Mathematical Programms with Equilibrium Constraints: A sequential optimality condition, new constraint qualifications

Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, is a difficult class of constrained optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications (CQs). Thus, the standard KKT conditions are not necessary satisfied by minimizers and the convergence assumptions of many standard methods for solving constrained optimization problems are not fulfilled. This makes it necessary, both from a theoretical and numerical point of view, to consider suitable optimality conditions, tailored CQs, and specially designed algorithms for solving MPECs. In this paper, we present a new sequential optimality condition useful for the convergence analysis for several relaxation methods for solving MPECs. We also introduce a companion CQ for M-stationary that is weaker than the recently introduced MPEC relaxed constant positive linear dependence (MPEC-RCPLD). Relations between the old and new CQs as well as the algorithmic consequences will be discussed.