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

  • Johannes O. RoysetEmail author
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Abstract

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.

Keywords

Hypo-convergence Attouch-Wets distance Approximation theory Solution stability Stochastic optimization Epi-splines Rate of convergence 

Mathematics Subject Classification

90C15 Stochastic programming 62G07 Density estimation 62G08 Nonparametric regression 62G15 Tolerance and confidence regions 

Notes

Acknowledgements

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.

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Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Operations Research DepartmentNaval Postgraduate SchoolMontereyUSA

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