Convergence analysis of an augmented Lagrangian method for mathematical programs with complementarity constraints

In this paper, a mathematical program with complementarity constraints (MPCC) is reformulated as a nonsmooth constrained optimization problem by using the Fischer-Burmeister function. An augmented (proximal) Lagrangian method is applied to tackle the resulting constrained optimization problem. The augmented Lagrangian problems are in general nonsmooth. We derive first- and second-order optimality conditions for the augmented Lagrangian problems using an approximate smooth variational principle and establish that the limit point of a sequence of points that satisfy the second-order necessary optimality conditions of the augmented Lagrangian problems is a strongly stationary point of the original MPCC if the limit point is feasible to MPCC, and the linear independence constraint qualification for MPCC and the upper level strict complementarity condition hold at the limit point.