On the Weak Stationarity Conditions for Mathematical Programs with Cardinality Constraints: A Unified Approach

In this paper, we study a class of optimization problems, called Mathematical Programs with Cardinality Constraints (MPCaC). This kind problem is generally difficult to deal with, because it involves constraint that not continuous neither convex, but provides sparse solutions. Thereby reformulate MPCaC in suitable way, by modeling as mixed-integer and then addressing its counterpart, which will be referred relaxed problem. We investigate the analyzing classical constraints two cases: linear nonlinear. case, propose general approach present discussion Guignard Abadie qualifications, proving case every minimizer satisfies Karush–Kuhn–Tucker (KKT) conditions. On other hand, nonlinear show some standard qualifications may violated. Therefore, cannot assert about KKT points. Motivated find for problem, define new weaker stationarity conditions, proposing unified approach.