On inner calmness*, generalized calculus, and derivatives of the normal cone mapping

TL;DR

This paper introduces new inner calmness* conditions for set-valued mappings, develops generalized calculus rules, and applies these to compute derivatives of the normal cone mapping for sensitivity analysis.

Contribution

It proposes novel inner calmness* notions, establishes their properties for polyhedral maps, and derives an exact chain rule for graphical derivatives of normal cone mappings.

Findings

Polyhedral maps satisfy inner calmness*.

New rules for generalized differential calculus are developed.

Derivatives of the normal cone mapping are explicitly computed.

Abstract

In this paper, we study continuity and Lipschitzian properties of set-valued mappings, focusing on inner-type conditions. We introduce new notions of inner calmness* and, its relaxation, fuzzy inner calmness*. We show that polyhedral maps enjoy inner calmness* and examine (fuzzy) inner calmness* of a multiplier mapping associated with constraint systems in depth. Then we utilize these notions to develop some new rules of generalized differential calculus, mainly for the primal objects (e.g. tangent cones). In particular, we propose an exact chain rule for graphical derivatives. We apply these results to compute the derivatives of the normal cone mapping, essential e.g. for sensitivity analysis of variational inequalities.