  We introduce a class of stochastic algorithms for minimizing weakly convex
functions over proximally smooth sets. As their main building blocks, the
algorithms use simplified models of the objective function and the constraint
set, along with a retraction operation to restore feasibility. All the proposed
methods come equipped with a finite time efficiency guarantee in terms of a
natural stationarity measure. We discuss consequences for nonsmooth
optimization over smooth manifolds and over sets cut out by weakly-convex
inequalities.
