Formulas for Calculating Generalized Differentials with Respect to a Set and Their Applications

This paper provides formulas for calculating Fréchet 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 the ones of Fréchet and limiting normal cones and the limiting coderivative. By using these new formulas, we state explicit necessary optimality conditions with respect to a set for optimization problems with equilibrium constraints under certain qualification conditions. Some examples are also presented.