<s>
Linear	B-Algorithm
programming	I-Algorithm
(	O
LP	O
)	O
,	O
also	O
called	O
linear	B-Algorithm
optimization	I-Algorithm
,	O
is	O
a	O
method	O
to	O
achieve	O
the	O
best	O
outcome	O
(	O
such	O
as	O
maximum	O
profit	O
or	O
lowest	O
cost	O
)	O
in	O
a	O
mathematical	O
model	O
whose	O
requirements	O
are	O
represented	O
by	O
linear	O
relationships	O
.	O
</s>
<s>
Linear	B-Algorithm
programming	I-Algorithm
is	O
a	O
special	O
case	O
of	O
mathematical	O
programming	O
(	O
also	O
known	O
as	O
mathematical	O
optimization	O
)	O
.	O
</s>
<s>
More	O
formally	O
,	O
linear	B-Algorithm
programming	I-Algorithm
is	O
a	O
technique	O
for	O
the	O
optimization	O
of	O
a	O
linear	O
objective	O
function	O
,	O
subject	O
to	O
linear	O
equality	O
and	O
linear	O
inequality	B-Application
constraints	I-Application
.	O
</s>
<s>
A	O
linear	B-Algorithm
programming	I-Algorithm
algorithm	O
finds	O
a	O
point	O
in	O
the	O
polytope	O
where	O
this	O
function	O
has	O
the	O
smallest	O
(	O
or	O
largest	O
)	O
value	O
if	O
such	O
a	O
point	O
exists	O
.	O
</s>
<s>
Here	O
the	O
components	O
of	O
x	O
are	O
the	O
variables	O
to	O
be	O
determined	O
,	O
c	O
and	O
b	O
are	O
given	O
vectors	O
(	O
with	O
indicating	O
that	O
the	O
coefficients	O
of	O
c	O
are	O
used	O
as	O
a	O
single-row	O
matrix	B-Architecture
for	O
the	O
purpose	O
of	O
forming	O
the	O
matrix	B-Architecture
product	O
)	O
,	O
and	O
A	O
is	O
a	O
given	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
inequalities	O
Ax≤b	O
and	O
x	O
≥	O
0	O
are	O
the	O
constraints	B-Application
which	O
specify	O
a	O
convex	O
polytope	O
over	O
which	O
the	O
objective	O
function	O
is	O
to	O
be	O
optimized	O
.	O
</s>
<s>
Linear	B-Algorithm
programming	I-Algorithm
can	O
be	O
applied	O
to	O
various	O
fields	O
of	O
study	O
.	O
</s>
<s>
Industries	O
that	O
use	O
linear	B-Algorithm
programming	I-Algorithm
models	O
include	O
transportation	O
,	O
energy	O
,	O
telecommunications	O
,	O
and	O
manufacturing	O
.	O
</s>
<s>
It	O
has	O
proven	O
useful	O
in	O
modeling	O
diverse	O
types	O
of	O
problems	O
in	O
planning	B-Application
,	O
routing	B-Protocol
,	O
scheduling	B-Application
,	O
assignment	B-Algorithm
,	O
and	O
design	O
.	O
</s>
<s>
The	O
problem	O
of	O
solving	O
a	O
system	O
of	O
linear	O
inequalities	O
dates	O
back	O
at	O
least	O
as	O
far	O
as	O
Fourier	O
,	O
who	O
in	O
1827	O
published	O
a	O
method	O
for	O
solving	O
them	O
,	O
and	O
after	O
whom	O
the	O
method	O
of	O
Fourier	B-Algorithm
–	I-Algorithm
Motzkin	I-Algorithm
elimination	I-Algorithm
is	O
named	O
.	O
</s>
<s>
In	O
1939	O
a	O
linear	B-Algorithm
programming	I-Algorithm
formulation	I-Algorithm
of	O
a	O
problem	O
that	O
is	O
equivalent	O
to	O
the	O
general	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
was	O
given	O
by	O
the	O
Soviet	O
mathematician	O
and	O
economist	O
Leonid	O
Kantorovich	O
,	O
who	O
also	O
proposed	O
a	O
method	O
for	O
solving	O
it	O
.	O
</s>
<s>
About	O
the	O
same	O
time	O
as	O
Kantorovich	O
,	O
the	O
Dutch-American	O
economist	O
T	O
.	O
C	O
.	O
Koopmans	O
formulated	O
classical	O
economic	O
problems	O
as	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
In	O
1941	O
,	O
Frank	O
Lauren	O
Hitchcock	O
also	O
formulated	O
transportation	O
problems	O
as	O
linear	B-Algorithm
programs	I-Algorithm
and	O
gave	O
a	O
solution	O
very	O
similar	O
to	O
the	O
later	O
simplex	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
From	O
1946	O
to	O
1947	O
George	O
B	O
.	O
Dantzig	O
independently	O
developed	O
general	O
linear	B-Algorithm
programming	I-Algorithm
formulation	I-Algorithm
to	O
use	O
for	O
planning	B-Application
problems	O
in	O
the	O
US	O
Air	O
Force	O
.	O
</s>
<s>
In	O
1947	O
,	O
Dantzig	O
also	O
invented	O
the	O
simplex	B-Algorithm
method	I-Algorithm
that	O
,	O
for	O
the	O
first	O
time	O
efficiently	O
,	O
tackled	O
the	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
in	O
most	O
cases	O
.	O
</s>
<s>
When	O
Dantzig	O
arranged	O
a	O
meeting	O
with	O
John	O
von	O
Neumann	O
to	O
discuss	O
his	O
simplex	B-Algorithm
method	I-Algorithm
,	O
Neumann	O
immediately	O
conjectured	O
the	O
theory	O
of	O
duality	B-Algorithm
by	O
realizing	O
that	O
the	O
problem	O
he	O
had	O
been	O
working	O
in	O
game	O
theory	O
was	O
equivalent	O
.	O
</s>
<s>
In	O
the	O
post-war	O
years	O
,	O
many	O
industries	O
applied	O
it	O
in	O
their	O
daily	O
planning	B-Application
.	O
</s>
<s>
Dantzig	O
's	O
original	O
example	O
was	O
to	O
find	O
the	O
best	O
assignment	B-Algorithm
of	O
70	O
people	O
to	O
70	O
jobs	O
.	O
</s>
<s>
The	O
computing	O
power	O
required	O
to	O
test	O
all	O
the	O
permutations	O
to	O
select	O
the	O
best	O
assignment	B-Algorithm
is	O
vast	O
;	O
the	O
number	O
of	O
possible	O
configurations	O
exceeds	O
the	O
number	O
of	O
particles	O
