Regularity Properties of Constrained Set-Valued Mappings

ABSTRACT In this paper, we present a general result concerning the Lipschitz regularity of a certain type of set-valued maps often found in constrained optimization and control problems. The class of multifunctions examined in this paper is characterized by means of a set of Lipschitz continuous constraint functions defined on some Lipschitz manifold. The proof of the regularity result for this class of multifunctions is based on a quantitative version of the Implicit Function Theorem for Lipschitzian maps which provides estimates for the neighborhoods where the implicit map can be defined. KEYWORDS. Multifunctions, Lipschitz regularity, Implicit Function Theorem, Generalized Jacobian.