Consider the problem of constructing an experimental design that is optimal for a given statistical model with respect to a chosen criterion. To address this problem, the literature typically provides a single solution; sometimes, however, there exists a rich set of optimal designs. The knowledge of this set allows the experimenter to select an optimal design based on its secondary properties. In this paper, we show that the set of all optimal approximate designs often corresponds to a polytope. The polytope can be fully represented by the set of its vertices, which we call vertex optimal designs. We prove that these optimal designs possess unique characteristics, such as small supports, and suggest their potential applications. We also demonstrate that for a variety of models, the vertex optimal designs can be computed using rational arithmetic with perfect accuracy. Consequently, we enumerate all vertex optimal designs for several standard multifactor regression models.

翻译：考虑为给定统计模型基于选定准则构建最优实验设计的问题。文献通常针对该问题提供单一解，但有时存在丰富的最优设计集合。对此集合的认知能使实验者根据次要性质选择最优设计。本文证明，所有最优近似设计的集合通常对应一个多面体。该多面体可通过其顶点集合完全表征，我们将这些顶点称为顶点最优设计。我们证明这些最优设计具有独特特性（如小支撑集），并揭示其潜在应用价值。同时证明对于多类模型，顶点最优设计可通过有理算术精确计算。最终，我们枚举了若干标准多因子回归模型的所有顶点最优设计。