CHARACTERIZING AND TESTING SUBDIFFERENTIAL REGULARITY FOR PIECEWISE SMOOTH OBJECTIVE FUNCTIONS

Functions defined by evaluation programs involving smooth elementals and absolute values as well as the maxand min-operator are piecewise smooth. Using piecewise linearization we derived in [7] for this class of nonsmooth functions φ first and second order conditions for local optimality (MIN). They are necessary and sufficient, respectively. These generalizations of the classical KKT and SSC theory assumed that the given representation of φ satisfies the LinearIndependence-Kink-Qualification (LIKQ). In this paper we relax LIKQ to the Mangasarin-FromovitzKink-Qualification (MFKQ) and develop a constructive condition for a local convexity concept, i.e., the convexity of the local piecewise linearization on a neighborhood. As a consequence we show that this first order convexity (FOC) is always required by subdifferential regularity (REG) as defined in [20], and is even equivalent to it under MFKQ. Whereas it was observed in [7] that testing for MIN is polynomial under LIKQ, we show here that even under this strong kink qualification, testing for FOC and thus REG is co-NP complete. We conjecture that this is also true for testing MFKQ itself.