LRS: Vertexes of a polytope giving halfspace definition

In LatticeDesign: Lattice-Based Space-Filling Designs

Vertexes of a polytope giving halfspace definition

Description

Computes the radius, widths, and vertexes of a polytope giving halfspace definition. The program is a R shell of LRS (v.5.1a with lrsmp.h), a reverse search vertex enumeration program/CH package in C which is developed by David Avis <http://cgm.cs.mcgill.ca/~avis/C/lrs.html>. Consider the problem of Ax<=b, where A is an n*p matrix, x is a p-vector, and b is an n-vector. Please make sure that the solution of x is nonempty and bounded. Then the nonequalities give the halfspace definition of a polytope. Also make sure that A and b are rational numbers.

Usage

LRS(numerator,denominator);

Arguments

numerator	The numerators of cbind(b,A), an n*(p+1) matrix of integer numbers.
denominator	The denominators of cbind(b,A), an n*(p+1) matrix of integer numbers.

Details

This function computes the radius, widths, and vertexes of a polytope giving halfspace definition. It is used in constructing interleaved lattice-based minimax distance designs. Currently only tested when the maximum values of numerators and denominators are below 2^20. If the nonequalities are not defined by rational numbers, round-up to small rational numbers is needed before calling the function. The computation is slow for large p but very fast for slow p. Avoid redundant nonequalities may accelerate the calculation.

Value

The value returned from the function is a list containing the following components:

Radius	The maximum L2 distance of vertexes to the origin.
MaxValue	The maximum k-dimensional value of the vertexes, for k from 1 to p.
Vertexes	The vertexes of the polytope.

References

Avis, David. LRS, http://cgm.cs.mcgill.ca/~avis/C/lrs.html.

He, Xu (2017). "Interleaved lattice-based minimax distance designs", Biometrika, 104(3): 713-725.

See Also

InterleavedMinimaxD.

Examples

num = matrix(0,5,3)
den = matrix(1,5,3)
num[1,2] = -1;  den[1,2] = 2; 
num[1,1] = 1;   den[1,1] = 8;
num[2,3] = -1;
num[2,1] = 1;   den[2,1] = 2;
num[3,2] = -1;  den[3,2] = 4;
num[3,3] = -1;  den[3,3] = 2;
num[3,1] = 5;   den[3,1] = 32;
num[4,2] = 1; 
num[4,1] = 0;
num[5,3] = 1;
num[5,1] = 0; 
LRS(num,den)

LatticeDesign documentation built on Nov. 13, 2022, 9:06 a.m.