<s>
In	O
the	O
special	O
case	O
when	O
there	O
is	O
no	O
penalty	O
to	O
be	O
minimized	O
,	O
and	O
when	O
the	O
goal	O
is	O
to	O
design	O
a	O
stable	O
routing	O
policy	O
in	O
a	O
multi-hop	O
network	O
,	O
the	O
method	O
reduces	O
to	O
backpressure	B-Protocol
routing	I-Protocol
.	O
</s>
<s>
The	O
function	O
L(t )	O
is	O
typically	O
defined	O
as	O
the	O
sum	O
of	O
the	O
squares	O
of	O
all	O
queue	O
sizes	O
at	O
time	O
t	O
,	O
and	O
is	O
called	O
a	O
Lyapunov	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
Lyapunov	B-Algorithm
drift	I-Algorithm
is	O
defined	O
:	O
</s>
<s>
Like	O
backpressure	B-Protocol
routing	I-Protocol
,	O
this	O
method	O
typically	O
does	O
not	O
require	O
knowledge	O
of	O
the	O
probability	O
distributions	O
for	O
job	O
arrivals	O
and	O
network	O
mobility	O
.	O
</s>
<s>
When	O
the	O
method	O
reduces	O
to	O
greedily	O
minimizing	O
the	O
Lyapunov	B-Algorithm
drift	I-Algorithm
.	O
</s>
<s>
This	O
results	O
in	O
the	O
backpressure	B-Protocol
routing	I-Protocol
algorithm	O
originally	O
developed	O
by	O
Tassiulas	O
and	O
Ephremides	O
(	O
also	O
called	O
the	O
max-weight	O
algorithm	O
)	O
.	O
</s>
<s>
However	O
,	O
the	O
mathematical	O
techniques	O
can	O
be	O
applied	O
to	O
optimization	O
and	O
control	O
for	O
other	O
stochastic	O
systems	O
,	O
including	O
renewable	O
energy	O
allocation	O
in	O
smart	B-Architecture
power	I-Architecture
grids	I-Architecture
and	O
inventory	O
control	O
for	O
product	O
assembly	O
systems	O
.	O
</s>
<s>
To	O
stabilize	O
the	O
queues	O
,	O
define	O
the	O
Lyapunov	B-Algorithm
function	I-Algorithm
L(t )	O
as	O
a	O
measure	O
of	O
the	O
total	O
queue	O
backlog	O
on	O
slott	O
:	O
</s>
<s>
The	O
modifications	O
can	O
use	O
either	O
place-holder	O
backlog	O
or	O
Last-in-First-Out	O
(	O
LIFO	B-Application
)	O
scheduling	O
.	O
</s>
<s>
When	O
implemented	O
for	O
non-stochastic	O
functions	O
,	O
the	O
drift-plus-penalty	O
method	O
is	O
similar	O
to	O
the	O
dual	B-Algorithm
subgradient	I-Algorithm
method	I-Algorithm
of	O
convex	O
optimization	O
theory	O
,	O
with	O
the	O
exception	O
that	O
its	O
output	O
is	O
a	O
time	O
average	O
of	O
primal	O
variables	O
,	O
rather	O
than	O
the	O
primal	O
variables	O
themselves	O
.	O
</s>
<s>
There	O
are	O
many	O
important	O
cases	O
where	O
these	O
variables	O
do	O
not	O
converge	O
to	O
the	O
optimal	O
solution	O
,	O
and	O
never	O
even	O
get	O
near	O
the	O
optimal	O
solution	O
(	O
this	O
is	O
the	O
case	O
for	O
most	O
linear	B-Algorithm
programs	I-Algorithm
,	O
as	O
shown	O
below	O
)	O
.	O
</s>
<s>
Consider	O
the	O
special	O
case	O
of	O
a	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
suppose	O
that	O
xmin	O
,	O
1	O
=	O
0	O
,	O
xmax	O
,	O
1	O
=	O
1	O
,	O
and	O
suppose	O
that	O
all	O
optimal	O
solutions	O
to	O
the	O
linear	B-Algorithm
program	I-Algorithm
have	O
x1	O
=	O
3/4	O
.	O
</s>
