Set-Convergence and Its Application: A Tutorial
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 set-convergence 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.
Cite
@article{arxiv.2002.09774,
title = {Set-Convergence and Its Application: A Tutorial},
author = {Johannes O. Royset},
journal= {arXiv preprint arXiv:2002.09774},
year = {2020}
}