Volume computation and sampling

VolEsti is a C++ library for volume approximation and sampling of convex bodies (e.g. polytopes) with an R interface.

- R Interface

---

Install Rcpp package

• Install package-dependencies: Rcpp, RcppEigen, BH, lpSolveAPI.

1. Then use devtools package to install volesti Rcpp package. In folder /root/R-prog Run:

Rcpp::compileAttributes()  
library(devtools)  
devtools::build()  
devtools::install()  
library(volesti)  

2. You can use Rstudio as well to open volesti.Rproj and then click build source Package and then Install and Restart in Build at the menu bar.

Generate CRAN version

To generate the CRAN version of the R package follow the instructions below:

1. From the command line navigate to folder /cran_gen. Then Run:

source('genCRANpkg.R')  

2. Open genCRANpkg.R script with Rstudio and run it.

Run volesti from R

• The main function is volume(). It can be used to approximate the volume of a convex polytope given as a set of linear inequalities or a set of vertices (d-dimensional points) or as a Minkowski sum of segments (zonotope). There are two algorithms that can be used. The first is SequenceOfBalls and the second is CoolingGaussian (see References).
• The function sample_points() can be used to sample points from a convex polytope with uniform or spherical gaussian target distribution.
• The function round_polytope() can be used to round a convex polytope.
• The function rand_rotate() can be used to apply a random rotation to a convex polytope.

For more details you can read the documentation in folder /inst/doc.

Create pdf documentation from Rd files

• Install volesti library.
• In R mode (or in Rstudio) Run

pack = "volesti"  
path = find.package(pack)  
system(paste(shQuote(file.path(R.home("bin"), "R")),  
    "CMD", "Rd2pdf", shQuote(path)))

• The pdf will be created and saved in R-proj folder.
• We give such a documentation in /R-proj/doc folder.

- C++ Interface

---

Compile C++ sources and run tests

To compile the C++ code you have to specify the path to external library liblpsolve55.so, by running, in folder test:

cmake -DLP_SOLVE=_PATH_TO_LIB_FILE_ .  
make  

For example: -DLP_SOLVE=/usr/lib/lpsolve/liblpsolve55.so

You can run the tests by cmake test or ctest -jK where K the number of CPU threads. By adding the option --verbose to ctest you get more information about the tests, e.g. time per test, volume computed and the name of the polytope or convex body.

Polytope input

The current version of the software assumes that the polytope is given in the form of linear inequalities i.e. {x \in R^d : Ax <= b} where A is a matrix of dimension m x d and b a vector of dimension m or as a set of m vertices {\in R^d} or as a Minkowski sum of m segments {\in R^d}. The input is described in an .ine-file (H-polytopes) or in a .ext file (V-polytopes or zonotopes). The .ine file is described as follows:

various comments  
begin  
m d+1 numbertype  
b -A  
end  
various options  

The .ext file is described as follows:

various comments  
begin  
m d numbertype  
1 v_1  
.. ...  
1 v_m  
end  
various options  

In V-polytope case v_i are vertices and in zonotope case they are segments.

This filestype (or similar) is used by a number of other software in polyhedral computation (e.g. cdd, vinci, latte). In the current version of the software, the options are treated as comments and the numbertype as C++ double type.
If your input has equality constraints then you have to transform it in the form that only contain linear inequalities which described above by using some other software. We recommend to use latte https://www.math.ucdavis.edu/~latte for this transformation.

Run volesti from command line

After successful compilation you can use the software by command line. For example, the following command ./vol -h will display a help message about the program's available options.

Example

To estimate the volume of the 10-dimensional hypercube first prepare the file cube10.ine as follows:

cube10.ine  
H-representation  
begin  
 20 11 real  
 1 1 0 0 0 0 0 0 0 0 0  
 1 0 1 0 0 0 0 0 0 0 0  
 1 0 0 1 0 0 0 0 0 0 0  
 1 0 0 0 1 0 0 0 0 0 0  
 1 0 0 0 0 1 0 0 0 0 0  
 1 0 0 0 0 0 1 0 0 0 0  
 1 0 0 0 0 0 0 1 0 0 0  
 1 0 0 0 0 0 0 0 1 0 0  
 1 0 0 0 0 0 0 0 0 1 0  
 1 0 0 0 0 0 0 0 0 0 1  
 1 -1 0 0 0 0 0 0 0 0 0  
 1 0 -1 0 0 0 0 0 0 0 0  
 1 0 0 -1 0 0 0 0 0 0 0  
 1 0 0 0 -1 0 0 0 0 0 0  
 1 0 0 0 0 -1 0 0 0 0 0  
 1 0 0 0 0 0 -1 0 0 0 0  
 1 0 0 0 0 0 0 -1 0 0 0  
 1 0 0 0 0 0 0 0 -1 0 0  
 1 0 0 0 0 0 0 0 0 -1 0  
 1 0 0 0 0 0 0 0 0 0 -1  
end  
input_incidence  

Then to use SequenceOfBalls (SOB) algorithm run the following command:

./vol -f1 cube_10.ine  

which returns 17 numbers:
d m #experiments exactvolOr-1 approxVolume [.,.] #randPoints walkLength meanVol [minVol,maxVol] stdDev errorVsExact maxminDivergence time timeChebyshevBall

To use CoolingGaussian (CG) algorithm run the following command:

./vol -f1 cube_10.ine -CG  

which returns the same output as before.