in	O
the	O
observable	O
universe	O
.	O
</s>
<s>
However	O
,	O
it	O
takes	O
only	O
a	O
moment	O
to	O
find	O
the	O
optimum	O
solution	O
by	O
posing	O
the	O
problem	O
as	O
a	O
linear	B-Algorithm
program	I-Algorithm
and	O
applying	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
theory	O
behind	O
linear	B-Algorithm
programming	I-Algorithm
drastically	O
reduces	O
the	O
number	O
of	O
possible	O
solutions	O
that	O
must	O
be	O
checked	O
.	O
</s>
<s>
The	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
was	O
first	O
shown	O
to	O
be	O
solvable	O
in	O
polynomial	O
time	O
by	O
Leonid	O
Khachiyan	O
in	O
1979	O
,	O
but	O
a	O
larger	O
theoretical	O
and	O
practical	O
breakthrough	O
in	O
the	O
field	O
came	O
in	O
1984	O
when	O
Narendra	O
Karmarkar	O
introduced	O
a	O
new	O
interior-point	B-Algorithm
method	I-Algorithm
for	O
solving	O
linear-programming	O
problems	O
.	O
</s>
<s>
Linear	B-Algorithm
programming	I-Algorithm
is	O
a	O
widely	O
used	O
field	O
of	O
optimization	O
for	O
several	O
reasons	O
.	O
</s>
<s>
Many	O
practical	O
problems	O
in	O
operations	O
research	O
can	O
be	O
expressed	O
as	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Certain	O
special	O
cases	O
of	O
linear	B-Algorithm
programming	I-Algorithm
,	O
such	O
as	O
network	B-Algorithm
flow	I-Algorithm
problems	I-Algorithm
and	O
multicommodity	O
flow	B-Algorithm
problems	I-Algorithm
,	O
are	O
considered	O
important	O
enough	O
to	O
have	O
much	O
research	O
on	O
specialized	O
algorithms	O
.	O
</s>
<s>
A	O
number	O
of	O
algorithms	O
for	O
other	O
types	O
of	O
optimization	O
problems	O
work	O
by	O
solving	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
as	O
sub-problems	O
.	O
</s>
<s>
Historically	O
,	O
ideas	O
from	O
linear	B-Algorithm
programming	I-Algorithm
have	O
inspired	O
many	O
of	O
the	O
central	O
concepts	O
of	O
optimization	O
theory	O
,	O
such	O
as	O
duality	B-Algorithm
,	O
decomposition	O
,	O
and	O
the	O
importance	O
of	O
convexity	O
and	O
its	O
generalizations	O
.	O
</s>
<s>
Likewise	O
,	O
linear	B-Algorithm
programming	I-Algorithm
was	O
heavily	O
used	O
in	O
the	O
early	O
formation	O
of	O
microeconomics	O
,	O
and	O
it	O
is	O
currently	O
utilized	O
in	O
company	O
management	O
,	O
such	O
as	O
planning	B-Application
,	O
production	O
,	O
transportation	O
,	O
and	O
technology	O
.	O
</s>
<s>
Standard	O
form	O
is	O
the	O
usual	O
and	O
most	O
intuitive	O
form	O
of	O
describing	O
a	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
is	O
usually	O
expressed	O
in	O
matrix	B-Architecture
form	O
,	O
and	O
then	O
becomes	O
:	O
</s>
<s>
Other	O
forms	O
,	O
such	O
as	O
minimization	O
problems	O
,	O
problems	O
with	O
constraints	B-Application
on	O
alternative	O
forms	O
,	O
and	O
problems	O
involving	O
negative	O
variables	O
can	O
always	O
be	O
rewritten	O
into	O
an	O
equivalent	O
problem	O
in	O
standard	O
form	O
.	O
</s>
<s>
This	O
problem	O
can	O
be	O
expressed	O
with	O
the	O
following	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
in	O
the	O
standard	O
form	O
:	O
</s>
<s>
In	O
matrix	B-Architecture
form	O
this	O
becomes	O
:	O
</s>
<s>
Linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
can	O
be	O
converted	O
into	O
an	O
augmented	O
form	O
in	O
order	O
to	O
apply	O
the	O
common	O
form	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
This	O
form	O
introduces	O
non-negative	O
slack	B-Algorithm
variables	I-Algorithm
to	O
replace	O
inequalities	O
with	O
equalities	O
in	O
the	O
constraints	B-Application
.	O
</s>
<s>
The	O
problems	O
can	O
then	O
be	O
written	O
in	O
the	O
following	O
block	B-Algorithm
matrix	I-Algorithm
form	O
:	O
</s>
<s>
where	O
are	O
the	O
newly	O
introduced	O
slack	B-Algorithm
variables	I-Algorithm
,	O
are	O
the	O
decision	O
variables	O
,	O
and	O
is	O
the	O
variable	O
to	O
be	O
maximized	O
.	O
</s>
<s>
where	O
are	O
(	O
non-negative	O
)	O
slack	B-Algorithm
variables	I-Algorithm
,	O
representing	O
in	O
this	O
example	O
the	O
unused	O
area	O
,	O
the	O
amount	O
of	O
unused	O
fertilizer	O
,	O
and	O
the	O
amount	O
of	O
unused	O
pesticide	O
.	O
</s>
<s>
In	O
matrix	B-Architecture
form	O
this	O
becomes	O
:	O
</s>
<s>
Every	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
,	O
referred	O
to	O
as	O
a	O
primal	O
problem	O
,	O
can	O
be	O
converted	O
into	O
a	O
dual	O
problem	O
,	O
which	O
provides	O
an	O
upper	O
bound	O
to	O
the	O
optimal	O
value	O
of	O
the	O
primal	O
problem	O
.	O
</s>
<s>
In	O
matrix	B-Architecture
form	O
,	O
we	O
can	O
express	O
the	O
primal	O
problem	O
as	O
:	O
</s>
<s>
There	O
are	O
two	O
ideas	O
fundamental	O
to	O
duality	B-Algorithm
theory	O
.	O
</s>
<s>
One	O
is	O
the	O
fact	O
that	O
(	O
for	O
the	O
symmetric	O
dual	O
)	O
the	O
dual	O
of	O
a	O
