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
