Improvements on removing non-optimal support points in D-optimum design algorithms

TL;DR

This paper enhances the inequality used in D-optimum design algorithms to more effectively eliminate non-optimal support points, thereby accelerating the search for optimal experimental designs.

Contribution

It introduces a refined lower bound on support points in D-optimum design, improving the efficiency of design algorithms by removing non-optimal points more effectively.

Findings

New lower bound on support points in D-optimum design

The bound is proven to be optimal in a certain sense

Algorithm acceleration through improved point removal

Abstract

We improve the inequality used in Pronzato [2003. Removing non-optimal support points in D-optimum design algorithms. Statist. Probab. Lett. 63, 223-228] to remove points from the design space during the search for a $D$-optimum design. Let $\xi$ be any design on a compact space $\mathcal{X} \subset \mathbb{R}^m$ with a nonsingular information matrix, and let $m+\epsilon$ be the maximum of the variance function $d(\xi,\mathbf{x})$ over all $\mathbf{x} \in \mathcal{X}$. We prove that any support point $\mathbf{x}_{*}$ of a $D$-optimum design on $\mathcal{X}$ must satisfy the inequality $d(\xi,\mathbf{x}_{*}) \geq m(1+\epsilon/2-\sqrt{\epsilon(4+\epsilon-4/m)}/2)$. We show that this new lower bound on $d(\xi,\mathbf{x}_{*})$ is, in a sense, the best possible, and how it can be used to accelerate algorithms for $D$-optimum design.