dual	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
is	O
the	O
original	O
primal	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Additionally	O
,	O
every	O
feasible	O
solution	O
for	O
a	O
linear	B-Algorithm
program	I-Algorithm
gives	O
a	O
bound	O
on	O
the	O
optimal	O
value	O
of	O
the	O
objective	O
function	O
of	O
its	O
dual	O
.	O
</s>
<s>
The	O
weak	B-Algorithm
duality	I-Algorithm
theorem	O
states	O
that	O
the	O
objective	O
function	O
value	O
of	O
the	O
dual	O
at	O
any	O
feasible	O
solution	O
is	O
always	O
greater	O
than	O
or	O
equal	O
to	O
the	O
objective	O
function	O
value	O
of	O
the	O
primal	O
at	O
any	O
feasible	O
solution	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
duality	I-Algorithm
theorem	O
states	O
that	O
if	O
the	O
primal	O
has	O
an	O
optimal	O
solution	O
,	O
x*	O
,	O
then	O
the	O
dual	O
also	O
has	O
an	O
optimal	O
solution	O
,	O
y*	O
,	O
and	O
cTx*	O
=	O
bTy*	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
program	I-Algorithm
can	O
also	O
be	O
unbounded	O
or	O
infeasible	O
.	O
</s>
<s>
Duality	B-Algorithm
theory	O
tells	O
us	O
that	O
if	O
the	O
primal	O
is	O
unbounded	O
then	O
the	O
dual	O
is	O
infeasible	O
by	O
the	O
weak	B-Algorithm
duality	I-Algorithm
theorem	O
.	O
</s>
<s>
See	O
dual	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
for	O
details	O
and	O
several	O
more	O
examples	O
.	O
</s>
<s>
A	O
covering	B-Algorithm
LP	I-Algorithm
is	O
a	O
linear	B-Algorithm
program	I-Algorithm
of	O
the	O
form	O
:	O
</s>
<s>
such	O
that	O
the	O
matrix	B-Architecture
A	O
and	O
the	O
vectors	O
b	O
and	O
c	O
are	O
non-negative	O
.	O
</s>
<s>
The	O
dual	O
of	O
a	O
covering	B-Algorithm
LP	I-Algorithm
is	O
a	O
packing	O
LP	O
,	O
a	O
linear	B-Algorithm
program	I-Algorithm
of	O
the	O
form	O
:	O
</s>
<s>
such	O
that	O
the	O
matrix	B-Architecture
A	O
and	O
the	O
vectors	O
b	O
and	O
c	O
are	O
non-negative	O
.	O
</s>
<s>
Covering	O
and	O
packing	O
LPs	O
commonly	O
arise	O
as	O
a	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
of	O
a	O
combinatorial	O
problem	O
and	O
are	O
important	O
in	O
the	O
study	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
LP	B-Algorithm
relaxations	I-Algorithm
of	O
the	O
set	O
packing	O
problem	O
,	O
the	O
independent	O
set	O
problem	O
,	O
and	O
the	O
matching	O
problem	O
are	O
packing	O
LPs	O
.	O
</s>
<s>
The	O
LP	B-Algorithm
relaxations	I-Algorithm
of	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
,	O
the	O
vertex	O
cover	O
problem	O
,	O
and	O
the	O
dominating	O
set	O
problem	O
are	O
also	O
covering	O
LPs	O
.	O
</s>
<s>
Finding	O
a	O
fractional	O
coloring	O
of	O
a	O
graph	O
is	O
another	O
example	O
of	O
a	O
covering	B-Algorithm
LP	I-Algorithm
.	O
</s>
<s>
In	O
this	O
case	O
,	O
there	O
is	O
one	O
constraint	B-Application
for	O
each	O
vertex	O
of	O
the	O
graph	O
and	O
one	O
variable	O
for	O
each	O
independent	O
set	O
of	O
the	O
graph	O
.	O
</s>
<s>
Let	O
(	O
w1	O
,	O
w2	O
,...,	O
wm	O
)	O
denote	O
the	O
corresponding	O
primal	O
slack	B-Algorithm
variables	I-Algorithm
,	O
and	O
let	O
(	O
z1	O
,	O
z2	O
,...,	O
zn	O
)	O
denote	O
the	O
corresponding	O
dual	O
slack	B-Algorithm
variables	I-Algorithm
.	O
</s>
<s>
So	O
if	O
the	O
i-th	O
slack	B-Algorithm
variable	I-Algorithm
of	O
the	O
primal	O
is	O
not	O
zero	O
,	O
then	O
the	O
i-th	O
variable	O
of	O
the	O
dual	O
is	O
equal	O
to	O
zero	O
.	O
</s>
<s>
Likewise	O
,	O
if	O
the	O
j-th	O
slack	B-Algorithm
variable	I-Algorithm
of	O
the	O
dual	O
is	O
not	O
zero	O
,	O
then	O
the	O
j-th	O
variable	O
of	O
the	O
primal	O
is	O
equal	O
to	O
zero	O
.	O
</s>
<s>
Likewise	O
,	O
if	O
there	O
is	O
slack	O
in	O
the	O
dual	O
(	O
shadow	O
)	O
price	O
non-negativity	O
constraint	B-Application
requirement	O
,	O
i.e.	O
,	O
the	O
price	O
is	O
not	O
zero	O
,	O
then	O
there	O
must	O
be	O
scarce	O
supplies	O
(	O
no	O
"	O
leftovers	O
"	O
)	O
.	O
</s>
<s>
Geometrically	O
,	O
the	O
linear	O
constraints	B-Application
define	O
the	O
feasible	O
region	O
,	O
which	O
is	O
a	O
convex	O
polyhedron	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
function	I-Algorithm
is	O
a	O
convex	O
function	O
,	O
which	O
implies	O
that	O
every	O
local	O
minimum	O
is	O
a	O
global	O
minimum	O
;	O
similarly	O
,	O
a	O
linear	B-Algorithm
function	I-Algorithm
is	O
a	O
concave	O
function	O
,	O
which	O
implies	O
that	O
every	O
local	O
maximum	O
is	O
a	O
global	O
maximum	O
.	O
</s>
<s>
First	O
,	O
if	O
the	O
constraints	B-Application
are	O
inconsistent	O
,	O
then	O
no	O
feasible	O
solution	O
exists	O
:	O
For	O
instance	O
,	O
the	O
constraints	B-Application
x≥2	O
and	O
x≤1	O
cannot	O
be	O
satisfied	O
jointly	O
;	O
in	O
this	O
case	O
,	O
we	O
say	O
that	O
the	O
LP	O
is	O
infeasible	O
.	O
</s>
<s>
Otherwise	O
,	O
if	O
a	O
feasible	O
solution	O
exists	O
and	O
