Approximations of semicontinuous functions with applications to stochastic optimization and statistical estimation

Abstract

The article of record as published may be found at https://doi.org/10.1007/s10107-019-01413-zUpper 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.This work in supported in parts by DARPA under Grants HR0011-14-1-0060 and HR0011-8-34187, and Office of Naval Research (Science of Autonomy Program) under Grant N00014- 17-1-2372.This work in supported in parts by DARPA under Grants HR0011-14-1-0060 and HR0011-8-34187, and Office of Naval Research (Science of Autonomy Program) under Grant N00014- 17-1-2372