## Volume computation and sampling

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**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:
```r
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:  
```r
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.