if	O
the	O
constraint	B-Application
set	O
is	O
bounded	O
,	O
then	O
the	O
optimum	O
value	O
is	O
always	O
attained	O
on	O
the	O
boundary	O
of	O
the	O
constraint	B-Application
set	O
,	O
by	O
the	O
maximum	O
principle	O
for	O
convex	O
functions	O
(	O
alternatively	O
,	O
by	O
the	O
minimum	O
principle	O
for	O
concave	O
functions	O
)	O
since	O
linear	O
functions	O
are	O
both	O
convex	O
and	O
concave	O
.	O
</s>
<s>
However	O
,	O
some	O
problems	O
have	O
distinct	O
optimal	O
solutions	O
;	O
for	O
example	O
,	O
the	O
problem	O
of	O
finding	O
a	O
feasible	O
solution	O
to	O
a	O
system	O
of	O
linear	O
inequalities	O
is	O
a	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
in	O
which	O
the	O
objective	O
function	O
is	O
the	O
zero	O
function	O
(	O
that	O
is	O
,	O
the	O
constant	O
function	O
taking	O
the	O
value	O
zero	O
everywhere	O
)	O
.	O
</s>
<s>
Then	O
the	O
fundamental	O
theorem	O
of	O
linear	O
inequalities	O
implies	O
(	O
for	O
feasible	O
problems	O
)	O
that	O
for	O
every	O
vertex	O
x*	O
of	O
the	O
LP	O
feasible	O
region	O
,	O
there	O
exists	O
a	O
set	O
of	O
d	O
(	O
or	O
fewer	O
)	O
inequality	B-Application
constraints	I-Application
from	O
the	O
LP	O
such	O
that	O
,	O
when	O
we	O
treat	O
those	O
d	O
constraints	B-Application
as	O
equalities	O
,	O
the	O
unique	O
solution	O
is	O
x*	O
.	O
</s>
<s>
Thereby	O
we	O
can	O
study	O
these	O
vertices	O
by	O
means	O
of	O
looking	O
at	O
certain	O
subsets	O
of	O
the	O
set	O
of	O
all	O
constraints	B-Application
(	O
a	O
discrete	O
set	O
)	O
,	O
rather	O
than	O
the	O
continuum	O
of	O
LP	O
solutions	O
.	O
</s>
<s>
This	O
principle	O
underlies	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
for	O
solving	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
developed	O
by	O
George	O
Dantzig	O
in	O
1947	O
,	O
solves	O
LP	B-Algorithm
problems	I-Algorithm
by	O
constructing	O
a	O
feasible	O
solution	O
at	O
a	O
vertex	O
of	O
the	O
polytope	O
and	O
then	O
walking	O
along	O
a	O
path	O
on	O
the	O
edges	O
of	O
the	O
polytope	O
to	O
vertices	O
with	O
non-decreasing	O
values	O
of	O
the	O
objective	O
function	O
until	O
an	O
optimum	O
is	O
reached	O
for	O
sure	O
.	O
</s>
<s>
In	O
rare	O
practical	O
problems	O
,	O
the	O
usual	O
versions	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
may	O
actually	O
"	O
cycle	O
"	O
.	O
</s>
<s>
To	O
avoid	O
cycles	O
,	O
researchers	O
developed	O
new	O
pivoting	B-Algorithm
rules	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
is	O
quite	O
efficient	O
and	O
can	O
be	O
guaranteed	O
to	O
find	O
the	O
global	O
optimum	O
if	O
certain	O
precautions	O
against	O
cycling	O
are	O
taken	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
proved	O
to	O
solve	O
"	O
random	O
"	O
problems	O
efficiently	O
,	O
i.e.	O
</s>
<s>
However	O
,	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
has	O
poor	O
worst-case	B-General_Concept
behavior	O
:	O
Klee	O
and	O
Minty	O
constructed	O
a	O
family	O
of	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
for	O
which	O
the	O
simplex	B-Algorithm
method	I-Algorithm
takes	O
a	O
number	O
of	O
steps	O
exponential	O
in	O
the	O
problem	O
size	O
.	O
</s>
<s>
In	O
fact	O
,	O
for	O
some	O
time	O
it	O
was	O
not	O
known	O
whether	O
the	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
was	O
solvable	O
in	O
polynomial	O
time	O
,	O
i.e.	O
</s>
<s>
Like	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
of	O
Dantzig	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
is	O
a	O
basis-exchange	O
algorithm	O
that	O
pivots	O
between	O
bases	O
.	O
</s>
<s>
However	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
need	O
not	O
maintain	O
feasibility	O
,	O
but	O
can	O
pivot	O
rather	O
from	O
a	O
feasible	O
basis	O
to	O
an	O
infeasible	O
basis	O
.	O
</s>
<s>
The	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
does	O
not	O
have	O
polynomial	O
time-complexity	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Both	O
algorithms	O
visit	O
all2D	O
corners	O
of	O
a	O
(	O
perturbed	O
)	O
cube	O
in	O
dimensionD	O
,	O
the	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
,	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
finds	O
an	O
optimal	O
solution	O
by	O
traversing	O
the	O
edges	O
between	O
vertices	O
on	O
a	O
polyhedral	O
set	O
,	O
interior-point	B-Algorithm
methods	I-Algorithm
move	O
through	O
the	O
interior	O
of	O
the	O
feasible	O
region	O
.	O
</s>
<s>
This	O
is	O
the	O
first	O
worst-case	B-General_Concept
polynomial-time	O
algorithm	O
ever	O
found	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Leonid	O
Khachiyan	O
solved	O
this	O
long-standing	O
complexity	O
issue	O
in	O
1979	O
with	O
the	O
introduction	O
of	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
convergence	O
analysis	O
