  In this paper, 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 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 classical
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. Therefore, we cannot assert about KKT points.
Motivated to find a minimizer for the MPCaC problem, we define new and weaker
stationarity conditions, by proposing a unified approach that goes from the
weakest to the strongest stationarity.
