  In this paper, we study a generic direct-search algorithm in which the
polling directions are defined using random subspaces. Complexity guarantees
for such an approach are derived thanks to probabilistic properties related to
both the subspaces and the directions used within these subspaces. Our analysis
crucially extends previous deterministic and probabilistic arguments by
relaxing the need for directions to be deterministically bounded in norm. As a
result, our approach encompasses a wide range of new optimal polling strategies
that can be characterized using our subspace and direction properties. By
leveraging results on random subspace embeddings and sketching matrices, we
show that better complexity bounds are obtained for randomized instances of our
framework. A numerical investigation confirms the benefit of randomization,
particularly when done in subspaces, when solving problems of moderately large
dimension.
