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

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. Comment: 27 pages

Vitali’s generalized absolute differential calculus

The paper provides an analysis of Giuseppe Vitali’s contributions to differential geometry over the period 1923–1932. In particular, Vitali’s ambitious project of elaborating a generalized differential calculus regarded as an extension of Ricci-Curbastro tensor calculus is discussed in some detail. Special attention is paid to describing the origin of Vitali’s calculus within the context of Ernesto Pascal’s theory of forms and to providing an analysis of the process leading to a fully general notion of covariant derivative. Finally, the reception of Vitali’s theory is discussed in light of Enea Bortolotti and Enrico Bompiani’s subsequent works.

Restrictive metric regularity and generalized differential calculus in Banach spaces

We consider nonlinear mappingsf:X→Ybetween Banach spaces and study the notion ofrestrictive metric regularityoffaround some pointx¯, that is, metric regularity offfromXinto the metric spaceE=f(X). Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case whenfis strictly differentiable atx¯but its strict derivative∇f(x¯)is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions.