Relaxed Constant Positive Linear Dependence Constraint Qualification for Disjunctive Systems

The disjunctive system is a system involving a disjunctive set which is the union of finitely many polyhedral convex sets. In this paper, we introduce a notion of the relaxed constant positive linear dependence constraint qualification (RCPLD) for the disjunctive system. For a disjunctive system, our notion is weaker than the one we introduced for a more general system recently (J. Glob. Optim. 2020) and is still a constraint qualification. To obtain the local error bound for the disjunctive system, we introduce the piecewise RCPLD under which the error bound property holds if all inequality constraint functions are subdifferentially regular and the rest of the constraint functions are smooth. We then specialize our results to the ortho-disjunctive program, which includes the mathematical program with equilibrium constraints (MPEC), the mathematical program with vanishing constraints (MPVC) and the mathematical program with switching constraints (MPSC) as special cases. For MPEC, we recover MPEC-RCPLD, an MPEC variant of RCPLD and propose the MPEC piecewise RCPLD to obtain the error bound property. For MPVC, we introduce new constraint qualifications MPVC-RCPLD and the piecewise RCPLD, which also implies the local error bound. For MPSC, we show that both RCPLD and the piecewise RCPLD coincide and hence it leads to the local error bound.