We give a randomized algorithm to estimate the volume within a target relative error. The algorithm can compute a reasonable estimate for the volume of 100-dimensional bodies within a few minutes.
The implementation needs a description of a convex body as the intersection of halfspaces and an ellipsoid (only one of these must be non-empty, e.g. a polytope is only described by halfspaces). The algorithm works in a series of phases, which start out with a distribution very concentrated around a single point inside the body and slowly flatten to the uniform distribution (i.e. the volume). In each phase, a sequence of sample points from a Markov chain is used to estimate the change in volume. When a phase is sufficiently close to the uniform distribution, we will have an estimate for the volume.
In addition to the volume (i.e. uniform distribution), it can also estimate the volume of a spherical Gaussian restricted to a convex set. Also, it can compute approximate samples from a convex body, with respect to the uniform distribution or a spherical Gaussian.
This package was developed in collaboration with Santosh Vempala.
Consider citing http://mpc.zib.de/index.php/MPC/article/view/178/99
The following command sequence
% begin of code
c2diag = makeBody( 'cube', 2 );
c2diag.A_eq = [-1 1];
c2diag.b_eq = ;
c2diag_preprocessed = preprocess(c2diag);
vol = Volume(c2diag_preprocessed)
% end of code
gives the error:
% begin of error message
Error using *
Inner matrix dimensions must agree.
Error in ConvexBody (line 84)
K.b = P.b - K.A * P.p;
Error in Volume (line 29)
K = ConvexBody(P,E,eps,flags);
% end of error message
Is it possible to fix it?
Update citation text.
Connected matlab version to github version that is more updated.
Minor improvements to plotting functionality.
Added coordinate hit-and-run as the default Markov chain. Also, functionality to preprocess and compute the volume of a lower dimensional polytope.
New version not uploaded properly (again).
Added zonotope functionality. Also fixed small bugs with rounding and ellipsoids.
New version was not uploaded properly.
Added function to sample from a convex body. Also added Birkhoff polytope as a built-in convex body.