New Implementations of the Double Description Method

Abstract

A pair (A; R) of real matrices A and R is said to be a double description pair or simply DD pair if the relationship Ax 0 if and only if x = 3DR for some 0 holds. Clearly, for a pair (A; R) to be a DD pair, it is necessary that the column size of A is equal to the row size of R, say d. The term "double description" was introduced by Motzkin et al. [MRTT53], and it is quite natural in the sense that such a pair contains two different descriptions of the same polyhedral cone. We will call A a representation matrix and R a generating matrix for this cone that we denote by P (A). Minkowski's and Weyl's theorems suggest the two fundamental problems, one to construct a matrix R from a given matrix A and the converse. It is well known that these two problems are computationally equivalent, and it can be shown that (A; R) is a DD pair if and only if (R T ; A T ) is a DD pair. Thus, we concentrate on the first problem, to find a generating matrix R for a given A. Clearly a more appr..