To estimate the volume of a 10-dimensional V-cross polytope described in cross_10.ext as follows:

cross_10.ext  
V-representation  
begin  
 20 11 integer  
 1 1 0 0 0 0 0 0 0 0 0  
 1 0 1 0 0 0 0 0 0 0 0  
 1 0 0 1 0 0 0 0 0 0 0  
 1 0 0 0 1 0 0 0 0 0 0  
 1 0 0 0 0 1 0 0 0 0 0  
 1 0 0 0 0 0 1 0 0 0 0  
 1 0 0 0 0 0 0 1 0 0 0  
 1 0 0 0 0 0 0 0 1 0 0  
 1 0 0 0 0 0 0 0 0 1 0  
 1 0 0 0 0 0 0 0 0 0 1  
 1 -1 0 0 0 0 0 0 0 0 0  
 1 0 -1 0 0 0 0 0 0 0 0  
 1 0 0 -1 0 0 0 0 0 0 0  
 1 0 0 0 -1 0 0 0 0 0 0  
 1 0 0 0 0 -1 0 0 0 0 0  
 1 0 0 0 0 0 -1 0 0 0 0  
 1 0 0 0 0 0 0 -1 0 0 0  
 1 0 0 0 0 0 0 0 -1 0 0  
 1 0 0 0 0 0 0 0 0 -1 0  
 1 0 0 0 0 0 0 0 0 0 -1  
end  
hull  
incidence  

Run:

./vol -f2 cross_10.ext  

which returns the same output as before.

To estimate the volume of a 4-dimensional zonotope defined by the Minkowski sum of 8 segments described in zonotope_4_8.ext as follows:

zonotope_4_8.ext  
Zonotpe  
begin  
 8 5 real  
 1 0.981851 -0.188734 -0.189761 0.0812645  
 1 -0.0181493 0.811266 -0.189761 0.0812645  
 1 -0.0181493 -0.188734 0.810239 0.0812645  
 1 -0.0181493 -0.188734 -0.189761 1.08126  
 1 -0.177863 0.437661 -0.0861379 -0.674634  
 1 0.737116 -0.204646 -0.540973 -0.471883  
 1 -0.684154 0.262324 0.292341 -0.265955  
 1 -0.802502 -0.740403 0.0938152 0.0874131  
end  
hull  
incidence  

Run:

./vol -f3 zonotope_4_8.ext  

Flag -v enables the print mode.

Generate polytopes

You can use executable generator to generate polytopes (hypercubes, simplices, cross polytopes, skinny hypercubes (only in H-representation), product of two simplices (only in H-representation) and zonotoes. For example:

1. To generate a 10-dimensional hypercube in H-representation run:

./generate -cube -h -d 10

2. To generate a 20-dimensional simplex in V-representaion run:

./generate -simplex -v -d 20

3. To generate a 5-dimensional zonotope defined by 10 segments run:

./generate -zonotope -d 5 -m 10

Command ./generate -help will display a help message about the program's available options.

Sampling

You can sample from a convex polytope uniformly or from the spherical gaussian distribution. For example:

1. To sample uniformly from the 10-dimensional hypercube, run:

./vol -f1 cube_10.ine -rand -nsample 1000

Flag -nsample declares the number of points we wish to sample (default is 100).

2. To sample from the gaussian distribution, run:

./vol -f1 cube_10.ine -rand -nsample 1300 -gaussian -variance 1.5

Flag -variance declares the variance (default is 1.0). The center of the spherical gaussian is the Chebychev center for H-polytopes, or the origin for zonotopes. For V-polytopes is the chebychev center of the simplex that is defined by a random choice of d+1 vertices.

3. To sample from a zonotope described in zonotope.ext file run:

./vol -f3 zonotope.ext -rand -nsample 1500

For V-polytopes use flag -f2 before the .ext file. In all cases use flag -v to print the excecutional time.

Credits

Copyright (c) 2012-2018 Vissarion Fisikopoulos
Copyright (c) 2018 Apostolos Chalkis

You may redistribute or modify the software under the GNU Lesser General Public License as published by Free Software Foundation, either version 3 of the License, or (at your option) any later version. It is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY.

Main development by Vissarion Fisikopoulos while he was affiliated with University of Athens (UoA, Greece), University of Brussels (ULB, Belgium), and Chalkis Apostolos affiliated with University of Athens (UoA, Greece).

Publications

1. I.Z. Emiris and V. Fisikopoulos, Efficient random-walk methods for approximating polytope volume, In Proc. ACM Symposium on Computational Geometry, Kyoto, Japan, p.318-325, 2014.
2. I.Z. Emiris and V. Fisikopoulos, Practical polytope volume approximation, ACM Transactions on Mathematical Software, vol 44, issue 4, 2018.
3. L. Cales, A. Chalkis, I.Z. Emiris, V. Fisikopoulos, Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises, Proc. of Symposium on Computational Geometry, Budapest, Hungary, 2018.