Calmness and Calculus: Two Basic Patterns

TL;DR

This paper develops estimates for generalized derivatives of set-valued mappings, highlighting the roles of calmness and inner calmness*, and applies these to enhance calculus rules and optimality conditions in variational analysis.

Contribution

It introduces new estimates for generalized derivatives based on calmness and inner calmness*, improving calculus rules and applications in optimization.

Findings

Enhanced calculus rules for tangents and normals

New conditions for generalized derivatives of set-valued mappings

Applications to minimax optimization and semismoothness*

Abstract

We establish two types of estimates for generalized derivatives of set-valued mappings which carry the essence of two basic patterns observed troughout the pile of calculus rules. These estimates also illustrate the role of the essential assumptions that accompany these two patters, namely calmness on the one hand and (fuzzy) inner calmness* on the other. Afterwards, we study the relationship between and sufficient conditions for the various notions of (inner) calmness. The aforementioned estimates are applied in order to recover several prominent calculus rules for tangents and normals as well as generalized derivatives of marginal functions and compositions as well as Cartesian products of set-valued mappings under mild conditions. We believe that our enhanced approach puts the overall generalized calculus into some other light. Some applications of our findings are presented which exemplary address necessary optimality conditions for minimax optimization problems as well as the calculus related to the recently introduced semismoothness* property.