On Stationarity Conditions and Constraint Qualifications for Multiobjective Optimization Problems with Cardinality Constraints

The purpose of this paper is to develop Pareto optimality conditions and constraint qualifications (CQs) for Multiobjective Programs with Cardinality Constraints (MOPCaC). In general, such problems are difficult to solve, not only because they involve a cardinality constraint that is neither continuous nor convex, but also because there may be a potential conflict between the various objective functions. Thus, we reformulate the MOPCaC based on the problem with continuous variables, namely the relaxed problem. Furthermore, we consider different notions of optimality (weak/strong Pareto optimal solutions). Thereby, we define new stationarity conditions that extend the classical Karush-Kuhn-Tucker (KKT) conditions of the scalar case. Moreover, we also introduce new CQs, based on the recently defined multiobjective normal cone, to ensure compliance with such stationarity conditions. Important statements are illustrated by examples.