  We propose a class of subspace ascent methods for computing optimal
approximate designs that covers both existing as well as new and more efficient
algorithms. Within this class of methods, we construct a simple, randomized
exchange algorithm (REX). Numerical comparisons suggest that the performance of
REX is comparable or superior to the performance of state-of-the-art methods
across a broad range of problem structures and sizes. We focus on the most
commonly used criterion of D-optimality that also has applications beyond
experimental design, such as the construction of the minimum volume ellipsoid
containing a given set of data-points. For D-optimality, we prove that the
proposed algorithm converges to the optimum. We also provide formulas for the
optimal exchange of weights in the case of the criterion of A-optimality. These
formulas enable one to use REX for computing A-optimal and I-optimal designs.
