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An efficient implementation of the Double Description Method

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# cddlib/cddlib

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# cddlib

The C-library cddlib is a C implementation of the Double Description Method of Motzkin et al. for generating all vertices (i.e. extreme points) and extreme rays of a general convex polyhedron in Rd given by a system of linear inequalities:

P = { x=(x1, ..., xd)T : b - A·x ≥ 0 }

where A is a given m×d real matrix, b is a given m-vector and 0 is the m-vector of all zeros.

The program can be used for the reverse operation (i.e. convex hull computation). This means that one can move back and forth between an inequality representation and a generator (i.e. vertex and ray) representation of a polyhedron with cddlib. Also, cddlib can solve a linear programming problem, i.e. a problem of maximizing and minimizing a linear function over P.

## Report an Issue

If you find some problems or bugs, please kindly report them to our issue tracker.

## Documentation

The documentation is still incomplete but contains descriptions of the most important functions. Please look at the examples for how the library functions can be used in user's C-codes.

## H-representation & V-polyhedron

One convenient feature of cddlib is the ability to handle essentially any data. More precisely, it can generate an H-representation of a V-polyhedron which is not full-dimensional, and it can generate a V-representation of an H-polyhedron which has no extreme points.

## Numerical Problems

A little caution is in order. Many people have seen numerical problems when the floating version of cddlib is used. As we all know, floating-point computation might not give a correct answer, especially when an input data is very sensitive to a small perturbation. When some strange behavior is observed, it is always wise to create a rational approximation of the input (for example, one can replace 0.3333333 with 1/3) and to compute it with cddlib compiled with GMP Rational to see what a correct behavior should be. Whenever the time is not important, it is safer to use GMP rational arithmetic.

If you need speedy computation with floating-point arithmetic, you might want to play with the constant `dd_almostzero` defined in cdd.h:

``````#define dd_almostzero  1.0E-6
``````

This number is used to recognize whether a number is zero: a number whose absolute value is smaller than `dd_almostzero` is considered zero, and nonzero otherwise. You might want to change this to modify the behavior of cddlib. Another thing one can do is scaling. If the values in one column of an input is of smaller magnitude than those in another column, scale one so that they become comparable.

## Build the Latest Released Version

Download the most recent tarball from our Releases page and build cddlib with

``````tar zxf cddlib-*.tar.gz
cd cddlib-*
./configure
make
``````

To install cddlib to `/usr/local` type

``````sudo make install
``````

``````gcc -I/usr/local/include -L/usr/local/lib YOUR_CODE.c -lcdd
``````

or, if you want to use rational arithmetic provided by GMP, with

``````gcc -I/usr/local/include -L/usr/local/lib -DGMPRATIONAL YOUR_CODE.c -lcddgmp -lgmp
``````

## Build from the Source Code Repository

Similar to the above, you can build from our github repository with

``````git clone https://github.com/cddlib/cddlib.git
cd cddlib
./bootstrap
./configure
make
``````

## Contact the Author

The library cddlib is free software, but if cddlib turns out to be useful, please kindly send a note or paper to the author mentioning what purpose and how cddlib has been used for. The most powerful support for free software development is user's appreciation.

An efficient implementation of the Double Description Method

0.94m Latest
Dec 8, 2020

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