An Active-Set Newton Method for Mathematical Programs with Complementarity Constraints

For a mathematical program with complementarity constraints (MPCC), we propose an active-set Newton method, which has the property of local quadratic convergence under the MPCC linear independence constraint qualification (MPCC-LICQ) and the standard second-order sufficient condition (SOSC) for optimality. Under MPCC-LICQ, this SOSC is equivalent to the piecewise SOSC on branches of MPCC, which is weaker than the special MPCC-SOSC often employed in the literature. The piecewise SOSC is also more natural than MPCC-SOSC because, unlike the latter, it has an appropriate second-order necessary condition as its counterpart. In particular, our assumptions for local quadratic convergence are weaker than those required by standard SQP when applied to MPCC and are equivalent to assumptions required by piecewise SQP for MPCC. Moreover, each iteration of our method consists of solving a linear system of equations instead of a quadratic program. Some globalization issues of the local scheme are also discussed, and illustrative examples and numerical experiments are presented.