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