## 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.

### MSC:

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

### Keywords:

degree theory; Euclidean projection map; B-derivative
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