  In the area of statistical planning, there is a large body of theoretical
knowledge and computational experience concerning so-called optimal approximate
designs of experiments. However, for an approximate design to be executed in
practice, it must be converted into an exact, i.e., integer, design, which is
usually done via rounding procedures. Although rapid, rounding procedures have
many drawbacks; in particular, they often yield worse exact designs than
heuristics that do not require approximate designs at all.
  In this paper, we build on an alternative principle of utilizing optimal
approximate designs for the computation of optimal, or nearly-optimal, exact
designs. The principle, which we call ascent with quadratic assistance (AQuA),
is an integer programming method based on the quadratic approximation of the
design criterion in the neighborhood of the optimal approximate information
matrix.
  To this end, we present quadratic approximations of all Kiefer's criteria
with an integer parameter, including D- and A-optimality and, by a model
transformation, I-optimality. Importantly, we prove a low-rank property of the
associated quadratic forms, which enables us to apply AQuA to large design
spaces, for example via mixed integer conic quadratic solvers. We numerically
demonstrate the robustness and superior performance of the proposed method for
models under various types of constraints. More precisely, we compute optimal
size-constrained exact designs for the model of spring-balance weighing, and
optimal symmetric marginally restricted exact designs for the Scheffe mixture
model. We also show how can iterative application of AQuA be used for a
stratified information-based subsampling of large datasets under a lower bound
on the quality and an upper bound on the cost of the subsample.
