  The work presented here is motivated by the development of StoDARS, a
framework for large-scale stochastic blackbox optimization that not only is
both an algorithmic and theoretical extension of the stochastic directional
direct-search (SDDS) framework but also extends to noisy objectives a recent
framework of direct-search algorithms in reduced spaces (DARS). Unlike SDDS,
StoDARS achieves scalability by using~$m$ search directions generated in random
subspaces defined through the columns of Johnson--Lindenstrauss transforms
(JLTs) obtained from Haar-distributed orthogonal matrices. For theoretical
needs, the quality of these subspaces and the accuracy of random estimates used
by the algorithm are required to hold with sufficiently large, but fixed,
probabilities. By leveraging an existing supermartingale-based framework, the
expected complexity of StoDARS is proved to be similar to that of SDDS and
other stochastic full-space methods up to constants, when the objective
function is continuously differentiable. By dropping the latter assumption, the
convergence of StoDARS to Clarke stationary points with probability one is
established. Moreover, the analysis of the second-order behavior of the mesh
adaptive direct-search (MADS) algorithm using a second-order-like extension of
the Rademacher's theorem-based definition of the Clarke subdifferential
(so-called generalized Hessian) is extended to the StoDARS framework, making it
the first in a stochastic direct-search setting, to the best of our knowledge.
