  Upper semicontinuous (usc) functions arise in the analysis of maximization
problems, distributionally robust optimization, and function identification,
which includes many problems of nonparametric statistics. We establish that
every usc function is the limit of a hypo-converging sequence of piecewise
affine functions of the difference-of-max type and illustrate resulting
algorithmic possibilities in the context of approximate solution of
infinite-dimensional optimization problems. In an effort to quantify the ease
with which classes of usc functions can be approximated by finite collections,
we provide upper and lower bounds on covering numbers for bounded sets of usc
functions under the Attouch-Wets distance. The result is applied in the context
of stochastic optimization problems defined over spaces of usc functions. We
establish confidence regions for optimal solutions based on sample average
approximations and examine the accompanying rates of convergence. Examples from
nonparametric statistics illustrate the results.
