<s>
Quadratic	B-Algorithm
programming	I-Algorithm
(	O
QP	O
)	O
is	O
the	O
process	O
of	O
solving	O
certain	O
mathematical	O
optimization	O
problems	O
involving	O
quadratic	O
functions	O
.	O
</s>
<s>
Specifically	O
,	O
one	O
seeks	O
to	O
optimize	O
(	O
minimize	O
or	O
maximize	O
)	O
a	O
multivariate	O
quadratic	O
function	O
subject	O
to	O
linear	O
constraints	B-Application
on	O
the	O
variables	O
.	O
</s>
<s>
Quadratic	B-Algorithm
programming	I-Algorithm
is	O
a	O
type	O
of	O
nonlinear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
quadratic	B-Algorithm
programming	I-Algorithm
problem	O
with	O
variables	O
and	O
constraints	B-Application
can	O
be	O
formulated	O
as	O
follows	O
.	O
</s>
<s>
an	O
-dimensional	O
real	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
</s>
<s>
As	O
a	O
special	O
case	O
when	O
Q	O
is	O
symmetric	B-Algorithm
positive-definite	I-Algorithm
,	O
the	O
cost	O
function	O
reduces	O
to	O
least	O
squares	O
:	O
</s>
<s>
Conversely	O
,	O
any	O
such	O
constrained	O
least	O
squares	O
program	O
can	O
be	O
equivalently	O
framed	O
as	O
a	O
QP	O
,	O
even	O
for	O
generic	O
non-square	O
matrix	B-Architecture
.	O
</s>
<s>
When	O
minimizing	O
a	O
function	O
in	O
the	O
neighborhood	O
of	O
some	O
reference	O
point	O
,	O
is	O
set	O
to	O
its	O
Hessian	O
matrix	B-Architecture
and	O
is	O
set	O
to	O
its	O
gradient	O
.	O
</s>
<s>
A	O
related	O
programming	O
problem	O
,	O
quadratically	O
constrained	O
quadratic	B-Algorithm
programming	I-Algorithm
,	O
can	O
be	O
posed	O
by	O
adding	O
quadratic	O
constraints	B-Application
on	O
the	O
variables	O
.	O
</s>
<s>
interior	B-Algorithm
point	I-Algorithm
,	O
</s>
<s>
active	B-Algorithm
set	I-Algorithm
,	O
</s>
<s>
augmented	B-Algorithm
Lagrangian	I-Algorithm
,	O
</s>
<s>
conjugate	B-Algorithm
gradient	I-Algorithm
,	O
</s>
<s>
extensions	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
the	O
case	O
in	O
which	O
is	O
positive	B-Algorithm
definite	I-Algorithm
,	O
the	O
problem	O
is	O
a	O
special	O
case	O
of	O
the	O
more	O
general	O
field	O
of	O
convex	O
optimization	O
.	O
</s>
<s>
Quadratic	B-Algorithm
programming	I-Algorithm
is	O
particularly	O
simple	O
when	O
is	O
positive	B-Algorithm
definite	I-Algorithm
and	O
there	O
are	O
only	O
equality	O
constraints	B-Application
;	O
specifically	O
,	O
the	O
solution	O
process	O
is	O
linear	O
.	O
</s>
<s>
For	O
large	O
problems	O
,	O
the	O
system	O
poses	O
some	O
unusual	O
difficulties	O
,	O
most	O
notably	O
that	O
the	O
problem	O
is	O
never	O
positive	B-Algorithm
definite	I-Algorithm
(	O
even	O
if	O
is	O
)	O
,	O
making	O
it	O
potentially	O
very	O
difficult	O
to	O
find	O
a	O
good	O
numeric	O
approach	O
,	O
and	O
there	O
are	O
many	O
approaches	O
to	O
choose	O
from	O
dependent	O
on	O
the	O
problem	O
.	O
</s>
<s>
If	O
the	O
constraints	B-Application
do	O
n't	O
couple	O
the	O
variables	O
too	O
tightly	O
,	O
a	O
relatively	O
simple	O
attack	O
is	O
to	O
change	O
the	O
variables	O
so	O
that	O
constraints	B-Application
are	O
unconditionally	O
satisfied	O
.	O
</s>
<s>
where	O
has	O
dimension	O
of	O
minus	O
the	O
number	O
of	O
constraints	B-Application
.	O
</s>
<s>
Finding	O
such	O
entails	O
finding	O
the	O
null	B-Algorithm
space	I-Algorithm
of	O
,	O
which	O
is	O
more	O
or	O
less	O
simple	O
depending	O
on	O
the	O
structure	O
of	O
.	O
</s>
<s>
Under	O
certain	O
conditions	O
on	O
,	O
the	O
reduced	O
matrix	B-Architecture
will	O
be	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
write	O
a	O
variation	O
on	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
which	O
avoids	O
the	O
explicit	O
calculation	O
of	O
.	O
</s>
<s>
To	O
see	O
this	O
let	O
us	O
focus	O
on	O
the	O
case	O
where	O
and	O
is	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
For	O
positive	B-Algorithm
definite	I-Algorithm
,	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
solves	O
the	O
problem	O
in	O
(	O
weakly	O
)	O
polynomial	O
time	O
.	O
</s>
<s>
This	O
leads	O
to	O
the	O
formulation	O
of	O
a	O
mixed-integer	O
quadratic	B-Algorithm
programming	I-Algorithm
(	O
MIQP	O
)	O
problem	O
.	O
</s>
<s>
NameBrief	O
infoAIMMS	O
A	O
software	O
system	O
for	O
modeling	O
and	O
solving	O
