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

We propose a new concept of generalized di erentiation of setvalued maps that captures rst order information. This concept encompasses the standard notions of Fréchet di erentiability, strict di erentiability, 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 re ned to have a directional property similar to our concept of generalized di erentiation. Finally, we discuss the relationship between the robust form of generalization di erentiation and its one sided counterpart.