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

Authors:
C. H. Jeffrey Pang
Affiliations:
Massachusetts Institute of Technology, Department of Mathematics, Cambridge, Massachusetts 02139
Venue:
Mathematics of Operations Research
Year:
2011

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Abstract

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.