Various Lipschitz-like Properties for Functions and Sets I: Directional Derivative and Tangential Characterizations

Abstract. In this work we introduce for extended real valued functions, defined on a Banach space X, the concept of K directionally Lipschitzian behavior, where K is a bounded subset of X. For different types of sets K (e.g., zero, singleton, or compact), the K directionally Lipschitzian behavior recovers well-known concepts in variational analysis (locally Lipschitzian, directionally Lip-schitzian, or compactly epi-Lipschitzian properties, respectively). Characterizations of this notion are provided in terms of the lower Dini subderivatives. We also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the Bouligand tangent cones. The special case of convex functions and sets is also studied. Key words. directional derivative, tangent cone, directionally Lipschitzian function, compactly epi-Lipschitzian function, epi-Lipschitzian set, compactly epi-Lipschitzian set, multidirectional mea