Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity

TL;DR

This paper introduces a new generalized differentiation concept for set-valued maps that unifies various existing notions and enhances understanding of metric regularity and related properties.

Contribution

It develops a comprehensive framework for generalized differentiation of set-valued maps, including calculus rules and refined properties like metric regularity.

Findings

Unified differentiation framework for set-valued maps

Enhanced calculus rules and property relationships

Refined directional metric regularity analysis

Abstract

We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet 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 generalization differentiation and its one sided counterpart.