Scaled constraint qualifications for generalized equation constrained problems and application to nonsmooth mathematical programs with equilibrium constraints

In this paper, the notion of graphical derivatives is applied to define a new class of several well-known constraint qualifications for a nonconvex multifunction M at a point of its graph. This class is called as “scaled constraint qualifications”. The reason of this terminology is that these conditions ensure the existence of bounded KKT multiplier vectors with a proper upper bound. The relations between these constraint qualifications and stability properties of M are also investigated. New sharp necessary optimality conditions with bounded multiplier vectors are derived for an optimization problem with a generalized equation constraint. The results are adapted to nonsmooth general constrained problems and nonsmooth mathematical programs with equilibrium constraints.