Generalized Differentiation with Positively Homogeneous Maps: Applications in Set‐Valued Analysis and Metric Regularity

We propose a new concept of generalized differentiation of set‐valued maps that captures first‐order information. This concept encompasses the standard notions of Fréchet differentiability, strict differentiability, calmness and Lipschitz continuity in single‐valued maps, and the Aubin property and Lipschitz continuity in set‐valued maps. We present calculus rules, sharpen the relationship between the Aubin property and coderivatives, and study how metric regularity and open covering can be refined to have a directional property similar to our concept of generalized differentiation. Finally, we discuss the relationship between the robust form of generalized differentiation and its one‐sided counterpart.