# Convergence properties of a second order augmented Lagrangian method for mathematical programs with complementarity constraints

Mathematical Programs with Complementarity Constraints (MPCCs) are difficult optimization problems that do not satisfy the majority of the usual constraint qualifications (CQs) for standard nonlinear optimization. Despite this fact, classical methods behaves well when applied to MPCCs. Recently, Izmailov, Solodov and Uskov proved that first order augmented Lagrangian methods, under a natural adaption of the Linear Independence Constraint Qualification to the MPCC setting (MPCC-LICQ), converge to Strong stationary (S-stationary) points, if the multiplier sequence is bounded. If the multiplier sequence is not bounded, only Clarke stationary (C-stationary) points are recovered. In this paper we improve this result in two ways. For the case of bounded multipliers we are able replace the MPCC-LICQ assumption by the much weaker MPCC-Relaxed Positive Linear Dependence condition (MPCC-RCLPD). For the case with unbounded multipliers we show that a {\em second order} augmented Lagrangian method converges to points that are at least to Mordukhovich stationary (M-stationary) but we still need the more stringent MPCC-LICQ assumption. Numerical tests, validating the theory, are also presented.

## Citation

Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Brazil, April 2017.