Optimization problems, generalized equations, and a 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.

优化问题，广义方程式以及许多其他变分问题总是导致对集和集值映射及其近似的分析。我们回顾了集合收敛的核心概念，并解释了它在定义集合之间的接近性概念中的作用，尤其是对于函数的题词和集合值映射的图。这一发展导致了针对优化问题和广义方程的近似理论，对算法的构建产生了深远的影响。我们还介绍了集收敛在变分几何和次微分中的作用，并将其应用于最优条件。实例说明了集合收敛在稳定性分析，误差分析，算法构造，统计估计和概率论中的重要性。