The Polytope of Optimal Approximate Designs: Extending the Selection of Informative Experiments

TL;DR

This paper reveals that the set of all optimal approximate experimental designs forms a polytope, with vertex designs offering practical advantages, and demonstrates methods to compute these designs for common models, broadening experimental options.

Contribution

It introduces the concept of the polytope of optimal designs and identifies vertex designs as key elements, providing strategies for their computation and application.

Findings

Optimal designs form a polytope structure.

Vertex designs have small supports and unique properties.

Computational methods can determine these designs for common models.

Abstract

Consider the problem of constructing an experimental design, optimal for estimating parameters of a given statistical model with respect to a chosen criterion. To address this problem, the literature usually provides a single solution. Often, however, there exists a rich set of optimal designs, and the knowledge of this set can lead to substantially greater freedom to select an appropriate experiment. In this paper, we demonstrate that the set of all optimal approximate designs generally corresponds to a polytope. Particularly important elements of the polytope are its vertices, which we call vertex optimal designs. We prove that the vertex optimal designs possess unique properties, such as small supports, and outline strategies for how they can facilitate the construction of suitable experiments. Moreover, we show that for a variety of situations it is possible to construct the vertex optimal designs with the assistance of a computer, by employing error-free rational-arithmetic calculations. In such cases the vertex optimal designs are exact, often closely related to known combinatorial designs. Using this approach, we were able to determine the polytope of optimal designs for some of the most common multifactor regression models, thereby extending the choice of informative experiments for a large variety of applications.