The notions and certain fundamental characteristics of the proximal and limiting normal cones with respect to a set are first presented in this paper. We present the ideas of the limiting coderivative and subdifferential with respect to a set of multifunctions and singleton mappings, respectively, based on these normal cones. The necessary and sufficient conditions for the Aubin property with respect to a set of multifunctions are then described by using the limiting coderivative with respect to a set. As a result of the limiting subdifferential with respect to a set, we offer the requisite optimality criteria for local solutions to optimization problems. In addition, we also provide examples to demonstrate the outcomes.

翻译：本文首先给出了相对于某个集合的邻近法锥与极限法锥的概念及其若干基本特征。基于这些法锥，我们分别提出了关于多值函数和单值映射的相对于集合的极限余导数与次微分的概念。随后，利用相对于集合的极限余导数描述了多值函数关于集合的Aubin性质的充要条件。借助相对于集合的极限次微分，我们给出了优化问题局部解的必要最优性条件。此外，还通过实例验证了所得结论。