Geometric scaling, introduced by Schulz and Weismantel in 2002, solves the integer optimization problem $\max \{c\cdot x:x\in P\cap {\mathbb{Z}}^n\}$ by means of primal augmentations, where $P\subset {\mathbb{R}}^n$ is a polytope. We restrict ourselves to the important case when P is a 0/1-polytope. Schulz and Weismantel showed that no more than $O(n{\log }_{2}n{\Vert c\Vert }_{\infty })$ calls to an augmentation oracle are required. This upper bound can be improved to $O(n{\log }_2{\Vert c\Vert }_{\infty })$ using the early-stopping policy proposed in 2018 by Le Bodic, Pavelka, Pfetsch, and Pokutta. Considering both the maximum ratio augmentation variant of the method as well as its approximate version, we show that these upper bounds are essentially tight by maximizing over a n-dimensional simplex with vectors c such that ${\Vert c\Vert }_{\infty }$ is either n or $2^n$.

几何缩放由 Schulz 和 Weismantel 于 2002 年提出，通过原始增广的方式解决整数优化问题 $\max \{c\cdot x:x\in P\cap {\mathbb{Z}}^n\}$ ，其中 $P\subset {\mathbb{R}}^n$ 是多面体。我们将自己限制在 P 是 0/1 多胞形时的重要情况。 Schulz 和 Weismantel 表明，只需要 $O(n{\log }_{2}n{\Vert c\Vert }_{\infty })$ 次调用增强预言机。使用 Le Bodic、Pavelka、Pfetsch 和 Pokutta 于 2018 年提出的提前停止策略，可以将该上限改进为 $O(n{\log }_2{\Vert c\Vert }_{\infty })$ 。考虑到该方法的最大比率增强变体及其近似版本，我们通过最大化具有向量 c 的 n 维单纯形使得 ${\Vert c\Vert }_{\infty }$ 为 n 或 $2^n$ 。