Mathematical programs with equilibrium constraints: a sequential optimality condition, new constraint qualifications and algorithmic consequences

Mathematical programs with equilibrium constraints is a difficult class of constrained optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications. Thus, the Karush-Kuhn-Tucker conditions are not necessarily satisfied at minimizers, and the convergence assumptions of many methods for solving constrained optimization problems are not fulfilled. Thus, it is necessary, from a theoretical and numerical point of view, to consider suitable optimality conditions, tailored constraints qualifications, and designed algorithms for solving such optimization problems. In this paper, we present a new sequential optimality condition useful for the convergence analysis of several methods for solving mathematical programs with equilibrium constraints such as relaxations schemes, complementarity-penalty methods, and interior-relaxation methods. Furthermore, the weakest constraint qualification for M-stationarity associated with such sequential optimality condition is presented. Relations between the old and new constraint qualifications, as well as the algorithmic consequences, will be discussed.