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：由仿射不等式定义的多面体顶点计算问题称为顶点枚举。其逆问题（由极性等价）称为凸包问题。我们引入“近似顶点枚举”问题，即计算与给定仿射不等式定义的多面体相近的多面体的顶点。与精确顶点枚举不同，这两个多面体无需在组合结构上等价。针对该问题提出两种算法：第一种是Motzkin双重描述法的近似变体。仅在某些严格条件下（因实际原因不可接受），我们才能证明该方法对任意维数多面体的正确性。第二种方法称为捷径算法，基于平面图构造，仅适用于二维和三维多面体。我们证明了捷径算法的正确性。由此也得到近似双重描述法在二维和三维下的正确性，且无需高维情形仍需要的限制条件。研究表明，对于二维和三维情形，若使用不精确算术且计算误差不过高，两种算法仍保持正确。两种算法均已实现。数值示例通过表明近似问题通常比精确顶点枚举问题更易求解，佐证了近似顶点枚举问题的研究动机。关于近似双重描述法（无限制条件）对四维及以上多面体是否正确的问题仍悬而未决。