  This paper provides formulas for calculating of Fr\'{e}chet and limiting
normal cones with respect to a set of sets and the limiting coderivative with
respect to a set of set-valued mappings. These calculations are obtained under
some qualification constraints and are expressed in the similar forms of these
ones of Fr\'{e}chet and limiting normal cones and the limiting coderivative. By
using these obtained formulas, we state explicit necessary optimality
conditions with respect to a set for optimization problems with equilibrium
constraints under certain qualification conditions. Some illustrated examples
to obtained results are also established.
