Mathematical Programs with Equilibrium Constraints: Enhanced Fritz John-conditions, New Constraint Qualifications, and Improved Exact Penalty Results

Mathematical programs with equilibrium (or complementarity) constraints (MPECs for short) form a difficult class of optimization problems. The standard KKT conditions are not always necessary optimality conditions due to the fact that suitable constraint qualifications are often violated. Alternatively, one can therefore use the Fritz John-approach to derive necessary optimality conditions. While the usual Fritz John-conditions do not provide much information, we prove an enhanced version of the Fritz John-conditions. This version motivates the introduction of some new constraint qualifications (CQs) which can then be used in order to obtain, for the first time, a completely elementary proof of the fact that a local minimum is an M-stationary point under one of these CQs. We also show how these CQs can be used to obtain a suitable exact penalty result under weaker or different assumptions than those that can be found in the literature.