has	O
(	O
real-number	O
)	O
predecessors	O
,	O
notably	O
the	O
iterative	B-Algorithm
methods	I-Algorithm
developed	O
by	O
Naum	O
Z	O
.	O
Shor	O
and	O
the	O
approximation	B-Algorithm
algorithms	I-Algorithm
by	O
Arkadi	O
Nemirovski	O
and	O
D	O
.	O
Yudin	O
.	O
</s>
<s>
Khachiyan	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
was	O
of	O
landmark	O
importance	O
for	O
establishing	O
the	O
polynomial-time	O
solvability	O
of	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
was	O
not	O
a	O
computational	O
break-through	O
,	O
as	O
the	O
simplex	B-Algorithm
method	I-Algorithm
is	O
more	O
efficient	O
for	O
all	O
but	O
specially	O
constructed	O
families	O
of	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
However	O
,	O
Khachiyan	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
inspired	O
new	O
lines	O
of	O
research	O
in	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
In	O
1984	O
,	O
N	O
.	O
Karmarkar	O
proposed	O
a	O
projective	B-Algorithm
method	I-Algorithm
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
improved	O
on	O
Khachiyan	O
's	O
worst-case	B-General_Concept
polynomial	O
bound	O
(	O
giving	O
)	O
.	O
</s>
<s>
Karmarkar	O
claimed	O
that	O
his	O
algorithm	O
was	O
much	O
faster	O
in	O
practical	O
LP	O
than	O
the	O
simplex	B-Algorithm
method	I-Algorithm
,	O
a	O
claim	O
that	O
created	O
great	O
interest	O
in	O
interior-point	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Since	O
Karmarkar	O
's	O
discovery	O
,	O
many	O
interior-point	B-Algorithm
methods	I-Algorithm
have	O
been	O
proposed	O
and	O
analyzed	O
.	O
</s>
<s>
Formally	O
speaking	O
,	O
the	O
algorithm	O
takes	O
arithmetic	O
operations	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
,	O
where	O
is	O
the	O
number	O
of	O
constraints	B-Application
,	O
is	O
the	O
number	O
of	O
variables	O
,	O
and	O
is	O
the	O
number	O
of	O
bits	O
.	O
</s>
<s>
In	O
2015	O
,	O
Lee	O
and	O
Sidford	O
showed	O
that	O
,	O
it	O
can	O
be	O
solved	O
in	O
time	O
,	O
where	O
represents	O
the	O
number	O
of	O
non-zero	O
elements	O
,	O
and	O
it	O
remains	O
taking	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
.	O
</s>
<s>
In	O
2019	O
,	O
Cohen	O
,	O
Lee	O
and	O
Song	O
improved	O
the	O
running	O
time	O
to	O
time	O
,	O
is	O
the	O
exponent	O
of	O
matrix	B-Architecture
multiplication	O
and	O
is	O
the	O
dual	O
exponent	O
of	O
matrix	B-Architecture
multiplication	O
.	O
</s>
<s>
is	O
(	O
roughly	O
)	O
defined	O
to	O
be	O
the	O
largest	O
number	O
such	O
that	O
one	O
can	O
multiply	O
an	O
matrix	B-Architecture
by	O
a	O
matrix	B-Architecture
in	O
time	O
.	O
</s>
<s>
The	O
current	O
opinion	O
is	O
that	O
the	O
efficiencies	O
of	O
good	O
implementations	O
of	O
simplex-based	O
methods	O
and	O
interior	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
are	O
similar	O
for	O
routine	O
applications	O
of	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
However	O
,	O
for	O
specific	O
types	O
of	O
LP	B-Algorithm
problems	I-Algorithm
,	O
it	O
may	O
be	O
that	O
one	O
type	O
of	O
solver	O
is	O
better	O
than	O
another	O
(	O
sometimes	O
much	O
better	O
)	O
,	O
and	O
that	O
the	O
structure	O
of	O
the	O
solutions	O
generated	O
by	O
interior	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
versus	O
simplex-based	O
methods	O
are	O
significantly	O
different	O
with	O
the	O
support	O
set	O
of	O
active	O
variables	O
being	O
typically	O
smaller	O
for	O
the	O
latter	O
one	O
.	O
</s>
<s>
There	O
are	O
several	O
open	O
problems	O
in	O
the	O
theory	O
of	O
linear	B-Algorithm
programming	I-Algorithm
,	O
the	O
solution	O
of	O
which	O
would	O
represent	O
fundamental	O
breakthroughs	O
in	O
mathematics	O
and	O
potentially	O
major	O
advances	O
in	O
our	O
ability	O
to	O
solve	O
large-scale	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
In	O
Smale	O
's	O
words	O
,	O
the	O
third	O
version	O
of	O
the	O
problem	O
"	O
is	O
the	O
main	O
unsolved	O
problem	O
of	O
linear	B-Algorithm
programming	I-Algorithm
theory.	O
"	O
</s>
<s>
While	O
algorithms	O
exist	O
to	O
solve	O
linear	B-Algorithm
programming	I-Algorithm
in	O
weakly	O
polynomial	O
time	O
,	O
such	O
as	O
the	O
ellipsoid	B-Algorithm
methods	I-Algorithm
and	O
interior-point	B-Algorithm
techniques	I-Algorithm
,	O
no	O
algorithms	O
have	O
yet	O
been	O
found	O
that	O
allow	O
strongly	O
polynomial-time	O
performance	O
in	O
the	O
number	O
of	O
constraints	B-Application
and	O
the	O
number	O
of	O
variables	O
.	O
</s>
<s>
The	O
immense	O
efficiency	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
in	O
practice	O
despite	O
its	O
exponential-time	O
theoretical	O
performance	O
hints	O
that	O
there	O
may	O
be	O
variations	O
of	O
simplex	O
that	O
run	O
in	O
polynomial	O
or	O
even	O
strongly	O
polynomial	O