optimization	O
and	O
scheduling-type	O
problemsALGLIB	O
Dual	O
licensed	O
(	O
GPL/proprietary	O
)	O
numerical	O
library	O
(	O
C++	O
,	O
.NET	O
)	O
.AMPL	O
A	O
popular	O
modeling	O
language	O
for	O
large-scale	O
mathematical	O
optimization.APMonitor	O
Modeling	O
and	O
optimization	O
suite	O
for	O
LP	B-Algorithm
,	O
QP	O
,	O
NLP	B-Algorithm
,	O
MILP	B-Algorithm
,	O
MINLP	O
,	O
and	O
DAE	O
systems	O
in	O
MATLAB	B-Language
and	O
Python.Artelys	O
Knitro	O
An	O
Integrated	O
Package	O
for	O
Nonlinear	O
OptimizationCGAL	O
An	O
open	O
source	O
computational	O
geometry	O
package	O
which	O
includes	O
a	O
quadratic	B-Algorithm
programming	I-Algorithm
solver	O
.	O
</s>
<s>
CPLEX	B-Application
Popular	O
solver	O
with	O
an	O
API	O
(	O
C	O
,	O
C++	O
,	O
Java	O
,	O
.Net	O
,	O
Python	B-Language
,	O
Matlab	B-Language
and	O
R	B-Language
)	O
.	O
</s>
<s>
Free	O
for	O
academics.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.GAMS	O
A	O
high-level	O
modeling	O
system	O
for	O
mathematical	O
optimizationGNU	O
Octave	B-Language
A	O
free	O
(	O
its	O
licence	O
is	O
GPLv3	O
)	O
general-purpose	O
and	O
matrix-oriented	O
programming-language	O
for	O
numerical	O
computing	O
,	O
similar	O
to	O
MATLAB	B-Language
.	O
</s>
<s>
Quadratic	B-Algorithm
programming	I-Algorithm
in	O
GNU	B-Language
Octave	I-Language
is	O
available	O
via	O
its	O
qp	O
commandHiGHS	O
Open-source	O
software	O
for	O
solving	O
linear	B-Algorithm
programming	I-Algorithm
(	O
LP	B-Algorithm
)	O
,	O
mixed-integer	B-Algorithm
programming	I-Algorithm
(	O
MIP	O
)	O
,	O
and	O
convex	O
quadratic	B-Algorithm
programming	I-Algorithm
(	O
QP	O
)	O
modelsIMSL	O
A	O
set	O
of	O
mathematical	O
and	O
statistical	O
functions	O
that	O
programmers	O
can	O
embed	O
into	O
their	O
software	O
applications.IPOPT	O
IPOPT	B-Algorithm
(	O
Interior	B-Algorithm
Point	I-Algorithm
OPTimizer	O
)	O
is	O
a	O
software	O
package	O
for	O
large-scale	O
nonlinear	O
optimization.Maple	O
General-purpose	O
programming	O
language	O
for	O
mathematics	O
.	O
</s>
<s>
Solving	O
a	O
quadratic	O
problem	O
in	O
Maple	B-Language
is	O
accomplished	O
via	O
its	O
QPSolve	O
command.MATLAB	O
A	O
general-purpose	O
and	O
matrix-oriented	O
programming-language	O
for	O
numerical	O
computing	O
.	O
</s>
<s>
Quadratic	B-Algorithm
programming	I-Algorithm
in	O
MATLAB	B-Language
requires	O
the	O
Optimization	O
Toolbox	O
in	O
addition	O
to	O
the	O
base	O
MATLAB	B-Language
productMathematica	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	O
for	O
several	O
languages	O
(	O
C++	O
,	O
Java	O
,	O
.Net	O
,	O
Matlab	B-Language
and	O
Python	B-Language
)	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
quadratic	B-Algorithm
programming	I-Algorithm
problems	O
with	O
both	O
sparse	O
and	O
non-sparse	O
linear	O
constraint	O
matrices	O
,	O
together	O
with	O
routines	O
for	O
the	O
optimization	O
of	O
linear	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>
Quadratic	B-Algorithm
programming	I-Algorithm
is	O
available	O
via	O
the	O
solve_qp	O
function	O
or	O
by	O
calling	O
a	O
specific	O
solver	O
directly.R	O
(	O
Fortran	O
)	O
GPL	B-License
licensed	O
universal	O
cross-platform	O
statistical	O
computation	O
framework.SAS/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	I-Application
;	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	B-Language
System.SuanShu	O
an	O
open-source	O
suite	O
of	O
optimization	O
algorithms	O
to	O
solve	O
LP	B-Algorithm
,	O
QP	O
,	O
SOCP	B-Algorithm
,	O
SDP	O
,	O
SQP	O
in	O
JavaTK	O
Solver	O
Mathematical	O
modeling	O
and	O
problem	O
solving	O
software	O
system	O
based	O
on	O
a	O
declarative	O
,	O
rule-based	O
language	O
,	O
commercialized	O
by	O
Universal	O
Technical	O
Systems	O
,	O
Inc	O
..	O
TOMLABSupports	O
global	O
optimization	O
,	O
integer	B-Algorithm
programming	I-Algorithm
,	O
all	O
types	O
of	O
least	O
squares	O
,	O
linear	O
,	O
quadratic	O
and	O
unconstrained	O
programming	O
for	O
MATLAB	B-Language
.	O
</s>
<s>
TOMLAB	B-Application
supports	O
solvers	O
like	O
CPLEX	B-Application
,	O
SNOPT	B-Application
and	O
KNITRO.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	O
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>
