Piecewise smoothness, local invertibility, and parametric analysis of normal maps. (English) Zbl 0857.90122

Summary: This paper is concerned with properties of the Euclidean projection map onto a convex set defined by finitely many smooth, convex inequalities and affine equalities. Under a constant rank constraint qualification, we show that the projection map is piecewise smooth (PC\(^1\)) hence B(ouligand)-differentiable, or directionally differentiable; and a relatively simple formula is given for the B-derivative. These properties of the projection map are used to obtain inverse and implicit function theorems for associated normal maps, using a new characterization of invertibility of a PC\(^1\) function in terms of its B-derivative. An extension of the implicit function theorem which does not require local uniqueness is also presented. Degree theory plays a major role in the analysis of both the locally unique case and its extension.


90C30 Nonlinear programming
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J52 Nonsmooth analysis
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