Inverse functions of pseudo regular mappings and regularity conditions

Abstract

We show that, even for monotone directionally differentiable Lipschitz functionals on Hilbert spaces, basic concepts of generalized derivatives identify only particular pseudo regular (or metrically regular) situations. Thus, pseudo regularity of (multi-) functions will be investigated by other means, namely in terms of the possible inverse functions. In this way, we show how pseudo regularity for the intersection of multifunctions can be directly characterized and estimated under general settings and how contingent and coderivatives may be modified to obtain sharper regularity conditions. Consequences for a concept of stationary points as limits of Ekeland points in nonsmooth optimization will be studied, too.