An Active Set Method for Mathematical Programs with Linear Complementarity Constraints

We study mathematical programs with linear complementarity constraints (MPLCC) for which the objective function is smooth. Current nonlinear programming (NLP) based algorithms including regularization methods and decomposition methods generate only weak (e.g., Cor M-) stationary points that may not be first-order solutions to the MPLCC. Piecewise sequential quadratic programming methods enjoy stronger convergence properties, but need to solve expensive subproblems. Here we propose a primal-dual active set projected Newton method for MPLCCs, that maintains the feasibility of all iterates. At every iteration the method generates a working set for predicting the active set. The projected step direction on the subspace associated with this working set is determined by the current dual iterate, while other elements in the step direction are computed by a Newton system. The major cost of a subproblem involves one matrix factorization and is comparable to that of NLP based algorithms. Our method has strong convergence properties. In particular, under the MPLCC-linear independence constraint qualification, any accumulation point of the generated iterates is a B-stationary solution (i.e., a first-order solution) to the MPLCC. The asymptotic rate of convergence is quadratic under additional MPLCC-second-order sufficient conditions and strict complementarity.