Lipschitzian and Pseudo-Lipschitzian Inverse Functions and Applications to Nonlinear Optimization
Bernd Kummer · 2020
As regularity conditions for multifunctions, Lipschitzian and pseudo-Lipschitzian behaviour of inverse mappings are investigated. We verify by a successive approximation scheme, that pseudo-regularity induces the same property with respect to Lipschitzian perturbations. In the case that the original map is a proper function in finite dimension, we characterize the two regularity properties by generalized directional derivatives as well as in terms of an exact penalty function, and show that continuous selections of the inverse map play the crucial role for the equivalence of these regularities. In particular, KKT-points of nonlinear C 1 -optimization problems are investigated. It turns out that pseudo-regularity requires the LICQ constraint qualification, and implies (Lipschitzian) regularity in the convex case. For piecewise C 2 -problems, inequalities to zero-Lagrange multipliers are not essential for the problem of equivalence, and the both regularities do not coincide, in general. Finally, we study regularity (strong stability) when the functions involved have only locally Lipschitzian derivatives. The results are extended to critical points of similar problems, in particular to zeros of generalized equations.