近似 Vertex 计数 (Approximate Vertex Enumeration)

The problem to compute the vertices of a polytope given by affine inequalities is called vertex enumeration. The inverse problem, which is equivalent by polarity, is called the convex hull problem. We introduce `approximate vertex enumeration' as the problem to compute the vertices of a polytope which is close to the original polytope given by affine inequalities. In contrast to exact vertex enumerations, both polytopes are not required to be combinatorially equivalent. Two algorithms for this problem are introduced. The first one is an approximate variant of Motzkin's double description method. Only under certain strong conditions, which are not acceptable for practical reasons, we were able to prove correctness of this method for polytopes of arbitrary dimension. The second method, called graph algorithm, is based on constructing a plane graph and is restricted to polytopes of dimension 2 and 3. We prove correctness of the graph algorithm. As a consequence, we also obtain correctness of the approximate double description method, only for dimension 2 and 3 but without any restricting conditions as still required for higher dimensions. We show that for dimension 2 and 3 both algorithm remain correct if imprecise arithmetic is used and the computational error caused by imprecision is not too high. Both algorithms were implemented. The numerical examples motivate the approximate vertex enumeration problem by showing that the approximate problem is often easier to solve than the exact vertex enumeration problem. It remains open whether or not the approximate double description method (without any restricting condition) is correct for polytopes of dimension 4 and higher.

翻译：用于计算由芬氏不平等给出的多面体的顶端的问题被称为顶点计数。反向问题, 即极性等同的反向问题, 称为convex船体问题。我们引入了“ 近似顶点计数 ”, 作为计算一个极性极性原多面体的顶端的问题。与精确的顶点计数相比, 两个多面体都不需要更简单。为这一问题引入了两种算法。第一个是Motzkin双面描述法的近似变量。前者是双面描述法的双面描述法。只有在某些因实际原因无法接受的强条件下, 我们才能够证明这一方法对任意性多面体的顶点的正确性。第二个方法, 称为图形算法, 以平面图2 和 3 的多面值为限制。我们证明, 图形算法的正确性问题并不比任何双面描述法方法的正确性。因此, 我们对于第2和3级描述方法, 通常不作任何限制条件。我们证明, 高面算算法是否正确, 正确, 3 正确性算法是正确, 正确, 正确, 正确性平面正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确性算法性正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确正确正确, 正确, 正确, 正确, 正确正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确, 正确,