Key Takeaway: The convergence behavior of a parametric NLP regularizing mathematical programs with equilibrium constraints is consistent with the linear independence, upper level strict complementarity, and second order optimality conditions.

Abstract

We study the convergence behavior of a sequence of stationary points of a parametric NLP which regularizes a mathematical program with equilibrium constraints (MPEC) in the form of complementarity conditions. Accumulation points are feasible points of the MPEC; they are C-stationary if the MPEC linear independence constraint qualification holds; they are M-stationary if, in addition, an approaching subsequence satisfies second order necessary conditions, and they are B-stationary if, in addition, an upper level strict complementarity condition holds. These results complement recent results of Fukushima and Pang [Convergence of a smoothing continuation method for mathematical programs with equilibrium constraints, in Ill-posed Variational Problems and Regularization Techniques, Springer-Verlag, New York, 1999]. We further show that every local minimizer of the MPEC which satisfies the linear independence, upper level strict complementarity, and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric NLP.