time	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
and	O
its	O
variants	O
fall	O
in	O
the	O
family	O
of	O
edge-following	O
algorithms	O
,	O
so	O
named	O
because	O
they	O
solve	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
by	O
moving	O
from	O
vertex	O
to	O
vertex	O
along	O
edges	O
of	O
a	O
polytope	O
.	O
</s>
<s>
Essentially	O
,	O
these	O
methods	O
attempt	O
to	O
find	O
the	O
shortest	O
pivot	O
path	O
on	O
the	O
arrangement	O
polytope	O
under	O
the	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
If	O
all	O
of	O
the	O
unknown	O
variables	O
are	O
required	O
to	O
be	O
integers	O
,	O
then	O
the	O
problem	O
is	O
called	O
an	O
integer	B-Algorithm
programming	I-Algorithm
(	O
IP	O
)	O
or	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
(	O
ILP	O
)	O
problem	O
.	O
</s>
<s>
In	O
contrast	O
to	O
linear	B-Algorithm
programming	I-Algorithm
,	O
which	O
can	O
be	O
solved	O
efficiently	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
,	O
integer	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
are	O
in	O
many	O
practical	O
situations	O
(	O
those	O
with	O
bounded	O
variables	O
)	O
NP-hard	O
.	O
</s>
<s>
0	O
–	O
1	O
integer	B-Algorithm
programming	I-Algorithm
or	O
binary	O
integer	B-Algorithm
programming	I-Algorithm
(	O
BIP	O
)	O
is	O
the	O
special	O
case	O
of	O
integer	B-Algorithm
programming	I-Algorithm
where	O
variables	O
are	O
required	O
to	O
be	O
0	O
or	O
1	O
(	O
rather	O
than	O
arbitrary	O
integers	O
)	O
.	O
</s>
<s>
There	O
are	O
however	O
some	O
important	O
subclasses	O
of	O
IP	O
and	O
MIP	O
problems	O
that	O
are	O
efficiently	O
solvable	O
,	O
most	O
notably	O
problems	O
where	O
the	O
constraint	B-Application
matrix	B-Architecture
is	O
totally	B-Algorithm
unimodular	I-Algorithm
and	O
the	O
right-hand	O
sides	O
of	O
the	O
constraints	B-Application
are	O
integers	O
or	O
–	O
more	O
general	O
–	O
where	O
the	O
system	O
has	O
the	O
total	B-Algorithm
dual	I-Algorithm
integrality	I-Algorithm
(	O
TDI	O
)	O
property	O
.	O
</s>
<s>
Advanced	O
algorithms	O
for	O
solving	O
integer	B-Algorithm
linear	I-Algorithm
programs	I-Algorithm
include	O
:	O
</s>
<s>
if	O
the	O
problem	O
has	O
some	O
extra	O
structure	O
,	O
it	O
may	O
be	O
possible	O
to	O
apply	O
delayed	B-Algorithm
column	I-Algorithm
generation	I-Algorithm
.	O
</s>
<s>
A	O
linear	B-Algorithm
program	I-Algorithm
in	O
real	O
variables	O
is	O
said	O
to	O
be	O
integral	O
if	O
it	O
has	O
at	O
least	O
one	O
optimal	O
solution	O
which	O
is	O
integral	O
,	O
i.e.	O
,	O
made	O
of	O
only	O
integer	O
values	O
.	O
</s>
<s>
Likewise	O
,	O
a	O
polyhedron	O
is	O
said	O
to	O
be	O
integral	O
if	O
for	O
all	O
bounded	O
feasible	O
objective	O
functions	O
c	O
,	O
the	O
linear	B-Algorithm
program	I-Algorithm
has	O
an	O
optimum	O
with	O
integer	O
coordinates	O
.	O
</s>
<s>
As	O
observed	O
by	O
Edmonds	O
and	O
Giles	O
in	O
1977	O
,	O
one	O
can	O
equivalently	O
say	O
that	O
the	O
polyhedron	O
is	O
integral	O
if	O
for	O
every	O
bounded	O
feasible	O
integral	O
objective	O
function	O
c	O
,	O
the	O
optimal	O
value	O
of	O
the	O
linear	B-Algorithm
program	I-Algorithm
is	O
an	O
integer	O
.	O
</s>
<s>
Integral	O
linear	B-Algorithm
programs	I-Algorithm
are	O
of	O
central	O
importance	O
in	O
the	O
polyhedral	O
aspect	O
of	O
combinatorial	O
optimization	O
since	O
they	O
provide	O
an	O
alternate	O
characterization	O
of	O
a	O
problem	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
we	O
can	O
prove	O
that	O
a	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
is	O
integral	O
,	O
then	O
it	O
is	O
the	O
desired	O
description	O
of	O
the	O
convex	O
hull	O
of	O
feasible	O
(	O
integral	O
)	O
solutions	O
.	O
</s>
<s>
in	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
,	O
described	O
in	O
the	O
previous	O
section	O
,	O
variables	O
are	O
forcibly	O
constrained	O
to	O
be	O
integers	O
,	O
and	O
this	O
problem	O
is	O
NP-hard	O
in	O
general	O
,	O
</s>
<s>
in	O
an	O
integral	O
linear	B-Algorithm
program	I-Algorithm
,	O
described	O
in	O
this	O
section	O
,	O
variables	O
are	O
not	O
constrained	O
to	O
be	O
integers	O
but	O
rather	O
one	O
has	O
proven	O
somehow	O
that	O
the	O
continuous	O
problem	O
always	O
has	O
an	O
integral	O
optimal	O
value	O
(	O
assuming	O
c	O
is	O
integral	O
)	O
,	O
and	O
this	O
optimal	O
value	O
may	O
be	O
found	O
efficiently	O
since	O
all	O
polynomial-size	O
linear	B-Algorithm
programs	I-Algorithm
can	O
be	O
solved	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
One	O
common	O
way	O
of	O
proving	O
that	O
a	O
polyhedron	O
is	O
integral	O
is	O
to	O
show	O
that	O
it	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
.	O
</s>
<s>
There	O
are	O
other	O
general	O
methods	O
including	O
the	O
integer	O
decomposition	O
property	O
