Characterizing generalized derivatives of set-valued maps: Extending the tangential and normal approaches

TL;DR

This paper develops new characterizations of generalized derivatives for set-valued maps using graphical derivatives, normal cones, and coderivatives, extending classical criteria like Aubin and Mordukhovich in finite and infinite dimensions.

Contribution

It introduces novel characterizations of generalized derivatives via convexified and unconvexified graphical derivatives, extending classical criteria such as Aubin and Mordukhovich.

Findings

Characterizations in terms of graphical derivatives near the point of interest.

Simplified practical check in finite-dimensional cases.

Bijective relationship between convexified coderivative and generalized derivatives.

Abstract

For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in the sense of [20]. This result generalizes the Aubin criterion in [9]. A second characterization of these generalized derivatives is easier to check in practice, especially in the finite dimensional case. Finally, the third characterization in terms of limiting normal cones and coderivatives generalizes the Mordukhovich criterion in the finite dimensional case. The convexified coderivative has a bijective relationship with the set of possible generalized derivatives. We conclude by illustrating a few applications of our result.