Removal of Redundant Candidate Points for the Exact D-Optimal Design Problem

Abstract

One of the most common problems in statistical experimentation is computing D-optimal designs on large finite candidate sets. While optimal approximate (i.e., infinite-sample) designs can be efficiently computed using convex methods, constructing optimal exact (i.e., finite-sample) designs is a substantially more difficult integer-optimization problem. In this paper, we propose necessary conditions, based on approximate designs, that must be satisfied by any support point of a D-optimal exact design. These conditions enable rapid elimination of redundant candidate points without loss of optimality, thereby reducing memory requirements and runtime of subsequent exact design algorithms. In addition, we prove that for sufficiently large sample sizes, the supports of D-optimal exact designs are contained in a typically small maximum-variance set. We demonstrate the approach on randomly generated benchmark models with candidate sets up to 100 million points, and on commonly used constrained mixture models with up to one million points. The proposed approach reduces the initial candidate sets by several orders of magnitude, thereby making it possible to compute exact D-optimal designs for these problems via mixed-integer second-order cone programming, which provides optimality guarantees.