Partial Exact Penalty for Mathematical Programs with Equilibrium Constraints

It is well known that mathematical programs with equilibrium constraints (MPEC) violate the standard constraint qualifications such as Mangasarian–Fromovitz constraint qualification (MFCQ) and hence the usual Karush–Kuhn–Tucker conditions cannot be used as stationary conditions unless relatively strong assumptions are satisfied. This observation has led to a number of weaker stationary conditions, with Mordukhovich stationary (M-stationary) condition being the strongest among the weaker conditions. In nonlinear programming, it is known that MFCQ leads to an exact penalization. In this paper we show that MPEC GMFCQ, an MPEC variant of MFCQ, leads to a partial exact penalty where all the constraints except a simple linear complementarity constraint are moved to the objective function. The partial exact penalty function, however, is nonsmooth. By smoothing the partial exact penalty function, we design an algorithm which is shown to be globally convergent to an M-stationary point under an extended version of the MPEC GMFCQ.