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 shortcut algorithm, is based on constructing a plane graph and is restricted to polytopes of dimension 2 and 3. We prove correctness of the shortcut 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。 倒数问题, 也就是极值的极值, 被称作convex 船体问题。 我们引入了“ 近似顶点计数 ”, 因为它是计算多面体的顶点的问题。 与精确的顶点计数不同, 两个多面都不需要更直截了当地对等。 引入了两种关于这一问题的算法。 第一个是Motzkin双面描述法的近似变体, 称为convex 船体问题。 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, 称为捷径算算法, 以平面图2 和 3 的顶点为限制。 我们证明, 快捷算算法问题不正确。 因此, 我们也会获得近似双面描述方法的正确性描述, 仅针对第2和3级描述方法, 但不作任何限制条件的精确度, 也显示精确度, 正确性算法是正确性, 。 我们用直数级算法的次数级算法, 。 我们显示, 正确性平面的计算法是, 正确, 正确, 正确, 正确, 正确, 正确性平面 正确, 正确, 正确, 正确, 正确, 正确性, 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,