Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications

In this paper we analyse the Fritz John and Karush-Kuhn-Tucker conditions for a (Gateaux) differentiable nonconvex optimization problem with inequality constraints and a geometric constraint set. Fritz John conditions are characterized in terms of an alternative theorem which covers beyond standard situations; while characterizations of KKT conditions, without assuming constraints qualifications, are related to strong duality of a suitable linear approximation of the given problem and the properties of its associated image mapping. Such characterizations are suitable for dealing with some problems in structural optimization, where most of the known constraint qualifications fail. In particular, several examples are exhibited showing the usefulness and optimality, in a certain sense, of our results, which even provide much more information than those (including Mordukhovich normal cone or Clarke's one) appearing elsewhere. The case with a single inequality constraint is discussed in details by establishing a hidden convexity in the validity of the KKT conditions. We outline possible applications to a class of mathematical programs with equilibrium constraints (MPEC), as well as to vector equilibrium or quasi-variational inequality problems.