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.