and	O
total	B-Algorithm
dual	I-Algorithm
integrality	I-Algorithm
.	O
</s>
<s>
Other	O
specific	O
well-known	O
integral	O
LPs	O
include	O
the	O
matching	O
polytope	O
,	O
lattice	O
polyhedra	O
,	O
submodular	B-Algorithm
flow	I-Algorithm
polyhedra	O
,	O
and	O
the	O
intersection	O
of	O
two	O
generalized	O
polymatroids/g	O
-polymatroids	O
–	O
e.g.	O
</s>
<s>
Permissive	B-License
licenses	I-License
:	O
</s>
<s>
NameLicenseBrief	O
info	O
GekkoMIT	O
LicenseOpen-source	O
library	O
for	O
solving	O
large-scale	O
LP	O
,	O
QP	B-Algorithm
,	O
QCQP	O
,	O
NLP	B-Algorithm
,	O
and	O
MIP	O
optimization	O
GLOPApache	O
v2Google	O
's	O
open-source	O
linear	B-Algorithm
programming	I-Algorithm
solver	O
PyomoBSDAn	O
open-source	O
modeling	O
language	O
for	O
large-scale	O
linear	O
,	O
mixed	O
integer	O
and	O
nonlinear	B-Algorithm
optimization	I-Algorithm
SCIPApache	O
v2A	O
general-purpose	O
constraint	B-Application
integer	B-Algorithm
programming	I-Algorithm
solver	O
with	O
an	O
emphasis	O
on	O
MIP	O
.	O
</s>
<s>
Copyleft	B-License
(	O
reciprocal	O
)	O
licenses	O
:	O
</s>
<s>
NameLicenseBrief	O
infoALGLIBGPL	O
2+	O
an	O
LP	O
solver	O
from	O
ALGLIB	B-Library
project	O
(	O
C++	O
,	O
C#	O
,	O
Python	O
)	O
Cassowary	B-Application
constraint	B-Application
solverLGPLan	O
incremental	O
constraint	B-Application
solving	O
toolkit	O
that	O
efficiently	O
solves	O
systems	O
of	O
linear	O
equalities	O
and	O
inequalitiesCLPCPL	O
an	O
LP	O
solver	O
from	O
COIN-ORglpkGPL	O
GNU	B-Application
Linear	I-Application
Programming	I-Application
Kit	I-Application
,	O
an	O
LP/MILP	O
solver	O
with	O
a	O
native	O
C	O
API	B-General_Concept
and	O
numerous	O
(	O
15	O
)	O
third-party	O
wrappers	O
for	O
other	O
languages	O
.	O
</s>
<s>
Specialist	O
support	O
for	O
flow	B-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
MINTO	B-Application
(	O
Mixed	O
Integer	O
Optimizer	O
,	O
an	O
integer	B-Algorithm
programming	I-Algorithm
solver	O
which	O
uses	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
algorithm	I-Algorithm
)	O
has	O
publicly	O
available	O
source	O
code	O
but	O
is	O
not	O
open	O
source	O
.	O
</s>
<s>
Proprietary	B-Application
licenses	I-Application
:	O
</s>
<s>
NameBrief	O
infoAIMMS	O
A	O
modeling	O
language	O
that	O
allows	O
to	O
model	O
linear	O
,	O
mixed	O
integer	O
,	O
and	O
nonlinear	B-Algorithm
optimization	I-Algorithm
models	O
.	O
</s>
<s>
It	O
also	O
offers	O
a	O
tool	O
for	O
constraint	B-Application
programming	O
.	O
</s>
<s>
Algorithm	O
,	O
in	O
the	O
forms	O
of	O
heuristics	O
or	O
exact	O
methods	O
,	O
such	O
as	O
Branch-and-Cut	O
or	O
Column	B-Algorithm
Generation	I-Algorithm
,	O
can	O
also	O
be	O
implemented	O
.	O
</s>
<s>
The	O
tool	O
calls	O
an	O
appropriate	O
solver	O
such	O
as	O
CPLEX	B-Application
or	O
similar	O
,	O
to	O
solve	O
the	O
optimization	O
problem	O
at	O
hand	O
.	O
</s>
<s>
Academic	O
licenses	O
are	O
free	O
of	O
charge.ALGLIB	O
A	O
commercial	O
edition	O
of	O
the	O
copyleft	B-License
licensed	O
library	O
.	O
</s>
<s>
C++	O
,	O
C#	O
,	O
Python.AMPL	O
A	O
popular	O
modeling	O
language	O
for	O
large-scale	O
linear	O
,	O
mixed	O
integer	O
and	O
nonlinear	O
optimisation	O
with	O
a	O
free	O
student	O
limited	O
version	O
available	O
(	O
500	O
variables	O
and	O
500	O
constraints	B-Application
)	O
.Analytica	O
A	O
general	O
modeling	O
language	O
and	O
interactive	O
development	O
environment	O
.	O
</s>
<s>
Its	O
influence	O
diagrams	O
enable	O
users	O
to	O
formulate	O
problems	O
as	O
graphs	O
with	O
nodes	O
for	O
decision	O
variables	O
,	O
objectives	O
,	O
and	O
constraints	B-Application
.	O
</s>
<s>
Analytica	B-Language
Optimizer	O
Edition	O
includes	O
linear	O
,	O
mixed	O
integer	O
,	O
and	O
nonlinear	O
solvers	O
and	O
selects	O
the	O
solver	O
to	O
match	O
the	O
problem	O
.	O
</s>
<s>
It	O
also	O
accepts	O
other	O
engines	O
as	O
plug-ins	O
,	O
including	O
XPRESS	B-Application
,	O
Gurobi	O
,	O
Artelys	B-Application
Knitro	I-Application
,	O
and	O
MOSEK.APMonitor	O
API	B-General_Concept
to	O
MATLAB	B-Language
and	O
Python	O
.	O
</s>
<s>
Solve	O
example	O
Linear	B-Algorithm
Programming	I-Algorithm
(	O
LP	O
)	O
problems	O
through	O
MATLAB	B-Language
,	O
Python	O
,	O
or	O
a	O
web-interface.CPLEX	O
Popular	O
solver	O
with	O
an	O
API	B-General_Concept
for	O
several	O
programming	O
languages	O
,	O
and	O
also	O
has	O
a	O
modelling	O
language	O
and	O
works	O
with	O
AIMMS	B-Application
,	O
AMPL	B-Language
,	O
GAMS	B-Application
,	O
MPL	O
,	O
OpenOpt	O
,	O
OPL	B-Application
Development	I-Application
Studio	I-Application
,	O
and	O
TOMLAB	B-Application
.	O
</s>
<s>
Free	O
for	O
academic	O
use.Excel	O
Solver	O
Function	O
A	O
nonlinear	O
solver	O
adjusted	O
to	O
spreadsheets	O
in	O
which	O
function	O
evaluations	O
are	O
based	O
