Resumo

In this work, we study a class of optimization problems, called Mathematical Programs with Cardinality Constraints (MPCaC). This kind of problem is generally difficult to deal with, because it involves a constraint that is not continuous neither convex, but provides sparse solutions. Thereby we reformulate MPCaC in a suitable way, by modeling it as a mixed-integer problem and then addressing its continuous counterpart, which will be referred to as relaxed problem. We investigate the relaxed problem by analyzing the general constraints in two cases: linear and nonlinear. In the linear case, we propose a general approach and present a discussion of the Guignard and Abadie constraint qualifications, proving in this case that every minimizer of the relaxed problem satisfies the Karush-Kuhn-Tucker (KKT) conditions. On the other hand, in the nonlinear case, we show that some standard constraint qualifications may be violated. Motivated to find a minimizer for the MPCaC problem, we define new stationarity conditions, weaker than KKT, by proposing a unified approach that goes from the weakest to the strongest stationarity (within a certain range of conditions). However, these conditions are not optimality conditions. Thereby, we also propose an Approximate Weak stationarity (AW-stationarity) concept designed to deal with MPCaC problems. We proved that it is a legitimate optimality condition independently of any constraint qualification. Many research on sequential optimality conditions has been addressed for nonlinear constrained optimization in the last few years, some works in the context of Mathematical Programs with Complementarity Constraints (MPCC). However, as far as we know, no sequential optimality condition has been proposed for MPCaC problems. We also establish some relationships between our AW-stationarity and other usual sequential optimality conditions, such as AKKT, CAKKT and PAKKT.