# zbMATH — the first resource for mathematics

Normal maps induced by linear transformations. (English) Zbl 0777.90063
Summary: We study a certain piecewise linear manifold, which we call the normal manifold, associated with a polyhedral convex set, and a family of continuous functions, called normal maps, that are induced on this manifold by continuous functions from $${\mathbf R}^ n$$ to $${\mathbf R}^ n$$. These normal maps occur frequently in optimization and equilibrium problems, and the subclass of normal maps induced by linear transformations plays a key role.
Our main result is that the normal map induced by a linear transformation is a Lipschitzian homeomorphism if and only if the determinant of the map in each $$n$$-cell of the normal manifold has the same (nonzero) sign.

##### MSC:
 90C30 Nonlinear programming 49J52 Nonsmooth analysis 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
##### Keywords:
normal manifold; polyhedral convex set; normal maps
Full Text: