<s>
WORHP	B-Application
(	O
"	O
warp	O
"	O
,	O
an	O
acronym	O
for	O
"	O
We	O
Optimize	O
Really	O
Huge	O
Problems	O
"	O
)	O
,	O
also	O
referred	O
to	O
as	O
eNLP	B-Application
(	O
European	O
NLP	B-Algorithm
solver	O
)	O
by	O
ESA	O
,	O
is	O
a	O
mathematical	O
software	B-Library
library	I-Library
for	O
numerically	O
solving	O
large	O
scale	O
continuous	O
nonlinear	B-Algorithm
optimization	I-Algorithm
problems	O
.	O
</s>
<s>
WORHP	B-Application
is	O
a	O
hybrid	O
Fortran	O
and	O
C	B-Language
implementation	O
and	O
can	O
be	O
used	O
from	O
C/C	O
++	O
and	O
Fortran	O
programs	O
using	O
different	O
interfaces	O
of	O
varying	O
complexity	O
and	O
flexibility	O
.	O
</s>
<s>
There	O
are	O
also	O
interfaces	O
for	O
the	O
MATLAB	B-Language
,	O
CasADi	B-Application
and	O
AMPL	B-Language
modelling	O
environments	O
.	O
</s>
<s>
Even	O
problems	O
with	O
large	O
dimensions	O
and	O
can	O
be	O
solved	O
efficiently	O
,	O
if	O
the	O
problem	O
is	O
sufficiently	O
sparse	B-Algorithm
.	O
</s>
<s>
Cases	O
where	O
objective	O
and	O
constraints	O
cannot	O
be	O
evaluated	O
separately	O
,	O
or	O
where	O
constraints	O
can	O
be	O
evaluated	O
element-wise	O
can	O
be	O
exploited	O
by	O
WORHP	B-Application
to	O
increase	O
the	O
computational	O
efficiency	O
.	O
</s>
<s>
WORHP	B-Application
requires	O
the	O
first	B-Algorithm
derivative	I-Algorithm
(	O
Gradient	O
)	O
of	O
and	O
of	O
(	O
Jacobian	O
)	O
and	O
second	O
derivatives	B-Algorithm
(	O
Hessian	O
matrix	O
)	O
of	O
the	O
Lagrange	O
function	O
;	O
in	O
a	O
modelling	O
environment	O
like	O
AMPL	B-Language
,	O
these	O
are	O
provided	O
by	O
automatic	B-Algorithm
differentiation	I-Algorithm
methods	O
,	O
but	O
need	O
to	O
be	O
provided	O
by	O
the	O
caller	O
in	O
other	O
environments	O
.	O
</s>
<s>
First	O
and	O
second	O
derivatives	B-Algorithm
can	O
be	O
approximated	O
by	O
WORHP	B-Application
using	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
To	O
reduce	O
the	O
otherwise	O
prohibitively	O
high	O
number	O
of	O
necessary	O
function	O
evaluations	O
in	O
large	O
scale	O
sparse	B-Algorithm
problems	O
,	O
graph	O
colouring	O
theory	O
is	O
used	O
to	O
group	O
first	O
and	O
second	O
partial	O
derivatives	B-Algorithm
.	O
</s>
<s>
Second	O
derivatives	B-Algorithm
may	O
also	O
be	O
approximated	O
using	O
variations	O
of	O
the	O
classic	O
BFGS	B-Algorithm
method	I-Algorithm
,	O
including	O
block-diagonal	O
or	O
sparse	B-Algorithm
BFGS	B-Algorithm
matrices	O
.	O
</s>
<s>
The	O
NLP	B-Algorithm
level	O
of	O
WORHP	B-Application
is	O
based	O
on	O
SQP	B-Algorithm
,	O
while	O
the	O
quadratic	O
subproblems	O
are	O
solved	O
using	O
an	O
interior	B-Algorithm
point	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
approach	O
was	O
chosen	O
to	O
benefit	O
from	O
the	O
robustness	O
of	O
SQP	B-Algorithm
methods	O
and	O
the	O
reliable	O
runtime	O
complexity	O
of	O
IP	O
methods	O
,	O
since	O
traditional	O
active	B-Algorithm
set	I-Algorithm
methods	I-Algorithm
may	O
be	O
unsuitable	O
for	O
large-scale	O
problems	O
.	O
</s>
<s>
(	O
whose	O
inclusion	O
in	O
eNLP	B-Application
was	O
discontinued	O
after	O
2010	O
)	O
to	O
develop	O
a	O
European	O
NLP	B-Algorithm
solver	O
for	O
use	O
in	O
trajectory	O
optimisation	O
,	O
mission	O
analysis	O
and	O
aerospace	O
applications	O
in	O
general	O
.	O
</s>
<s>
The	O
development	O
of	O
WORHP	B-Application
is	O
led	O
by	O
the	O
and	O
scientists	O
of	O
the	O
at	O
the	O
University	O
of	O
Bremen	O
,	O
and	O
at	O
the	O
Bundeswehr	O
University	O
of	O
Munich	O
.	O
</s>
<s>
The	O
developers	O
stress	O
that	O
WORHP	B-Application
,	O
despite	O
its	O
academic	O
roots	O
,	O
is	O
intended	O
as	O
industrial-grade	O
tool	O
rather	O
than	O
an	O
academic	O
research	O
platform	O
.	O
</s>
<s>
and	O
ASTOS	B-Application
,	O
and	O
is	O
being	O
used	O
at	O
ESOC	O
and	O
ESTEC	O
.	O
</s>
