More detailed descriptions can be found within the problems code.
| Problem name and subroutine |
Description |
Graphical representation |
America
america.f |
The objective is to draw a map of America in
which the countries appear with areas that are
proportional to their real values.
[M, 2000] |
 |
Hard spheres
hardspheres.f
(See also the associated Kissing Number problem
modeled as a feasibility problem in
two different ways kissing.f
kissing2.f)
|
The Hard-Spheres Problem
is to maximize the minimum pairwise distance
between npun points on a sphere in
R^ndim.
[CS, 1999] |
 |
Hard cube
hardcube.f |
We consider the cube [-1,1]^{ndim} intersected
with the complement of the unitary sphere. We
want to find npun points in this set such that
the minimum distance between them is maximal. |
 |
Smallest enclosing ellipsoid
centered at the origin
ellipsoid.f
ellipsoid.dat |
Given npun fixed points in R^ndim, minimize
the volume of the ellipsoid, centered at the
origin, that fits them. Points can be generated
using the Cauchy distribution or given by the user.
[TY, 2006]
File ellipsoid.dat
corresponds to the coordinates
of 4278 points in R^3 that represent the atoms of a
protein of the Isoform alpha of the
Thyroid Hormone Receptor. |
 |
Location problem
location.f |
This is a variant of the family of location
problems introduced in [BMR, 2001].
In the original problem, given a set of np
disjoint polygons P1, P2, ..., Pnp in R^2
one wishes to find the point z1 in P1 that
minimizes the sum of the distances to the
other polygons.
In this variant we have, in addition to
the np polygons, nc
circles. Moreover, there is an ellipse
which has a non empty intersection with P1
and such that z1 must be inside the ellipse
and zi, i = 2, ..., np+nc must be outside.
The detailed formulation can be found in
[ABMS, 2007].
Here you can find the scripts to run ALGENCAN (runlocation-algencan)
and ALSPG
(
runlocation-alspg) in a set of small
problems. ALSPG
can also solve a set of very large problems
(
runlocation-alspg-2). See in the scripts
the suggested input data to reproduce the
numerical examples presented in [ABMS, 2007]. Last
but not least, here you can find the
ready-to-use version of
ALSPG to solve the
location problem. |
 |
Nonlinear second-order one-dimensional
problem with observed points
condor.f |
We generate (using subroutine evalu) a function
called usol as a Fourier polynomial with up to
q = 20 terms. A grid in [0, 2 pi] with up to 1000
points is established. We generate function fi(t)
in such a way that usol is a solution of the
discretization of the differential equation
u''(t) + [u'(t)]^2 + u(t) = fi(t). We select up to
1000 observation points equally spaced in [-2, 18]
and observe the solution usol at these observation
points, obtaining the observed solution uobs. Finally,
uint, the interpolation of uobs in the grid points is
computed. The optimization problem is to obtain the
solution of the discretized differential equation
which is closest, in the least-squares sense, to uint.
Instances with q = 20, 500 observation points and 1000
grid points are challenging. |
 |
Piecefit: a novel application of OVO
piecefit.f |
We defined usol, a piecewise linear function defined
on [0,4] with kinks in 1 and 3 and discotinuity in 2.
We discretized [0,4] with n points, including extremes.
We defined uobs, a 10% perturbation of usol. The
objective function is the sum of the p smaller squares
of the discretized second derivatives of x. p, the OVO
parameter, is given by the user. The restriction is that
the root mean squared deviation of the solution found
with respect to uobs is smaller than 0.2. In other words,
using LOVO, this code aims to find a piecewise harmonic
one-dimensional function. |
 |
Fit a continuous piecewise polynomial
pedazos4.f |
We generate a table (t_i, y_i) with 30 equally spaced
points between 1 and 30. The first 10 points are on a
line, the second 10 on a parabola and the third 10 on
a cubic. The points are 10% perturbed The overall
function is continuous at 10 and 20. We fit a line to
the first 10, a quadratic to the second 10 and a cubic
to the third ten, with the conditions that we preserve
continuity at 10 and 20. So, we have 9 unknowns and 2
equality constraints. The objective function is quadratic
and the constraints are linear. |
 |
Pass mountain
mountain1.f
mountain2.f |
Given a surface S(x,y) and initial and final
points pi, pf in R^2 the problem consists on
finding a path pi, p1, p2, ..., pN, pf from pi
to pf such that max_{1 <= k <= N} S(pk) is as
small as possible. Moreover, the distance between
consecutive points in the path must be less than
or equal to a prescribed tolerance. The problem
has n = 2 N + 1 variables and m = 2 N + 1 inequality
constraints, where N is the number of intermediate
points in the path. The considered surfaces are:
S(x,y) = sin( x * y ) + sin( x + y ) and S(x,y) =
sin(x) + cos(y). [H, 2004],
[MM, 2004] |
 |
Packing of circles within a circle
packccmn.f
(modeled as a feasibility problem
packccmn-feas.f) |
We wish to fit a fixed number of different-sized
circular items within a circular object of radius R
in such a way that the circular items do not overlap.
