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.