Set-Convergence and Its Application: A Tutorial

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Authors
Royset, Johannes O.
Subjects
set-convergence
epi-convergence
graphical convergence
weak convergence
stability
approximation theory
variational geometry
subdifferentiability
truncated Hausdorff distance
Advisors
Date of Issue
2020-02
Date
Publisher
ArXiv
Language
Abstract
Optimization problems, generalized equations, and the multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence and explain its role in defining a notion of proximity between sets, especially for epigraphs of functions and graphs of set-valued mappings. The development leads to an approximation theory for optimization problems and generalized equations with profound consequences for the construction of algorithms. We also introduce the role of setconvergence in variational geometry and subdifferentiability with applications to optimality conditions. Examples illustrate the importance of set-convergence in stability analysis, error analysis, construction of algorithms, statistical estimation, and probability theory.
Type
Preprint
Description
Series/Report No
Department
Operations Research (OR)
Organization
Identifiers
NPS Report Number
Sponsors
This work is supported by ONR (Operations Research) under N0001420WX00519 and AFOSR (Optimization and Discrete Mathematics) under F4FGA08272G001.
Funder
This work is supported by ONR (Operations Research) under N0001420WX00519 and AFOSR (Optimization and Discrete Mathematics) under F4FGA08272G001.
Format
14 p.
Citation
Royset, Johannes O. "Set-Convergence and Its Application: A Tutorial." arXiv preprint arXiv:2002.09774 (2020).
Distribution Statement
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.
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