<s>
Subgradient	B-Algorithm
methods	I-Algorithm
are	O
iterative	B-Algorithm
methods	I-Algorithm
for	O
solving	O
convex	O
minimization	O
problems	O
.	O
</s>
<s>
Originally	O
developed	O
by	O
Naum	O
Z	O
.	O
Shor	O
and	O
others	O
in	O
the	O
1960s	O
and	O
1970s	O
,	O
subgradient	B-Algorithm
methods	I-Algorithm
are	O
convergent	O
when	O
applied	O
even	O
to	O
a	O
non-differentiable	O
objective	O
function	O
.	O
</s>
<s>
When	O
the	O
objective	O
function	O
is	O
differentiable	O
,	O
sub-gradient	O
methods	O
for	O
unconstrained	O
problems	O
use	O
the	O
same	O
search	O
direction	O
as	O
the	O
method	O
of	O
steepest	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
Subgradient	B-Algorithm
methods	I-Algorithm
are	O
slower	O
than	O
Newton	O
's	O
method	O
when	O
applied	O
to	O
minimize	O
twice	O
continuously	O
differentiable	O
convex	O
functions	O
.	O
</s>
<s>
In	O
recent	O
years	O
,	O
some	O
interior-point	B-Algorithm
methods	I-Algorithm
have	O
been	O
suggested	O
for	O
convex	O
minimization	O
problems	O
,	O
but	O
subgradient	O
projection	O
methods	O
and	O
related	O
bundle	O
methods	O
of	O
descent	O
remain	O
competitive	O
.	O
</s>
<s>
Many	O
different	O
types	O
of	O
step-size	O
rules	O
are	O
used	O
by	O
subgradient	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
This	O
"	O
off-line	O
"	O
property	O
of	O
subgradient	B-Algorithm
methods	I-Algorithm
differs	O
from	O
the	O
"	O
on-line	O
"	O
step-size	O
rules	O
used	O
for	O
descent	O
methods	O
for	O
differentiable	O
functions	O
:	O
Many	O
methods	O
for	O
minimizing	O
differentiable	O
functions	O
satisfy	O
Wolfe	O
's	O
sufficient	O
conditions	O
for	O
convergence	O
,	O
where	O
step-sizes	O
typically	O
depend	O
on	O
the	O
current	O
point	O
and	O
the	O
current	O
search-direction	O
.	O
</s>
<s>
An	O
extensive	O
discussion	O
of	O
stepsize	O
rules	O
for	O
subgradient	B-Algorithm
methods	I-Algorithm
,	O
including	O
incremental	O
versions	O
,	O
is	O
given	O
in	O
the	O
books	O
by	O
Bertsekas	O
and	O
by	O
Bertsekas	O
,	O
Nedic	O
,	O
and	O
Ozdaglar	O
.	O
</s>
<s>
These	O
classical	O
subgradient	B-Algorithm
methods	I-Algorithm
have	O
poor	O
performance	O
and	O
are	O
no	O
longer	O
recommended	O
for	O
general	O
use	O
.	O
</s>
<s>
Contemporary	O
bundle-methods	O
often	O
use	O
"	O
level	B-Algorithm
control	O
"	O
rules	O
for	O
choosing	O
step-sizes	O
,	O
developing	O
techniques	O
from	O
the	O
"	O
subgradient-projection	O
"	O
method	O
of	O
Boris	O
T	O
.	O
Polyak	O
(	O
1969	O
)	O
.	O
</s>