on	O
the	O
recalculating	O
cells	O
.	O
</s>
<s>
Basic	O
version	O
available	O
as	O
a	O
standard	O
add-on	O
for	O
Excel.FortMPGAMSIMSL	O
Numerical	O
Libraries	O
Collections	O
of	O
math	O
and	O
statistical	O
algorithms	O
available	O
in	O
C/C	O
++	O
,	O
Fortran	O
,	O
Java	O
and	O
C#	O
/.NET	O
.	O
</s>
<s>
Optimization	O
routines	O
in	O
the	O
IMSL	B-Library
Libraries	O
include	O
unconstrained	O
,	O
linearly	O
and	O
nonlinearly	O
constrained	O
minimizations	O
,	O
and	O
linear	B-Algorithm
programming	I-Algorithm
algorithms.LINDO	O
Solver	O
with	O
an	O
API	B-General_Concept
for	O
large	O
scale	O
optimization	O
of	O
linear	O
,	O
integer	O
,	O
quadratic	O
,	O
conic	O
and	O
general	O
nonlinear	O
programs	O
with	O
stochastic	B-Algorithm
programming	I-Algorithm
extensions	O
.	O
</s>
<s>
It	O
also	O
has	O
a	O
statistical	O
sampling	O
API	B-General_Concept
to	O
integrate	O
Monte-Carlo	O
simulations	O
into	O
an	O
optimization	O
framework	O
.	O
</s>
<s>
It	O
has	O
an	O
algebraic	B-Application
modeling	I-Application
language	I-Application
(	O
LINGO	B-Language
)	O
and	O
allows	O
modeling	O
within	O
a	O
spreadsheet	O
(	O
What'sBest	O
)	O
.Maple	O
A	O
general-purpose	O
programming-language	O
for	O
symbolic	O
and	O
numerical	O
computing.MATLAB	O
A	O
general-purpose	O
and	O
matrix-oriented	O
programming-language	O
for	O
numerical	O
computing	O
.	O
</s>
<s>
Linear	B-Algorithm
programming	I-Algorithm
in	O
MATLAB	B-Language
requires	O
the	O
Optimization	B-Application
Toolbox	I-Application
in	O
addition	O
to	O
the	O
base	O
MATLAB	B-Language
product	O
;	O
available	O
routines	O
include	O
INTLINPROG	O
and	O
LINPROGMathcad	O
A	O
WYSIWYG	O
math	O
editor	O
.	O
</s>
<s>
It	O
has	O
functions	O
for	O
solving	O
both	O
linear	O
and	O
nonlinear	B-Algorithm
optimization	I-Algorithm
problems.Mathematica	O
A	O
general-purpose	O
programming-language	O
for	O
mathematics	O
,	O
including	O
symbolic	O
and	O
numerical	O
capabilities.MOSEK	O
A	O
solver	O
for	O
large	O
scale	O
optimization	O
with	O
API	B-General_Concept
for	O
several	O
languages	O
(	O
C++	O
,	O
java	O
,.	O
net	O
,	O
Matlab	B-Language
and	O
python	O
)	O
.NAG	O
Numerical	O
Library	O
A	O
collection	O
of	O
mathematical	O
and	O
statistical	O
routines	O
developed	O
by	O
the	O
Numerical	O
Algorithms	O
Group	O
for	O
multiple	O
programming	O
languages	O
(	O
C	O
,	O
C++	O
,	O
Fortran	O
,	O
Visual	O
Basic	O
,	O
Java	O
and	O
C#	O
)	O
and	O
packages	O
(	O
MATLAB	B-Language
,	O
Excel	B-Application
,	O
R	B-Language
,	O
LabVIEW	O
)	O
.	O
</s>
<s>
The	O
Optimization	O
chapter	O
of	O
the	O
NAG	O
Library	O
includes	O
routines	O
for	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
with	O
both	O
sparse	O
and	O
non-sparse	O
linear	O
constraint	B-Application
matrices	O
,	O
together	O
with	O
routines	O
for	O
the	O
optimization	O
of	O
quadratic	O
,	O
nonlinear	O
,	O
sums	O
of	O
squares	O
of	O
linear	O
or	O
nonlinear	O
functions	O
with	O
nonlinear	O
,	O
bounded	O
or	O
no	O
constraints	B-Application
.	O
</s>
<s>
The	O
NAG	O
Library	O
has	O
routines	O
for	O
both	O
local	O
and	O
global	O
optimization	O
,	O
and	O
for	O
continuous	O
or	O
integer	O
problems.OptimJ	O
A	O
Java-based	O
modeling	O
language	O
for	O
optimization	O
with	O
a	O
free	O
version	O
available.http://www.in-ter-trans.eu/resources/Zesch_Hellingrath_2010_Integrated	O
+	O
Production-Distribution	O
+	O
Planning.pdf	O
OptimJ	B-Application
used	O
in	O
an	O
optimization	O
model	O
for	O
mixed-model	O
assembly	O
lines	O
,	O
University	O
of	O
Münsterhttp://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/viewFile/1769/2076	O
OptimJ	B-Application
used	O
in	O
an	O
Approximate	O
Subgame-Perfect	O
Equilibrium	O
Computation	O
Technique	O
for	O
Repeated	O
GamesSAS/OR	O
A	O
suite	O
of	O
solvers	O
for	O
Linear	O
,	O
Integer	O
,	O
Nonlinear	O
,	O
Derivative-Free	O
,	O
Network	O
,	O
Combinatorial	O
and	O
Constraint	B-Application
Optimization	O
;	O
the	O
Algebraic	B-Application
modeling	I-Application
language	I-Application
OPTMODEL	O
;	O
and	O
a	O
variety	O
of	O
vertical	O
solutions	O
aimed	O
at	O
specific	O
problems/markets	O
,	O
all	O
of	O
which	O
are	O
fully	O
integrated	O
with	O
the	O
SAS	O
System.XPRESSSolver	O
for	O
large-scale	O
linear	B-Algorithm
programs	I-Algorithm
,	O
quadratic	B-Algorithm
programs	I-Algorithm
,	O
general	O
nonlinear	O
and	O
mixed-integer	O
programs	O
.	O
</s>
<s>
Has	O
API	B-General_Concept
for	O
several	O
programming	O
languages	O
,	O
also	O
has	O
a	O
modelling	O
language	O
Mosel	O
and	O
works	O
with	O
AMPL	B-Language
,	O
GAMS	B-Application
.	O
</s>
<s>
Free	O
for	O
academic	O
use.VisSim	O
A	O
visual	O
block	B-Application
diagram	I-Application
language	O
for	O
simulation	O
of	O
dynamical	O
systems	O
.	O
</s>
