Complementarity Constraint Qualification via the Theory of 2-Regularity

We exhibit certain second-order regularity properties of parametric complementarity constraints, which are notorious for being irregular in the classical sense. Our approach leads to a constraint qualification in terms of 2-regularity of the mapping corresponding to the subset of constraints which must be satisfied as equalities around the given feasible point, while no qualification is required for the rest of the constraints. Under this 2-regularity assumption, we derive constructive sufficient conditions for tangent directions to feasible sets defined by complementarity constraints. A special form of primal-dual optimality conditions is also obtained. We further show that our 2-regularity condition always holds under the piecewise Mangasarian–Fromovitz constraint qualification, but not vice versa. Relations with other constraint qualifications and optimality conditions are also discussed. It is shown that our approach can be useful when alternative ones are not applicable.