<p>One of the most common problems in statistical experimentation is computing D-optimal designs for linear or locally linearized models on large finite candidate sets. While optimal approximate designs can be efficiently computed using convex methods, constructing optimal exact designs with a prespecified total number of trials 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. We also prove that, for a sufficiently large number of trials, the support of every D-optimal exact design is contained in a set that typically coincides with the support of a D-optimal approximate design. We demonstrate the approach on randomly generated benchmark models with candidate sets of up to 100 million points and on commonly used constrained mixture models with up to 1 million points. The proposed approach reduces the initial candidate sets by several orders of magnitude, thereby making it possible to compute D-optimal exact designs for these problems via mixed-integer second-order cone programming, which provides optimality guarantees.</p>