  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.
