🤖 AI Summary
Computing exact D-optimal designs over large finite candidate sets is computationally prohibitive due to excessive memory and time requirements.
Method: We propose a redundancy elimination method grounded in necessary conditions for approximate designs. It integrates convex optimization to compute an approximate design, integer-constrained screening to identify candidate support points, and mixed-integer second-order cone programming to recover the exact optimal solution.
Contribution/Results: We establish, for the first time, a theoretical link between approximate and exact designs—proving that, asymptotically, the optimal support set is contained within the maximum-variance subset. This enables stepwise, order-of-magnitude compression of the candidate set. Our approach reduces candidate sets with tens of millions of points by several orders of magnitude, solves previously intractable large-scale instances, and delivers solutions with certified global optimality.
📝 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.