	O
</s>
<s>
Though	O
this	O
change	O
may	O
seem	O
trivial	O
,	O
the	O
problem	O
can	O
now	O
be	O
attacked	O
using	O
methods	O
of	O
constrained	B-Application
optimization	O
(	O
in	O
particular	O
,	O
the	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
)	O
,	O
and	O
the	O
objective	O
function	O
is	O
separable	O
in	O
x	O
and	O
y	O
.	O
</s>
<s>
The	O
dual	O
update	O
requires	O
solving	O
a	O
proximity	O
function	O
in	O
x	O
and	O
y	O
at	O
the	O
same	O
time	O
;	O
the	O
ADMM	B-Algorithm
technique	O
allows	O
this	O
problem	O
to	O
be	O
solved	O
approximately	O
by	O
first	O
solving	O
for	O
x	O
with	O
y	O
fixed	O
,	O
and	O
then	O
solving	O
for	O
y	O
with	O
x	O
fixed	O
.	O
</s>
<s>
Rather	O
than	O
iterate	O
until	O
convergence	O
(	O
like	O
the	O
Jacobi	B-Algorithm
method	I-Algorithm
)	O
,	O
the	O
algorithm	O
proceeds	O
directly	O
to	O
updating	O
the	O
dual	O
variable	O
and	O
then	O
repeating	O
the	O
process	O
.	O
</s>
<s>
Because	O
of	O
this	O
approximation	O
,	O
the	O
algorithm	O
is	O
distinct	O
from	O
the	O
pure	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
ADMM	B-Algorithm
can	O
be	O
viewed	O
as	O
an	O
application	O
of	O
the	O
Douglas-Rachford	O
splitting	O
algorithm	O
,	O
and	O
the	O
Douglas-Rachford	O
algorithm	O
is	O
in	O
turn	O
an	O
instance	O
of	O
the	O
Proximal	O
point	O
algorithm	O
;	O
details	O
can	O
be	O
found	O
here	O
.	O
</s>
<s>
There	O
are	O
several	O
modern	O
software	O
packages	O
that	O
solve	O
Basis	O
pursuit	O
and	O
variants	O
and	O
use	O
the	O
ADMM	B-Algorithm
;	O
such	O
packages	O
include	O
YALL1	O
(	O
2009	O
)	O
,	O
SpaRSA	O
(	O
2009	O
)	O
and	O
SALSA	O
(	O
2009	O
)	O
.	O
</s>
<s>
There	O
are	O
also	O
packages	O
that	O
use	O
the	O
ADMM	B-Algorithm
to	O
solve	O
more	O
general	O
problems	O
,	O
some	O
of	O
which	O
can	O
exploit	O
multiple	O
computing	O
cores	O
SNAPVX	O
(	O
2015	O
)	O
,	O
parADMM	O
(	O
2016	O
)	O
.	O
</s>
<s>
ADMM	B-Algorithm
is	O
originally	O
a	O
batch	O
method	O
.	O
</s>
<s>
The	O
alternating	O
direction	O
method	O
of	O
multipliers	O
(	O
ADMM	B-Algorithm
)	O
is	O
a	O
popular	O
method	O
for	O
online	O
and	O
distributed	O
optimization	O
on	O
a	O
large	O
scale	O
,	O
and	O
is	O
employed	O
in	O
many	O
applications	O
,	O
e.g.	O
</s>
<s>
ADMM	B-Algorithm
is	O
often	O
applied	O
to	O
solve	O
regularized	O
problems	O
,	O
where	O
the	O
function	O
optimization	O
and	O
regularization	O
can	O
be	O
carried	O
out	O
locally	O
,	O
and	O
then	O
coordinated	O
globally	O
via	O
constraints	O
.	O
</s>
<s>
Regularized	O
optimization	O
problems	O
are	O
especially	O
relevant	O
in	O
the	O
high	O
dimensional	O
regime	O
since	O
regularization	O
is	O
a	O
natural	O
mechanism	O
to	O
overcome	O
ill-poseidness	O
and	O
to	O
encourage	O
parsimony	O
in	O
the	O
optimal	O
solution	O
,	O
e.g.	O
,	O
sparsity	B-Algorithm
and	O
low	O
rank	O
.	O
</s>
<s>
Due	O
to	O
the	O
efficiency	O
of	O
ADMM	B-Algorithm
in	O
solving	O
regularized	O
problems	O
,	O
it	O
has	O
a	O
good	O
potential	O
for	O
stochastic	O
optimization	O
in	O
high	O
dimensions	O
.	O
</s>
<s>
Open	O
source	O
and	O
non-free/commercial	O
implementations	O
of	O
the	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
MINOS	B-Application
(	O
also	O
uses	O
an	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
for	O
some	O
types	O
of	O
problems	O
)	O
.	O
</s>
<s>
ALGENCAN	O
(	O
Fortran	O
implementation	O
of	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
with	O
safeguards	O
)	O
.	O
</s>
<s>
PyProximal	O
(	O
Python	O
implementation	O
of	O
augmented	B-Algorithm
Lagrangian	I-Algorithm
method	I-Algorithm
)	O
.	O
</s>
