PIECEWISE SMOOTHNESS, LOCAL INVERTIBILITY, AND PARAMETRIC ANALYSIS OFNORMAL, MAPS

This paper is concerned with properties of the Euclidean projection ma p onto a convex set defined by finitely many smooth, convex inequaliti es and affine equalities. Under a constant rank constraint qualificati on, we show that the projection map is piecewise smooth (PC1) hence B( ouligand)-differentiable, or directionally differentiable; and a relat ively simple formula is given for the B-derivative. These properties o f the projection map are used to obtain inverse and implicit function theorems for associated normal maps, using a new characterization of i nvertibility of a PC1 function in terms of its B-derivative. An extens ion of the implicit function theorem which does not require local uniq ueness is also presented. Degree theory plays a major role in the anal ysis of both the locally unique case and its extension.