[BMR, 2005] |
 |
Packing of circles within a rectangle
packcrmn-feas.f |
We wish to fit a fixed number of different-sized
circular items within a rectangular object of
dimension L x W in such a way that the circular
items do not overlap. In fact, the problem is
coded for any arbitrary dimension. Therefore, it
can also be used to fit spheres within a cuboid.
[BMR, 2005] |
No image available. |
Fix expiration dates for a content network location
cache problem
cache.f
cache.dat |
Mathematical programming model for fixing content location
cache expiration dates at aggregation nodes in a content
network, in order to maximize the information discovered by
the nodes of the networks, while taking into account
bandwidth constraints at the aggregation nodes. |
No image available. |
Contour I
contor.f |
Find the solution of the differential equation y'' = sin(y)
which is closest to a set of given data points. The data
(ti, yi) are such that yi is a 10 percent random perturbation
of a true discrete solution. The true discrete solution is
such that y(0)=y'(0)=1. |
No image available. |
Contour II
contor2.f |
Find the solution of the differential equation
y'' = a y^2 + b y + c which is closest to a set of given
data points, considering y, a, b and c as unknowns. The
data (ti, yi) are such that yi is a random [-1,1]
perturbation of a true discrete solution. The true discrete
solution is such that y(0) = y'(0) = 1. |
No image available. |
Simulation of Hartree-Fock
simfock.f |
Given kdim and ndim satisfying ndim <= kdim, the
problem is to minimize 0.5 x^T A x + b^T x subject to
mat(x) mat(x) = mat(x), trace[mat(x)] = ndim, and
mat(x) = mat(x)^T, where, n = kdim^2, x in R^n, and we
define mat(x) in R^{kdim x kdim} as the matrix whose
columnwise representation is x. Therefore, the problem
has kdim^2 variables, kdim^2 nonlinear equality constraints
and kdim * ( kdim - 1 ) / 2 + 1 linear equality
constraints. |
No image available. |
Simulation of Hartree-Fock II
simfock2.f |
This a different formulation of the problem above.
Given kdim and ndim satisfying ndim <= kdim, the problem
is to minimize 0.5 x^T A x + b^T x subject to M orthonornal,
where x is the columnwise representation of the matrix
X = M M^T, and M in R^{kdim x ndim}. Therefore, the problem
has ndim * kdim variables (the elements of matrix M) and ndim
+ ndim * ( ndim - 1 ) / 2 nonlinear equality constraints. |
No image available. |
Bratu
bratu3Db.f |
3D Boundary Value problem with known solution.
[ABMS, 2007] |
No image available. |
Bratu full
bratu3Db-full.f |
This is a different version of the problem
above. The distance of the solution to the origin
is minimized and the origin is the initial guess.
See [ABMS, 2007] or the
comments within the Fortran file for details. |
No image available. |
Packing of circular items within a circle minimizing its
area (or perimeter)
genpack-cc-mina.f |
Given a fixed number of fixed-size identical circles (called
items), the objective is to find the smallest circle that fits
the items without overlapping.
[BS, 2008] |
 |
Packing of circular items within a square minimizing its
area (or perimeter)
genpack-csq-mina.f |
Given a fixed number of fixed-size identical circles (called
items), the objective is to find the smallest square that fits
the items without overlapping.
[BS, 2008] |
 |
Balls
balls.f |
This problem consists of finding np points in R^ndim
such that the distance between any pair of them is
not less than 1 and the maximum distance is as small
as possible. |
 |
Elastic beam
vi8EA.f |
This problem consists in finding the thickness
distribution minimizing the compliance of an
elastic beam. |
No image available. |