Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints

Mathematical programs with nonlinear complementarity constraints are refor-mulated using better-posed but nonsmooth constraints. We introduce a class offunctions, parameterized by a real scalar, to approximate these nonsmooth prob-lems by smooth nonlinear programs. This smoothing procedure has the extrabenefits that it often improves the prospect of feasibility and stability of the con-straints of the associated nonlinear programs and their quadratic approximations.We present two globally convergent algorithms based on sequential quadratic pro-gramming, SQP, as applied in exact penalty methods for nonlinear programs.Global convergence of the implicit smooth SQP method depends on existence ofa lower-level nondegenerate (strictly complementary) limit point of the iterationsequence. Global convergence of the explicit smooth SQP method depends ona weaker property, i.e. existence of a limit point at which a generalized constraintqualification holds. We also discuss some practical matters relating to computerimplementations.